steel code check theory enu
Post on 14-Apr-2015
58 Views
Preview:
TRANSCRIPT
Theory
Steel Code Check
Information in this document is subject to change without notice. No part of this document may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic or mechanical, for any purpose, without the express written permission of the publisher.
SCIA Software is not responsible for direct or indirect damage as a result of imperfections in the documentation and/or software.
Copyright 2011 SCIA Group. All rights reserved.
Introduction Welcome to the Steel Code Check – Theoretical Background.
This document provides background information on the code checks according to different national and international regulations.
Version info
Documentation Title Steel Code Check – Theoretical Background
Release 2011.0
Revision 09/2011
TABLE OF CONTENTS
Introduction ....................................................................................................................... iv
Version info ....................................................................................................................... iv
EC3 – ENV 1993 .................................................................................................................. 2
EC3 – ENV code check ............................................................................................................... 2 Material properties ................................................................................................................... 2 Consulted articles .................................................................................................................... 3
Classification of sections .................................................................................................... 4 Effective cross-section properties for class 4 cross-section ............................................... 5 Section properties .............................................................................................................. 5 Bending moment ................................................................................................................ 5 Bending, shear and axial force ........................................................................................... 5 Torsion check ..................................................................................................................... 5 Built-in beams .................................................................................................................... 6 Compression members ...................................................................................................... 6 Lateral-torsional buckling ................................................................................................... 6 Use of diaphragms ............................................................................................................. 7 Shear Buckling check ........................................................................................................ 7 Shear buckling check for cold formed sections .................................................................. 7 Stability check for torsional buckling and torsional-flexural buckling .................................. 8 Bending and axial compression ....................................................................................... 10 Battened compression members ..................................................................................... 10
EC3 – ENV Fire Resistance ...................................................................................................... 12 Fire actions effect Efi ........................................................................................................ 12 Material properties ........................................................................................................... 12 Temperature analysis - Thermal actions .......................................................................... 13 Nominal temperature-time curve ...................................................................................... 13 Net heat flux ..................................................................................................................... 14 Steel Temperature ........................................................................................................... 15 Calculation model ............................................................................................................ 16 Code Check ..................................................................................................................... 16
Supported sections ................................................................................................................... 17
References ................................................................................................................................. 18
EC3 – EN 1993 .................................................................................................................. 19
EC3 – EN Code check ............................................................................................................... 19 Material properties ................................................................................................................. 19 Consulted articles .................................................................................................................. 20
Classification of sections .................................................................................................. 22 Effective cross-section properties for class 4 cross-section ............................................. 23 Section properties ............................................................................................................ 23 Bending, shear and axial force ......................................................................................... 23 Torsion check ................................................................................................................... 24 Built-in beams .................................................................................................................. 24 Compression members .................................................................................................... 24 Lateral-torsional buckling ................................................................................................. 25 Use of diaphragms ........................................................................................................... 30
Combined bending and axial compression ...................................................................... 31 Shear buckling check ....................................................................................................... 31 Battened compression members ..................................................................................... 32 Plate girders with sinusoidal corrugated webs ................................................................. 34 Moments on columns in simple construction.................................................................... 39 Scaffolding ....................................................................................................................... 41
EC3 – EN Fire Resistance ......................................................................................................... 47 Fire actions effect Efi .............................................................................................................. 47 Material properties ................................................................................................................. 48 Temperature analysis - Thermal actions ............................................................................... 48
Nominal temperature-time curve ...................................................................................... 48 Net heat flux ..................................................................................................................... 49
Steel Temperature ................................................................................................................. 50 Calculation model .................................................................................................................. 52 Code Check ........................................................................................................................... 52
EC3 – EN Cold-Formed ............................................................................................................. 53 Consulted articles .................................................................................................................. 53
Material properties ........................................................................................................... 55 Initial Shape ..................................................................................................................... 56 Geometrical Proportions .................................................................................................. 57 Effective Shape ................................................................................................................ 58 Section Checks ................................................................................................................ 64 Stability Checks ............................................................................................................... 76 Use of Diaphragms .......................................................................................................... 80 Special considerations for Purlins .................................................................................... 84
Supported sections ................................................................................................................... 91
References ................................................................................................................................. 92
DIN18800 ........................................................................................................................... 96
DIN18800 Code check ............................................................................................................... 96 Material properties ................................................................................................................. 96 Consulted articles .................................................................................................................. 97
Classification of sections ................................................................................................ 100 Net area properties ........................................................................................................ 100 Plastic interaction formula for RHS section .................................................................... 100 Plastic interaction formula for CHS section .................................................................... 104 Torsion check ................................................................................................................. 105 Built-in beams ................................................................................................................ 105 Calculation of the buckling length .................................................................................. 106 Torsional buckling .......................................................................................................... 106 Use of diaphragms ......................................................................................................... 107 LTB Check ..................................................................................................................... 108 Combined flexion for check method 2 ............................................................................ 111 Battened compression members ................................................................................... 112 Effective area properties ................................................................................................ 113 Shear buckling check ..................................................................................................... 114 Shear buckling check with buckling influence ................................................................ 114
Cold formed thin gauge members ....................................................................................... 114
Supported sections ................................................................................................................. 115
References ............................................................................................................................... 116
ONORM B 4300 ............................................................................................................... 118
ONORM B 4300 Code check ................................................................................................... 118 Material properties ............................................................................................................... 119 Consulted articles ................................................................................................................ 119
Supported sections ................................................................................................................. 120
References ............................................................................................................................... 121
NEN .................................................................................................................................. 123
NEN6770/6771 Code check ..................................................................................................... 123 Material properties ............................................................................................................... 123 Consulted articles ................................................................................................................ 124
Section properties .......................................................................................................... 126 Classification of sections ................................................................................................ 126 Effective cross-section properties for class 4 cross-section ........................................... 126 Torsion check ................................................................................................................. 127 Built-in beams ................................................................................................................ 127 Buckling length ............................................................................................................... 127 Lateral-torsional buckling ............................................................................................... 127 Use of diaphragms ......................................................................................................... 128 Battened compression members ................................................................................... 129 Shear buckling check ..................................................................................................... 130 Shear buckling check with buckling influence ................................................................ 130
NEN6072 - Fire Resistance ..................................................................................................... 131 Fire actions effect ................................................................................................................ 131 Material properties ............................................................................................................... 131 Nominal temperature-time curve ......................................................................................... 132 Steel Temperature ............................................................................................................... 132 Calculation model ................................................................................................................ 135 Code Check ......................................................................................................................... 135
Supported sections ................................................................................................................. 136
References ............................................................................................................................... 137
AISC – ASD : 1989 .......................................................................................................... 138
AISC - ASD Code check .......................................................................................................... 138 Classification of sections ..................................................................................................... 140 Section properties ................................................................................................................ 140 Buckling length .................................................................................................................... 140 Flexural Torsional Buckling .................................................................................................. 140 Lateral-torsional buckling ..................................................................................................... 140 Shear buckling check .......................................................................................................... 141
Supported sections ................................................................................................................. 142
References ............................................................................................................................... 142
AISC – LRFD : 2001 ........................................................................................................ 144
AISC - LRFD Code check ........................................................................................................ 144 Classification of sections ..................................................................................................... 146 Section properties ................................................................................................................ 146
Buckling length .................................................................................................................... 146 Lateral-torsional buckling ..................................................................................................... 146 Use of diaphragms .............................................................................................................. 147 Shear buckling check .......................................................................................................... 147
Supported sections ................................................................................................................. 147
References ............................................................................................................................... 148
ANSI/AISC 360-05:2005 .................................................................................................. 149
ANSI/AISC 360-05 Code check ............................................................................................... 149 Classification of sections ..................................................................................................... 151 Section properties ................................................................................................................ 151 Buckling length .................................................................................................................... 151 Lateral-torsional buckling ..................................................................................................... 151 Use of diaphragms .............................................................................................................. 151 Shear buckling check .......................................................................................................... 151
Supported sections ................................................................................................................. 152
References ............................................................................................................................... 152
AISI NAS S100-2007 ....................................................................................................... 153
AISI NAS S100-2007 Code check ........................................................................................... 153 Consulted articles ................................................................................................................ 153
Initial Shape ................................................................................................................... 155 Dimensional limits .......................................................................................................... 156 Effective Widths ............................................................................................................. 156 Properties of Sections .................................................................................................... 157 Tension Members .......................................................................................................... 158 Flexural Members .......................................................................................................... 158 Compression Members .................................................................................................. 167 Combined Compression and Bending ........................................................................... 170 Use of diaphragms ......................................................................................................... 170 2
nd Order using Appendix 2 ............................................................................................ 174
Lapped Purlin Design ..................................................................................................... 175
References ............................................................................................................................... 176
CM66 ................................................................................................................................ 178
CM66 Code check ................................................................................................................... 178 Consulted articles ................................................................................................................ 178
Section properties .......................................................................................................... 180 Plastic coefficient ........................................................................................................... 180 Compression members .................................................................................................. 180 Factor kf ......................................................................................................................... 180 LTB Check ..................................................................................................................... 180 Use of diaphragms ......................................................................................................... 180 Combined flexion ........................................................................................................... 181 Shear buckling check ..................................................................................................... 181
Supported sections ................................................................................................................. 181
References ............................................................................................................................... 182
CM66 - Additif 80 ............................................................................................................ 183
CM66 - Additif 80 Code check ................................................................................................ 183
Consulted articles ................................................................................................................ 183 Classification of sections ................................................................................................ 184 Section check ................................................................................................................. 184 Compression members .................................................................................................. 184 Lateral-torsional buckling ............................................................................................... 184 Use of diaphragms ......................................................................................................... 184
Supported sections ................................................................................................................. 185
References ............................................................................................................................... 185
BS5950-1:1990 ................................................................................................................ 186
BS5950-1:1990 Code Check ................................................................................................... 186 Material properties ............................................................................................................... 186 Consulted articles ................................................................................................................ 187
Classification of sections ................................................................................................ 189 Slender cross-section .................................................................................................... 189 Section properties .......................................................................................................... 189 Bending moment ............................................................................................................ 190 Bending, shear, axial force............................................................................................. 190 Lateral torsional buckling ............................................................................................... 190 Use of diaphragms ......................................................................................................... 191 Compression member .................................................................................................... 191 Shear buckling check ..................................................................................................... 191
Supported sections ................................................................................................................. 192
References ............................................................................................................................... 192
BS5950-1:2000 ................................................................................................................ 194
BS5950-1:2000 Code Check ................................................................................................... 194
SIA263 ............................................................................................................................. 195
SIA263 Code check ................................................................................................................. 195 Material properties ............................................................................................................... 195 Consulted articles ................................................................................................................ 195
Section classification ...................................................................................................... 197 Slender cross-section .................................................................................................... 197 Sections properties ........................................................................................................ 197 Lateral torsional buckling ............................................................................................... 197 Use of diaphragms ......................................................................................................... 198 Shear buckling ............................................................................................................... 198 Stability check ................................................................................................................ 198 Torsion check ................................................................................................................. 198 Built-in beams ................................................................................................................ 198
SIA263 - Fire Resistance ......................................................................................................... 198 Fire actions effect Efi ............................................................................................................ 198 Material properties ............................................................................................................... 199 Temperature analysis - Thermal actions ............................................................................. 199 Nominal temperature-time curve ......................................................................................... 199 Net heat flux ........................................................................................................................ 199 Steel Temperature ............................................................................................................... 200 Calculation model ................................................................................................................ 201 Code Check ......................................................................................................................... 201
Supported sections ................................................................................................................. 201
References ............................................................................................................................... 202
GBJ 17-88 ........................................................................................................................ 203
The GBJ 17-88 code check ..................................................................................................... 203 Material properties ............................................................................................................... 203 Consulted articles ................................................................................................................ 204
Section properties .......................................................................................................... 205 Shear buckling check ..................................................................................................... 206 Buckling curves .............................................................................................................. 206 Buckling length ............................................................................................................... 206 Lateral torsional buckling ............................................................................................... 206 Local stability of compressed members ......................................................................... 206 Shear buckling check ..................................................................................................... 207
Supported sections ................................................................................................................. 207
References ............................................................................................................................... 208
Korean steel code check ............................................................................................... 209
The Korean steel code check ................................................................................................. 209 Material properties ............................................................................................................... 209 Consulted articles ................................................................................................................ 209
Section classification ...................................................................................................... 210 Section properties .......................................................................................................... 211 Buckling length ............................................................................................................... 211 Lateral torsional buckling ............................................................................................... 212 Combined stresses ........................................................................................................ 213 Shear buckling check ..................................................................................................... 213
Supported sections ................................................................................................................. 214
References ............................................................................................................................... 214
BSK 99 ............................................................................................................................. 216
BSK 99 Code check................................................................................................................. 216 Consulted articles ................................................................................................................ 218
Classification of sections ................................................................................................ 219 Effective cross-section properties for class 3 cross-section ........................................... 219 Section properties .......................................................................................................... 219 Section check ................................................................................................................. 219 Compression members .................................................................................................. 219 Stability check for torsional buckling and torsional-flexural buckling .............................. 220 Lateral-torsional buckling ............................................................................................... 221 Use of diaphragms ......................................................................................................... 222 Shear force ( shear buckling) ......................................................................................... 222
Supported sections ................................................................................................................. 223
References ............................................................................................................................... 224
IS 800 ............................................................................................................................... 225
IS:800 Code check .................................................................................................................. 225 Material properties ............................................................................................................... 225 Consulted articles ................................................................................................................ 225
Classification of sections ................................................................................................ 226
Section properties .......................................................................................................... 227 Section check ................................................................................................................. 227 Compression members .................................................................................................. 227 Stability check for torsional buckling and torsional-flexural buckling .............................. 227 Lateral-torsional buckling ............................................................................................... 229 Use of diaphragms ......................................................................................................... 230
Supported sections ................................................................................................................. 230
References ............................................................................................................................... 231
EAE code check ............................................................................................................. 232
Material properties ............................................................................................................... 232 Consulted articles ................................................................................................................ 233
Classification of sections ................................................................................................ 235 Effective cross-section properties for class 4 cross-section ........................................... 235 Section properties .......................................................................................................... 235 Torsion check ................................................................................................................. 235 Built-in beams ................................................................................................................ 235 Compression members .................................................................................................. 236 Lateral-torsional buckling ............................................................................................... 236 Use of diaphragms ......................................................................................................... 236 Combined bending and axial compression .................................................................... 237 Shear buckling check ..................................................................................................... 237
Supported sections ................................................................................................................. 237
References ............................................................................................................................... 238
Calculation of buckling ratio ......................................................................................... 240
Introduction to the calculation of buckling ratio .................................................................. 240
Calculation buckling ratio – general formula ........................................................................ 240
Calculation buckling ratios for crossing diagonals ............................................................. 242 Continuous compression diagonal, supported by continuous tension diagonal ................... 242 Continuous compression diagonal, supported by pinned tension diagonal ......................... 243 Pinned compression diagonal, supported by continuous tension diagonal .......................... 243 Continuous compression diagonal, supported by continuous compression diagonal .......... 244 Continuous compression diagonal, supported by pinned compression diagonal ................. 244 Pinned compression diagonal, supported by continuous compression diagonal ................. 245
Calculation of critical Euler force for VARH elements ......................................................... 246 Definitions ............................................................................................................................ 246 Calculation of the critical Euler force ................................................................................... 246
Calculation buckling ratio for lattice tower members .......................................................... 248 Leg with symmetrical bracing .............................................................................................. 249 Leg with intermediate transverse support ............................................................................ 249 Leg with staggered bracing.................................................................................................. 250 Single Bracing ..................................................................................................................... 250 Single Bracing with SBS (Secondary Bracing System) ....................................................... 251 Cross bracing ...................................................................................................................... 251 Cross bracing with SBS ....................................................................................................... 252 K Bracing ............................................................................................................................. 253 Horizontal Bracing ............................................................................................................... 253
Horizontal Bracing with SBS ................................................................................................ 254 Discontinuous Cross bracing with horizontal member ......................................................... 255
Calculation of buckling ratio – From Stability Analysis ....................................................... 256
References ............................................................................................................................... 256
Calculation of moment factors for LTB ....................................................................... 258
Introduction to the calculation of moment factors ............................................................... 258
Calculation moment factors ................................................................................................... 258 Moment distribution generated by q load ............................................................................. 258 Moment distribution generated by F load............................................................................. 260 Moment line with maximum at the start or at the end of the beam ...................................... 261
References ............................................................................................................................... 261
LTBII: Lateral Torsional Buckling 2nd Order Analysis .............................................. 262
Introduction to LTBII ............................................................................................................... 262
Eigenvalue solution Mcr ......................................................................................................... 262
2nd
Order analysis ................................................................................................................... 264
Supported National Codes ..................................................................................................... 264
Supported Sections ................................................................................................................ 265
Loadings .................................................................................................................................. 267
Imperfections ........................................................................................................................... 267 Initial bow imperfection v0 for DIN and ONORM ................................................................. 268 Initial bow imperfection v0 for EC-EN and EAE ................................................................... 268 Initial bow imperfections v0 and w0 for other supported codes ........................................... 269
LTB Restraints ......................................................................................................................... 270
Diaphragms ............................................................................................................................. 271
Linked Beams .......................................................................................................................... 272
Limitations and Warnings ...................................................................................................... 273
References ............................................................................................................................... 274
Profile conditions for code check ................................................................................ 275
Introduction to profile characteristics ................................................................................... 275
Data for general section stability check ................................................................................ 275
Data depending On the profile shape .................................................................................... 276 I section ............................................................................................................................... 276 RHS ..................................................................................................................................... 277 CHS ..................................................................................................................................... 278 Angle section ....................................................................................................................... 279 Channel section ................................................................................................................... 280 T section .............................................................................................................................. 281 Full rectangular section........................................................................................................ 282 Full circular section .............................................................................................................. 283 Asymmetric I section ........................................................................................................... 284 Z section .............................................................................................................................. 285 General cold formed section ................................................................................................ 286 Cold formed Angle section................................................................................................... 288 Cold formed Channel section .............................................................................................. 289
Cold formed Z section ......................................................................................................... 290 Cold formed C section ......................................................................................................... 291 Cold formed Omega section ................................................................................................ 292 Cold formed C section eaves beam ..................................................................................... 293 Cold formed C Plus section ................................................................................................. 294 Cold formed ZED section..................................................................................................... 295 Cold formed ZED section asymmetric lips ........................................................................... 296 Cold formed ZED section inclined lip ................................................................................... 297 Cold formed Sigma section .................................................................................................. 298 Cold formed Sigma section stiffened ................................................................................... 299 Cold formed Sigma Plus section .......................................................................................... 300 Cold formed Sigma section eaves beam ............................................................................. 301 Cold formed Sigma Plus section eaves beam ..................................................................... 302 Cold formed ZED section both lips inclined ......................................................................... 303 Cold formed I-Plus section................................................................................................... 304 Cold formed IS-Plus section ................................................................................................ 305 Cold formed Sigma section asymmetric .............................................................................. 306 Rail type KA ......................................................................................................................... 307 Rail type KF ......................................................................................................................... 308 Rail type KQ ........................................................................................................................ 309
Warping check ................................................................................................................ 310
Stress check ............................................................................................................................ 310 Calculation of the direct stress due to warping .................................................................... 311 Calculation of the shear stress due to warping .................................................................... 314
Plastic Check ........................................................................................................................... 316
Standard diagrams for warping torque, bimoment and the St.Venant torsion .................. 319 Torsion fixed ends, warping free ends, local torsional loading Mt ........................................ 320 Torsion fixed ends, warping fixed ends, local torsional loading Mt ...................................... 321 Torsion fixed ends, warping free ends, distributed torsional loading mt .............................. 323 Torsion fixed ends, warping fixed ends, distributed torsional loading mt ............................. 324 One end free, other end torsion and warping fixed, local torsional loading Mt .................... 325 One end free, other end torsion and warping fixed, distributed torsional loading mt ........... 326
Decomposition of arbitrary torsion line ................................................................................ 327 Decomposition for situation 1 and situation 3 ...................................................................... 328 Decomposition for situation 2 .............................................................................................. 328
References ............................................................................................................................... 328
Check of numerical sections ........................................................................................ 330
Stress check ............................................................................................................................ 330
Use of diaphragms ......................................................................................................... 331
Adaptation of torsional constant ........................................................................................... 331
References ............................................................................................................................... 332
Section check for built-in beams (IFB, SFB, THQ sections) ...................................... 334
Introduction ............................................................................................................................. 334
Reduction of plastic moment capacity due to plate bending .............................................. 334
Plastic interaction formula for single bending and shear force .......................................... 337
Plastic check for plate in bending ......................................................................................... 338
Stress check for slim floor beams ......................................................................................... 339 Normal stress check ............................................................................................................ 339 Shear stress check in plate.................................................................................................. 339 Torsion check due to unbalanced loading ........................................................................... 340
References ............................................................................................................................... 342
Effective cross-section properties for lattice tower angle members........................ 343
Effective cross-section properties for compressed lattice tower angle members ............ 343
References ............................................................................................................................... 344
Scia Engineer Steel Code Check Theoretical Background
1
Scia Engineer Steel Code Check Theoretical Background
2
EC3 – ENV 1993
EC3 – ENV code check
The beam elements are checked according to the regulations given in
Eurocode 3
Design of steel structures
Part 1 - 1 : General rules and rules for buildings
ENV 1993-1-1:1992
Material properties
For standard steel grades, the yield strength fy and tensile strength fu are defined according to the
thickness of the element (see Ref. 1, art.3.2.2.1.)
(fy, fu in N/mm², t in mm)
t<=40 t<=40 40<t<=100 40<t<=100 100<t<=250 100<t<=250
fy fu fy fu fy fy
S235
S 235
235 360 215 340 175 320
S275
S 275
275 430 255 410 205 380
S355
S 355
355 510 335 490 275 450
S420
S 420
420 520 390 520
S460
S 460
460 550 430 550
Remark : For cold formed section, the values for fy and fu are not influenced by the previous table
Remark : For cold formed sections, the average yield strength fya can be used (by setting the proper data flag in the Cross Section input dialog).
The average yield strength is determined as follows:
Scia Engineer Steel Code Check Theoretical Background
3
with fyb the tensile yield strength = fy
fu the tensile ultimate strength
t the material thickness
Ag the gross cross-sectional area
k is a coefficient depending on the type of forming :
k = 0.7 for cold rolling
k = 0.5 for other methods of forming
n the number of 90° bends in the section
Consulted articles
The cross-section is classified according to Table 5.3.1. (class 1,2,3 or 4). The section is checked for tension (art. 5.4.3.), compression (art. 5.4.4.), shear (art. 5.4.6.) and the combination of bending, shear and axial force (art. 5.4.9.).
For the stability check, the beam element is checked according to art.5.5.. The following criteria are considered :
for compression : art. 5.5.1.
for lateral torsional buckling : art. 5.5.2.
for bending and axial compression : art. 5.5.4.
The shear buckling resistance is checked using the simple post-critical method from art. 5.6.3.
A more detailed overview for the used articles is given for part 5.3., 5.4., 5.5. and 5.6. in the following table. The chapters marked with “x” are consulted. The chapters marked with (*) have a supplementary explanation the following chapters.
5.3. Classification of cross sections
5.3.1. Basis x
5.3.2. Classification x
5.3.3. Cross-section requirements for plastic global analysis
5.3.4. Cross-section requirements when elastic global analysis is used
5.3.5. Effective cross-section properties for class 4 cross-section x (*)
5.3.6. Effects of transverse forces on webs
5.4. Resistance of cross-sections
5.4.1. General x
5.4.2. Section properties (*)
5.4.3. Tension x
5.4.4. Compression x
Scia Engineer Steel Code Check Theoretical Background
4
5.4.5. Bending moment x (*)
5.4.6. Shear x
5.4.7. Bending and shear x
5.4.8. Bending and axial force x
5.4.9. Bending, shear and axial force x (*)
5.4.10. Transverse forces on webs
5.5. Buckling resistance of members
5.5.1. Compression members x (*)
5.5.2. Lateral-torsional buckling x (*)
5.5.3. Bending and axial tension
5.5.4. Bending and axial compression x (*)
5.6. Shear buckling resistance
5.6.1. Basis x
5.6.2. Design methods
5.6.3. Simple post-critical method x
5.6.4. Tension field method
5.6.5. Intermediate transverse stiffeners
5.6.6. Welds
5.6.7. Interaction between shear force, bending moment and axial force x
5.9. Built-up compression members
5.9.3. Battened compression members
5.9.3.1. Application x(*)
5.9.3.2. Constructional details
5.9.3.3. Second moment of inertia x
5.9.3.4. Chord forces ar mid-length x
5.9.3.5. Buckling resistance of chords x
5.9.3.6. Moments and shear due to battening x
Classification of sections
For each intermediary section, the classification is determined and the proper section check is performed. The classification can change for each intermediary point.
For each load case/combination, the critical section classification over the member is used to perform the stability check. So, the stability section classification can change for each load case/combination.
However, for non-prismatic sections, the stability section classification is determined for each intermediary section.
Scia Engineer Steel Code Check Theoretical Background
5
Effective cross-section properties for class 4 cross-section
The calculation of the effective area is performed with the direct method (sigma_d = fy,k).
For each intermediary section, the classification (and if necessary, the effective area ) is determined and the proper section check is performed. The classification (and effective area) can change for each intermediary point. The most critical check is displayed on the screen.
For each load case and combination, the most critical effective area properties are saved :
Aeff is the effective area of the cross section when subject to uniform compression. Weff is the effective section modulus of the cross-section when subject only to moment about the relevant axis. eN is the shift of the relevant centroidal axis when the cross section is subject to uniform compression.
With these critical properties, the stability check is performed.
For non-prismatic elements, the effective area properties are calculated on each intermediary section, also for the stability check.
For angle sections, see chapter 'Effective cross-section properties for compressed lattice tower angle members'.
Section properties
5.4.2.2 : The net area properties are only taken into account in the Tension Check in case of lattice tower angle sections with bolted diagonal connections if the LTA functionality has been activated. For more information, reference is made to the Theoretical Background Bolted Diagonal Connections. In all other cases the net area properties are not taken into account.
5.4.2.3 : The shear lag effects are neglected .
Bending moment
5.4.5.3 : The holes for fasteners are neglected.
Bending, shear and axial force
The reduced design plastic resistance moment for the interaction of bending, shear and axial force,
is taken from Table 5.17. Ref. 2
Torsion check
For the cross section check inclusive torsion and warping, we refer to Chapter 'Warping check'.
Scia Engineer Steel Code Check Theoretical Background
6
Built-in beams
For built-in beam sections (IFB, SFB, THQ sections), proper section checks are performed, taking into account the local plate bending. See Chapter „Section check for built-in beams (IFB, SFB, THQ sections)‟
Compression members
5.5.1.5 For the calculation of the buckling length, we refer to chapter "Calculation of buckling ratio"
The buckling properties for a VARH element are calculated by using the critical Euler force for this member (see chapter “Calculation of critical Euler force for VARH elements”).
The buckling curves for steel grade S420 and S460 are taken from Ref.[5], Annex D.
Lateral-torsional buckling
For I sections (symmetric and asymmetric), RHS (Rectangular Hollow Section) sections and CHS (Circular Hollow Section) sections, the elastic critical moment for LTB Mcr is given by the general
formula F.2. Annex F Ref. 1. For the calculation of the moment factors C1, C2 and C3 we refer to "Calculation of moment factors for LTB".
For the other supported sections, the elastic critical moment for LTB Mcr is given by
z2
t
z2
z2
EI
L²GI
I
Iw
L
EIMcr
with E the modulus of elasticity
G the shear modulus
L the length of the beam between points which have lateral restraint (= lLTB)
Iw the warping constant
It the torsional constant
Iz the moment of inertia about the minor axis
See also Ref. 3, part 7 and in particular part 7.7. for channel sections.
Haunched sections (I+Ivar, Iw+Plvar, Iw+Iwvar, Iw+Ivar, I+Iwvar) and composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
For advanced Lateral-torsional buckling analysis, see chapter „LTBII: Lateral Torsional Buckling 2nd Order Analysis‟.
Scia Engineer Steel Code Check Theoretical Background
7
Use of diaphragms
See Chapter 'Adaptation of torsional constant'.
Shear Buckling check
Composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
Shear buckling check for cold formed sections
See Ref.[4] 5.8 :
The shear resistance of the web Vw,Rd shall be taken as the lesser of the shear buckling resistance Vb,Rd and the plastic shear resistance Vpl,Rd.
The shear resistance of the web should be checked if:
The shear buckling resistance Vb,Rd is given by
The plastic shear resistance Vpl,Rd is given by
with w
the relative web slenderness
fyb the basic yield strength
fy the average yield strength
sw the web length
t the web thickness
E the modulus of elasticity
fbv the shear buckling strength
M0 the partial safety factor for resistance of cross-sections where failure is caused by yielding (=1.1)
M1 the partial safety factor for resistance of cross-sections where failure is caused by buckling (=1.1)
E
f
t
s346.0
f
f83.0
ybww
_
1M
0M
y
ybw
_
1M
bvwRd,b
ftsV
3
ftsV
0M
yw
Rd,pl
Scia Engineer Steel Code Check Theoretical Background
8
The value for fbv is given by :
fbv
<1.40
w
ybf
48.0
1.40
Remarks:
For an arbitrary composed section, the total Vb,Rd and Vpl,Rd is taken as the sum of resistance of
each web, where the angle (teta) is larger than 45° (see figure)
The basic yield strength is taken equal to the average yield strength.
Stability check for torsional buckling and torsional-flexural buckling
See Ref.[4] 6.2.3.
The design buckling resistance Nb,Rd for torsional or torsional-flexural buckling shall be obtained using buckling curve b, and with relative slenderness given by :
w
_
²
f67.0
w
_
yb
Scia Engineer Steel Code Check Theoretical Background
9
with A the ratio Aeff/A (see Ref.[1] 5.5)
fyb the basic yield strength
cr the critical stress
cr,T the elastic critical stress for torsional buckling
cr,TF
the elastic critical stress for torsional-flexural buckling
G the shear modulus
E the modulus of elasticity
IT the torsion constant of the gross section
CM the warping constant
iy the radius of gyration about yy-axis
iz the radius of gyration about zz-axis
lT the buckling length of the member for torsional buckling
y0 the position of the shear center
ly the buckling length for flexural buckling about the yy-axis
Scia Engineer Steel Code Check Theoretical Background
10
Bending and axial compression
When the torsional buckling and/or the torsional-flexural buckling is governing, the formula (6.12) from Ref.[4], article 6.5.2. is applied.
Battened compression members
The following section pairs are supported as battened compression member :
(1) 2I
(2) 2Uo
(3) 2Uc
Two links (battens) are used.
Scia Engineer Steel Code Check Theoretical Background
11
The following additional checks are performed:
- buckling resistance check around weak axis of single chord with Nf,Sd
- section check of single chord, using internal forces :
4
aVM
2
V V
N N
sG
sG
SDf,G
- section check of single batten, using the internal forces :
4
aVM
2h
aV T
s
0
s
For the calculation of Vs, the value of Ms is increased with the value of the internal force Mzz.
l
a
ho
Scia Engineer Steel Code Check Theoretical Background
12
EC3 – ENV Fire Resistance
Fire actions effect Efi
The design effects of actions for the fire situation Efi,d,t are taken from the results of the analysis. It is recommended to use the accidental combination rules, for calculating the internal forces used in the fire resistance check.
The accidental combination is given by
)f(AQQG dj,kj,21,k1,1kGA
with Gk characteristic values of permanent actions
Qk,1 characteristic value of the (main) variable action
Qk,j characteristic values of the other variable actions
Af(d) design values of actions from fire exposure
GA partial safety factor for permanent actions in the accidental situation
1,1 2,j combination coefficients
Material properties
The material properties are depending on the steel temperature.
Strength and deformation properties :
a
,a,E
y
,p,p
y
,y,y
EE
k
f
fk
f
fk
The variation in function of the steel temperature of the value for yield strength ky,, proportional limit
kp, and modulus of elasticity kE, is given by tables in Ref.[6], table 3.1.
For cold formed members ky, is taken from Ref.[7], table III.2.5.
Scia Engineer Steel Code Check Theoretical Background
13
In the simplified calculation method, the following default properties are considered to be constant during the analysis:
unit mass a 7850 kg/m³
thermal elongation l/l 14 x 10-6
(a-20)
thermal conductivity a 45 W/mK
Temperature analysis - Thermal actions
In this part, the nominal temperature-time curves and the related net heat flux are described. See Ref.[8], Section 4, and Ref.[7], II.2.2.
Nominal temperature-time curve
The following temperature-time curves can be selected :
with t time in [min]
g gas temperature in [°C]
c the coefficient of heat transfer by convection
ISO 834 curve
external fire curve
Scia Engineer Steel Code Check Theoretical Background
14
hydrocarbon curve
smoldering fire curve
during 21 minutes, followed by the standard ISO 834 curve
Net heat flux
r,netr,nc,netc,nd,net hhh
with hnet,d the net heat flux
hnet,c the convective heat flux
hnet,r the radiative heat flux
n,c factor depending on NAD [1.0]
n,r factor depending on NAD [1.0]
with configuration factor [1.0]
res resultant emissivity
= f m
f emissivity related to fire compartment
= [0.800]
m emissivity related to surface material
= [0.625]
r = g
gas temperature in [°C]
m surface temperature of member in [°C]
c coefficient of heat transfer by convection
Scia Engineer Steel Code Check Theoretical Background
15
Steel Temperature
The increase of temperature a,t in an unprotected steel member during a time interval t
thc
V/Ad,net
aa
mt,a
with Am the exposed surface area per unit length [m²/m]
V the volume of the member per unit length [m³/m]
The factor Am/V should not be taken as less than 10m-1
ca the specific heat of steel [J/kgK]
hnet,d the net heat flux per unit area [W/m²]
t the time interval [seconds]
The value should not be taken as more than 5 seconds
a the unit mass of steel [kg/m³]
The increase of temperature a,t in an insulated steel member during a time interval t
with Ap the area of fire protection material per unit length [m²/m]
V the volume of the member per unit length [m³/m]
ca the specific heat of steel [J/kgK]
cp the specific heat of fire protection material [J/kgK]
dp the thickness of the fire protection material [m]
t the time interval [seconds]
The value should not be taken as more than 30 seconds
a the unit mass of steel [kg/m³]
p the unit mass of fire protection [kg/m³]
a,t the steel temperature at time t
g,t the ambient gas temperature at time t
g,t the increase of the ambient gas temperature during the time interval
p the thermal conductivity of the fire protection material [W/mK]
The value a,t 0.0
For the increase of temperature a,t in an insulated steel member with intumescent coating, we refer to the NEN specifications, Chapter 'Steel Temperature'.
Scia Engineer Steel Code Check Theoretical Background
16
Calculation model
The calculation can be performed in 2 domains :
- strength domain
- temperature/time domain
In the strength domain, the strength Rfi,d,t(unity check) is calculated after a given time t (e.g. strength
after 45 min). In the temperature/time domain, the critical steel temperature cr,d is computed. From this critical temperature, the fire resistance time tfi,d is calculated (the time domain).
Code Check
The section and stability checks (buckling, lateral torsional buckling) are performed according to the regulations given in 'ENV 1993-1-2:1995' and/or 'Model Code on Fire Engineering - ECCS N° 111'. The checks are performed in the resistance domain or in the temperature/time domain..
Torsional buckling and shear buckling are not considered.
For each member, the classification of the cross section, the section check and the stability check are performed.
The following checks are executed :
EC3-1-2 :
- classification of cross section : art. 4.2.2.
- resistance for tension members : art. 4.2.3.1
- resistance for compression members (class 1,2 or 3) : art. 4.2.3.2.
- resistance for beams (class 1,2) : art. 4.2.3.3.
- resistance for beams (class 3) : art.4.2.3.4.
- resistance for members (class 1,2,3) subject to bending and compression : art. 4.2.3.5.
- critical temperature : art. 4.2.4.
ECCS Model Code on Fire Engineering
- resistance for tension members : art. III.5.2.
- resistance for compression members (class 1,2 or 3) : art. III.5.3.
- resistance for beams (class 1,2) : art. III.5.4.
- resistance for beams (class 3) : art. III.5.5.
- resistance for members (class 1,2,3) subject to bending and compression : art. III.5.6.
- resistance for members (class 4) : art. III.5.7.
- critical temperature : art. III.5.8.
Scia Engineer Steel Code Check Theoretical Background
17
Supported sections
I Symmetric I shapes (IPE, HEA, HEB, ….)
RHS Rectangular Hollow Section
CHS Circular Hollow Section
L Angle section
U Channel section
T T section
PPL Asymmetric I shapes
Z Z section
RS Rectangular section
Cold formed section
COM Composed section in PRIMAWIN
O Solid tube
NUM Numerical section
The necessary data conditions for these sections are described in chapter "Profile conditions for code check
".
The COM and NUM sections are not read out of the profile library.
I
RHS
CHS
L
U
T
PPL
RS
Z
O
COM
NUM
Classification x x x x x x x x (1) x (1) (1) (1)
Section check class 1 x x x
Section check class 2 x x x
Section check class 3 x x x x x x x x x x x x x
Section check class 4 x x x x x x
Stability check class 1
x x x
Stability check class 2
x x x
Stability check class 3
x x x x x x x x x x x x x
Stability check class 4
x x x x x x
Shear buckling check x x x x
(1) sections are classified as class 3 cross section by default.
Scia Engineer Steel Code Check Theoretical Background
18
References
1 Eurocode 3
Design of steel structures
Part 1 - 1 : General rules and rules for buildings
ENV 1993-1-1:1992, 1992
2 Essentials of Eurocode 3
Design Manual for Steel Structures in Building
ECCS - N° 65, 1991
3 R. Maquoi
ELEMENTS DE CONSTRUCTIONS METALLIQUE
Ulg , Faculté des Sciences Appliquées, 1988
[4] ENV 1993-1-3:1996
Eurocode 3 : Design of steel structures
Part 1-3 : General rules
Supplementary rules for cold formed thin gauge members and sheeting
CEN 1996
[5] Eurocode 3
Design of steel structures
Part 1 - 1/ A1 : General rules and rules for buildings
ENV 1993-1-1:1992/A1, 1994
[6] Eurocode 3
Design of steel structures
Part 1 - 2 : General rules - Structural fire design
ENV 1993-1-2:1995, 1995
[7] Model Code on Fire Engineering
ECCS - N° 111
May 2001
[8] Eurocode 1
Basis of design and actions on structures
Part 2-2 : Actions on structures - Actions on structures exposed to fire
ENV 1991-2-2:1995
Scia Engineer Steel Code Check Theoretical Background
19
EC3 – EN 1993
EC3 – EN Code check
The beam elements are checked according to the regulations given in:
Eurocode 3
Design of steel structures
Part 1 - 1: General rules and rules for buildings
EN 1993-1-1:2005
Corrigendum
EN 1993-1-1:2005/AC:2006
Corrigendum
EN 1993-1-1:2005/AC:2009
Material properties
For standard steel grades, the yield strength fy and tensile strength fu are defined according to the
thickness of the element (see Ref. 1, table 3.1.)
Within the material properties the rules for reduction of the yield strength in function of the thickness can be edited.
Remark: For cold formed sections, the reductions of the yield strength in function of the thickness are not applied.
Scia Engineer Steel Code Check Theoretical Background
20
Consulted articles
The beam elements are checked according to the regulations given in "Eurocode 3: Design of steel structures - Part 1-1: General rules and rules for buildings - EN 1993-1-1:2005".
The cross-sections are classified according to Table 5.2. All classes of cross-sections are included. For class 4 sections (slender sections) the effective section is calculated in each intermediary point, according to prEN 1993-1-5:2003, Chapter 4.4 .
The stress check is taken from art. 6.2.: the section is checked for tension (art. 6.2.3.), compression (art. 6.2.4.), bending (art. 6.2.5.), shear (art. 6.2.6.), torsion (art.6.2.7.) and combined bending, shear and axial force (art. 6.2.8., art.6.2.9. and art.6.2.10.).
The stability check is taken from art. 6.3.: the beam element is checked for buckling (art. 6.3.1.), lateral torsional buckling (art. 6.3.2.), combined bending and axial compression (art. 6.3.3.) and battened compression members (art. 6.4).
The shear buckling is checked according to EN 1993-1-5:2006, Chapter 5.
For I sections, U sections and cold formed sections warping can be considered.
A check for critical slenderness and torsion moment is also included.
For integrated beams, the local plate bending is taken into account for the plastic moment capacity and the bending stresses in the section. The out-of-balance loading is checked.
A more detailed overview for the used articles is given in the following table. The chapters marked with “x” are consulted. The chapters marked with (*) have a supplementary explanation in the following chapters.
EN 1993-1-1
5.5 Classification of cross section (*)
5.5.1. Basis x
5.5.2. Classification x
6. Ultimate limit states
6.1. General x
6.2. Resistance of cross-sections
6.2.1 General
x
6.2.2 Section properties x(*)
6.2.3 Tension x
6.2.4 Compression x
6.2.5 Bending moment x
6.2.6 Shear x
6.2.7 Torsion x(*)
6.2.8 Bending and shear x
6.2.9 Bending and axial force x
6.2.10 Bending, shear and axial force x
6.3. Buckling resistance of members
6.3.1 Uniform members in compression
x(*)
6.3.2 Uniform members in bending x
6.3.3 Uniform members in bending and axial compression x(*)
Scia Engineer Steel Code Check Theoretical Background
21
6.4 Uniform built-up compression members
6.4.1 General
6.4.3 Battened compression members
Annex A:Method 1:Interaction factors kij for interaction formula in 6.3.3.(4)
x
Annex B:Method 2:Interaction factors kij for interaction formula in 6.3.3.(4)
x
EN 1993-1-3
6.1.2. Axial tension x
6.1.3. Axial compression x
6.1.5. Shear force x
6.1.6. Torsional moment x
EN 1993-1-5
4.4. Plate elements without longitudinal stiffeners x
5. Resistance to shear
5.1. Basis x
5.2. Design resistance x
5.3. Contribution from webs x
5.4. Contribution from flanges x
5.5. Verification x
7.1. Interaction between shear force, bending moment and axial force
x
Scia Engineer Steel Code Check Theoretical Background
22
Classification of sections
For each intermediary section, the classification is determined and the proper section check is performed. The classification can change for each intermediary point.
For each load case/combination, the critical section classification over the member is used to perform the stability check since stability effects are related to the whole member and not to a single cross-section.
To determine this critical classification, all sections in the Ly and Lz system lengths of the buckling
system are checked and the worst classification is used as the critical. Note that only sections on the actual member are used so in case the system length spans multiple members, only the sections of the actual member are used to determine the critical classification.
For non-prismatic sections, the stability section classification is determined for each intermediary section.
The classification according to EN 1993-1-1 Table 5.2 is done using the general formulation for
„parts subjected to bending and compression‟. In this way the beneficial effect of tension is also accounted for.
- For example, for class 1, in case of bending combined with a tensile axial force the cross-
section is less subjected to compression thus leading to an value smaller than 0,5. This in
turn leads to higher classification limits than the case for „part subject to bending‟.
- For example, for class 1, in case of an asymmetrical section in bending, depending on the geometry of the section the cross-section is less subjected to compression thus leading to an
value smaller than 0,5. This in turn leads to higher classification limits than the case for
„part subject to bending‟.
Specifically for sheet welded Iw and Iwn cross-sections the weld size as inputted in the cross-
section is accounted for in the classification as follows:
With: Hw Web height
a Weld size
Scia Engineer Steel Code Check Theoretical Background
23
Effective cross-section properties for class 4 cross-section
The calculation of the effective area is performed with the direct method (sigma_d = fy,k).
For each intermediary section, the classification (and if necessary, the effective area ) is determined and the proper section check is performed. The classification (and effective area) can change for each intermediary point. The most critical check is displayed on the screen.
For each load case and combination, the most critical effective area properties are saved :
Aeff is the effective area of the cross section when subject to uniform compression. Weff is the effective section modulus of the cross-section when subject only to moment about the relevant axis. eN is the shift of the relevant centroidal axis when the cross section is subject to uniform compression.
With these critical properties, the stability check is performed.
For non-prismatic elements, the effective area properties are calculated on each intermediary section, also for the stability check.
Section properties
The net area properties are not taken into account.
The shear lag effects are neglected.
Bending, shear and axial force
Plastic Interaction for CHS
The 2009 correction sheet to EN 1993-1-1 Ref.[11] specifies an interaction formula for CHS
sections. This formula however does not include the influence of shear. Table 5.17. Ref. 15 specifies the full interaction formula including also shear.
In addition for CHS sections the resultant shear force and resultant moment are determined.
The moment resistance is determined taking into account this resultant shear force.
The unity check then becomes the following:
Plastic Interaction for other sections
By default the plastic combined interaction check according to EN 1993-1-1 article 6.2.9.1 formula (6.41) is only executed for I-sections, full rectangular and RHS sections since Eurocode only gives formulas to reduce the plastic bending resistance of these sections in case of a normal force.
However, in case there is no normal force, no reduction of the plastic bending resistance for normal force is needed and thus the plastic interaction can be used also for other cross-sections.
This is used specifically for the following cross-sections:
Angle section (Form code 4)
U-section (Form code 5)
T-section (Form code 6)
Asymmetric I-section (Form code 101)
In case these sections are classified as class 1 or 2 and no normal force is present the combined interaction is checked according to formula (6.41) with Alpha and Beta set to 1.00.
Scia Engineer Steel Code Check Theoretical Background
24
Torsion check
For the cross section check including torsion and warping, we refer to Chapter 'Warping check'.
Built-in beams
For built-in beam sections (IFB, SFB, THQ sections), proper section checks are performed, taking into account the local plate bending. See Chapter „Section check for built-in beams (IFB, SFB, THQ sections)‟.
Compression members
For the calculation of the buckling length, we refer to chapter "Calculation of buckling ratio"
The buckling properties for a VARH element are calculated by using the critical Euler force for this member (see chapter “Calculation of critical Euler force for VARH elements”).
Buckling curves are determined according to EN 1993-1-1 Table 6.2.
In case the cross-section of the member is not listed in this table, the buckling curves are taken from the user inputted values in the cross-section properties.
For non-prismatic members with cross-sections that are not listed in Table 6.2 all generated
sections will receive the user inputted values of the buckling curves of the first section in the span.
Scia Engineer Steel Code Check Theoretical Background
25
Lateral-torsional buckling
For I sections (symmetric and asymmetric), and Rectangular Hollow Sections (RHS), the elastic critical moment for Lateral-Torsional Buckling Mcr is given by the general formula F.2. Annex F Ref.
10.
For any other section, the elastic critical moment for Lateral-Torsional Buckling Mcr is given by:
With E Modulus of elasticity
G Shear modulus
L Length of the beam between points which have lateral restraint (= lLTB)
Iw Warping constant
It Torsional constant
Iz Moment of inertia about the minor axis
C1 Moment factor which by default is taken as 1,00. Within the Steel setup it can be set to use the calculated value of C1 instead of 1,00.
See also Ref. 4, part 7 and in particular part 7.7. for channel sections.
Circular hollow sections (CHS) are taken as non-susceptible to Lateral Torsional Buckling.
Rectangular hollow sections are classified as non-susceptible to Lateral Torsional Buckling if the following condition is fulfilled (Ref.[9] pp.119).
With: h Height of RHS section
b Width of RHS section
Relative slenderness for weak axis flexural buckling
Haunched sections (I+Ivar, Iw+Plvar, Iw+Iwvar, Iw+Ivar, I+Iwvar) and composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
For advanced Lateral-torsional buckling analysis, see chapter „LTBII: Lateral Torsional Buckling 2nd Order Analysis‟.
Scia Engineer Steel Code Check Theoretical Background
26
Determination of the moment factors C1, C2 and C3
The coefficients C1, C2 and C3 can be calculated according to three different methods:
ENV 1993-1-1 Annex F
ECCS 119/Galea
Lopez, Yong, Serna
By default the method according to ECCS 119/Galea is applied.
The following paragraphs give information on these methods.
ENV 1993-1-1 Annex F
When this setting is chosen, the moment factors are determined according to ENV 1993-1-1 Annex F Ref.[10].
Detailed information can be found in chapter "Calculation of moment factors for LTB".
ECCS 119/Galea
When this setting is chosen, the moment factors are determined according to ECCS 119 Annex B Ref.[9].
The figures given in this reference for C1 and C2 in case of combined loading originate from Ref.[28]
which in fact also gives the tabulated values of those figures as well as an extended range.
The actual moment distribution is compared with several standard moment distributions. These standard moment distributions are moment lines generated by a distributed q load, a nodal F load, or where the moment line is maximum at the start or at the end of the beam.
The standard moment distribution which is closest to the actual moment distribution, is taken for the calculation of the factors C1 and C2.
Linear Moment
In case of a linear moment diagram the C1 coefficient is determined using formula (301) of ECCS 119
Annex B Ref.[9].
The coefficient C2 is taken as zero in this case.
Point Loading
In case of Point loading the coefficients C1 and C2 are calculated using tables 5-8 of Galea Ref.[28].
A double interpolation is used for intermediate values.
Line Loading
In case of Line loading the coefficients C1 and C2 are calculated using tables 1-4 of Galea Ref.[28].
A double interpolation is used for intermediate values.
In case k differs from 1.00 the C1 and C2 values determined from Galea Ref.[28] are overruled by
the values from ECCS 119 Annex B Ref.[9] tables 63 and 64
For all cases the factor C3 is taken from ECCS 119 Annex B Ref.[9] tables 63 and 64. The C3 value is determined based on the case of which the C1 value most closely matches the table value.
Scia Engineer Steel Code Check Theoretical Background
27
The table for C3 uses the value f which is taken as 0 by default.
For asymmetrical I-sections (Form code 101) f is calculated as follows:
Ifc and Ift concern the moments of inertia of the compression ( c ) and tension ( t ) flange about the
minor axis.
For this method f should be within the following range:
When this is not the case f is set to the respective limit and a warning is given.
I-section Cantilevers
ECCS 119 Annex B Ref.[9] tables 65 to 68 give values for C1, C2 and C3 for I-section cantilevers.
These coefficients are used only in case the following conditions are met:
- The member concerns a cantilever.
A cantilever is defined as a member at the end of a buckling system which has free ends for both buckling about the y-y and z-z axis. In addition, the LTB length should correspond to the full system length of the buckling system.
- The cross-section is an I-section (Form code 1) or Asymmetric I-section (Form code 101).
This method differentiates between „warping prevented‟ and „warping free‟ at the fixed end. This setting is taken from the buckling system.
This method uses the value f which is calculated as specified above.
For this method f should be within the following range:
When this is not the case f is set to the respective limit and a warning is given.
This method uses the coefficient which is defined as follows:
With: L System length for LTB
E Modulus of Young
G Shear modulus
Iz Inertia about the weak axis
It Torsion constant
hs Distance defined as follows:
Form Code 1: H - t
Form Code 101: H – 0,5 * tt – 0,5 * tb
Scia Engineer Steel Code Check Theoretical Background
28
should be within the following range:
When this is not the case is set to the respective limit and a warning is given.
In addition this method should be used in combination with k equal to 2,00 and kw equal to 1,00
When this is not the case an additional warning is given.
Scia Engineer Steel Code Check Theoretical Background
29
Lopez, Yong, Serna
When this setting is chosen, the moment factors are determined according to Lopez, Yong, Serna Ref.[29].
When using this method the coefficients C2 and C3 are set to zero.
The coefficient C1 is calculated as follows:
With: k1 Taken equal to kw
k2 Taken equal to kw
M1, M2, M3, M4, M5 The moments My determined on the buckling system in the given sections as shown on the above figure.
These moments are determined by dividing the beam into 10 parts (11 sections) and interpolating between these sections.
Mmax The maximal moment My along the LTB system.
This method is only supported in case both k and kw equal 0.50 or 1.00
Scia Engineer Steel Code Check Theoretical Background
30
Modified design rule for LTB of Channel sections
In case this setting is activated within the Steel Setup, the reduction factor for Lateral-Torsional Buckling of Channel sections is determined according to Ref.[22].
More specifically the calculation is done as follows:
This Modified design rule is applied only in case the following conditions are met:
- The section concerns a Channel section (Form Code 5)
- The General Case is used for LTB (Not the Rolled and Equivalent Welded Case) - 15 <= Lltb/h <= 40 (with Lltb the LTB length and h the cross-section height)
Correction factor kc
In case Lateral-Torsional Buckling curves for the „Rolled and equivalent welded‟ case are used according to EN 1993-1-1 article 6.3.2.3 the correction factor kc can be determined in two ways:
By default kc is taken from Table 6.6
Alternatively, kc can be determined from C1 as follows Ref.[30]:
Use of diaphragms
See Chapter 'Adaptation of torsional constant'.
Scia Engineer Steel Code Check Theoretical Background
31
Combined bending and axial compression
For prismatic members the value My,Ed is the maximum value of the bending moment around the strong axis in the member. The value Mz,Ed is the maximum value of the bending moment around the weak axis in the member.
For non-prismatic sections, the values My,Ed and Mz,Ed are the concurrent bending moments for each intermediary section.
When Torsional(-Flexural) buckling is governing the z value in equation (6.62) is taken
as the minimum of z and TF.
Interaction Method 1 – Annex A
For Czz the corrected formula given in correction sheet EN 1993-1-1:2005/AC:2009 Ref.[11] is used:
Interaction Method 2 – Annex B
Doubly symmetric I sections which have a reduction factor for Lateral Torsional Buckling LT equal to 1,00 are classified as non-susceptible to torsional deformations.
Circular hollow sections are classified as non-susceptible to torsional deformations.
Rectangular hollow sections are classified as non-susceptible to torsional deformations if the following condition is fulfilled (Ref.[9] pp.119).
With: h Height of RHS section
b Width of RHS section
Relative slenderness for weak axis flexural buckling
Shear buckling check
Composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
Scia Engineer Steel Code Check Theoretical Background
32
Battened compression members
The following section pairs are supported as battened compression member:
(1) 2I
(2) 2Uo
(3) 2Uc
This specifically concerns hot rolled sections i.e. cold-formed pair sections are not supported.
Battened compression members are evaluated according to EN 1993-1-1 article 6.4.1 and 6.4.3.
Two links (battens) are used.
The following additional checks are performed:
Section check of chord as beam in field between battens
This check is executed according to articles 6.4.3.1 & 6.2.9.1 using the following internal forces:
4
aVM
2
V V
N N
EdG
Ed
G
Edch,G
With: Nch,Ed Chord force according to formula (6.69)
VEd Shear force in the built-up member according to formula (6.70)
a Distance between battens
For I-sections a classification is made which thus supports both an elastic or plastic interaction. For U-sections always an elastic interaction is made.
Scia Engineer Steel Code Check Theoretical Background
33
Buckling check of chord
This concerns a weak axis buckling check of a single chord according to articles 6.4.3.1 & 6.3.1.1 using chord force Nch,Ed.
Section check of single batten
An elastic section check of a single batten is executed according to articles 6.4.3.1, 6.2.9.2 & 6.2.6 using the following forces:
4
aVM
2h
aV T
Ed
0
Ed
With: VEd Shear force in the built-up member according to formula (6.70)
a Distance between battens
h0 Distance between centroids of chords
l
a
ho
Scia Engineer Steel Code Check Theoretical Background
34
Plate girders with sinusoidal corrugated webs
Plate girders with sinusoidal corrugated webs (“SIN beams”) are covered in EN 1993-1-5 Annex D. The method given in this chapter is specified in Ref.[33]. Background information can be found in Ref.[16].
The check is executed for sheet welded cross-sections of type Iw c and Iwn c. The corrugations are taken
to be perpendicular to the upper flange. The dimensioning of corrugated web girders is executed for the in plane effects NEd, Vz,Ed and My,Ed.
Transformation of internal forces
For every point of the plate girder the chord forces N,og and N,ug are found by transformation. These chord forces are still parallel to the member axis while the shear force is orthogonal to the axis. The following angles are defined: - α = the slope of the lower chord against the upper chord - β = the angle between the centre line and chords.
The shear force Vz is decomposed into a corrugation-parallel component V* and an axis-parallel component N(V)*.
Scia Engineer Steel Code Check Theoretical Background
35
N(V)* can be added directly to the calculated normal force N. The chord forces can now be determined as
follows:
With: A,og Area of the upper flange A,ug Area of the lower flange H,steg Web height t,og Thickness of the upper flange t,ug Thickness of the lower flange
From the chord forces the chord-parallel components and the corrugation-parallel components are determined. For the upper chord this becomes:
For the lower chord the following intermediate step is used:
The actual force in the lower chord is then:
The actual component of the shear force can then be written as:
Scia Engineer Steel Code Check Theoretical Background
36
The chord forces Nog* and Nug* are now known. By summation of the V* and V(Nog)* and V(Nug)*
components the total shear force is obtained.
Resistance of sinusoidal corrugated web girders
The normal force and bending moment are taken by the flanges while the shear force is taken by the corrugated web.
Flanges
For the flanges the following limits are checked:
- Yielding - Local buckling - Global buckling
Yielding
NRd,yield = bf * tf * fy / M0
With: bf Flange width
tf Flange thickness
fy Yield strength
M0 Partial safety factor
Local buckling
Local buckling of the compression flange is checked according to EN 1993-1-5 article 4.4.
To avoid local buckling the slenderness is limited to 0,748. By substituting this into the formula for the slenderness the following limit is obtained for the width:
For a sinusoidal corrugated web member the total flange width thus becomes:
The resistance for local buckling can then be written out as:
NRd,local = b * tf * fy / M0
Scia Engineer Steel Code Check Theoretical Background
37
Global buckling
Global buckling of the compression flange (Lateral-Torsional Buckling) is checked according to EN 1993-1-1 article 6.3.2.4:
This is written out to the following resistance for the compression flange:
With: b Flange width
t Flange thickness
fy Yield strength
E Modulus of Young
Lc Length between lateral restraints (LTB length)
kc Correction factor according to EN 1993-1-1 Table 6.6
The design value can then be written out as:
NRd,global = NRk / M1
With: M1 Partial safety factor
Web
For the web the shear resistance is determined according to EN 1993-1-5 Annex D article D2.2:
Where c is taken as the lesser of the reduction factors for local buckling c,l and global buckling
c,g.
According to Ref.[34] it was found by testing and FEM that no local buckling occurs for all actually produced beams with sinusoidal corrugated webs. Therefore only the reduction factor for global
buckling c,g needs to be accounted for.
Scia Engineer Steel Code Check Theoretical Background
38
With: fy Yield strength
E Modulus of Young
Poisson ratio
tw Web thickness
hw Web depth
Iz Second moment of area of one corrugation of length w, calculated as:
a3 Height of a sinus wave
Taken as 40 mm for tw < 3 mm
Taken as 43 mm for tw ≥ 3 mm
w Length of the projection of a half wave
s Unfolded length of a half wave
Taken as 178 mm for tw < 3 mm
Taken as 182 mm for tw ≥ 3 mm
Scia Engineer Steel Code Check Theoretical Background
39
Moments on columns in simple construction
This NCCI presents a method for determining the moments on columns in simple construction due to the eccentricity of the beam-to-column joints. This method is intended for braced frames with nominally pinned joints. The method is detailed in Ref.[31] and [32].
Conditions
In case the setting is activated in the Steel Setup the additional moments will be calculated on columns in which the following conditions are satisfied:
- The column cross-section concerns an I-section (Form code 1) or RHS section (Form code 2)
- The column has structural type Column, Gable column or Secondary column
- The column is uniform i.e. does not have arbitrary sections or haunches
- Only connected beams with structural type Beam or Rafter are accounted for. In addition these
beams should have a hinge at the side where they are connected to the column.
- There can maximally be two connected beams in the same plane in the same node. These two
connected beams must have the same X-axis direction of their LCS.
Additional moments
When the above conditions are satisfied the additional moments are calculated in the following way:
With: Rb1,Ed Shear force in the considered plane in the connected beam at the
specified distance
h Profile height for an I-section
Profile height or width for an RHS-section
tw Web thickness for an I-section
Scia Engineer Steel Code Check Theoretical Background
40
The distribution of the additional moments to the upper and lower column sections is carried out in
proportion to their stiffness, except where the ratio of the stiffnesses (I/L) does not exceed 1.5,
when the moments may be shared equally. This is illustrated on the following picture:
With: MU Distributed moment to the upper column section
ML Distributed moment to the lower column section
IU Inertia in the considered plane of the upper column section
IL Inertia in the considered plane of the lower column section
LU System length in the considered plane of the upper column section
LL System length in the considered plane of the lower column section
These additional moments are then added to the sections in the column just above and just below the
connected beam.
The simplified procedure given in this chapter allows to account for eccentricities without specifically adding these eccentricities in the calculation model. In case however an actual member eccentricity is defined on the column member the above procedure will not be used since additional moments will already be generated during the analysis.
Scia Engineer Steel Code Check Theoretical Background
41
Scaffolding
The scaffolding member and coupler check are implemented according to EN 12811-1 Ref.[23].
The following paragraphs give detailed information on these checks.
Scaffolding member check for tubular members
The check is executed specifically for circular hollow sections (Form code 3) and Numerical
sections in case the proper setting is activated in the Steel Setup.
The check is executed according to Equation 9 given in EN 12811-1 article 10.3.3.2. However, the
EN 12811-1 only gives an interaction equation in case of a low shear force.
Since the EN 12811-1 is based entirely on DIN 4420-1 Teil 1 Ref.[26] the interaction formulas according to Tabelle 7 of DIN 4420-1 Teil 1 are applied in case of a large shear force.
The interaction equations are summarised as follows:
Conditions Interaction for tubular member
and
and
and
and
Scia Engineer Steel Code Check Theoretical Background
42
With: M
V
Npld
Vpld
Mpld
A Area of the cross-section
Wel Elastic section modulus
Wpl Plastic section modulus
N Normal force
Vy Shear force in y direction
Vz Shear force in z direction
My Bending moment about the y axis
Mz Bending moment about the z axis
fy Yield strength of the material
Safety factor taken as M0 of EN 1993-1-1
As specified in EN 12810 Ref.[25] & 12811 Ref.[23] the scaffolding check for tubular members assumes the use of a 2
nd order analysis including imperfections.
In case these conditions are not set the default EN 1993-1-1 check should be applied instead.
Scia Engineer Steel Code Check Theoretical Background
43
Scaffolding coupler check
The scaffolding couplers according to EN 12811-1 Annex C Ref.[23] are provided by default within
Scia Engineer.
The interaction check of the couplers is executed according to EN 12811-1 article 10.3.3.5.
The interaction equations are summarised as follows:
Coupler type Interaction equation
Right angle coupler
Friction sleeve
With: Fsk Characteristic Slipping force
Taken as Nxk and Vzk of the coupler properties
2Fsk = Nxk + Vzk
Fpk Characteristic Pull-apart force
Taken as Vyk of the coupler properties
MBk Characteristic Bending moment
Taken as Myk of the coupler properties
N Normal force
Vy Shear force in y direction
Vz Shear force in z direction
My Bending moment about the y axis
Safety factor taken as M0 of EN 1993-1-1 for steel couplers
Safety factor taken as M1 of EN 1999-1-1 for aluminium couplers
Scia Engineer Steel Code Check Theoretical Background
44
Manufacturer couplers
In addition to the scaffolding couplers listed above, specific manufacturer couplers are provided within Scia Engineer.
The interaction checks of these couplers are executed according to the respective validation reports.
Cuplock
The cuplock coupler which connects a ledger and a standard is described in Zulassung Nr. Z-8.22-208 Ref.[35].
The interaction equations are summarised as follows:
Cuplock Coupler Interaction equation
Interaction 1
Interaction 2
With: Nxk Taken from the coupler properties
Myk Taken from the coupler properties
Mxk Taken from the coupler properties
N Normal force in the ledger
My Bending moment about the y axis
Mx Torsional moment about the x axis
Nv Normal force in a connecting vertical diagonal
Angle between connecting vertical diagonal and standard
Safety factor taken as M0 of EN 1993-1-1 for steel couplers
Safety factor taken as M1 of EN 1999-1-1 for aluminium couplers
Scia Engineer Steel Code Check Theoretical Background
45
Layher Variante II & K2000+
The Layher coupler which connects a ledger and a standard is described in Zulassung Nr. Z-8.22-64 Ref.[36]. Both Variante II and Variante K2000+ are provided.
Layher Coupler
Interaction equation
Interaction 1 Variante II:
Variante K2000+:
Interaction 2
With: NR,d = Nxk /
With Nxk taken from the coupler properties
My,R,d = Myk /
With Myk taken from the coupler properties
MT,R,d = Mxk /
With Mxk taken from the coupler properties
Scia Engineer Steel Code Check Theoretical Background
46
Vz,R,d = Vzk /
With Vzk taken from the coupler properties
N Normal force in the ledger
(+) This index indicates a tensile force
Vy Shear force in y direction
Vz Shear force in z direction
My Bending moment about the y axis
Mx Torsional moment about the x axis
Nv Normal force in a connecting vertical diagonal
Angle between connecting vertical diagonal and standard
e = 2,75 cm for Variante II
= 3,30 cm for Variante K2000+
eD = 5,7 cm for Variante II and Variante K2000+
= 1,26 cm for Variante II
= 1,41 cm for Variante K2000+
Safety factor taken as M0 of EN 1993-1-1 for steel couplers
Safety factor taken as M1 of EN 1999-1-1 for aluminium couplers
Scia Engineer Steel Code Check Theoretical Background
47
EC3 – EN Fire Resistance
The beam elements are checked according to the regulations given in
Eurocode 3
Design of steel structures
Part 1 - 2 : General rules – Structural fire design
EN 1993-1-2:2005
Corrigendum
EN 1993-1-2:2005/AC:2005
Corrigendum
EN 1993-1-2:2005/AC:2009
Fire actions effect Efi
The design effects of actions for the fire situation Efi,d,t are taken from the results of the analysis. It is recommended to use the accidental combination rules, for calculating the internal forces used in the fire resistance check.
The accidental combination is given by (see EN 1990 – Ref[5])
Eq. 6.11b Gk,j + P + Ad+ (1,l or 2,l)Qk,l+ 2,iQk,i
The choice between 1,l or 2,l is done by the user. Default is 1,l.
with Gk,j characteristic value of permanent action j
P relevant representative value of prestressing action
Qk,l characteristic value of leading variable action l
Qk,i characteristic value of accompanying variable action i
Ad design value of the accidental action
1,l 2,l combination coefficients
Scia Engineer Steel Code Check Theoretical Background
48
Material properties
The material properties are depending on the steel temperature.
Strength and deformation properties :
a
,a,E
y
,p,p
y
,y,y
EE
k
f
fk
f
fk
The variation in function of the steel temperature of the value for yield strength ky,, proportional limit
kp, and modulus of elasticity kE, is given by tables in ref.[6], table 3.1.
For cold formed members ky, is taken from Ref.[7]; table III.2.5.
In the simplified calculation method, the following default properties are considered to be constant during the analysis :
unit mass a 7850 kg/m³
thermal elongation l/l 14 x 10-6
(a-20)
thermal conductivity a 45 W/mK
Temperature analysis - Thermal actions
In this part, the nominal temperature-time curves and the related net heat flux are described. See Ref.[8], Section 3, and Ref.[7], II.2.2.
Nominal temperature-time curve
The following temperature-time curves can be selected :
with t time in [min]
g gas temperature in [°C]
c the coefficient of heat transfer by convection
Scia Engineer Steel Code Check Theoretical Background
49
ISO 834 curve
external fire curve
hydrocarbon curve
smoldering fire curve
during 21 minutes, followed by the standard ISO 834 curve
user defined temperature-time curve
Net heat flux
rnetcnetdnet hhh ,,,
with hnet,d the net heat flux
hnet,c the convective heat flux
hnet,r the radiative heat flux
Scia Engineer Steel Code Check Theoretical Background
50
with configuration factor [1.0]
res resultant emissivity
= f m
f emissivity related to fire compartment
= [1.00]
m emissivity related to surface material
= [0.70]
r = g
gas temperature in [°C]
m surface temperature of member in [°C]
c coefficient of heat transfer by convection
Steel Temperature
The increase of temperature a,t in an unprotected steel member during a time interval t
thc
VAk dnet
aa
mshta ,,
/
with Am the exposed surface area per unit length [m²/m]
V the volume of the member per unit length [m³/m]
The factor Am/V should not be taken as less than 10m-1
ca the specific heat of steel [J/kgK]
hnet,d the net heat flux per unit area [W/m²]
t the time interval [seconds]
The value should not be taken as more than 5 seconds
a the unit mass of steel [kg/m³]
ksh correction factor for the shadow effect [1.0]
The correction factor is calculated for I sections only
The increase of temperature a,t in an insulated steel member during a time interval t
Scia Engineer Steel Code Check Theoretical Background
51
with Ap the area of fire protection material per unit length [m²/m]
V the volume of the member per unit length [m³/m]
ca the specific heat of steel [J/kgK]
cp the specific heat of fire protection material [J/kgK]
dp the thickness of the fire protection material [m]
t the time interval [seconds]
The value should not be taken as more than 30 seconds
a the unit mass of steel [kg/m³]
p the unit mass of fire protection [kg/m³]
a,t the steel temperature at time t
g,t the ambient gas temperature at time t
g,t the increase of the ambient gas temperature during the time interval
p the thermal conductivity of the fire protection material [W/mK]
The value a,t 0.0
For the increase of temperature a,t in an insulated steel member with intumescent coating, we refer to the NEN specifications, Chapter 'Steel Temperature'.
Scia Engineer Steel Code Check Theoretical Background
52
Calculation model
The calculation can be performed in 2 domains :
- strength domain
- temperature/time domain
In the strength domain, the strength Rfi,d,t(unity check) is calculated after a given time t (e.g. strength
after 45 min). In the temperature/time domain, the critical steel temperature cr,d is computed. From this critical temperature, the fire resistance time tfi,d is calculated (the time domain).
Code Check
The section and stability checks (buckling, lateral torsional buckling) are performed according to the regulations given in 'EN 1993-1-2:2005'. The checks are performed in the resistance domain or in the temperature/time domain..
Torsional buckling and shear buckling are not considered.
For each member, the classification of the cross section, the section check and the stability check are performed.
The following checks are executed :
- classification of cross section : art. 4.2.2.
- resistance for tension members : art. 4.2.3.1
- resistance for compression members (class 1,2 or 3) : art. 4.2.3.2.
- resistance for beams (class 1,2) : art. 4.2.3.3.
- resistance for beams (class 3) : art.4.2.3.4.
- resistance for members (class 1,2,3) subject to bending and compression : art. 4.2.3.5.
- check for class 4 sections : Annex E
Scia Engineer Steel Code Check Theoretical Background
53
EC3 – EN Cold-Formed
The members are checked according to the regulations given in:
Eurocode 3
Design of steel structures
Part 1 - 3: Supplementary rules for cold-formed members and sheeting
EN 1993-1-3:2006
Corrigendum
EN 1993-1-3:2006/AC:2009
Eurocode 3
Design of steel structures
Part 1 - 5: Plated Structural elements
EN 1993-1-5:2006
Corrigendum
EN 1993-1-5:2006/AC:2009
Consulted articles
An overview for the used articles is given in the following table. The articles marked with “x” are consulted. The articles marked with (*) have a supplementary explanation in the following paragraphs.
Article Title
1 Introduction X
2 Basis of design X
3 Materials
3.1 General X
3.2 Structural Steel X(*)
5 Structural Analysis
5.1 Influence of rounded corners X(*)
Scia Engineer Steel Code Check Theoretical Background
54
5.2 Geometrical proportions X(*)
5.3 Structural modelling for analysis X
5.5 Local and distortional buckling
5.5.1 General
5.5.2 Plane elements without stiffeners
5.5.3 Plane elements with edge or intermediate stiffeners
5.5.3.1 General
5.5.3.2 Plane elements with edge stiffeners
5.5.3.3 Plane elements with intermediate stiffeners
X
X(*)
X(*)
X(*)
X(*)
6 Ultimate Limit States
6.1 Resistance of cross-sections
6.1.1 General
6.1.2 Axial Tension
6.1.3 Axial Compression
6.1.4 Bending moment
6.1.4.1 Elastic and elastic-plastic resistance with yielding at the compressed flange
6.1.5 Shear Force
6.1.6 Torsional Moment
6.1.7 Local Transverse Forces
6.1.8 Combined Tension and Bending
6.1.9 Combined Compression and Bending
6.1.10 Combined shear force, axial force and bending moment
6.1.11 Combined Bending moment and local load or support reaction
X
X(*)
X(*)
X(*)
X(*)
X(*)
X(*)
X(*)
X(*)
X(*)
X(*)
6.2 Buckling Resistance
6.2.1 General
6.2.2 Flexural buckling
6.2.3 Torsional buckling and torsional-flexural buckling
6.2.4 Lateral Torsional buckling of members subject to bending
6.2.5 Bending and axial compression
X
X(*)
X(*)
X(*)
X(*)
6.3 Bending and axial tension X(*)
10 Special considerations for purlins, liner trays and sheetings
10.1 Beams restrained by sheeting
10.1.1 General
10.1.2 Calculation methods
10.1.3 Design criteria
10.1.4 Design resistance
10.1.5 Rotational restraint given by sheeting
10.1.5.1 Lateral spring stiffness
X(*)
X
X
X(*)
X(*)
Scia Engineer Steel Code Check Theoretical Background
55
As specified in EN 1993-1-3: 1.1(3) the code does not apply to cold -formed CHS (FC 3) and RHS (FC 2) sections. For these form codes the default EN 1993-1-1 provisions apply.
Haunches, arbitrary members and cross-sections without initial shapes are not supported for the EN 1993-1-3 code check. In this case the default EN 1993-1-1 code check is executed.
The checks are executed according to the principal axis in accordance with EN 1993-1-3 art. 1.5.1(4) NOTE except where stated otherwise.
Material properties
The steel grades given within EN 1993-1-3 Table 3.1b are available in the default Material Library of
Scia Engineer.
Average Yield Strength
The average yield strength is supported according to EN 1993-1-3 art. 3.2.2.
The average yield strength is applied in the following resistance calculations:
- Axial Tension
- Axial Compression
- Bending Moment
- Torsional moment
- Flexural Buckling
- Torsional (-Flexural) Buckling
- Purlin design – Cross-section resistance
The average yield strength is calculated using Ag of the Initial shape.
Steel Core Thickness
The steel core thickness is supported according to EN 1993-1-3 art. 3.2.4.
The steel core thickness is only available for the following sections:
- Cross-section which have form code FC 111 – FC 126 & FC 129 - Cold-formed pair sections (2CFUo, 2CFUc, 2CFCo, 2CFCc, 2CFLT)
The ranges for the core thickness are set ‘for sheeting and members’.
Form codes 172 & 128 are not supported for the Steel Core Thickness.
Scia Engineer Steel Code Check Theoretical Background
56
Initial Shape
For a cross-section with material Steel and fabrication set to Cold-Formed, the Initial Shape can be defined.
For a General cross-section the „Thin-walled representation‟ has to be used to be able to define the Initial Shape.
The thin-walled cross-section parts can have the following types:
F Fixed Part – No reduction is needed
I Internal cross-section part
SO Symmetrical Outstand
UO Unsymmetrical Outstand
Parts can also be specified as reinforcement:
None Not considered as reinforcement
RUO Reinforced Unsymmetrical Outstand (edge stiffener)
RI Reinforced Intermediate (intermediate stiffener)
DEF Double Edge Fold (edge stiffener)
ROU and DEF reinforcement types can be set only to elements of type SO or UO.
RI types can be set only to elements of type I or UO or SO.
For general cross-sections neighbouring elements of type RI are seen as one stiffener for the
calculation of the stiffener area and inertia.
The initial shape is supported for the following cross-section types:
- Standard profile library cross-sections
- Cold formed Pair cross-sections of profile library sections
- General thin-walled sections
- General sections with thin-walled representation
- Thin-walled geometric sections
Scia Engineer Steel Code Check Theoretical Background
57
- All other sections which support the centerline and do not have roundings
For standard profile library cross-sections, the flat parts are taken between the roundings. The roundings are set as fixed parts.
For predefined sections without roundings, the initial shape is based on the centreline dimensions i.e. the flat parts are taken between the intersection points of the centrelines.
For standard profile library cross-sections and pair sections the Initial Shape is generated automatically. Within this automatic generation the stiffeners are handled as follows:
o For the following form codes edge stiffeners are automatically set as RUO FC 114 Cold formed C-section FC 115 Cold formed Omega section FC 116 Cold formed C-Section eaves beam FC 118 Cold formed ZED section FC 119 Cold formed ZED section asymmetric lips FC 120 Cold formed ZED section inclined lip FC 121 Cold formed Sigma section FC 124 Cold formed Sigma section eaves beam FC 126 Cold formed ZED section both lips inclined FC 129 Cold formed Sigma section asymmetric
o For the following form codes edge stiffeners are automatically set as DEF
FC 117 Cold formed C-Plus section FC 122 Cold formed Sigma section stiffened FC 123 Cold formed Sigma-Plus section FC 125 Cold formed Sigma-Plus section eaves beam FC 127 Cold formed I-Plus section FC 128 Cold formed IS-Plus section
o For the following form codes internal stiffeners are automatically set as RI
FC 121 Cold formed Sigma section FC 124 Cold formed Sigma section eaves beam FC 122 Cold formed Sigma section stiffened FC 123 Cold formed Sigma-Plus section FC 125 Cold formed Sigma-Plus section eaves beam FC 128 Cold formed IS-Plus section FC 129 Cold formed Sigma section asymmetric
Geometrical Proportions
The Geometrical proportions are checked according to EN 1993-1-3 art. 5.2(1) Table 5.1.
The limits for edge stiffeners (c) and double edge folds (d) are checked in case the correct stiffener type (RUO or DEF) has been set in the initial shape.
The limit ratio‟s given in EN 1993-1-3 art. 5.2(2) are checked. Lip dimensions c and d are however
always accounted for and will not be ignored.
In addition the limit for the internal radius given in EN 1993-1-3 art. 5.1(6) is checked.
For general sections, the geometrical proportions are checked for elements I, UO and SO using their respective part lengths. Flanges including RI stiffeners are thus considered part by part and not as one whole flange.
Scia Engineer Steel Code Check Theoretical Background
58
Effective Shape
Influence of rounded corners
Within Scia Engineer the exact method is applied i.e. all properties and dimensions are determined including the influence of rounded corners.
The approximate procedure given in EN 1993-1-3 art. 5.1(3) and following is thus not supported.
Notional widths
The notional widths are specified in EN 1993-1-3 art. 5.1 and Figure 5.1.
The initial shape elements are taken between the roundings (i.e. internal dimensions w).
The notional widths bp are then calculated as follows:
For an internal element (I)
bp = w + rm * sin ( left / 2) + rm * sin ( right / 2)
For an outstand element (UO, SO)
bp = w + rm * sin ( / 2)
In addition to the notional with bp, for each element the centerline length lc is determined as follows:
For an internal element (I)
lc = bp + gr,left + gr,right
With gr,left = rm * [tan ( left / 2) - sin ( left / 2)]
gr,right = rm * [tan ( right / 2) - sin ( right / 2) ]
Scia Engineer Steel Code Check Theoretical Background
59
For an outstand element (UO, SO)
lc = bp + gr
With gr = rm * [tan ( / 2) - sin ( / 2)]
General procedure for one element
By default, EN 1993-1-3 specifies that the stress f (com,Ed) to be used for the effective section
calculation should be taken as fyb/M0
The reduction of an element is in general given by:
beff = p * b
With: beff Effective width
p Reduction factor
b Full width
Step 1:
For the given stress f the normal stress over the rectangular plate element of the initial
geometrical shape is calculated. These stresses are calculated based on the nominal width bp.
beg : normal stress at start point of rectangular shape – compression stress is positive
end : normal stress at end point of rectangular shape – compression stress is positive
If the rectangular shape is completely under tension, i.e. beg and end are both tensile stresses, no reduction is needed, p = 1.0
Scia Engineer Steel Code Check Theoretical Background
60
Step 2: Determine f1 and f2: in case
f1 = beg
f2 = end
in case
f1 = end
f2 = beg
Step 3: Calculate the stress gradient :
Step 4: If = 1 the element is under uniform compression, else the element is under stress gradient.
Depending on the stress gradient and the element type, the effective width can be calculated as specified in the following paragraphs.
EN 1993-1-3 art. 6.1.4.2 concerning the plastic reserve of the tension flange is not supported i.e. always an elastic stress distribution is used.
Internal Compression Elements
The effective width of internal compression elements is calculated according to EN 1993-1-5 art. 4.4 and Table 4.1.
This applies to elements of type I.
The notional width bp is used as
Outstand Compression Elements
The effective width of outstand compression elements is calculated according to EN 1993-1-5 art. 4.4 and Table 4.2.
This applies to elements of type UO and SO
The notional width bp is used as
When activating the checkbox “Use Lambda,p,red 4.4(4)” the reduced slenderness is
determined using com,Ed as the maximal compressive stress f1 f2 in the element.
When activating the checkbox “Use Annex E E.1(1)” the formulas given in Annex E
are used to determine the reduction factor .
Scia Engineer Steel Code Check Theoretical Background
61
Plane Elements with Edge Stiffeners
The procedure for determining the effective width/thickness of elements with edge stiffeners is given in EN 1993-1-3 art. 5.5.3.2 and art. 5.5.3.1.
This applies to elements of type RUO and DEF
General remarks regarding the stiffness K of the edge stiffener given in formula (5.10b) .
hw is taken as lc (centreline length) of the biggest adjacent element. Adjacent
elements are those elements connected to the flange. For typical cross-sections, there is only one adjacent element, the web.
For Sigma sections, hw is taken as the sum of the centreline lengths of the web
elements.
This concerns the following form codes: FC 121 Cold formed Sigma section FC 124 Cold formed Sigma section eaves beam FC 122 Cold formed Sigma section stiffened FC 123 Cold formed Sigma-Plus section FC 125 Cold formed Sigma-Plus section eaves beam FC 128 Cold formed IS-Plus section FC 129 Cold formed Sigma section asymmetric
General Cross-section: hw for stiffener:
o Elements connected to the stiffener are not accounted for since they are considered as flanges
o Elements connected to those flanges are all accounted for in case they are of type I and the summation is made of the lengths of these elements
o Roundings are not accounted for
General Cross-section: In case there is only one edge stiffener kf is taken as zero. (i.e. no interaction between two flanges since there is only one
flange).
General Cross-section: In case there are two edge stiffeners kf is determined by default. (i.e. interaction between the two flanges is accounted
for).
General Cross-section: In case there are more than two edge stiffeners The same logic is followed as for a single stiffener. The factor kf is thus taken as
zero.
The formula for K given in the EN 1993-1-3 is based purely on simple sections with two flanges. In case of more complex cross-sections, the only exact procedure is to perform a numerical analysis (finite strip method) to determine the critical stresses for local and distortional buckling. This is referenced as the „Advanced Procedure‟ given in art. 5.5.1(7).
Critical stresses for local and distortional buckling obtained from a numerical analysis can be inputted in the cross-section manager.
The reduced effective area of the stiffener As,red according to art 5.5.3.2(11) is
calculated using com,Ed = fyb/M0.
Scia Engineer Steel Code Check Theoretical Background
62
Plane Elements with Intermediate Stiffeners
The procedure for determining the effective width/thickness of elements with intermediate stiffeners is given in EN 1993-1-3 art. 5.5.3.3 and art. 5.5.3.1.
This applies to elements of type RI
The stiffness K of the internal stiffener is determined from formula (5.11):
The reduced effective area of the stiffener As,red according to art 5.5.3.3(10) is
calculated using com,Ed = fyb/M0.
General procedure of Effective Shape calculation
The general procedure which combines the effective section calculation of plane elements without and plane elements with stiffeners is given in EN 1993-1-3 art. 5.5.2(3) and art. 5.5.3.
This procedure can be written out as follows:
Step 1: The effective width of the flanges and edge/intermediate stiffeners within the flanges are calculated based on gross section properties. This includes the optional iterative procedure for the edge/intermediate stiffeners as specified in art. 5.5.3.2(10) and art. 5.5.3.3(9).
Step 2: This partially effective shape of the previous step is used to determine the stress gradient and effective width of the web. This includes the optional iterative procedure for the intermediate stiffeners as specified in art. 5.5.3.3(9).
Step 3: The end result of the previous two steps is the effective cross-section and its properties can be calculated.
Step 4: This process can now be optionally iterated using the stress ratio based on the effective cross-section in place of the gross cross-section.
Both iteration procedures (iteration of stiffeners and iteration of the full cross-section) can be set in the Steel Setup.
Scia Engineer Steel Code Check Theoretical Background
63
Manufacturer provided effective section properties
In case in the Steel Setup the option „Use manufacturer provided effective section properties‟is
activated, effective section properties from the manufacturer are taken from the Effective Section Library instead of calculated by EN 1993-1-3.
The following properties can be defined in the Effective Section Library:
Property Description
fy [MPa] Steel grade for which the effective properties have been derived
Aeff [mm^2] Effective Area for compression
eN,y [mm] Shift of centroid in y direction for compression
eN,z [mm] Shift of centroid in z direction for compression
Ieff,y My+ [mm^4] Effective moment of inertia about the y-y axis for a positive moment My
Weff,y My+ [mm^3] Effective section modulus to the extreme fiber about the y-y axis for a positive moment My
eM,z My+ [mm] Shift of centroid in z direction for a positive moment My
Ieff,y My- [mm^4] Effective moment of inertia about the y-y axis for a negative moment My
Weff,y My- [mm^3] Effective section modulus to the extreme fiber about the y-y axis for a negative moment My
eM,z My- [mm] Shift of centroid in z direction for a negative moment My
Ieff,z Mz+ [mm^4] Effective moment of inertia about the z-z axis for a positive moment Mz
Weff,z Mz+ [mm^3] Effective section modulus to the extreme fiber about the z-z axis for a positive moment Mz
eM,y Mz+ [mm] Shift of centroid in y direction for a positive moment Mz
Ieff,z Mz- [mm^4] Effective moment of inertia about the z-z axis for a negative moment Mz
Weff,z Mz- [mm^3] Effective section modulus to the extreme fiber about the z-z axis for a negative moment Mz
eM,y Mz- [mm] Shift of centroid in y direction for a negative moment Mz
In case the yield strength used for the cross-section does not match any of the yield strengths defined in the Effective Section Library the default EN 1993-1-3 calculation will be used.
Scia Engineer Steel Code Check Theoretical Background
64
Section Checks
Axial Tension
The Axial Tension Check is executed according to EN 1993-1-3 art. 6.1.2.
The net section resistance Fn,Rd is taken as:
With Anet taken equal to Ag since bolt holes are not accounted for.
Axial Compression
The Axial Compression Check is executed according to EN 1993-1-3 art. 6.1.3.
The choice between formula (6.2) and (6.3) is made by comparing the gross area Ag from the initial shape with the effective area Aeff of the effective shape for compression.
The gross area Ag used in the formulas is taken from the cross-section manager.
This comparison using the initial shape property is of importance for the following reasons: - Profile Library sections can have different gross properties compared to the initial shape since the gross properties come from certain sources (books, tables, …) and are mostly rounded off. - For general cross-sections the gross shape can differ from the initial shape since the initial shape concerns a thin walled representation.
Each element on which a distortional buckling reduction factor d is applied is seen as „stiffened‟.
All other elements are seen as „plane‟.
Scia Engineer Steel Code Check Theoretical Background
65
Bending Moment
The Bending Moment Check is executed according to EN 1993-1-3 art. 6.1.4.1.
The choice between formula (6.4) and (6.5) is made by comparing the elastic section modulus Wel from the initial shape with the effective section modulus Weff of the effective shape for bending.
The elastic section modulus Wel used in the formulas is taken from the cross-section manager.
Note: This comparison using the initial shape property is of importance for the following reasons: - Profile Library sections can have different gross properties compared to the initial shape since the gross properties come from certain sources (books, tables, …) and are mostly rounded off. - For general cross-sections the gross shape can differ from the initial shape since the initial shape concerns a thin walled representation.
An element of type I is seen as „plane‟.
An element of type UO or SO is seen as „outstand‟.
As indicated in EN 1993-1-3 art. 6.1.4.1(2) formula (6.5) is only applied in case:
There is only single bending i.e. My OR Mz
There is no torsion i.e. Mx = 0
There is no Torsional (-Flexural) buckling i.e. TF = 1,00
There is no Lateral Torsional buckling i.e. LTB = 1,00
There is no Distortional buckling i.e. all reinforcement types of the cross-section elements should be „none‟ or, in case there are stiffeners, they should not be in compression.
The angle between the web and flange exceeds 60°.
In case formula (6.5) should be applied but the above conditions are not fulfilled, formula (6.6) is
applied.
EN 1993-1-3 art. 6.1.4.2 concerning the plastic reserve of the tension flange is not supported i.e.
always an elastic stress distribution is used.
EN 1993-1-3 art. 6.1.4.3 concerning the effects of shear lag is not supported.
Scia Engineer Steel Code Check Theoretical Background
66
Shear Force
The Shear Force Check is executed according to EN 1993-1-3 art. 6.1.5.
General
The shear resistance is calculated for each „web‟ element separately and the cross-section resistance is taken as the sum of these element resistances.
Only elements with element types I, UO and SO are accounted for.
In addition, elements with stiffener type RUO or DEF are not accounted for.
Formula (6.8) is rewritten as follows for both directions:
With: i = Angle of element i related to the principal y-y axis
lc,i = Centreline length of element i
By default the Shear Check is executed „without stiffening at the support‟ In case Local Transverse Forces data are inputted which have the checkbox „No Local Transverse Forces Check‟ activated, the Shear Check in those sections is executed „with stiffening at the support‟.
Elements without Internal stiffeners
The centreline length lc,i for each element i is taken from the Initial shape.
The angle i for each element i is determined as the angle related to the principal y-y axis.
The relative web slenderness for each element i is determined according to formula (6.10a).
The slant height sw,i is taken as the notional width bp,i of the element under consideration as
indicated on the following picture.
Scia Engineer Steel Code Check Theoretical Background
67
Sections with Internal stiffeners
Special considerations are required for cross-sections with internal stiffeners (Type RI).
The following picture illustrates a web with internal stiffener:
The internal stiffener and connected elements are seen as „one web‟. This „composed‟ web is seen as „one‟ element i in the shear calculation.
For such a „composed‟ web, the different distances are determined as follows:
The slant height sw is taken as the distance between - The starting point of the nominal width bp,i of the first element in the web. - The end point of the nominal width bp,i of the last element in the web.
The total developed slant height sd is taken as the sum of the nominal widths bp,i of all the
elements in the web.
The slant height sp concerns the notional width bp,i of the largest plane web element.
The relative web slenderness is determined according to formula (6.10b).
The inertia of the stiffener(s) Is is taken from the Initial shape
Scia Engineer Steel Code Check Theoretical Background
68
The centreline length lc of this composed web is calculated as follows:
In case the first or last element of the composed web has element type SO or UO:
lc = sw + gr
With gr = rm * [tan ( / 2) - sin ( / 2)]
If the first element is an outstand, gr is taken as gr at the end point of the last
element. If the last element is an outstand, gr is taken as gr at the starting point of the first
element. Reference is made to Notional widths.
In case both the first and last element of the composed web has element type I:
lc = sw + gr,first + gr,end With gr,first taken as gr at the starting point of the first element. gr,end taken as gr at the end point of the last element.
The angle of the „composed‟ web concerns the angle of the centreline length lc relative to the
principal y-y axis.
Neighboring connected elements are seen as one „web‟. A typical example of this is a sigma section: the web has two internal stiffeners which both are connected to the same internal element. As such they are recognized as forming one web.
Torsional Moment
The Combined Stress Check including Torsion and Warping is executed according to EN 1993-1-3 art. 6.1.6.
Regarding warping reference is made to Warping check.
The average yield strength fya in all three formulas (6.11a), (6.11b), (6.11c) will only be used in
case for all three force components separately (N, My, Mz) the average yield strength may be used (Aeff = Ag ; Weff,y = Wel,y ; Weff,z = Wel,z).
Local Transverse Forces
The local transverse forces check is executed according to EN 1993-1-3 art 6.1.7 and following.
The check is executed on the positions where there is a jump in the Vz shear force diagram.
Remarks:
The shear force diagram of both the actual member as well as adjacent members is evaluated. Adjacent members are defined as members which are in the same buckling system.
The Flange Condition depends on the definition of the initial shape. In case there is an element with reinforcement type ROU or DEF the setting is taken as „Stiffened ‟.
Scia Engineer Steel Code Check Theoretical Background
69
The distances for One-flange/Two-flange and End/Interior are evaluated taking into account adjacent members. Adjacent members are defined as members which are in the same buckling system.
In case the cross-section has multiple webs, for determining the load condition the maximal web height is used.
As opposed to EN 1993-1-3 art.6.1.7.2(4), the exact inputted bearing length ss will be used
at all times i.e. the simplification of using the minimal length for both opposing loads is not supported.
As indicated on EN 1993-1-3 Figure 6.6, the local transverse force resistance is taken relative to the support, not according to the principal z-axis. Therefore FEd, is determined
according to the LCS axis system and not according to the principal axis system!
General Procedure
This paragraph specifies the general procedure to determine the local transverse web resistance which is applied for any type of cross-section except for FC 115 (Cold formed Omega).
In case the cross-section has any element with stiffener type RI, the procedure for stiffened webs is applied first.
In a first step the web height hw is determined for each „web‟ element:
Only elements of type I are accounted for. In addition elements with stiffener types RUO and DEF are not accounted for.
For each of those elements i the centreline length lc,i is read from the Initial shape
For each of those elements i the angle i is determined as the angle of the element relative to the horizontal axis (based on Figure 6.6).
In addition, only elements with an angle i ≥ 45° are accounted for.
The web height for each element i is calculated as:
In case none of the cross-section elements fulfil the above conditions, the local transverse forces check is not supported for the cross-section.
When hw,i is determined, the local transverse resistance Rw,Rd,i for each of those elements is determined based on EN 1993-1-3 art.6.1.7.2
The final cross-section resistance is taken as the sum of the individual element resistances.
By default, the local transverse resistance Rw,Rd,i is determined using EN 1993-1-3 Figure 6.7a & 6.7b.
The following table shows the relation between the loading conditions and the cases defined in the tables.
Loading Condition Table Case
End One Flange (EOF) 6.7a a) i)
Interior One Flange (IOF) 6.7a a) ii)
End Two Flange (ETF) 6.7b b) i)
Interior Two Flange (ITF) 6.7b b) ii)
Scia Engineer Steel Code Check Theoretical Background
70
In case Web rotation prevented was set using Local Transverse Forces data instead of EN 1993-1-3 Figure 6.7a & 6.7b the formulas given in EN 1993-1-3 art. 6.1.7.2(4) are used.
The following table shows the relation between the loading conditions and the cases defined in this article.
Loading Condition Article Case
End One Flange (EOF) art. 6.1.7.2(4) a) i)
Interior One Flange (IOF) art. 6.1.7.2(4) a) ii)
End Two Flange (ETF) art. 6.1.7.2(4) b) i)
Interior Two Flange (ITF) art. 6.1.7.2(4) b) ii)
Omega Sections
Specifically for FC 115 (Cold formed Omega) cross-sections the special procedure for sections with two or more unstiffened webs is applied. The local transverse resistance Rw,Rd,i for each of those webs is determined according to EN 1993-1-3 art. 6.1.7.3.
Other cross-sections with two or more unstiffened webs will always be calculated according to the General Procedure, not this special procedure.
The value of in EN 1993-1-3 art. 6.1.7.3(5) is taken for „liner trays and hat sections‟.
The following table shows the relation between the loading conditions and the categories defined in EN 1993-1-3 Figure 6.9.
Loading Condition Category
End One Flange (EOF) 1
Interior One Flange (IOF) 1
End Two Flange (ETF) 1
Interior Two Flange (ITF) 2
Figure 6.9 does not directly specify ETF. However since two flange loading is specified as category 1 and End loading is also specified as category 1, the combined condition of ETF is considered as category 1.
According to [27] to use la = 10 mm for the end support reaction force (category 1) results in a very conservative resistance. A modification is given for case 2 and 3 of Figure 6.9: la = c + Ss. By activating the setting “Use la correction in (6.18)” this modification is applied.
Scia Engineer Steel Code Check Theoretical Background
71
Stiffened Webs
This paragraph outlines the special procedure in case of stiffened webs according to EN 1993-1-3 art. 6.1.7.4.
This method is used only in case there are one or more elements with stiffener type RI
The procedure consists of four steps.
Step 1: Creating ‘composed’ webs
In a first step, „composed‟ webs are created using the same procedure as outlined in Sections with Internal stiffeners.
This includes the determination of the centreline length lc,i of those „composed‟ webs.
Step 2: Evaluation of ‘composed’ webs
The special procedure outlined in EN 1993-1-3 art. 6.1.7.4 is only valid under certain conditions.
Therefore, each „composed‟ web is evaluated to see if it meets the following requirements:
There is one or more elements with stiffener type RI
Each RI element should have element type I (i.e. it is at both sides connected to
other elements signifying it‟s a fold instead of a stiffener).
Elements connected to this RI element should not have stiffener type RI. This
implies that the procedure is not applied in case of neighbouring stiffener elements i.e. elements forming „one‟ big stiffener.
Composed webs which do not meet these requirements are further evaluated in step 3.
Composed webs which meet all requirements are further evaluated in step 4.
Example:
All four sections have „composed‟ webs.
Section A contains two RI stiffeners which are connected. The web thus does not meet the
requirements.
Section B contains a single RI stiffener which meets all the requirements. This stiffener is thus a
„true‟ two fold stiffener so the special article applies.
Section C contains several RI stiffeners however not all match the requirements (one is an outstand,
others are connected etc). The web thus does not meet the requirements.
Section D has a composed web which contains two RI stiffeners. Both meet all the requirements
and are thus „true‟ two fold stiffeners. The special article applies.
Scia Engineer Steel Code Check Theoretical Background
72
Step 3: Composed webs which do NOT meet the requirements
For composed webs which do not meet the requirements, the special article is not valid. The local transverse force resistance of these webs will be determined according to the General Procedure
In this case, the centre line length lc,i of the composed web is used in the determination of hw.
The angle i is determined as the angle of the centre line length relative to the horizontal axis.
Step 4: Composed webs which meet all requirements
For composed webs which meet all requirements, the special procedure outlined in EN 1993-1-3 art. 6.1.7.4 is applied.
The „system line‟ of this web is taken as the centre line length lc,i.
The eccentricity e is determined at each end of an RI within the „composed‟ web. Eccentricity emin and emax are then taken as the min and max value for the considered composed web.
In case the limit specified in formula (6.21) is not fulfilled, the special article is not applied and the
composed web is considered as a web which does not meet all requirements. For such a web the procedure outlined in step 3 is applied.
For the developed width of the loaded flange bd any RI stiffeners of element Type I are always included, independent of their angle. RI stiffeners of element Type UO or SO are always ignored.
Connected flange elements which have a relative angle > 135° are accounted for as „one‟ flange for
the determination of bd.
In case there is no connected flange, for example when using a general section, then bd is
considered as zero. Practically this implies that there is no limit for a,s.
The data is then used to determine a,s according to formula (6.22).
The Rw,Rd,i value of the composed web is then calculated as:
Rw,Rd,i = a,s * Rw,Rd,i,general
With Rw,Rd,i,general calculated according to the General Procedure
The value of hw,i for this composed web is calculated using the centre line lc,i of the composed web
as outlined in step 3.
Combined Tension and Bending
The Combined Tension and Bending Check is executed according to EN 1993-1-3 art. 6.1.8.
The bending resistances are determined using the section moduli Weff of the effective shapes for
bending.
Scia Engineer Steel Code Check Theoretical Background
73
Combined Compression and Bending
The Combined Compression and Bending Check is executed according to EN 1993-1-3 art. 6.1.9.
Additional moments due to the shift in neutral axis are calculated at the beginning of the check and added to the internal forces.
This ensures specific bending checks are executed also in case there is no initial moment but only an additional moment.
The shifts in neutral axis eNy and eNz are read directly from the effective shape for compression.
As specified in EN 1993-1-3 art. 6.1.3(3) additional moments are only accounted for in case they
lead to an unfavourable check result.
The bending resistances are determined using the section moduli Weff of the effective shapes for
bending.
Combined Shear Force, Axial Force and Bending Moment
The Combined Shear Force, Axial Force and Bending Moment Check is executed according to EN 1993-1-3 art. 6.1.10.
In the following paragraphs formula (6.27) is written out for both directions.
Shear Vy
In case of shear Vy formula (6.27) is written out as follows:
Remarks:
Mf,Rd is taken as zero in case of Vy
(In case of weak axis bending, the „web‟ becomes a „flange‟. Since there is only a single „flange‟ in that case, the moment resistance of this flange is negligible. In addition, in case of more webs like in a box section EN 1993-1-5 art. 7.1 (5) specifies Mf,Rd = 0. Therefore, as a general conservative approach for Vy the value of Mf,Rd is taken as 0.)
Shear Vz
In case of shear Vz formula (6.27) is written out as follows:
Scia Engineer Steel Code Check Theoretical Background
74
Remarks:
According to [Ref.16] pp70 Mf,Rd is calculated as follows:
This is generalised in the following way:
a) Only elements with element types I, UO and SO are accounted for
b) Only elements which have an angle with the principal y-y axis which is 45°
are considered In case there is only one or none of such element, Mf,Rd = 0.
c) Of these elements, the one with the lowest beff is considered. The width beff
concerns the effective with of this element, read from the effective shape for bending.
d) Af = beff * t with t the thickness of the considered element.
e) Next only elements which have an angle with the principal y-y axis which is > 45°are considered.
In case there are no such elements, set Mf,Rd = 0.
f) Of these elements, the one with the highest value of lc * sin() is considered, with lc the centreline length of the element.
g) hf = lc * sin()
h) Mf,Rd is now be calculated as:
According to [Ref.16] pp70 Mpl,Rd is calculated as follows:
with Wpl read from the gross section properties.
Scia Engineer Steel Code Check Theoretical Background
75
Combined Bending Moment and local Load/Support Reaction
The Combined Bending Moment and local Load/Support Reaction Check is executed according to EN 1993-1-3 art. 6.1.11.
In formula (6.28c) the internal force MEd is taken as the actual moment in the section considered, not
the moment at the edge of the support.
Scia Engineer Steel Code Check Theoretical Background
76
Stability Checks
Flexural Buckling
The Flexural Buckling Check is executed according to EN 1993-1-3 art. 6.2.2 and EN 1993-1-1 art. 6.3.1.
Table 6.3 regarding the buckling curves is revised as follows:
Form code Description about axis Curve
1 I section y-y
z-z
a
b
101 Asymmetric I section y-y
z-z
a
b
114 Cold formed C section any b
116 Cold formed C-Section eaves beam any b
117 Cold formed C-Plus section any b
118 Cold formed ZED section any b
119 Cold formed ZED section asymmetric lips any b
120 Cold formed ZED section inclined lip any b
121 Cold formed Sigma section any b
122 Cold formed Sigma section stiffened any b
123 Cold formed Sigma-Plus section any b
124 Cold formed Sigma section eaves beam any b
125 Cold formed Sigma-Plus section eaves beam any b
126 Cold formed ZED section both lips inclined any b
127 Cold formed I-Plus section y-y
z-z
a
b
128 Cold formed IS-Plus section y-y
z-z
a
b
129 Cold formed Sigma section asymmetric any b
2CFCo with a = 0 y-y
z-z
a
b
2CFCc with a = 0 Closed section rule 6.2.2(3)
2CFUo with a = 0 y-y
z-z
a
b
2CFUc with a = 0 Closed section rule 6.2.2(3)
2CFLT with a = 0 any c
Any other section any c
All other sections fall in the „other cross-section‟ case of curve c for any axis.
For the calculation of the buckling length, we refer to chapter "Calculation of buckling ratio"
Scia Engineer Steel Code Check Theoretical Background
77
Torsional (-Flexural) Buckling
The Torsional (-Flexural) Buckling Check is executed according to EN 1993-1-3 art. 6.2.3 and EN 1993-1-1 art. 6.3.1.4.
The buckling curve for torsional (-flexural) buckling is taken as the z-z buckling curve according to the table given in Flexural Buckling.
The value of the elastic critical load Ncr is taken as the smallest of Ncr,T (Torsional buckling) and Ncr,TF (Torsional-Flexural buckling).
Calculation of Ncr,T
The elastic critical load Ncr,T for torsional buckling is calculated according to Ref.[17].
With: E Modulus of Young
G Shear modulus
It Torsion constant
Iw Warping constant
lT Buckling length for the torsional buckling mode
y0 and z0 Coordinates of the shear center with respect to the centroid
iy radius of gyration about the strong axis
iz radius of gyration about the weak axis
Scia Engineer Steel Code Check Theoretical Background
78
Calculation of Ncr,TF
The elastic critical load Ncr,TF for torsional flexural buckling is calculated according to Ref.[17].
Ncr,TF is taken as the smallest root of the following cubic equation in N:
0
With: Ncr,y Critical axial load for flexural buckling about the y-y axis
Ncr,z Critical axial load for flexural buckling about the z-z axis
Ncr,T Critical axial load for torsional buckling
Lateral Torsional Buckling
The Lateral Torsional Buckling Check is executed according to EN 1993-1-3 art. 6.2.4 and EN 1993-1-1 art. 6.3.2.2.
For additional information reference is made to Lateral-torsional buckling.
For information regarding the influence of diaphragms on the Lateral Torsional Buckling Check reference is made to Use of Diaphragms.
Bending and Axial Compression
For determining the Combined Bending and Axial Compression check according to EN 1993-1-3 art. 6.2.5 EN 1993-1-3 allows two possibilities:
Use the EN 1993-1-1 interaction according to article 6.3.3
Use the alternative according to EN 1993-1-3 article 6.2.5(2)
The choice between these two methods is set in the Steel Setup.
Interaction according to EN 1993-1-1
The interaction is executed according to EN 1993-1-1 art. 6.3.3 using interaction factors from Annex A & B.
In both Method 1 (Annex A) and Method 2 (Annex B) the cold - formed sections are seen as ‘class 3 or 4’.
Alternative interaction according to EN 1993-1-3
The interaction is executed according to EN 1993-1-3 art. 6.2.5(2).
Nb,Rd is taken as the lowest value of
the flexural buckling resistance about the y-y axis
the flexural buckling resistance about the z-z axis
the torsional (-flexural) buckling resistance
Scia Engineer Steel Code Check Theoretical Background
79
Formula (6.36) includes the strong axis bending resistance Mb,Rd. There is however no indication for a weak axis bending moment. Therefore, in case a weak axis bending moment is present, this interaction cannot be applied and the general interaction according to EN 1993-1-1 is applied.
Bending and Axial Tension
The Combined Bending and Tension Check is executed according to EN 1993-1-3 art. 6.3.
The code specifies that the same equations as for compression should be used. These interaction equations are however not fully valid in case of tension.
The purpose of the interaction check for bending and tension is to check the stresses at the compression fiber. In the AISI NAS 2007 Ref.[18] code the following formula is given in article C5:
This formula is rewritten using EC-EN notations as follows:
With: Mb,y,Rd The Lateral Torsional Buckling resistance.
Mc,z,Rd,com The moment resistance for the compression fiber in case of Mz.
Nt,Rd The Tension Resistance
Scia Engineer Steel Code Check Theoretical Background
80
Use of Diaphragms
The influence of a diaphragm is outlined in the following diagram.
Scia Engineer Steel Code Check Theoretical Background
81
First of all the lateral stiffness S of the diaphragm is determined and compared to the required stiffness Serf.
The lateral stiffness S is calculated according to Ref.19,3.5 and Ref.20,3.3.4.
L
K+K
10a.=S
s
21
4
with a The frame distance
Ls The length of diaphragm
K1 Diaphragm factor K1
K2 Diaphragm factor K2
The required stiffness Serf is determined according to EN 1993-1-3 art. 10.1.1
In case S < Serf the member is seen as Inadequately braced. In this case, when the diaphragm is
located on the compression side, the Lateral Torsional Buckling check is executed using the augmented torsional stiffness It. Reference is made to Adaptation of torsional constant.
G
lvorhCII
2
2
tid,t
with l The LTB length
G The shear modulus
vorhC The actual rotational stiffness of diaphragm
As specified in art. 10.1.1 the shear stiffness S is replaced by 0,2 S in case the diaphragm is connected every second rib only.
In case S ≥ Serf the member is seen as Fully braced. In this case, a first test is executed to evaluate if the special purlin checks according to EN 1993-1-3 Chapter 10 can be applied.
Scia Engineer Steel Code Check Theoretical Background
82
More specifically, this chapter is applied only in case the cross-section concerns a Z, C, or U
section:
Form code Description
5 Channel section
102 Rolled Z section
112 Cold formed channel
113 Cold formed Z
114 Cold formed C section
116 Cold formed C-Section eaves beam
117 Cold formed C-Plus section
118 Cold formed ZED section
119 Cold formed ZED section asymmetric lips
120 Cold formed ZED section inclined lip
121 Cold formed Sigma section
122 Cold formed Sigma section stiffened
123 Cold formed Sigma-Plus section
124 Cold formed Sigma section eaves beam
125 Cold formed Sigma-Plus section eaves beam
126 Cold formed ZED section both lips inclined
129 Cold formed Sigma section asymmetric
The code specifies that the chapter is also valid for hat (Omega) sections however in all further paragraphs; no specific formulas are given for Omega sections. For
example the free flange geometry is described only for Z, C and sections, not for Omega sections. Therefore, Omega sections are not supported for this special chapter.
In case the cross-section does not match any of the above, the default checks are executed. Since the member is seen as fully braced, no Lateral Torsional Buckling check needs to be executed in case the diaphragm is located on the compression side.
In case the cross-section does match the list of set form codes, a second test is executed. More specifically, the special purlin checks according to EN 1993-1-3 Chapter 10 can be applied only in
case:
The dimensional limits of article 10.1.1(1) are satisfied
The section is only loaded by N, Vz, My
Chapter 10 specifies only checks related to in plane effects N, Vz and My. Therefore, in case of other loading components, the special articles are not valid and the default checks will be applied.
Scia Engineer Steel Code Check Theoretical Background
83
For a section which meets all requirements, the following is done:
Reduced default Checks are executed i.e. not all default checks will be executed.
Special purlin checks according to Chapter 10
More specifically, the following „default‟ checks will be executed:
Section Check Article
Axial tension 6.1.2
Axial compression 6.1.3
Bending moment 6.1.4
Shear force 6.1.5
Torsional moment NOT
Local Transverse Forces 6.1.7
Combined tension and bending NOT
Combined compression and bending NOT
Combined shear, axial force and bending moment 6.1.10
Combined Bending and Local Transverse Force 6.1.11
Stability Check Article
Flexural buckling only for y-y 6.2.2
Torsional and Torsional-Flexural buckling NOT
Lateral-Torsional buckling NOT
Bending and axial compression NOT
Bending and axial tension NOT
The Torsional moment check will never occur in this case since the prerequisite is to have only N, Vz, My.
The combined axial and bending checks are not executed since they are replaced by the special purlin checks.
The flexural buckling check is executed for y-y buckling in accordance with EN 1993-1-3 art. 10.1.4.2(2).
Torsional buckling and Lateral-torsional buckling are prohibited by the fully braced diaphragm. The compression in the free flange is included in the special purlin checks.
The combined stability checks are not executed since they are replaced by the special purlin checks.
In contrast to art. 10.1.3.3(2) the Local Transverse Load Check and its interaction with the bending moment is executed even if the support reaction is a tensile force.
Scia Engineer Steel Code Check Theoretical Background
84
Special considerations for Purlins
As outlined in Use of Diaphragms for a section which meets all requirements, special purlin checks according to EN 1993-1-3 Chapter 10 will be executed:
Diaphragm on the compression side
Cross-section resistance according to EN 1993-1-3 art. 10.1.4.1
In case of compression in the free flange also Stability of the free flange according to EN 1993-1-3 art. 10.1.4.2
Diaphragm on the tension side
Cross-section resistance according to EN 1993-1-3 art. 10.1.4.1
Stability of the free flange according to EN 1993-1-3 art. 10.1.4.2
Resistance of Cross-Section
The Resistance of the Cross-Section is determined according to EN 1993-1-3 art. 10.1.4.1.
Since this check concerns a separate formula for each flange (10.3a) and (10.3b) the effective section modulus Weff,y is determined for each flange separately.
The average yield strength will only be used in case for both force components separately (N, My) the average yield strength may be used (Aeff = Ag ; Weff,y = Wel,y).
Definition of the free flange geometry
The dimension h is taken as the full cross section height.
The properties of the free flange are calculated according to the z-z axis of the full cross-section.
The following table shows the supported cross-sections including the contributing web height.
Form code Description Contributing web
5 Channel section 1/5 h
102 Rolled Z section 1/5 h
112 Cold formed channel 1/5 h
113 Cold formed Z 1/5 h
114 Cold formed C section 1/5 h
116 Cold formed C-Section eaves beam 1/5 h
117 Cold formed C-Plus section 1/5 h
118 Cold formed ZED section 1/5 h
119 Cold formed ZED section asymmetric lips 1/5 h
120 Cold formed ZED section inclined lip 1/5 h
121 Cold formed Sigma section 1/6 h
122 Cold formed Sigma section stiffened 1/6 h
123 Cold formed Sigma-Plus section 1/6 h
Scia Engineer Steel Code Check Theoretical Background
85
124 Cold formed Sigma section eaves beam 1/6 h
125 Cold formed Sigma-Plus section eaves beam 1/6 h
126 Cold formed ZED section both lips inclined 1/5 h
129 Cold formed Sigma section asymmetric 1/6 h
As the code indicates in Figure 10.2, for sigma sections the rounding which leads to the web depression is also accounted for in the height of the free flange. Therefore, to generalize this principle, within Scia Engineer the rounding between the flange and the web is always accounted for in the free flange height (for all section types).
Determination of the equivalent lateral load
The equivalent lateral load on the free flange qh,Ed is determined from the vertical load qEd on the purlin using formula (10.4).
For any given moment diagram, the equivalent vertical line load qEd is determined as the line load
which results in approximately the same bending moment diagram..
The factor kh is determined according to EN 1993-1-3 Figure 10.3.
For kh0, the general formula for Z,C or sections is applied. The formula for a simple Z-section is not supported.
For Gravity loading, the vertical loading is assumed to be positioned at the outside of the web. For Uplift loading the vertical loading is assumed to be positioned exactly in the middle of the flange width.
For Gravity loading the general formula including the shear center distance e is used.
For Uplift loading the general formula including the shear center distance f is used. In case of a symmetrical Z section this distance will become a.
The load qh,Ed is given a positive sign in case it follows the same convention as shown in the code.
The load is given a negative sign in case it points in the other direction.
Determination of the lateral bending moment
Table 10.1 provides the formulas to determine Mfz,Ed for specific positions within the beams: at the
ends (e) and at the position of the maximal moment (m).
Within Scia Engineer however, the check is executed in different sections. Therefore, the values of Mfz,Ed need to be known in each section.
To this end, as indicated in the code in EN 1993-1-3 art. 10.1.4.1(7), the general equations have
been derived using the theory of beams on an elastic Winkler foundation.
Scia Engineer Steel Code Check Theoretical Background
86
The differential equation for the displacement of a beam on elastic foundation loaded by a line load is written out as follows Ref.[21]:
With: E Section modulus
I Bending stiffness
L Member length, taken as La
q Line load, taken as qh,Ed
K Foundation stiffness, taken as lateral spring stiffness K
A,B,C,D Integration constants
The integration constants are determined depending on the boundary conditions for the cases given in Table 10.1.
Using the beam equation with the second derivative of the displacement the equation for the bending moment Mfz,Ed is obtained and leads to the following solutions:
Scia Engineer Steel Code Check Theoretical Background
87
Solution for a beam on elastic Winkler foundation with Hinged end conditions
Solution for a beam on elastic Winkler foundation with Hinged-Fixed end conditions
Solution for a beam on elastic Winkler foundation with Fixed end conditions
Scia Engineer Steel Code Check Theoretical Background
88
The determination of a hinged or fixed end for Mfz,Ed is done as follows:
A single span member is always considered to have hinged ends. A single span member is defined as a member with only one part in the buckling system for Ly.
An LTB restraint is always considered as a fixed end.
For multi-span members, the ends of the buckling system for Ly are considered as hinged. The internal points of the buckling system for Ly are considered as fixed.
As specified in EN 1993-1-3 art. 10.1.4.1(5) in case the free flange is in tension Mfz,Ed is taken equal
to zero. To determine if the free flange is in tension or compression the following stress is calculated:
(My,Ed / Weff,y,free flange) + (Ned / Aeff)
In case this stress results in tension, the free flange is considered to be in tension.
In case this stress results in compression, the free flange is considered to be in compression.
The sign of Mfz,Ed determines the tension/compression side of the free flange and thus determines which Wfz is used in the check.
The limit of R 40 given in art. 10.1.4.1(6) does not apply since the general Winkler theory is used instead of table 10.1.
Determination of the distance between anti-sag bars
The code defines anti-sag bars as bars which provide lateral rigid support to the free flange. Within Scia Engineer, LTB restraints are thus seen as anti-sag bars.
In case LTB restraints are defined at the free flange, the length La is taken as the length between these restraints. In case there are no LTB restraints defined at the free flange, La is read from the
buckling system.
Determination of the lateral spring stiffness
The lateral spring stiffness K is determined according to EN 1993-1-3 art. 10.1.5.1(4).
The developed height of the purlin web hd is taken as the total developed slant height sd used in the
Shear Check, as described in Shear Force.
The rotational restraint CD is taken as vorhC, the rotational stiffness of the diaphragm, as described
in Adaptation of torsional constant.
The dimension bmod depends on the direction of the equivalent horizontal load qh,Ed and the type of
cross-section. According to the code this depends if the load brings the purlin into contact with the sheeting at the purlin web or at the tip of the purlin flange.
Scia Engineer Steel Code Check Theoretical Background
89
This is clarified in the following picture:
The distance a i.e. position of the fastener is taken as 0,5 b. The fastener is thus assumed to be
positioned in the middle of the flange.
Buckling Resistance of the Free Flange
The Buckling Resistance of the Free Flange is determined according to EN 1993-1-3 art. 10.1.4.2.
To determine if the free flange is in tension or compression the following stress is calculated:
(My,Ed / Weff,y,free flange) + (Ned / Aeff)
In case this stress results in tension, the free flange is considered to be in tension.
In case this stress results in compression, the free flange is considered to be in compression.
For a free flange in tension the buckling resistance does not need to be checked.
For determining the buckling length lfz of the free flange a difference is made between gravity
loading (downward –z loading) and uplift loading (upward +z loading).
Gravity Loading
In case of downward –z loading the buckling length of the free flange is determined according to formula (10.9).
The i factors are determined according to EN 1993-1-3 Table 10.2a.
Art. 10.1.4.2(4) is not supported.
Scia Engineer Steel Code Check Theoretical Background
90
Uplift Loading
In case of upward +z loading the buckling length of the free flange is determined according to formula (10.9).
The i factors are determined according to EN 1993-1-3 Table 10.2b.
The method according to art. 10.1.4.2(6) & (7) is not supported.
General Notes
For both loading types, Tables 10.2a & b differentiate between „simple span‟, „end span‟ and „intermediate span‟. This is based on the Ly system length.
In case the member under consideration has only one part for Ly then it is considered as „simple
span‟.
When the member has more parts for Ly it is considered as multi-span. For a multi-span, sections
located in the first or last part of the system length are considered as „end span‟. Sections located in the other parts are considered as „intermediate span‟.
Table 10.2a does not specify „simple span‟. The values for a „simple span‟ are taken equal as an „end span‟.
The „number of anti-sag‟ bars used in Tables 10.2a & b concerns the number of LTB restraints
defined on the actual member. Only LTB restraints at the side of the free flange are accounted for in this „number‟.
EN 1993-1-3 art. 10.1.4.2(5) specifies a method for the buckling length in case of a „relatively large
axial force‟. Within Scia Engineer this is quantified using a limit value, which is set in the Steel Setup.
In case the axial load is considered as large, the method described in EN 1993-1-3 art. 10.1.4.2(5)
is applied.
This procedure applies to both gravity and uplift loading using Table 10.2a and 10.2b respectively.
Scia Engineer Steel Code Check Theoretical Background
91
Supported sections
I Symmetric I shapes (IPE, HEA, HEB, ….)
RHS Rectangular Hollow Section
CHS Circular Hollow Section
L Angle section
U Channel section
T T section
PPL Asymmetric I shapes
Z Z section
RS Rectangular section
Cold formed section
COM Composed section in PRIMAWIN
O Solid tube
NUM Numerical section
The necessary data conditions for these sections are described in chapter "Profile conditions for code check
".
The COM and NUM sections are not read out of the profile library.
I
RHS
CHS
L
U
T
PPL
RS
Z
O
COM
NUM
Classification x x x x x x x x (1) x (1) (1) (1)
Section check class 1 x x x
Section check class 2 x x x
Section check class 3 x x x x x x x x x x x x x
Section check class 4 x x x x x x
Stability check class 1
x x x
Stability check class 2
x x x
Stability check class 3
x x x x x x x x x x x x x
Stability check class 4
x x x x x x
Shear buckling check x x x x
(1) Sections are classified as class 3 cross section by default.
Scia Engineer Steel Code Check Theoretical Background
92
References
1 Eurocode 3
Design of steel structures
Part 1 - 1 : General rules and rules for buildings
EN 1993-1-1:2005
[2] Eurocode 3
Design of steel structures
Part 1-3: General rules
Supplementary rules for cold-formed members and sheeting
EN 1993-1-3:2006
3 Eurocode 3
Design of steel structures
Part 1.5 : Plated structural elements
EN 1993-1-5 : 2006
4 R. Maquoi
ELEMENTS DE CONSTRUCTIONS METALLIQUE
Ulg , Faculté des Sciences Appliquées, 1988
[5] EN 1990
Eurocode – Basis of structural design
EN 1990:2002 E
[6] Eurocode 3
Design of steel structures
Part 1 - 2 : General rules - Structural fire design
EN 1993-1-2:2005
[7] Model Code on Fire Engineering
ECCS - N° 111
May 2001
[8] Eurocode 1
Actions on structures
Part 1-2 : General Actions - Actions on structures exposed to fire
prEN 1991-1-2:2002
[9] Rules for Member Stability in EN 1993-1-1
Background documentation and design guidelines
ECCS - N° 119
2006
Scia Engineer Steel Code Check Theoretical Background
93
[10] Eurocode 3
Design of steel structures
Part 1 - 1/ A1 : General rules and rules for buildings
ENV 1993-1-1:1992/A1, 1994
[11] Eurocode 3
Design of steel structures
Part 1 - 1 : General rules and rules for buildings
EN 1993-1-1:2005/AC:2009 Corrigendum
[12] Eurocode 3
Design of steel structures
Part 1 - 2 : General rules - Structural fire design
EN 1993-1-2:2005/AC:2009 Corrigendum
[13] Eurocode 3
Design of steel structures
Part 1-3: General rules
Supplementary rules for cold-formed members and sheeting
EN 1993-1-3:2006/AC:2009 Corrigendum
[14] Eurocode 3
Design of steel structures
Part 1.5 : Plated structural elements
EN 1993-1-5 : 2006/AC:2009 Corrigendum
[15] Essentials of Eurocode 3
Design Manual for Steel Structures in Building
ECCS - N° 65, 1991
[16] Commentary and Worked Examples to EN 1993-1-5 “Plated Structural Elements”
Johansson B., Maquoi R., Sedlacek G., Müller C., Beg D.,
JRC - ECCS, 2007.
[17] SN001a-EN-EU
NCCI: Critical axial load for torsional and flexural torsional buckling modes
Access Steel, 2006
www.access-steel.com
[18] AISI S100-2007
North American Specification for the Design of Cold-Formed Steel Structural Members
2007 edition
[19] E. Kahlmeyer
Stahlbau nach DIN 18 800 (11.90)
Werner-Verlag, Düsseldorf
Scia Engineer Steel Code Check Theoretical Background
94
[20] Beuth-Kommentare
Stahlbauten
Erläuterungen zu DIN 18 800 Teil 1 bis Teil 4, 1.Auflage
Beuth Verlag, Berlin-Köln 1993
[21] D. Vandepitte
Berekening van Constructies
Boekdeel 1 pp522
www.berekeningvanconstructies.be
[22] Design rule for Lateral Torsional Buckling of Channel Sections
A-2007.9 O-2007.21
Karin de Louw
2007
[23] EN 12811-1
Temporary works equipment
Part 1: Scaffolds – performance requirements and general design
2004
[24] EN 12810-1
Façade scaffolds made of prefabricated components
Part 1: Products specifications
2004
[25] EN 12810-2
Façade scaffolds made of prefabricated components
Part 2: Particular methods of structural design
2004
[26] DIN 4420 Teil 1
Arbeits- und Schutzgerüste
Allgemeine Regelungen, Sicherheitstechnische Anforderungen, Prüfungen
Dezember 1990
[27] Corrections and amendments to EN 1993-1-3
Meeting of ECCS-TWG 7.5
T. Höglund
2010
[28] Déversement élastique d‟une poutre à section bi-symétrique soumise à des moments d‟extrémité et une charge répartie ou concentrée.
Y. Galéa
CTICM, Construction Métallique, n° 2-2002.
Scia Engineer Steel Code Check Theoretical Background
95
[29] Lateral-Torsional buckling of steel beams:
A general expression for the moment gradient factor.
A. López, D. J. Yong, M. A. Serna
Stability and Ductility of Steel Structures, 2006.
[30] SC001a-EN-EU
Code commentary: Collection No. 1
Access-Steel, 2007.
[31] SN005a-EN-EU
Determination of moments on columns in simple construction
Access-Steel, 2005.
[32] Steel Building Design
Medium Rise Braced Frames
SCI PUBLICATION P365.
[33] Target specification Dimensioning Profiles
ZEMAN & CO. GmbH
Wien, 2006.
[34] New proposals for EN 1993-1-5, Annex D:
Plate girders with corrugated webs.
H. Pasternak, J. Robra, G. Kubieniec
IABSE-FIB Conference, Dubrovnik, 2010.
[35] Zulassung Nr. Z-8.22-208
Modulsystem "CUPLOK"
Deutsches Institut für Bautechnik, 2006.
[36] Zulassung Nr. Z-8.22-64
Modulsystem "Layher-Allround"
Deutsches Institut für Bautechnik, 2008.
Scia Engineer Steel Code Check Theoretical Background
96
DIN18800
DIN18800 Code check
The beam elements are checked according to the regulations given in
DIN 18800 Teil 1
Stahlbauten
Bemessung und Konstruktion
DK 693.814.014.2, November 1990
DIN 18800 Teil 2
Stahlbauten
Stabilitätsfälle, Knicken von Stäben und Stabwerken
DK 693.814.074.5, November 1990
DIN 18800 Teil 3
Stahlbauten
Stabilitätsfälle, Plattenbeulen
DK 693.814.073.1, November 1990
Material properties
For standard steel grades, the yield strength fy and tensile strength fu are defined according to the
thickness of the element (see Ref. 1, Tab.1)
The standard steel grades are :
(fy, fu in N/mm², t in mm)
t<=40 t<=40 40<t<=80 40<t<=80
fy fu fy fu
S235
S 235
St 37-2
240 360 215 360
S275
S 275
280 430 255 430
S355
S 355
St 52-3
360 510 325 510
Scia Engineer Steel Code Check Theoretical Background
97
t<=40 t<=40 40<t<=100 40<t<=100
fy fu fy fu
S420
S 420
420 520 390 520
S460
S 460
460 550 430 550
Consulted articles
For the section check, the cross section is classified according to DIN18800 Teil I, Table 12,13,14,15 and 18.. Depending on this classification, the section is checked as slender section, EL/EL (elastic/elastic), as EL/PL (elastic/plastic) or as PL/PL (plastic/plastic).
For the EL/EL check, DIN18800 Teil I, Element (746), (747), (748), (749), (750) are used.
The EL/PL check takes the rules from DIN18800 Teil I, Element (756), (757) and Table (16) ,(17). The PL/PL check is done according to DIN18800 Teil I, Element (758), Table (16),(17).
The slender cross section is checked according to DIN18800 Teil 2, Element (715).
For the stability check, the beam element is checked according to DIN18800 Teil 2 for buckling, lateral torsional buckling and bending and compression. The following criteria are used :
compression : Element (304),(306)
lateral torsional buckling : Element (311),(309)
bending and axial compression : Element (313),(321),(322)
bending (LTB) and compression : Element (320),(323)
For slender sections, the following criteria are used :
calculation of effective area : Element (705),(706),(708),(709),(712),(713)
buckling check : Element (715),(716),(718),(719)
LTB check : Element (725),(726),(728),(729)
For the shear buckling check, the beam element is checked according to DIN18800 Teil 3. The following criteria are used : Element (113), (504), (602),(603)
A more detailed overview for the used articles is given for the relevant parts following table. The chapters marked with “x” are consulted. The chapters marked with (*) have a supplementary explanation the following chapters.
Scia Engineer Steel Code Check Theoretical Background
98
Teil 1
7.5. Verfahren beim Tragsicherheitsnachweis Nachweise (*)
7.5.1. Abgrenzungskriterien und Detailregelungen (*)
7.5.2. Nachweis nach dem Verfahren Elastisch-Elastisch
(745)………………………………………………………………………………
(746) ………………………………………………………………………………
(747) ………………………………………………………………………………
(748) ………………………………………………………………………………
(749) ………………………………………………………………………………
(750) ………………………………………………………………………………
x
x
x
x
x
x
x
Nachweis nach dem Verfahren Elastisch-Plastisch
(753) ………………………………………………………………………………
(756) ………………………………………………………………………………
(757) ………………………………………………………………………………
x
x
x
x
Nachweis nach dem Verfahren Plastisch-Plastisch
(758) ………………………………………………………………………………
x
x
Teil 2
3.2. Planmässig mittiger Druck
3.2.1. Biegeknicken
(304) ………………………………………………………………………………
x
x
x (*)
3.2.2. Biegedrillknicken
(306) ………………………………………………………………………………
x
x (*)
3.3. Einachsige Biegung ohne Normalkraft
3.3.1. Allgemeines
(307) ………………………………………………………………………………
x
x
x
3.3.2. Behinderung der Verformung
(309) ………………………………………………………………………………
x
x (*)
3.3.3. Nachweis des Druckgurtes als Druckstab
3.3.4. Biegedrillknicken
(311) ………………………………………………………………………………
x
x (*)
3.4. Einachsige Biegung mit Normalkraft
3.4.1. Stäbe mit geringer Normalkraft
(312) ………………………………………………………………………………
x
x
x
3.4.2. Biegeknicken
(314) ………………………………………………………………………………
x
x
3.4.3. Biegedrillknicken
(320) ………………………………………………………………………………
x
x
3.5. Zweiachsige Biegung mit oder ohne Normalkraft
3.5.1. Biegeknicken
(321) ………………………………………………………………………………
(322) ………………………………………………………………………………
x
x
x
x(*)
3.5.2. Biegedrillknicken
(323) ………………………………………………………………………………
x
x
Scia Engineer Steel Code Check Theoretical Background
99
4. Mehrteilige, einfeldrige Stäbes
4.1. Allgemeines
4.2. Häufig verwendete Formelzeichnen
(404) ………………………………………………………………………………
4.3. Ausweichen rechtwinklig zur stofffreien Achse
(405) ………………………………………………………………………………
(406)……………………………………………………………………………….
(408)……………………………………………………………………………….
(409)……………………………………………………………………………….
x(*)
x
x
x
x
x
7. Planmässig gerade Stäbe mit ebenen dünnwandigen Quenschnittsteilen
7.1. Allgemeines
(701) ………………………………………………………………………………
(702) ………………………………………………………………………………
(704) ………………………………………………………………………………
x
x
x
x
x
7.2. Berechnungsgrundlage
(705) ………………………………………………………………………………
(706) ………………………………………………………………………………
(707) ………………………………………………………………………………
(708) ………………………………………………………………………………
(709) ………………………………………………………………………………
x
x
x
x
x
x
7.3. Wirksame Breite beim Verfahren Elastisch-Elastisch
(711) ………………………………………………………………………………
(712) ………………………………………………………………………………
(713) ………………………………………………………………………………
x
x
x (*)
x
7.4. Wirksame Breite beim Verfahren Elastisch-Plastisch
7.5. Biegeknicken
7.5.1. Spannungsnachweis beim Verfahren Elastisch-Elastisch
(715) ………………………………………………………………………………
x
x
x
7.5.2. Vereinfachte Nachweise
(716) ………………………………………………………………………………
(718) ………………………………………………………………………………
(719) ………………………………………………………………………………
(721) ………………………………………………………………………………
x
x
x
x
x
7.6. Biegedrillknicken
(722) ………………………………………………………………………………
(723) ………………………………………………………………………………
(725) ………………………………………………………………………………
(726) ………………………………………………………………………………
(728) ………………………………………………………………………………
(729) ………………………………………………………………………………
x
x
x
x
x
x
x
Teil 3
5. Nachweise
(504) ………………………………………………………………………………
(*)
x
Scia Engineer Steel Code Check Theoretical Background
100
6. Abminderungsfaktoren
(601) ………………………………………………………………………………
(602) ………………………………………………………………………………
x
x
x
Classification of sections
For each intermediary section, the classification is determined and the proper section check is performed. The classification can change for each intermediary point.
For each load case/combination, the critical section classification over the member is used to perform the stability check. So, the stability section classification can change for each load case/combination.
However, for non-prismatic sections, the stability section classification is determined for each intermediary section.
Net area properties
The net area properties are not taken into account .
The holes for fasteners are neglected.
Plastic interaction formula for RHS section
b
s/2
h
AG
AS/2
Scia Engineer Steel Code Check Theoretical Background
101
For RHS section, classified as Plastic-Plastic or Elastic-Plastic, the plastic interaction formula according to Ref.[13], can be selected.
Used variable :
A sectional area
AS = s h
AG = (A-AS)/2.0
Wel,y elastic section modulus around y axis
Wel,z elastic section modulus around z axis
fy,d yield strength
y,d shear strength
Vz,pl,Rd = AS y,d
Vy,pl,Rd = 2AG y,d
NSd normal force
My,Sd bending moment around y axis
Mz,Sd bending moment around z axis
Vy,Sd shear force in y direction
Vz,Sd shear force in z direction
MT,Sd torsional moment
2
Rd,pl,z
Sd,T
Sd,z
z
z
Rd,pl,z
Sd,T
Sd,z
V
b
MV
1else
0.14
1
V
b
MV
if
Scia Engineer Steel Code Check Theoretical Background
102
2
Rd,pl,y
Sd,T
Sd,y
y
y
Rd,pl,y
Sd,T
Sd,y
V
h
MV
1else
0.14
1
V
h
MV
if
Ar= zAS + 2yAG
r
Sz
A
A
Npl,Rd = Ar fy,d
ydy,elRd,plRd,pl,y fW25.1,hN
4
2minM
ydz,elRd,plRd,pl,z fW25.1,bN
4
1minM
Rd,pl
Sd
N
Nn
Rd,pl,y
Sd,y
yM
Mm
Rd,pl,z
Sd,z
zM
Mm
The following interaction formula are checked :
Scia Engineer Steel Code Check Theoretical Background
103
Scia Engineer Steel Code Check Theoretical Background
104
Plastic interaction formula for CHS section
For CHS section, classified as Plastic-Plastic or Elastic-Plastic, the plastic interaction formula according to Ref.[14], Tafel 6.74, is used :
selQ,plQ,pl
srQ,pl
r
2
pl
v
pl
v
pl
v
spl
2
z
2
yv
2
z
2
yv
plQ
vQ,pl
v
W25.1,Nd
minM
AN
dtA
Q
Q1:
4
1
Q
Q
1:4
1
Q
Q
3
dt2Q
MMM
QQQ
1
2N
Ncos
1
M
M
with Qy,Qz internal shear force
Nv internal normal force
My,Mz internal bending moments
s yield strength
d,t dimensions from CHS
Wel elastic section modulus
Scia Engineer Steel Code Check Theoretical Background
105
t
d
Torsion check
For the cross section check inclusive torsion and warping, we refer to Chapter 'Warping check'.
The stability check (DIN 18800 T2, formula 28 & 30) for doubly symmetric I section becomes (Ref.[9], pp. 259) :
)30(0.1kM
MMk
M
M
N
N
)28(0.1kM
MMk
M
M
N
N
z
d,z,pl
w,zz
y
d,y,plM
y
d,plz
z
d,z,pl
w,zz
y
d,y,pl
y
d,pl
with Mz,w
h
M2 w
Mw bimoment (see chapter 'Standard diagrams for warping torque, bimoment and the St.Venant torsion')
kz = 1.50 In case there is no compression force kz is taken as 1.00 (Ref.[9], pp. 270).
Built-in beams
For built-in beam sections (IFB, SFB, THQ sections), proper section checks are performed, taking into account the local plate bending. See Chapter „Section check for built-in beams (IFB, SFB, THQ sections)‟
Scia Engineer Steel Code Check Theoretical Background
106
Calculation of the buckling length
For the calculation of the buckling length, we refer to chapter "Calculation of buckling ratio".
The buckling properties for a VARH element are calculated by using the critical Euler force for this member (see “Calculation of critical Euler force for VARH elements”).
The buckling curves for steel grade S420 and S460 are taken from Ref.[10], Annex D.
Torsional buckling
The slenderness for torsional buckling vi is given by (see Ref.6 , 7.5):
with l0 the torsional buckling length, refers to the input value for the system length lyz
lz the system length for buckling around zz-axis
Remark : the z-axis refers to the axis which goes through the shear force centre.
z refers to the buckling ratio around the zz-axis
Remark : the z-axis refers to the axis which goes through the shear force centre.
0 refers to end warping and is input by the value kxy
zM the shear center
iy the radius of gyration around major axis
iz the radius of gyration around minor axis
ip² = iy² + iz²
iM² = ip² + zM²
Iw the warping constant
Iz the moment of inertia around minor axis
It the torsional constant
With this slenderness vi and the buckling curve c, the reduction factor is calculated.
Scia Engineer Steel Code Check Theoretical Background
107
Use of diaphragms
(see also Ref.7,3.5 and Ref.8,3.3.4.)
The shear stiffness S for diaphragm is calculated as follows:
L
K+K
10a.=S
s
21
4
with a the frame distance
Ls the length of diaphragm
K1 factor K1
K2 factor K2
The torsional constant It is adapted with the stiffness of the diaphragms:
G
lvorhCII
2
2
tid,t
with l the LTB length
G the shear modulus
vorh
C
the actual rotational stiffness of diaphragm
Scia Engineer Steel Code Check Theoretical Background
108
LTB Check
For aysmmetric I sections, RHS (Rectangular Hollow Section) sections and CHS (Circular Hollow Section) sections, the elastic critical moment for LTB Mcr is given by the general formula F.2. Annex
F Ref. 4. For the calculation of the moment factors C1, C2 and C3 we refer to "Calculation of moment factors for LTB".
Depending on the input of the basic data, Mcr for symmetric I sections is given by the general
formula F.2. Annex F Ref. 4, by the DIN formula (19), or by formula according to Ref.[11] "Roik, Carl, Lindner, Biegetorsionsprobleme gerader dünnwandiger Stäbe, Verlag von Wilhelm Ernst & Sohn, 1972".
DIN formula (19) :
with l,l0 the LTB length
z refers to rotational end-restraint „in plan’ (about the z-z local axis).
0 refers to end warping
zp the point of load application
Iw the warping constant
Iz the moment of inertia around minor axis
It the torsional constant
A the sectional area
E the modulus of elasticity
vi the slenderness for torsional buckling ( see above)
the moment factor ( equivalent for factor C1)
Scia Engineer Steel Code Check Theoretical Background
109
Roik, Carl & Lindner
z
tw
p
2
pzcry,ki
I
I²l039.0Ic
²
z5²c
²
z5
²l
²EIMM
with moment factor according to Roik, Carl, Lindner
E modulus of elasticity
I
z moment of inertia around weak axis zz
l system length for LTB
z
p application point for loading, negative value is on top and has negative influence
I
w warping constant
I
t torsional constant
The factor is supported for the following cases (described in Ref.[11], tables 5.13, 5.14, 5.15, 5.18, 5.19, 5.20, 5.21, 5.22, 5.23, 5.24, 5.25, 5.26, 5.27, 5.28, 5.29, 5.30, 5.33) :
Linear moment distribution :
Moment line according to distributed loading
Scia Engineer Steel Code Check Theoretical Background
110
Moment line according to concentrated loading
Scia Engineer Steel Code Check Theoretical Background
111
For the other supported sections, the elastic critical moment for LTB Mcr is given by
z
2
t
2
z
2
z
2
EI
GIL
I
Iw
L
EIMcr
with E the modulus of elasticity
G the shear modulus
L the length of the beam between points which have lateral restraint (= lLTB)
Iw the warping constant
It the torsional constant
Iz the moment of inertia about the minor axis
See also Ref. 5, part 7 and in particular part 7.7. for channel sections.
Haunched sections (I+Ivar, Iw+Plvar, Iw+Iwvar, Iw+Ivar, I+Iwvar) and composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
For full rectangular sections the value of n according to DIN 18800-2 tabelle 9 is taken as 1,5 according to Ref.[8] pp 175.
For advanced Lateral-torsional buckling analysis, see chapter „LTBII: Lateral Torsional Buckling 2nd Order Analysis‟.
Combined flexion for check method 2
The value My is the maximum value of the bending moment around the strong axis in the member. The value Mz is the maximum value of the bending moment around the weak axis in the member.
For non-prismatic sections, the values My and Mz are the concurrent bending moments for each intermediary section.
Scia Engineer Steel Code Check Theoretical Background
112
Battened compression members
The following section pairs are supported as battened compression member :
(1) 2I
(2) 2Uo
(3) 2Uc
Two links (battens) are used.
The following additional checks are performed :
- buckling resistance check around weak axis of single chord with NG
- section check of single chord, using internal forces (Ref.[7], pp.88-95) :
4
amaxVM
2
maxV V
W
A)
l
asin(Mmax
2
N N
y
G
y
G
*
z
GzG
- section check of single batten, using the internal forces (Ref.[7], pp.88-95) :
2
TeM
2h
amaxV T
y
y
For the calculation of maxVy, the value of Mz is increased with the value of the internal force Mzz.
Scia Engineer Steel Code Check Theoretical Background
113
l
a
hy
e
Effective area properties
The calculation of the effective area is performed with the direct method (sigma_d = fy,k) according to the El-El procedure (DIN18800 T2, 7.3.).
For each intermediary section, the classification (and if necessary, the effective area ) is determined and the proper section check is performed. The classification (and effective area) can change for each intermediary point. The most critical check is displayed on the screen.
For each load case and combination, the most critical effective area properties are saved. The most critical effective area properties are the effective area properties on the position where the appropriate moment of inertia is the minimum.
With these critical properties, the stability check is performed.
For non-prismatic elements, the effective area properties are calculated on each intermediary section, also for the stability check.
Scia Engineer Steel Code Check Theoretical Background
114
Shear buckling check
Composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
Shear buckling check with buckling influence
The influence of the buckling effect into the shear buckling control, is neglected when there is a
bending moment present.
It means that k=1 if <0.9. See also Ref.[3], Element 503.
Cold formed thin gauge members
The following table includes a list of DASt-Richtlinie 016 (Ref.[12]) elements which are implemented in Scia Engineer by using the related DIN18800 T2 (Ref.[2]) element.
Supported elements from
DASt - Richtlinie 016
Covered by DIN 18800 T2 elements
Remarks
3.7.1. Grenzzustand der Tragfähigkeit
328 Tab.26
329 712
330 712
333 Tab.27
335 706
4.3.1. Biegemomententragfähigkeit
404 715
4.4. Biegedrillknicken biegebeanspruchter Bauteile
4.4.3. Allgemeiner Nachweis
421 311
422 311
423 725, 726
4.5. Druckbeanspruchte einteilige Stäbe
4.5.1. Allgemeines
429 708-710
430 708-710
431 708-710
432 708-710
Scia Engineer Steel Code Check Theoretical Background
115
433 708-710
434 708-710
4.5.2. Planmäig mittiger Druck
435 716 AD
ef is not used
436 manual input / input in profile library for KSL
437 723
438 723
4.5.3. Einachsige Biegung mit Druck
440 707
441 718
442 728
4.5.3. Zweiachsige Biegung mit Druck
443 707
444 721 AD
ef is not used
445 729
Supported sections
I Symmetric I shapes (IPE, HEA, HEB, ….)
RHS Rectangular Hollow Section (RHS)
CHS Circular Hollow Section (CHS)
L Angle section
U Channel section
T T section
PPL Asymmetric I shapes
RS Rectangular section
Cold formed section
COM Composed section in PRIMAWIN
O Solid tube
NUM Numerical section
The necessary data conditions for these sections are described in chapter "Profile conditions for code check
".
The COM and NUM sections are not read out of the profile library.
Scia Engineer Steel Code Check Theoretical Background
116
I
RHS CHS L U T PPL RS O COM NUM
Classification x x x x x x x x x (1) (1) (1)
Section check PL-PL x x
Section check EL-PL x x
Section check EL-EL x x x x x x x x x x x x
Section check slender section
x x x x x x
Stability check x x x x x x x x x x x x
Stability check slender section
x x x x x x
Shear buckling check x x x x
(1) sections are classified as EL-EL cross section by default.
References
1 DIN 18800 Teil 1
Stahlbauten
Bemessung und Konstruktion
DK 693.814.014.2, November 1990
2 DIN 18800 Teil 2
Stahlbauten
Stabilitätsfälle, Knicken von Stäben und Stabwerken
DK 693.814.074.5, November 1990
3 DIN 18800 Teil 3
Stahlbauten
Stabilitätsfälle, Plattenbeulen
DK 693.814.073.1, November 1990
[4] Eurocode 3
Design of steel structures
Part 1 - 1 : General rules and rules for buildings
ENV 1993-1-1:1992, 1992
[5] R. Maquoi
ELEMENTS DE CONSTRUCTIONS METALLIQUE
Ulg , Faculté des Sciences Appliquées, 1988
[6] G. Hünersen, E. Fritzsche
Stahlbau in Beispielen
Berechnungspraxis nach DIN 18 800 Teil 1 bis Teil 3 (11.90)
Werner-Verlag, Düsseldorf 1991
Scia Engineer Steel Code Check Theoretical Background
117
[7] E. Kahlmeyer
Stahlbau nach DIN 18 800 (11.90)
Werner-Verlag, Düsseldorf
[8] Beuth-Kommentare
Stahlbauten
Erläuterungen zu DIN 18 800 Teil 1 bis Teil 4, 1.Auflage
Beuth Verlag, Berlin-Köln 1993
[9] Stahlbau Kalender 1999
DSTV
Ernst & Sohn, 1999
[10] Eurocode 3
Design of steel structures
Part 1 - 1/ A1 : General rules and rules for buildings
ENV 1993-1-1:1992/A1, 1994
[11] Roik, Carl, Lindner
Biegetorsionsprobleme gerader dünnwandiger Stäbe
Verlag von Wilhelm Ernst & Sohn
1972
[12] DASt-Richtlinie 016
Bemessung und konstruktive Gestaltung von Tragwerken aus dünnwandigen kaltgeformted Bauteilen
Stahlbau-Verlagsgesellschaft - 1992
[13] H. Rubin,
Interaktionsbeziehungen für doppeltsymmetrische I- und Kasten-Querschnitte bei zweiachsiger Biegung und Normalkraft
Der Stahlbau 5/1978, 6/1978
[14] Stahl im Hochbau
14. Auflage, Band I / Teil 2
1986, Verlag Stahleisen mbH, Düsseldorf
Scia Engineer Steel Code Check Theoretical Background
118
ONORM B 4300
ONORM B 4300 Code check
The beam elements are checked according to the regulations given in
ÖNORM B 4300-1
Stahlbau
Berechnung und Konstruktion der Tragwerke
Bemessung nach Grenzzuständen
DK 624.014.2.046, März 1994
ÖNORM B 4300-2
Stahlbau
Knicken von Stäben und Stabwerken
Bedingungen für die gemeinsame Anwendung von DIN 18 800 Teil 2 und ÖNORM B 4300-1
DK 624.014.2.075.2, April 1994
ÖNORM B 4300-3
Plattenbeulen
Bedingungen für die gemeinsame Anwendung von DIN 18 800 Teil 3 und ÖNORM B 4300-1
DK 624.014.2.075.4, April 1994
DIN 18800 Teil 1
Stahlbauten
Bemessung und Konstruktion
DK 693.814.014.2, November 1990
DIN 18800 Teil 2
Stahlbauten
Stabilitätsfälle, Knicken von Stäben und Stabwerken
DK 693.814.074.5, November 1990
DIN 18800 Teil 3
Stahlbauten
Stabilitätsfälle, Plattenbeulen
DK 693.814.073.1, November 1990
Scia Engineer Steel Code Check Theoretical Background
119
Material properties
For standard steel grades, the yield strength fy and tensile strength fu are defined according to the
thickness of the element (see Ref. 1, 2.1. and Ref. 4, Tab.1)
The standard steel grades are:
(fy, fu in N/mm², t in mm)
t<=40 t<=40 40<t<=80 40<t<=80
fy fu fy fu
St 360
S235
S 235
240 360 215 360
St 430
S275
S 275
280 430 255 430
St 510
S355
S 355
360 510 325 510
t<=40 t<=40 40<t<=100 40<t<=100
fy fu fy fu
S420
S 420
420 520 390 520
S460
S 460
460 550 430 550
Consulted articles
For the section check, the cross section is classified according to ONORM B 4300-1 Tab.3,4,5 and to DIN18800 Teil I, Table 15,18. Depending on this classification, the section is checked as slender section, EL/EL (elastic/elastic), as EL/PL (elastic/plastic) or as PL/PL (plastic/plastic).
For the EL/EL check, ONORM B 4300-1 Art. 5.2. is used. (The 7% increase of the moment of
inertia is taken into account for rolled I section - see Ref. 1, Art. 5.2.5.4.).
The EL/PL check takes the rules from DIN18800 Teil I, Element (756), (757) and Table (16) ,(17). The PL/PL check is done according to DIN18800 Teil I, Element (758), Table (16),(17).
The slender cross section is checked according to DIN18800 Teil 2, Element (715).
Scia Engineer Steel Code Check Theoretical Background
120
For the stability check, the beam element is checked according to DIN18800 Teil 2 for buckling, lateral torsional buckling and bending and compression. The following criteria are used :
compression : Element (304),(306)
lateral torsional buckling : Element (311),(309)
bending and axial compression : Element (313),(321),(322)
bending (LTB) and compression : Element (320),(323)
For slender sections, the following criteria are used :
calculation of effective area : Element (705),(706),(708),(709),(712),(713)
buckling check : Element (715),(716),(718),(719)
LTB check : Element (725),(726),(728),(729)
For the shear buckling check, the beam element is checked according to DIN18800 Teil 3. The following criteria are used : Element (113), (504), (602),(603)
A more detailed overview for the used articles is given in "DIN18800 Code check".
Supported sections
I Symmetric I shapes (IPE, HEA, HEB, ….)
RHS Rectangular Hollow Section (RHS)
CHS Circular Hollow Section (CHS)
L Angle section
U Channel section
T T section
PPL Asymmetric I shapes
RS Rectangular section
Cold formed section
COM Composed section in PRIMAWIN
O Solid tube
NUM Numerical sections
The necessary data conditions for these sections are described in chapter "Profile conditions for code check
".
The COM and NUM sections are not read out of the profile library.
Scia Engineer Steel Code Check Theoretical Background
121
I
RHS
CHS
L
U
T
PPL
RS
O
COM
NUM
Classification x x x x x x x x x (1) (1) (1)
Section check PL-PL x
Section check EL-PL x
Section check EL-EL x x x x x x x x x x x x
Section check slender section
x x x x x x
Stability check x x x x x x x x x x x x
Stability check slender section
x x x x x x
Shear buckling check x x x x
(1) sections are classified as EL-EL cross section by default.
References
1 ÖNORM B 4300-1
Stahlbau
Berechnung und Konstruktion der Tragwerke
Bemessung nach Grenzzuständen
DK 624.014.2.046, März 1994
2 ÖNORM B 4300-2
Stahlbau
Knicken von Stäben und Stabwerken
Bedingungen für die gemeinsame Anwendung von DIN 18 800 Teil 2 und ÖNORM B 4300-1
DK 624.014.2.075.2, April 1994
3 ÖNORM B 4300-3
Plattenbeulen
Bedingungen für die gemeinsame Anwendung von DIN 18 800 Teil 3 und ÖNORM B 4300-1
DK 624.014.2.075.4, April 1994
[4] DIN 18800 Teil 1
Stahlbauten
Bemessung und Konstruktion
DK 693.814.014.2, November 1990
[5] DIN 18800 Teil 2
Stahlbauten
Stabilitätsfälle, Knicken von Stäben und Stabwerken
DK 693.814.074.5, November 1990
Scia Engineer Steel Code Check Theoretical Background
122
[6] DIN 18800 Teil 3
Stahlbauten
Stabilitätsfälle, Plattenbeulen
DK 693.814.073.1, November 1990
Scia Engineer Steel Code Check Theoretical Background
123
NEN
NEN6770/6771 Code check
The beam elements are checked according to the regulations given in
Staalconstructies TGB 1990
Basiseisen en basisrekenregels voor overwegend statisch belaste constructies
NEN 6770, december 1991
Staalconstructies TGB 1990
Stabiliteit
NEN 6771, december 1991-januari 2000
Material properties
For standard steel grades, the yield strength fy and tensile strength fu are defined according to the
thickness of the element (see Ref. 1, art.9.1.2.1.1.)
The standard steel grades are :
(fy, fu in N/mm², t in mm)
t<=40 t<=40 40<t<=100 40<t<=100 100<t<=250 100<t<=250
fy fu fy fu fy fy
S235
S 235
235 360 215 340 175 320
S275
S 275
275 430 255 410 205 380
S355
S 355
355 510 335 490 275 450
S420
S 420
420 520 390 520
S460
S 460
460 550 430 550
Remark : For cold formed section, the values for fy and fu are not influenced by the previous table.
Scia Engineer Steel Code Check Theoretical Background
124
Consulted articles
The cross section is classified according to NEN 6771 Table 1. (Class 1, 2, 3 or 4).
The section is checked on following criteria:
Tension: NEN 6770 Art. 11.2.1., NEN 6771 Art. 11.2.1.
Compression: NEN 6770 Art. 11.2.2., NEN 6771 Art. 11.2.2.
Shear: NEN 6770 Art. 11.2.4., NEN 6771 Art. 11.2.4.
Bending, shear and axial force: NEN 6770 Art. 11.3., NEN 6771 Art. 11.3.
For the stability check, the element is checked on following criteria:
Compression: NEN 6771 Art.12.1.1.1/ 12.1.2./12.1.3.
Lateral torsional buckling : NEN 6771 Art.12.2.
Bending and axial compression: NEN 6771 Art.12.3.
Shear buckling : NEN 6771 Art.13.8. / 13.9.
A more detailed overview for the used articles is given for NEN6770 part 11,12 and NEN6771 part 10,11,12,13. The chapters marked with “x” are consulted. The chapters marked with (*) have a supplementary explanation the following chapters.
NEN6770
11.Toetsing van de doorsnede
11.1. Algemeen
x
x
11.2. Enkelvoudige krachten en momenten
11.2.1. Axiale trek
x
x
11.2.2. Axiale druk x
11.2.3. Buiging
11.2.4. Afschuiving x
11.2.5. Torsie x
11.3. Combinaties van krachten en momenten
11.3.1. Enkele buiging met normaalkracht en afschuiving
x
x
11.3.2. Dubbele buiging met normaalkracht en afschuiving x
11.4. Vloeicriterium x
11.5. De invloed van de boutgaten (*)
NEN6771
10.2.4. Doorsneden x (*)
11.Toetsing van de doorsnede
11.1. Algemeen
x
x
11.2. Enkelvoudige krachten en momenten
11.2.1. Axiale trek
x
x
11.2.2. Axiale druk x
Scia Engineer Steel Code Check Theoretical Background
125
11.2.3. Buiging
11.2.4. Afschuiving x
11.2.5. Torsie
11.3. Combinaties van krachten en momenten x
12. Toetsing van de stabiliteit
12.1. Op druk belaste staven
12.1.1. Knikstabiliteit
x
x
x (*)
12.1.2. Torsiestabiliteit x
12.1.3. Torsieknikstabiliteit x
12.1.4. Verend gesteunde staven
12.1.5. Staven in vakwerken
12.1.6. Samengestelde staven
12.1.6.1 Algemeen
12.1.6.2. Benodigde grootheden
12.1.6.3. Toetsing van het middenveld van de samengestelde staaf
12.1.6.4. Toetsing van de eindvelden van de samengestelde staaf
12.1.6.4.2 Staven met raamwerkverband
x(*)
x
x
x
x
x
12.2. Op buiging belaste staven(kipstabiliteit)
12.2.1. Toepassingsgebied
xx
x
12.2.2. Toetsingsregel x
12.2.3. Ongesteunde lengte
12.2.4. Opleggingen en zijdelingse steunen
12.2.5. Het theoretisch elastische kipmoment x (*)
12.3. Op druk en buiging belaste staven
12.3.1. Knikstabiliteit
x
x
12.3.2. Torsiestabilteit x
12.3.3. Torsieknikstabiliteit x
12.4. Op trek en buiging belaste staven
13. Toetsing van de plooistabiliteit
13.1. Algemeen
x
x
13.2. Geometrie van het verstijfde en onverstijfde plaatveld x
13.3. Geometrie van de verstijvingen
13.4. Belasting in het vlak van het plaatveld
13.4.1. Normaalspanning in langsrichting
x
x
13.4.2. Schuifspanningen x
13.4.3. Normaalspanningen in dwarsrichting
13.4.4. Platen in en loodrecht op hun vlak belast
13.5. Belasting op verstijvingen
13.6. Ideële kritieke plooispanning van een onverstijfd plaatveld x
13.7. De plooispanning van een onverstijfd plaatveld
13.7.1. Bepaling van de relatieve slankheid van het plaatveld
x
x
Scia Engineer Steel Code Check Theoretical Background
126
13.7.2. De plooispanning voor een onverstijfd plaatveld met als opleggingen dwarsverstijving(en) en/of randen
x
13.7.3. De plooispanning voor een onverstijfd plaatveld met ten minste een langsverstijving als oplegging
13.8. Eisen waaraan plaatvelden en verstijvingen moeten voldoen
13.8.1. Onverstijfd plaatveld
x
x
13.8.2. Dwarsverstijvingen
13.8.3. Langsverstijvingen
13.8.4. Stijfheidseisen te stellen aan langs- en dwarsverstijvingen
13.8.5. Doorsnedecontrole voor langs- en dwarsverstijvingen
13.9. Interactie tussen plooi en knik
13.9.1. Algemeen
x (*)
x
13.9.2. Constructies opgebouwd uit plaatvelden al of niet verstijfd met dwarsverstijvingen
x
13.9.3. Constructies opgebouwd uit plaatvelden verstijfd met langsverstijvingen en/of niet verstijfd met dwarsverstijvingen
13.9.4. Berekeningen van de dwarsverstijvingen
Section properties
The influence of the bore hole is neglected.
Classification of sections
For each intermediary section, the classification is determined and the proper section check is performed. The classification can change for each intermediary point.
For each load case/combination, the critical section classification over the member is used to perform the stability check. So, the stability section classification can change for each load case/combination.
However, for non-prismatic sections, the stability section classification is determined for each intermediary section.
Effective cross-section properties for class 4 cross-section
The calculation of the effective area is performed with the direct method (sigma_d = fy,k).
For each intermediary section, the classification (and if necessary, the effective area ) is determined and the proper section check is performed. The classification (and effective area) can change for each intermediary point. The most critical check is displayed on the screen.
For each load case and combination, the most critical effective area properties are saved :
Aeff is the effective area of the cross section when subject to uniform compression. Weff is the effective section modulus of the cross-section when subject only to moment about the relevant axis. eN is the shift of the relevant centroidal axis when the cross section is subject to uniform compression.
With these critical properties, the stability check is performed.
Scia Engineer Steel Code Check Theoretical Background
127
For non-prismatic elements, the effective area properties are calculated on each intermediary section, also for the stability check.
For angle sections, see chapter 'Effective cross-section properties for compressed lattice tower angle members'.
Torsion check
For the cross section check inclusive torsion and warping, we refer to Chapter 'Warping check'.
Built-in beams
For built-in beam sections (IFB, SFB, THQ sections), proper section checks are performed, taking into account the local plate bending. See Chapter „Section check for built-in beams (IFB, SFB, THQ sections)
Buckling length
For the calculation of the buckling length, we refer tochapter "Calculation of buckling ratio".
The buckling properties for a VARH element are calculated by using the critical Euler force for this member (see “Calculation of critical Euler force for VARH elements”).
The buckling curves for steel grade S420 and S460 are taken from Ref.[5], Annex D.
Lateral-torsional buckling
For symmetric I sections and RHS (Rectangular Hollow Section) sections, the elastic critical moment
for LTB Mcr is given by the formula of Ref 2, part 12.2.5.. When the factor > 5000, the elastic
critical moment for LTB Mcr is given by the general formula in EC3, Annex F, F.2. Ref 3. For asymmetric I sections, the elastic critical moment for LTB Mcr is given by the general formula in
EC3, Annex F, F.2. Ref 3.
For the calculation of the moment factors C1, C2 and C3 we refer to Ref.[7], tables 9 (case 1), 10 and 11.
For the other supported sections, the elastic critical moment for LTB Mcr is given by
z
2
t
2
z
2
z
2
EI
GIL
I
Iw
L
EIMcr
with E the modulus of elasticity
G the shear modulus
L the length of the beam between points which have lateral restraint (= lLTB)
Iw the warping constant
It the torsional constant
Iz the moment of inertia about the minor axis
Scia Engineer Steel Code Check Theoretical Background
128
See also Ref. 4, part 7 and in particular part 7.7. for channel sections.
Haunched sections (I+Ivar, Iw+Plvar, Iw+Iwvar, Iw+Ivar, I+Iwvar) and composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
For advanced Lateral-torsional buckling analysis, see chapter „LTBII: Lateral Torsional Buckling 2nd Order Analysis‟.
Use of diaphragms
See Chapter 'Adaptation of torsional constant'.
Scia Engineer Steel Code Check Theoretical Background
129
Battened compression members
The following section pairs are supported as battened compression member :
(1) 2I
(2) 2Uo
(3) 2Uc
Two links (battens) are used.
The following additional checks are performed :
- buckling resistance check around weak axis of single chord with Nf,s;d
- section check of single chord, using internal forces :
4
aQM
2
Q V
N N
f;s;d
G
f;s;d
G
f;s;dG
- section check of single batten, using the internal forces :
4
aQM
2h
aQ V
ds;f;
ds;k;
0
ds;f;
ds;k;
For the calculation of Qf;s;d, the value of My;s;d is increased with the value of the internal force Mzz.
Scia Engineer Steel Code Check Theoretical Background
130
l
a
ho
Shear buckling check
Composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
Shear buckling check with buckling influence
The influence of the buckling effect into the shear buckling control, is neglected when there is a
bending moment present, i.e. if <0.9.
Scia Engineer Steel Code Check Theoretical Background
131
NEN6072 - Fire Resistance For more info, reference is made to to Ref.[8], Ref.[9].
Fire actions effect
The design effects of actions for the fire situation are taken from the results of the analysis. It is recommended to use the special combination rules according to Ref.[10], NEN6702 6.2.2., for calculating the internal forces used in the fire resistance check.
This special combination is given by
rep;aa;frep;iiq;frepg;f FQG
with Grep characteristic values of permanent actions
Qi characteristic value of the variable action
Fa;rep design values of special action (from fire exposure)
f;g partial safety factor for permanent actions in the special combination
=1.0
f;q partial safety factor for variable actions in the special combination
=1.0
f;a partial safety factor for special actions in the special combination
=1.0
I the 'momentaaan' factor for the variable action
Material properties
The yield strength is depending on the steel temperature :
d;yd;;y ff
The variation in function of the steel temperature of the value for yield strength is given by :
- =1.0 when a 400° C
- when 400°C < a 1200° C
Scia Engineer Steel Code Check Theoretical Background
132
with
2.39
482a
a steel temperature in °C
fy;d design value for yield strength at room temperature
fy;;d design value for yield strength at increased temperature
The following default properties are considered to be constant during the analysis :
unit mass a 7850 kg/m³
thermal elongation l/l 14 x 10-6
(a-20)
thermal conductivity a 45 W/mK
Nominal temperature-time curve
The standard temperature-time (ISO 834) curve is used :
)1t8(log34520 10g
with t time in [min]
g gas temperature in [°C]
Steel Temperature
The increase of temperature a in an unprotected steel member during a time interval t
with Am the exposed surface area per unit length [m²/m]
V the volume of the member per unit length [m³/m]
P = Am/V
t gas temperature in [°C]
Scia Engineer Steel Code Check Theoretical Background
133
a steel temperature [°C]
ca the specific heat of steel [J/kgK]
t the time interval [seconds]
a the unit mass of steel [kg/m³]
r resultant emissivity
= 0.5
c coefficient of heat transfer by convection
= 25 W/(m²K)
The increase of temperature a in an insulated (non intumescent coating) steel member during a time
interval t
with Ap the area of fire protection material per unit length [m²/m]
V the volume of the member per unit length [m³/m]
Pi = Ap/V
ca the specific heat of steel [J/kgK]
ci the specific heat of fire protection material [J/kgK]
di the thickness of the fire protection material [m]
t the time interval [seconds]
The value should not be taken as more than 30 seconds
a the unit mass of steel [kg/m³]
i the unit mass of fire protection [kg/m³]
a the steel temperature at time t
t the ambient gas temperature at time t
t the increase of the ambient gas temperature during the time interval
i;d;ef the thermal conductivity of the fire protection material [W/mK]
Scia Engineer Steel Code Check Theoretical Background
134
The increase of temperature a in an insulated (intumescent coating) steel member during a time interval
t
with Ap the area of fire protection material per unit length [m²/m]
V the volume of the member per unit length [m³/m]
Pi = Ap/V
ca the specific heat of steel [J/kgK]
Kd;ef coefficient of heat transfer of the intumescent coating
t the time interval [seconds]
The value should not be taken as more than 30 seconds
a the unit mass of steel [kg/m³]
a the steel temperature at time t
t the ambient gas temperature at time t
i;d;ef the thermal conductivity of the fire protection material [W/mK]
Scia Engineer Steel Code Check Theoretical Background
135
Calculation model
The calculation can be performed in 2 domains :
- strength domain
- temperature/time domain
In the strength domain, the strength (unity check) is calculated after a given time t (e.g. strength
after 45 min). In the temperature/time domain, the critical steel temperature a,cr is computed. From this critical temperature, the fire resistance time is calculated (the time domain).
The critical steel temperature a,cr is given by :
with degree of utilization at time t=0
correction factor
= 1.00 for tension elements
= 1.00 for beams, statically determined, 4 side exposure
= 0.70 for beams, statically determined, 3 side exposure
= 0.85 for beams, statically undetermined, 4 side exposure
= 0.60 for beams, statically undetermined, 3 side exposure
= 1.20 for compression elements (inclusive the buckling check)
= 1.20 for compression and bending elements (inclusive the buckling and LTB check)
Code Check
The section and stability checks (buckling, lateral torsional buckling) are performed according to the regulations given in NEN6770/6771, adapted with the yield strength for the increased temperature and the correction factor. The checks are performed in the resistance domain or in the temperature/time domain. Shear buckling is not considered.
Scia Engineer Steel Code Check Theoretical Background
136
Supported sections
I Symmetric I shapes (IPE, HEA, HEB, ….)
RHS Rectangular Hollow Section (RHS)
CHS Circular Hollow Section (CHS)
L Angle section
U Channel section
T T section
PPL Asymmetric I shapes
Z Z section
RS Rectangular section
Cold formed section
COM Composed section in PRIMAWIN
O Solid tube
NUM Numerical section
The necessary data conditions for these sections are described in chapter "Profile conditions for code check
". The COM and NUM sections are not read out of the profile library.
I
RHS
CHS
L
U
T
PPL
RS
Z
O
COM
NUM
Classification x x x x x x x x (1) x (1) (1) (1)
Section check class 1 x x x
Section check class 2 x x x
Section check class 3 x x x x x x x x x x x x x
Section check class 4 x x x x x x
Stability check class 1
x x x
Stability check class 2
x x x
Stability check class 3
x x x x x x x x x x x x x
Stability check class 4
x x x x x x
Shear buckling check x x x x
(1) Sections are classified as class 3 cross section by default.
Scia Engineer Steel Code Check Theoretical Background
137
References 1 Staalconstructies TGB 1990
Basiseisen en basisrekenregels voor overwegend statisch belaste constructies
NEN 6770, december 1991
2 Staalconstructies TGB 1990
Stabiliteit
NEN 6771, december 1991
3 Eurocode 3
Design of steel structures
Part 1 - 1 : General rules and rules for buildings
ENV 1993-1-1:1992, 1992
[4] R. Maquoi
ELEMENTS DE CONSTRUCTIONS METALLIQUE
Ulg , Faculté des Sciences Appliquées, 1988
[5] Eurocode 3
Design of steel structures
Part 1 - 1/ A1 : General rules and rules for buildings
ENV 1993-1-1:1992/A1, 1994
[6] ENV 1993-1-3:1996
Eurocode 3 : Design of steel structures
Part 1-3 : General rules
Supplementary rules for cold formed thin gauge members and sheeting
CEN 1996
[7] Staalconstructies TGB 1990
Stabiliteit
NEN 6771, januari 2000
[8] NEN 6072
Rekenkundige bepaling van de brandwerendheid van bouwdelen
Staalconstructies
December 1991
[9] NEN 6072/A2 - Wijzigingsblad
Rekenkundige bepaling van de brandwerendheid van bouwdelen
Staalconstructies
December 2001
[10] NEN 6702
Belastingen en vervormingen TGB 1990
December 1991
Scia Engineer Steel Code Check Theoretical Background
138
AISC – ASD : 1989
AISC - ASD Code check
The beam elements are checked according to the regulations given in
Manual of Steel Construction
Allowable Stress Design
Part 5 : Specification and Codes
AISC, Ninth Edition, 1989
The cross section is classified according to Table B5.1. (compact, non compact, or slender section).
The member is checked on following criteria:
tension : D1
compression : E2, E3
flexural members : F1,F2,F3,F4
plate girders : G2
combined forces : H1,H2
A more detailed overview for the used articles of the relevant parts is given in the following table. The chapters marked with “x” are consulted. The chapters marked with (*) have a supplementary explanation the following chapters.
B. DESIGN REQUIREMENTS
B1. Gross Area x
B2. Net Area (*)
B3. Effective Area
B4. Stability
B5. Local Buckling
1.Classification of Steel Sections
2.Slender Compression Elements
(*)
x
x
B6. Rotational Restraint at Points of Support
B7. Limiting Slenderness Ratios x
B8. Simple Spans
B9. End Restraint
B10. Proportions of Beams and Girders
B11. Proportioning of Crane Girders
Scia Engineer Steel Code Check Theoretical Background
139
D. TENSION MEMBERS
D1. Allowable Stress x (*)
D2. Built-up members
D3. Pin-Connected Members
E. COLUMN AND OTHER COMPRESSION MEMBERS
E1. Effective Length and Slenderness Ratio x (*)
E2. Allowable Stress x
E3. Flexural-torsional Buckling x (*)
E4. Built-up Members
E5. Pin-Connected Compression Members
E6. Column Web Shear
F. BEAMS AND OTHER FLEXURAL MEMBERS (*)
F1. Allowable Stress : Strong Axis Bending of I-Shaped Members and Channels
1.Members with Compact Sections
2.Members with Non-Compact Sections
3.Members with Compact or Non-Compact Sections with Unbraded Length Greater then Lc
x
x
x
x
F2. Allowable Stress : Weak Axis Bending of I-Shaped Members, Solid Bars and Rectangular Plates
1.Members with Compact Sections
2.Members with Non-Compact Sections
x
x
x
F3. Allowable Stress : Bending of Box Members, Rectangular Tubes and Circular Tubes
1.Members with Compact Sections
2.Members with Non-Compact Sections
x
x
x
F4. Allowable Shear Stress x
F5. Transverse Stiffeners
F6. Built-up Members
F7. Web-tapered Members
G. PLATE GIRDERS
G1. Web Slenderness Limitations
G2. Allowable Bending Stress x
G3. Allowable Shear Stress with Tension Field Action
G4. Transverse Stiffeners
G5. Combined Shear and Tension Stress
H. COMBINED STRESSES
H1. Axial Compression and Bending x
H2. Axial Tension and Bending x
Scia Engineer Steel Code Check Theoretical Background
140
APPENDIX B. DESIGN REQUIREMENTS
B5. Local Buckling x
Classification of sections
For each intermediary section, the classification is determined..
For each load case/combination, the critical section classification over the member is used to perform the code check. However, for non-prismatic sections, the section classification is determined for each intermediary section.
Section properties
The influence of the bore hole is neglected, i.e. only the gross area is used.
Buckling length
For the calculation of the buckling length, we refer to "Calculation of buckling ratio".
The buckling properties for a VARH element are calculated by using the critical Euler force for this member (see “Calculation of critical Euler force for VARH elements”).
Flexural Torsional Buckling
The slenderness ratio for flexural torsional buckling (KL/r)e is given by
Fe
E
r
KL
e
See Ref. 1, Commentary Chapter E1.
The calculation of Fe is given in Ref. 2, Appendix E.
Lateral-torsional buckling
For I sections and channel sections, the allowable LTB stress is given in F1.
For RHS (Rectangular Hollow Section) sections and CHS (Circular Hollow Section) , the allowable LTB stress is given in F3.
For angle sections with symmetrical legs, the allowable LTB stress is given in Ref. 1, pp.309-314, “Specification for allowable stress - Design of single-angle members”.
Scia Engineer Steel Code Check Theoretical Background
141
For the other supported sections, the elastic critical moment for LTB Mcr is given by
z
2
t
2
z
2
z
2
EI
GIL
I
Iw
L
EIMcr
with E the modulus of elasticity
G the shear modulus
L the length of the beam between points which have lateral restraint (= lLTB)
Iw the warping constant
It the torsional constant
Iz the moment of inertia about the minor axis
See also Ref. 4, part 7.
With this moment Mcr, the critical LTB stress LTB is calculated :
y
crLTB
I
M
with Iy the moment of inertia about the major axis
The slenderness ratio for LTB LTB, is given by
LTBLTB
E
The allowable LTB stress is calculated using the slenderness LTB with the formulas given in
Ref.1, E2.
See also Ref. 5, Bijlage E.
Haunched sections (I+Ivar, Iw+Plvar, Iw+Iwvar, Iw+Ivar, I+Iwvar) and composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
Shear buckling check
Composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
Scia Engineer Steel Code Check Theoretical Background
142
Supported sections
I Symmetric I shapes (IPE, HEA, HEB, ….)
RHS Rectangular Hollow Section (RHS)
CHS Circular Hollow Section (CHS)
L Angle section
U Channel section
T T section
PPL Asymmetric I shapes
RS Rectangular section
Cold formed section
COM Composed section in PRIMAWIN
O Solid tube
NUM Numerical section
The necessary data conditions for these sections are described in chapter "Profile conditions for code check
". The COM and NUM sections are not read out of the profile library.
I
RHS
CHS
L
U
T
PPL
RS
O
COM
NUM
Classification x x x x x x x x x (1)
(1) (1)
Compact section x x x x x
Non-compact section
x x x x x x x x x x x x
Slender section x x x x x x
Shear buckling check
x x x
(1) sections are classified as non-compact section by default.
References
1 Manual of Steel Construction
Allowable Stress Design
AISC, Ninth Edition, 1989
2 Manual of Steel Construction
Load & Resistance Factor Design
AISC, First Edition, 1986
Scia Engineer Steel Code Check Theoretical Background
143
3 Manual of Steel Construction
Load & Resistance Factor Design
AISC, Volume I, Second Edition, 1995
[4] R. Maquoi
ELEMENTS DE CONSTRUCTIONS METALLIQUE
Ulg , Faculté des Sciences Appliquées, 1988
[5] NBN B 51-001
Stalen Bouwconstructies
BIN, 5e uitg. April 1977
Scia Engineer Steel Code Check Theoretical Background
144
AISC – LRFD : 2001
AISC - LRFD Code check
The beam elements are checked according to the regulations given in
AISC – Manual of steel construction
Load and Resistance Factor Design
Part 16 Specifications and Codes
Third Edition
2001
The cross section is classified according to Table B5.1. (compact, non compact, or slender section).
The member is checked on following criteria :
tension : D1
compression : E2, E3, Appendix E3
flexural members : F1,Appendix F1, Appendix F2
plate girders : Appendix G2, Appendix G3, Appendix G5
combined forces : H1,H2
A more detailed overview for the used articles of the relevant parts is given in the following table. The chapters marked with “x” are consulted. The chapters marked with (*) have a supplementary explanation the following chapters.
B. DESIGN REQUIREMENTS
B1. Gross Area x
B2. Net Area (*)
B3. Effective Area for Tension Members
B4. Stability
B5. Local Buckling
1.Classification of Steel Sections
2.Slender Compression Elements
3.Slender-Element Compression Sections
(*)
x
x
x
B6. Bracing at Support
B7. Limiting Slenderness Ratios x
B8. Simple Spans
Scia Engineer Steel Code Check Theoretical Background
145
B9. End Restraint
B10. Proportions of Beams and Girders
D. TENSION MEMBERS
D1. Design Tensile Strength x (*)
D2. Built-up members
D3. Pin-Connected Members and Eyebars
E. COLUMN AND OTHER COMPRESSION MEMBERS
E1. Effective Length and Slenderness Limitations
1.Effective Length
2.Design by Plastic Analysis
x
x (*)
E2. Design Compressive Strength for Flexural Buckling x
E3. Design Compressive Strength for Flexural-Torsional Buckling x
E4. Built-up Members
E5. Pin-Connected Compression Members
F. BEAMS AND OTHER FLEXURAL MEMBERS (*)
F1. Design for Flexure
1.Yielding
2.Lateral-Torsional Buckling
x
x
x
F2. Design for Shear x
F3. Web-tapered Members
F4. Beams and Girders with Web Openings
G. PLATE GIRDERS x
H. MEMBERS UNDER COMBINED FORCES AND TORSION
H1. Symmetric Members Subject to Bending and Axial Force x
H2. Unsymmetric Members and Members under Torsion and Combined Torsion, Flexure, Shear and/or Axial Force
x
H3. Alternative Interaction Equation for Members under Combined Stress
APPENDIX B. DESIGN REQUIREMENTS
B5. Local Buckling x
APPENDIX E. COLUMN AND OTHER COMPRESSION MEMBERS
E3. Design Compressive Strength for Flexural-Torsional Buckling x
APPENDIX F. BEAMS AND OTHER FLEXURAL MEMBERS
F1. Design for Flexure x
F2. Design for Shear x
Scia Engineer Steel Code Check Theoretical Background
146
F3. Web-tapered Members
APPENDIX G. PLATE GIRDERS
G1. Limitations
G2. Design Flexural Strength x(*)
G3. Design Shear Strength with Tension Field Action x(*)
G4. Transverse Stiffeners
G5. Flexure-Shear Interaction x(*)
Classification of sections
For each intermediary section, the classification is determined..
For each load case/combination, the critical section classification over the member is used to perform the code check. However, for non-prismatic sections, the section classification is determined for each intermediary section.
Section properties
The influence of the bore hole is neglected, i.e. only the gross area is used.
Buckling length
For the calculation of the buckling length, we refer to "Calculation of buckling ratio".
The buckling properties for a VARH element are calculated by using the critical Euler force for this member (see “Calculation of critical Euler force for VARH elements”).
Lateral-torsional buckling
For I sections, channel sections, RHS (Rectangular Hollow Section) sections, T sections, rectangular sections, and asymmetric I sections, the critical LTB moment is given in F1 and Appendix F1.
For angle sections with symmetrical legs, the critical LTB moment is given in Ref. 1, pp.281-288, “Specification for Load and Resistance Factor Design of Single-Angle members”.
For the other supported sections, the elastic critical moment for LTB Mcr is given by
z
2
t
2
z
2
z
2
EI
GIL
I
Iw
L
EIMcr
with E the modulus of elasticity
G the shear modulus
L the length of the beam between points which have lateral restraint (= lLTB)
Iw the warping constant
It the torsional constant
Iz the moment of inertia about the minor axis
Scia Engineer Steel Code Check Theoretical Background
147
See also Ref. 2, part 7.
Haunched sections (I+Ivar, Iw+Plvar, Iw+Iwvar, Iw+Ivar, I+Iwvar) and composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
Use of diaphragms
See Chapter 'Adaptation of torsional constant'.
Shear buckling check
Composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
Supported sections
I Symmetric I shapes (IPE, HEA, HEB, ….)
RHS Rectangular Hollow Section (RHS)
CHS Circular Hollow Section (CHS)
L Angle section
U Channel section
T T section
PPL Asymmetric I shapes
RS Rectangular section
Cold formed section
COM Composed section in PRIMAWIN
O Solid tube
NUM Numerical section
The necessary data conditions for these sections are described in chapter "Profile conditions for code check
".
The COM and NUM sections are not read out of the profile library.
I
RHS
CHS
L
U
T
PPL
RS
O
COM
NUM
Classification x x x x x x x x x (1) (1) (1)
Compact section x x x x x
Non-compact section
x x x x x x x x x x x x
Slender section x x x x x x
Shear buckling check
x x x
Scia Engineer Steel Code Check Theoretical Background
148
(1) sections are classified as non-compact section by default.
References
1 AISC – Manual of steel construction
Load and Resistance Factor Design
Third Edition
2001
2 R. Maquoi
ELEMENTS DE CONSTRUCTIONS METALLIQUE
Ulg , Faculté des Sciences Appliquées, 1988
Scia Engineer Steel Code Check Theoretical Background
149
ANSI/AISC 360-05:2005
ANSI/AISC 360-05 Code check
The beam elements are checked according to the regulations given in
ANSI/AISC 360-05
Specifications for Structural Steel Buildings
2005
The steel code check can be executed according to either ASD or LRFD provisions.
The cross section is classified according to Table B4.1. (compact, non compact, or slender section).
The member is checked on following criteria:
tension : Chapter D
compression : Chapter E
flexural members :Chapter F
shear : Chapter G
combined forces : Chapter H
A more detailed overview for the used articles of the relevant parts is given in the following table. The chapters marked with “x” are consulted. The chapters marked with (*) have a supplementary explanation the following chapters.
B. DESIGN REQUIREMENTS
B2. Loads and Load Combination x
B3. Design Basis
1.Required Strength
2.Limit States
3.Design for Strength using LRFD
4.Design for Strength using ASD
x
x
B4. Classification of Sections for Local Buckling x
D. DESIGN OF MEMBERS FOR TENSION
D1. Slenderness Limitation x
D2. Tensile Strength x
D3. Area Determination x(*)
Scia Engineer Steel Code Check Theoretical Background
150
E. DESIGN OF MEMBERS FOR COMPRESSION
E1. General Provisions x
E2. Slenderness Limitations and Effective Length x(*)
E3. Compressive Strength for Flexural Buckling of members without Slender Elements
x
E4. Compressive Strength for Torsional and Flexural-Torsional Buckling of members without Slender Elements
x
E7. Members with Slender Elements x
F. DESIGN FOR MEMBERS FOR FLEXURE
F1. General Provisions x
F2. Doubly Symmetric Compact I-Shaped Members and Channels Bent about their Major Axis
x
F3. Doubly Symmetric I-Shaped Members with Compact Webs and Noncompact or Slender Flanges Bent about Their Major Axis
x
F4. Other I-Shaped Members with Compact or Noncompact Webs Bent about Their Major Axis
x
F5. Doubly Symmetric and Singly Symmetric I-Shaped Members with Slender Webs Bent about Their Major Axis
x
F6. I-Shaped Members and Channels Bent about Their Minor Axis x
F7. Square and Rectangular HSS and Box-Shaped Members x
F8. Round HSS x
F9. Tees and Double Angle Loaded in Plane of Symmetry x
F10. Single Angle x
F11. Reactangular Bars and Rounds x
F12. Unsymmetrical Shapes
G. DESIGN OF MEMBERS FOR SHEAR
G1. General Provisions x
G2. Members with Unstiffened or Stiffened Webs x
G4. Single Angles x
G5. Rectangular HSS and Box Members x
G6. Round HSS x
G7. Weak Axis Shear in Singly and Doubly Symmetric Shapes x
H. DESIGN OF MEMBERS FOR COMBINED FORCES AND TORSION
H1. Doubly and Singly Symmetric Members Subject to Flexure and Axial Force x
H2. Unsymmetric and Other Members Subject to Flexure and Axial Force x
H3. Members Under Torsion and Combined Torsion and Combined Stress x
Scia Engineer Steel Code Check Theoretical Background
151
Classification of sections
For each intermediary section, the classification is determined..
For each load case/combination, the critical section classification over the member is used to perform the code check. However, for non-prismatic sections, the section classification is determined for each intermediary section.
Section properties
The influence of the bore hole is neglected, i.e. only the gross area is used.
Buckling length
For the calculation of the buckling length, we refer to "Calculation of buckling ratio".
The buckling properties for a VARH element are calculated by using the critical Euler force for this member (see “Calculation of critical Euler force for VARH elements”).
Lateral-torsional buckling
Haunched sections (I+Ivar, Iw+Plvar, Iw+Iwvar, Iw+Ivar, I+Iwvar) and composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
Use of diaphragms
See Chapter 'Adaptation of torsional constant'.
Shear buckling check
Composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
Scia Engineer Steel Code Check Theoretical Background
152
Supported sections
I Symmetric I shapes (IPE, HEA, HEB, ….)
RHS Rectangular Hollow Section (RHS)
CHS Circular Hollow Section (CHS)
L Angle section
U Channel section
T T section
PPL Asymmetric I shapes
RS Rectangular section
Cold formed section
COM Composed section in PRIMAWIN
O Solid tube
NUM Numerical section
The necessary data conditions for these sections are described in chapter "Profile conditions for code check
". The COM and NUM sections are not read out of the profile library.
I
RHS
CHS
L
U
T
PPL
RS
O
COM
NUM
Classification x x x x x x x x x (1) (1) (1)
Compact section x x x x x
Non-compact section
x x x x x x x x x x x x
Slender section x x x x x x
Shear buckling check
x x x x x x
(1) Sections are classified as non-compact section by default.
References
1 ANSI/AISC 360-05
Specifications for Structural Steel Buildings
2005
Scia Engineer Steel Code Check Theoretical Background
153
AISI NAS S100-2007
AISI NAS S100-2007 Code check
The beam elements are checked according to the regulations given in:
AISI S100-2007
North American Specification for the Design of Cold-Formed Steel Structural Members
2007 edition
AISI S100-07-E1
Errata to North American Specification for the Design of Cold-Formed Steel Structural Members
2007 edition February 20, 2008
Amended September 25, 2008
Amended June 4, 2009
AISI S100-07/S1-09
Supplement No. 1 to the North American Specification for the Design of Cold-Formed Steel Structural Members, 2007 edition August, 2009
AISI S100-07/S2-10
Supplement No. 2 to the North American Specification for the Design of Cold-Formed Steel Structural Members, 2007 edition February, 2010
The steel code check is supported for the United States provisions and can be executed according to either ASD or LRFD principles. The Canadian LSD method is not supported.
Consulted articles
An overview for the used articles is given in the following table. The articles marked with “x” are consulted. The articles marked with (*) have a supplementary explanation in the following paragraphs.
Article Title
A General Provisions
A4 Allowable Strength Design X
A5 Load and resistance Factor Design X
B Elements
B1 Dimensional Limits and Considerations X(*)
Scia Engineer Steel Code Check Theoretical Background
154
B2
Effective Widths of Stiffened Elements
B2.1 Uniformly Compressed Stiffened Elements
B2.3 Webs and Other Stiffened Elements under Stress Gradient
X(*)
X(*)
B3 Effective Widths of Unstiffened Elements
B3.1 uniformly Compressed Unstiffened Elements
B3.2 Unstiffened Elements and Edge Stiffeners with Stress Gradient
X(*)
X(*)
B4 Effective Width of Uniformly Compressed Elements with a Simple Lip Edge Stiffener
X(*)
C Members
C1 Properties of Sections X(*)
C2 Tension Members X(*)
C3 Flexural members
C3.1 Bending
C3.1.1 Nominal Section Strength
C3.1.2 Lateral-Torsional Buckling Strength
C3.1.3 Flexural Strength of Closed Cylindrical Tubular Members
C3.1.4 Distortional Buckling Strength
X(*)
X(*)
X(*)
X(*)
C3.2 Shear
C3.2.1 Shear Strength of Webs without Holes
C3.3 Combined Bending and Shear
X(*)
X(*)
C3.4 Web Crippling
C3.4.1 Web Crippling Strength of Webs without Holes
C3.5 Combined Bending and Web Crippling
X(*)
X(*)
C3.6 Combined Bending and Torsional Loading X(*)
C4 Concentrically Loaded Compression Members
C4.1 Nominal Strength for Yielding, Flexural, Flexural-Torsional and Torsional Buckling
C4.2 Distortional Buckling Strength
X(*)
X(*)
C5 Combined Axial Load and Bending
C5.1 Combined Tensile Axial Load and Bending
C5.2 Combined Compressive Axial Load and Bending
X
X(*)
D Structural Assemblies and Systems
D6 Metal Roof and Wall Systems
D6.1 Purlins, Girts and Other Members
D6.1.1 Flexural Members Having One Flange Through-Fastened to Deck of Sheeting
D6.1.3 Compression Members Having One Flange Through-Fastened to Deck of Sheeting
X(*)
X(*)
Scia Engineer Steel Code Check Theoretical Background
155
Appendix 2 Second-Order Analysis
2.1 General requirements
X(*)
Haunches, arbitrary members and cross-sections without initial shapes are not supported for the AISI NAS S100-2007 code check. In this case the default AISC 2005 code check is executed.
Initial Shape
For a cross-section with material Steel and fabrication set to Cold-Formed, the Initial Shape can be defined.
For a General cross-section the „Thin-walled representation‟ has to be used to be able to define the Initial Shape.
The thin-walled cross-section parts can have the following types:
F Fixed Part – No reduction is needed
I Internal cross-section part
SO Symmetrical Outstand
UO Unsymmetrical Outstand
Parts can also be specified as reinforcement:
None Not considered as reinforcement
RUO Reinforced Unsymmetrical Outstand (edge stiffener)
ROU reinforcement types can be set only to elements of type SO or UO.
The initial shape is supported for the following cross-section types:
- Standard profile library cross-sections
- Cold formed Pair cross-sections of profile library sections
- General thin-walled sections
- General sections with thin-walled representation
- Thin-walled geometric sections
- All other sections which support the centerline and do not have roundings
For standard profile library cross-sections, the flat parts are taken between the roundings. The roundings are set as fixed parts.
For predefined sections without roundings, the initial shape is based on the centreline dimensions i.e. the flat parts are taken between the intersection points of the centrelines.
Scia Engineer Steel Code Check Theoretical Background
156
Dimensional limits
Dimensional limits are supported according to article B1.1 and B1.2.
Article B1.1 (a) (1) for a simple lip is checked for an internal element (I) connected to a stiffener
(RUO).
Article B1.1 (a) (2) is checked for an internal element (I).
Article B1.1 (a) (3) is checked for an outstand element (UO or SO).
Articles B1.1 (b) concerning flange curling and (c) concerning shear lag effects are not supported.
Article B1.2 (a) is checked for web elements under stress gradient. Webs are defined as elements
perpendicular (tolerance +/-45°) to the axis of bending.
Effective Widths
Uniformly Compressed Stiffened elements
The effective width of Uniformly Compressed Stiffened elements is calculated according to article B2.1 (a) Strength Determination.
More specifically, this concerns elements of type I with stress gradient = 1
Serviceability Determination is not supported.
Webs and Other Stiffened Elements under Stress Gradient
The effective width of Webs and Other Stiffened elements under stress gradient is calculated according to article B2.3 (a) Strength Determination.
More specifically, this concerns elements of type I with stress gradient ≠ 1
Serviceability Determination is not supported.
Uniformly Compressed Unstiffened elements
The effective width of Uniformly Compressed Unstiffened elements is calculated according to article B3.1 (a) Strength Determination.
More specifically, this concerns elements of type SO or UO (with or without reinforcement type
RUO) with stress gradient = 1
Serviceability Determination is not supported.
Scia Engineer Steel Code Check Theoretical Background
157
Unstiffened elements and Edge Stiffeners with Stress Gradient
The effective width of Unstiffened elements and Edge Stiffeners with Stress Gradient is calculated according to article B3.2 (a) Strength Determination.
More specifically, this concerns elements of type SO or UO (with or without reinforcement type
RUO) with stress gradient ≠ 1
The alternative methods for unstiffened C-sections are not supported.
Serviceability Determination is not supported.
Effective width of Uniformly Compressed elements with a Simple Lip Edge Stiffener
The effective width of Uniformly Compressed elements with a Simple Lip Edge Stiffener is calculated according to article B4 (a) Strength Determination.
More specifically, this concerns elements of type I with stress gradient = 1 which are connected to a fixed element (rounding) which in turn is connected to an element of type UO or SO with reinforcement type RUO.
Serviceability Determination is not supported.
Effective section properties can never be bigger than gross section properties (for example in case of manually inputted gross section properties which have been rounded down).
Properties of Sections
Deductions for holes, openings and cut-outs are not supported.
Scia Engineer Steel Code Check Theoretical Background
158
Tension Members
The tensile strength is determined according to article C2.
For yielding in the gross section:
For rupture in the net section:
With: Fy Yield strength
Fu Tensile strength
Ag Gross area of cross-section
An Net area of cross-section
Since deductions for holes, openings … are not supported An = Ag.
Flexural Members
Nominal Section Strength
The nominal section strength is determined according to article C3.1.1. More specifically Procedure I - Based on Initiation of Yielding is applied.
Lateral Torsional Buckling Open Section
The Lateral Torsional Buckling strength for open sections is determined according to article C3.1.2.1 (a).
For diaphragms reference is made to “Use of diaphragms”. The simplified formulas of article C3.1.2.1 (b) are not supported.
Doubly symmetric sections
For Doubly symmetric sections formula (C3.1.2.1-4) is used for either axis.
This applies to the following form codes:
1 (Symmetric I shape)
7 (Rectangular section)
11 (Solid tube)
In addition this applies to the cold formed pair sections 2CFUo, 2CFUc, 2CFCo, 2CFCc
Scia Engineer Steel Code Check Theoretical Background
159
Formula (C3.1.2.1-4) is rewritten as follows:
Remarks:
For x-x bending the LTB length is used instead of the effective length KyLy.
For y-y bending Kx is taken as the buckling ratio about the x-axis and Lx the system
length for buckling about the x-axis.
The equation for r0 is expanded to allow any type of cross-section:
Cb for x-x bending is calculated according to formula (C3.1.2.1-6)
Cb for y-y bending is taken as unity.
Scia Engineer Steel Code Check Theoretical Background
160
Point symmetric sections
For Point symmetric sections formula (C3.1.2.1-5) is used for either axis.
This applies to the following form codes:
102 (Z section)
113 (Cold formed Z section)
118 (Cold formed ZED section)
119 (Cold formed ZED section asymmetrical lips)
120 (Cold formed ZED section inclined lip)
126 (Cold formed ZED section both lips inclined)
Formula (C3.1.2.1-5) is rewritten as follows:
The same remarks are valid as for doubly symmetric sections.
Singly symmetric sections
For Singly symmetric sections formula (C3.1.2.1-4) is used for bending about the x-x axis and formula (C3.1.2.1-10) for bending about the y-y axis.
This applies to the following form codes:
5 (Channel section)
112 (Cold formed Channel section)
114 (Cold formed C section)
117 (Cold formed C-Plus section)
121 (Cold formed Sigma section)
122 (Cold formed Sigma section stiffened)
123 (Cold formed Sigma-Plus section)
Formulas (C3.1.2.1-4) and (C3.1.2.1-10) are written as follows:
Scia Engineer Steel Code Check Theoretical Background
161
The same remarks are valid as for doubly symmetric sections.
The parameter j is calculated using the formula for C-sections given in Ref. [4].
Other section types
All other cross-sections which are not covered by the previous paragraphs are considered to be doubly symmetric, except for the following form codes:
2 (Rectangular Hollow Section)
3 (Circular Hollow Section)
Lateral Torsional Buckling Box Section
The Lateral Torsional Buckling strength for box sections is determined according to article C3.1.2.2.
This applies to the following form code:
2 (Rectangular Hollow Section)
In addition this applies to the cold formed pair sections 2CFUc and 2CFCc with distance a = 0 mm
Formulas (C3.1.2.2-1) and (C3.1.2.2-2) are rewritten as follows:
The same remarks are valid as for open doubly symmetric sections.
Flexural Strength Closed Cylindrical Tubular members
The Flexural Strength of Closed Cylindrical Tubular members is determined according to article C3.1.3.
This applies to the following form code:
3 (Circular Hollow Section)
In case the diameter to thickness ratio D/t exceeds the limit 0,441 E/Fy the check is not executed and a warning is issued on the output.
Scia Engineer Steel Code Check Theoretical Background
162
Distortional Buckling Strength
For both bending axis the distortional buckling strength is determined according to article C3.1.4. More specifically the general Procedure (a) is followed using formula (C3.1.4-6).
The check is executed in case the following conditions are met:
The cross-section has at least one element with reinforcement type RUO
For the given bending moment in the section, at least one of these elements is in compression
More specifically this implies that, if the stiffener is in compression distortional buckling can occur (even if the flange itself is partially in tension). This is in accordance with the distortional buckling shapes for weak axis bending of typical C-sections obtained using numerical analysis Ref. [9].
Remarks:
The unbraced length Lm is taken as the LTB length and this for both bending axis.
In case a diaphragm is positioned on the compression side and the diaphragm provides full bracing, the member is regarded as continuously restrained and Lm = Lcr.
The rotational stiffness of the restraining element is by default taken as zero.
In case a diaphragm is located on the compression side, is taken as the
rotational stiffness vorhC of the diaphragm.
For diaphragms reference is made to Use of diaphragms.
For calculating the compression flange properties, the default Scia Engineer axis convention is used (x-y axis system located at the centroid of the flange, with the x-axis measured positive to the right from the centroid and the y-axis positive up from the centroid)
The elastic section modulus of the full unreduced section relative to the extreme fiber in first yield Sfy is taken as Sfy,x for x-x bending and Sfy,y for y-y bending.
In determining the stress gradient in the web, pure symmetrical bending is assumed. This implies that for x-x bending this parameter equals 2 and for y-y bending this parameter equals zero.
The distance b0 for a standard profile library section is taken as the width property.
For a general section this is taken as the summation of the Internal (I) parts of the flange.
The distance h0 for a standard profile library section is taken as the height property.
For a Sigma section (Form Code 121 – 125) this is taken as the (full) height property. For a general section this is taken as the maximal height of the „web‟ elements. Web elements are defined as elements with an angle > 45° to the horizontal axis.
When there is no „web‟ element (i.e. CHS section ), distortional buckling is not checked.
Flanges are defined as elements with angle < 45° to the horizontal axis. Connected flange elements which have a relative angle > 135° are accounted for as „one‟ flange for distortional buckling.
Scia Engineer Steel Code Check Theoretical Background
163
For cross-sections with roundings, the flange/web junction is taken to be at the intersection between the flange/web rounding and the flat part of the flange.
The thickness t is taken as the smallest thickness of the cross-section elements.
For Omega sections (Form Code 115) the top flange is not seen as flange for distortional buckling.
Shear
The shear strength is determined according to article C3.2.1.
In the calculation of Aw only elements with element types I, UO and SO are accounted for. In addition, elements with reinforcement type ROU are not accounted for.
For each element i the shear area Aw,i is calculated as follows:
With: i = The number (ID) of the element.
xend = End position of element i .
xbeg = Begin position of element i.
t = Thickness of element i.
= Angle of element i to the horizontal x-x axis
In addition, for each element i the nominal shear stress Fv,i is calculated.
The shear strength of the element then becomes Vn,i = Aw,i * Fv,i
The nominal shear strength Vn for the cross section is taken as the sum of the Vn,i of the related
elements.
Transverse stiffeners are not supported, therefore the shear buckling coefficient kv is taken as 5,34.
AISI NAS S100-2007 does not give provisions to calculate the shear resistance for circular hollow sections (Form Code 3). Therefore the default AISC 2005 provisions are used in this case.
Combined Bending and Shear
The combined bending and shear check is determined according to articles C3.3.1 and C3.3.2.
Transverse stiffeners are not supported; therefore the equations for unreinforced webs are used.
Scia Engineer Steel Code Check Theoretical Background
164
Web Crippling Strength
The web crippling strength is determined according to article C3.4.1.
More specifically the general equation (C3.4.1-1) is applied.
The alternative given in equation (C3.4.1-2) is not supported.
The web crippling check is executed on the positions where there is a jump in the Vy shear force
diagram.
Remarks:
The shear force diagram of both the actual member as well as adjacent members is evaluated. Adjacent members are defined as members which are in the same buckling system.
The angle between the plane of the web and the plane of the bearing surface is taken as 90°.
The Flange Conditions depend on the definition of the initial shape. In case there is an element with reinforcement type ROU the setting is taken as „Stiffened or Partially Stiffened
Flanges‟.
The distances for One-flange/Two-flange and End/Interior are evaluated taking into account adjacent members. Adjacent members are defined as members which are in the same buckling system.
The following paragraphs specify the supported cross-section types.
Built-Up Sections
For built-up sections table C3.4.1-1 is used.
This applies to cold formed pair sections 2CFUo and 2CFCo with distance a = 0 mm and the
following form codes:
127 (Cold formed I-Plus section)
128 (Cold formed IS-Plus section)
Since these pair sections consist of two webs the resistance of the full section is obtained by adding the values of each web.
Single Web Channel and C-Sections
For single web channel and C-sections table C3.4.1-2 is used.
This applies to the following form codes:
5 (Channel section)
112 (Cold formed Channel section)
114 (Cold formed C section)
116 (Cold formed C section eaves beam)
117 (Cold formed C-Plus section)
In addition this applies to the following pair sections:
2CFUc and 2CFCc
2CFUo and 2CFCo with distance a > 0 mm
Scia Engineer Steel Code Check Theoretical Background
165
Since the pair sections consist of two webs the resistance of the full section is obtained by adding the values of each web.
Single Web Z-Sections
For single web Z-sections table C3.4.1-3 is used.
This applies to the following form codes:
102 (Z section)
113 (Cold formed Z section)
118 (Cold formed ZED section)
119 (Cold formed ZED section asymmetrical lips)
120 (Cold formed ZED section inclined lip)
126 (Cold formed ZED section both lips inclined)
Single Hat Sections
For single hat sections table C3.4.1-4 is used.
This applies to the following form code:
115 (Cold formed Omega section)
Since these sections consist of two webs the resistance of the full section is obtained by adding the values of each web.
Other Sections
For any other cross-section types as those listed in the previous paragraphs no web crippling check is executed.
In addition table C3.4.1-5 is not supported.
Scia Engineer Steel Code Check Theoretical Background
166
Combined Bending and Web Crippling
The combined bending and web crippling check is determined according to articles C3.5.1 and C3.5.2.
Requirement (a) is applied to the following form codes/sections:
5 (Channel section)
112 (Cold formed Channel section)
114 (Cold formed C section)
116 (Cold formed C section eaves beam)
117 (Cold formed C-Plus section)
102 (Z section)
113 (Cold formed Z section)
118 (Cold formed ZED section)
119 (Cold formed ZED section asymmetrical lips)
120 (Cold formed ZED section inclined lip)
126 (Cold formed ZED section both lips inclined)
115 (Cold formed Omega section)
2CFUc and 2CFCc
2CFUo and 2CFCo with distance a > 0 mm
Requirement (b) is applied to the following form codes/sections:
2CFUo and 2CFCo with distance a = 0 mm
Requirement (c) is applied to the following form codes/sections in case the check is executed within
a lapped zone:
102 (Z section)
113 (Cold formed Z section)
118 (Cold formed ZED section)
119 (Cold formed ZED section asymmetrical lips)
120 (Cold formed ZED section inclined lip)
126 (Cold formed ZED section both lips inclined)
Remarks:
The exception given for requirement (a) is not supported.
In case a lapped Z section does not meet the limits for requirement (c) the provisions of requirement (a) are applied instead.
For requirement (c) it is assumed that conditions (1), (2), (3) & (4) are fulfilled.
Scia Engineer Steel Code Check Theoretical Background
167
Combined Bending and Torsion
Combined bending and torsion loading is evaluated according to article C3.6.
In each fiber of the cross-section the bending stresses Sigma Mx and Sigma My are calculated.
These stresses are based on the effective cross-sectional properties and calculated in the fibers of the gross cross-section. In addition, in each fiber the shear stress due to torsion Tau t is calculated based on gross section
properties.
Using these stresses, the R factor is calculated according to equation (C3.6-1) using the following
expressions:
f bending = Sigma Mx + Sigma My
f torsion = Tau t
f bending + f torsion = (composed stress)
The critical fiber is taken as the fiber with the biggest composed stress.
The increase of the R factor in case of C-sections is not supported.
For diaphragms reference is made to “Use of diaphragms”.
Compression Members
Nominal axial strength
The nominal axial strength is determined according to article C4.1 using Fn = Fy.
Flexural Buckling
The stress Fe for flexural buckling is determined according to article C4.1.1.
For the calculation of the effective length factor, reference is made to “Calculation buckling ratio – general formula”.
In case an LTB restraint of type „Both‟ is inputted, it specifies that both the top and bottom flange are held into position. As such, this point is seen as a fixed point for weak axis buckling. This implies that the system length Ly is taken between the LTB restraints of type „Both‟ and the member ends. In addition the effective length factor ky is set to 1,00.
For diaphragms reference is made to “Use of diaphragms”.
Scia Engineer Steel Code Check Theoretical Background
168
Torsional (-Flexural) Buckling
The stress Fe for torsional (-flexural) buckling is determined according to the general method given
in Ref. [7]. Doubly symmetric and hollow sections are taken as not subject to torsional (-flexural) buckling.
This applies to the following form codes:
1 (Symmetric I shape)
2 (Rectangular Hollow Section)
3 (Circular Hollow Section)
For any other section the stress Fe is taken as the smallest of Sigma,t and Sigma,TF
Sigma,t = Ncr,T / Ag
Sigma,TF = Ncr,TF / Ag
With: Ncr,T Critical axial load for torsional buckling
Ncr,TF Critical axial load for torsional-flexural buckling
Ag Gross section area
Determination of Ncr,T
The elastic critical load Ncr,T for torsional buckling is calculated according to Ref.[7].
With: E Modulus of Young
G Shear modulus
J Torsion constant
Cw Warping constant
lT Buckling length for the torsional buckling mode
x0 and y0 Coordinates of the shear center with respect to the centroid
rx radius of gyration about the x-x axis
ry radius of gyration about the y-y axis
Scia Engineer Steel Code Check Theoretical Background
169
Determination of Ncr,TF
The elastic critical load Ncr,TF for torsional flexural buckling is calculated according to Ref.[7].
Ncr,TF is taken as the smallest root of the following cubic equation in N:
0
With: Ncr,x Critical axial load for flexural buckling about the x-x axis
Ncr,y Critical axial load for flexural buckling about the y-y axis
Ncr,T Critical axial load for torsional buckling
The smallest value of Fe (flexural, torsional and torsional-flexural buckling) is used for calculating Fn according to article C4.1.
For diaphragms reference is made to “Use of diaphragms”.
Closed Cylindrical Tubular sections
The axial strength for closed cylindrical tubular sections is determined according to article C4.1.5.
This applies to the following form code:
3 (Circular Hollow Section)
In case the diameter to thickness ratio D/t exceeds the limit 0,441 E/Fy the check is not executed and a warning is issued on the output.
Distortional Buckling Strength
The distortional buckling strength is determined according to article C4.2. More specifically the general Procedure (a) is followed using formula (C4.2-6).
The check is executed in case the cross-section has at least one element with reinforcement type RUO.
Remarks:
The same remarks are valid as for distortional buckling of flexural members.
The elastic distortional buckling stress Fd is determined for each flange separately. The minimal Fd is taken as the limiting value of the cross-section.
Because of this separate determination, a diaphragm on only one flange is accounted for in the Fd calculation of that specific flange.
In addition, this procedure allows stiffened flanges of unequal dimensions.
For diaphragms reference is made to “Use of diaphragms”.
Scia Engineer Steel Code Check Theoretical Background
170
Combined Compression and Bending
The combined compression and bending check is executed according to article C5.2.
The shifts ex and ey of the neutral axis are determined for the required compressive axial strength.
The additional moments due to these shifts are then calculated by multiplying the required compressive axial strength with these respective shifts.
The special provisions for angle sections apply for the following form codes:
4 (Angle section) 111 (Cold formed Angle section)
In case of 2nd
order analysis, reference is made to “2nd Order using Appendix 2”.
Use of diaphragms
Diaphragms are used specifically in conjunction with article D6.1 concerning purlin and girt design.
The lateral stiffness S for a diaphragm is calculated as follows in case the bolt pitch of the diaphragm
is set as „br‟: (Ref.11,3.5 and Ref.12,3.3.4.):
With a Frame distance
Ls Diaphragm length
K1 Diaphragm stiffness factor K1
K2 Diaphragm stiffness factor K2
For a bolt pitch of „2br‟ the shear stiffness S is replaced by 0,2 S (Ref.11 p22).
For the rotational stiffness vorhC of a diaphragm reference is made to “Adaptation of torsional
constant”.
The available lateral strength S is compared to the required lateral strength Serf Ref.[8]:
With E Modulus of Young
CW Warping constant of the purlin
L LTB length of the purlin
G Shear modulus
J Torsion constant of the purlin
Iy Second moment or area about the y-y axis of the purlin
h Height of the purlin
Scia Engineer Steel Code Check Theoretical Background
171
In case the available lateral strength S is higher than or equal to the required strength Serf, the diaphragm is providing sufficient stiffness and the purlin is seen as fully braced.
In case the available lateral strength S is lower than the required strength Serf, the diaphragm is not providing sufficient stiffness and the purlin is seen as inadequately braced.
The influence of a diaphragm on different checks (bending, compression and torsion) is outlined in the following overview.
Bending
Diaphragm on the compression flange
The lateral stiffness S is calculated and compared to the required stiffness Serf.
In case S ≥ Serf the member is taken as fully braced.
As a result no LTB check is required for bending about the x-x axis.
Distortional buckling still needs to be checked. For distortional buckling is taken as vorhC.
See Ref.[2] pp 47 “Since the distortional buckling has an intermediate buckling half wavelength; the distortional buckling still needs to be considered even for braced members.”
In case S < Serf the member is seen as inadequately braced.
As a result the LTB check for bending about the x-x axis is executed using the augmented torsional stiffness J.
Reference is made to “Adaptation of torsional constant”.
Distortional buckling still needs to be checked. For distortional buckling is taken as vorhC.
Diaphragm on the tension flange
The lateral stiffness S is calculated and compared to the required stiffness Serf.
In case S ≥ Serf the member is taken as fully braced on the tension flange.
In this case article D6.1.1 is applied.
As a result no LTB check is required for bending about the x-x axis.
In addition, no distortional buckling check is required.
In case S < Serf or in case the limits of article D6.1.1 are not met, the member is seen as
inadequately braced.
As a result the LTB check for bending about the x-x axis is executed by default, without an increased torsional stiffness J.
In addition distortional buckling is checked taking as zero.
Scia Engineer Steel Code Check Theoretical Background
172
Compression
Diaphragm on one flange
The lateral stiffness S is calculated and compared to the required stiffness Serf.
In case S ≥ Serf the member is taken as fully braced.
In this case article D6.1.3 is applied.
As a result no distortional buckling check is required.
In case S < Serf or in case the limits of article D6.1.3 are not met, the member is seen as
inadequately braced .
As a result the default compression checks are executed.
In addition distortional buckling will be checked taking as zero.
Diaphragm on both flanges
In this case the specifications of the previous step apply using the largest lateral stiffness S of both
diaphragms.
Torsion
Diaphragm on any flange
The lateral stiffness S is calculated and compared to the required stiffness Serf.
In case S ≥ Serf the member is taken as fully braced against torsion.
In this case the reduction due to torsion is not applied.
In case S < Serf, the member is taken as inadequately braced.
As a result the reduction for torsion is determined by default.
Scia Engineer Steel Code Check Theoretical Background
173
Flexural members having one flange through-fastened to sheeting
The nominal flexural strength is determined according to article D6.1.1.
This article is only applied in case the following conditions are met:
The member is in bending about the x-x axis
The diaphragm is located on the tension flange
The diagram is through fastened
The lateral stiffness S ≥ Serf
The conditions for article D6.1.1 are met
Remarks:
The article is only valid for C and Z sections with edge stiffeners (i.e. elements with reinforcement type ROU).
This applies to the following form codes: 114 (Cold formed C-section) 116 (Cold formed C-section eaves beam) 117 (Cold formed C-Plus section) 118 (Cold formed ZED section) 119 (Cold formed ZED section asymmetric lips) 120 (Cold formed ZED section inclined lip) 126 (Cold formed ZED section both lips inclined)
For determining the R factor a difference is made between simple span and continuous spans. This difference is based on the system length Lx. When the member under consideration has only one part for Lx it is taken as simple span. When the member has more parts for Lx it is taken as continuous span.
The article is not applied for cantilevers. A cantilever is defined as a member at the end of a buckling system which has free ends for both buckling about the x-x and y-y axis.
In addition, the article is not applied for continuous beams in the region between inflection points adjacent to a support.
It is assumed that conditions (8), (9), (10), (11), (12) & (13) are fulfilled.
The correction factor r for compressed insulation is not supported.
Compression members with one flange through-fastened to sheeting
The compressive strength is determined according to article D6.1.3.
This article is only applied in case the following conditions are met:
The member is in compression
The diaphragm is located on one or both flanges
The diagram is through fastened
The lateral stiffness S ≥ Serf
The conditions for article D6.1.3 are met
Scia Engineer Steel Code Check Theoretical Background
174
Remarks:
The article is only valid for C and Z sections with edge stiffeners (i.e. elements with reinforcement type ROU). This applies to the following form codes: 114 (Cold formed C-section) 116 (Cold formed C-section eaves beam) 117 (Cold formed C-Plus section) 118 (Cold formed ZED section) 119 (Cold formed ZED section asymmetric lips) 120 (Cold formed ZED section inclined lip) 126 (Cold formed ZED section both lips inclined)
The fastener distance x is taken as 0,5.
It is assumed that conditions (7) & (8) are fulfilled.
2nd Order using Appendix 2
In case the proper setting is activated in the steel setup, the provisions according to article 2.1 of Appendix 2 are applied.
More specifically, when the check is executed for a non-linear combination the following changes are applied:
Effective length factor Kx is set to 1,00
Effective length factor Ky is set to 1,00
x for article C5.2 is taken as 1,00
y for article C5.2 is taken as 1,00
Cmx for article C5.2 is taken as 1,00
Cmy for article C5.2 is taken as 1,00
Article 2.2 of Appendix 2 is not supported.
Scia Engineer Steel Code Check Theoretical Background
175
Lapped Purlin Design
For the analysis, the purlin line is considered prismatic i.e. the increased stiffness due to the doubled cross-section within the lap is ignored Ref.[5].
Since the lap length is defined along the member axis, it is important to specify a sufficient „number of sections on average member‟ in the Solver Setup when using overlaps.
Combined Strength
The strength within the lapped zones is taken as the sum of the strengths of the individual members Ref.[4].
The use of the combined strength of the individual members is applied for the following checks:
- Nominal Bending Check
- Shear Check
- Combined Bending and Shear Check
- Web crippling Check
- Combined Bending and Web Crippling Check
- Bending – Distortional Buckling Check
For distortional buckling, the distortional buckling stress Fd is calculated for the critical flange i.e. the flange resulting in the lowest Fd value.
The following equations are then used: Mcrd = (Sfsection 1 + Sfsection 2) * Fd My = (Sfysection 1 + Sfysection 2) * Fy
Special considerations for Lateral Torsional Buckling
Within a lapped zone, at the bottom flange the LTB check depends on the Bottom flange fully braced setting within the Overlap data.
In case this setting is activated it implies the bottom flange within the lapped zone is fully fixed and thus no LTB occurs. This has the following implications:
- Within the lapped zone, in case the bottom flange is in compression, no LTB check is executed.
- Outside of the lapped zone the LTB length is taken to the end of the lap.
Scia Engineer Steel Code Check Theoretical Background
176
Diaphragm on the tension flange
In case the following conditions are met:
- Diaphragm on the top flange which provides full bracing - Setting Bottom flange fully braced activated in the overlap data
- The top flange is in tension
By default it would imply article D6.1.1 should be applied however this article is only valid in case the
compression flange is free. Since in this case the compression flange is fully braced this article is not applied and the nominal bending strength is used.
References
[1] AISI S100-2007
North American Specification for the Design of Cold-Formed Steel Structural Members
2007 edition
[2] AISI S100-2007-C
Commentary on North American Specification for the Design of Cold-Formed Steel Structural Members
2007 edition
[3] AISI S100-07-E1
Errata to North American Specification for the Design of Cold-Formed Steel Structural Members
2007 edition February 20, 2008
Amended September 25, 2008
Amended June 4, 2009
[4] AISI SG03-2
Cold-Formed Steel Design Manual
2002 edition
[5] G. J. Hancock, T. M. Murray, D. S. Ellifritt
Cold-Formed Steel Structures to the AISI Specification
Marcel Dekker, Inc., 2001
[6] A Gerhsi, R. Landolfo, F.M. Mazzolani
Design of Metallic cold formed thin-walled members
Spon Press, London, UK, 2002
[7] SN001a-EN-EU
NCCI: Critical axial load for torsional and flexural torsional buckling modes
Access Steel, 2006
www.access-steel.com
Scia Engineer Steel Code Check Theoretical Background
177
[8] EN 1993-1-3:2006
Eurocode 3 - Design of steel structures
Part 1-3: General rules - Supplementary rules for cold-formed members and sheeting
CEN, 2006
[9] Schafer, B.W., Ádány, S.
Buckling analysis of cold-formed steel members using CUFSM: conventional and constrained finite strip methods.
Eighteenth International Specialty Conference on Cold-Formed Steel Structures,
Orlando, FL. October 2006.
[10] J. Schikowski
Stabilisierung von Hallenbauten unter besonderer Berücksichtigung der Scheibenwirkung von Trapez- und Sandwichelementdeckungen, 1999 http://www.jschik.de/
[11] E. Kahlmeyer
Stahlbau nach DIN 18 800 (11.90)
Werner-Verlag, Düsseldorf
[12] Beuth-Kommentare
Stahlbauten
Erläuterungen zu DIN 18 800 Teil 1 bis Teil 4, 1.Auflage
Beuth Verlag, Berlin-Köln 1993
[13] AISI S100-07/S1-09
Supplement No. 1 to the North American Specification for the Design of Cold-Formed Steel Structural Members, 2007 edition August, 2009
[14] AISI S100-07/S2-10
Supplement No. 2 to the North American Specification for the Design of Cold-Formed Steel Structural Members, 2007 edition February, 2010
Scia Engineer Steel Code Check Theoretical Background
178
CM66
CM66 Code check
The beam elements are checked according to the regulations given in
Règles de calcul des constrcutions en acier
ITBTP / CTICM
Régles CM Decembre 1966
Editions Eyrolles 1982
Consulted articles
The cross-section is checked for tension (art. 3,1), bending (art. 3,2.) and shear (art. 3,3.).
For the stability check, the following criteria are considered:
for compression : art. 3,4.
for compression and bending : art. 3,5
for lateral torsional buckling : art. 3,6.
for double bending and axial compression : art. 3,7.
for shear buckling : art 5,212
A more detailed overview for the used articles is given for the relevant parts in the following table. The chapters marked with “x” are consulted. The chapters marked with (*) have a supplementary explanation the following chapters.
3 Règles générales concernant les calculs de résistance et de déformation
3,0 Données numériques x
3,1 Pièces soumises à traction simple x (*)
3,2 Pièces soumises à flexion simple ou déviée x
3,21 Flexion simple x(*)
3,22 Flexion déviée
3,3 Effet de l‟effort tranchant dans les pièces fléchies x
Scia Engineer Steel Code Check Theoretical Background
179
3,4 Pièces soumises à la compression – flambement simple
3,40 Généralités x(*)
3,41 Pièces comprimées a parois pleines x
3,42 Pièces composées a treilis
3,43 Pièces composées a traverses de liaison
3,44 Conditions spéciales imposées aux éléments comprimés a parois minces x
3,5 Pièces soumises à compression avec flexion dans le plan de flambement
3,50 Principe x
3,51 Coefficient d‟amplification des contraintes de flexion x (*)
3,52 Vérfication des pièces a parois pleines
x
3,53 Vérification des pièces composées à treilis
3,54 Vérification des pièces composées à traverses de liaison
3,6 Déversement en flexion simple
3,60 Généralités x
3,61 Pièces symétriquement chargées et appuyées
3,611 Poutres à äme pleine x(*)
3,612 Poutres à treilis
3,62 Cas des piéces soumises à deux moments différents au droit des appuis x(*)
3,63 Cas des poutrelles en console parfaitement encastrées
3,64 Coeffcients utilisés pour la détermination de kd
3,641 Coefficient D x
3,642 Coefficient C x(*)
3,643 Coefficient B x(*)
3,7 Flexion composée
3,70 Domaine d‟application x
3,71 Notations x
3,72 Principe des vérifications x
3,73 Formules enveloppes pour les pièces à parois pleines x (*)
3,8 Flambement dans les systémes hyperstatiques
3,9 Déformations x
5 Règles spéciales à certains éléments
5,212 Poutres composées à âme pleine – âmes x
Scia Engineer Steel Code Check Theoretical Background
180
Section properties
The net area properties are not taken into account .
Plastic coefficient
The plastic coefficients are calculated according to the Ref.[1], 13,212 (Valeurs du coefficient ψ d‟adaptation plastique).
Compression members
For the calculation of the buckling length, we refer to "Calculation of buckling ratio".
The buckling properties for a VARH element are calculated by using the critical Euler force for this member (see “Calculation of critical Euler force for VARH elements”).
Factor kf
The factor kf is calculated using the formula given in Ref[1], 3,516
3;1
lM
A172.125.0
k
2
med
M
f
If Mmed ≈ 0.0, the formula 3,513 is used : 3.1
25.0k f
LTB Check
The LTB check is performed for symmetric I sections. For other cross sections the factor kd=1.0.
For the calculation of the coefficient C, we refer to "Calculation of moment factors for LTB".
The coefficient B is calculated by interpolating the table for B given in Ref[1] 3,643, and using the calculated C value with table for C given in Ref[1] 3,642.
Haunched sections (I+Ivar, Iw+Plvar, Iw+Iwvar, Iw+Ivar, I+Iwvar) and composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
Use of diaphragms
See Chapter 'Adaptation of torsional constant'.
Scia Engineer Steel Code Check Theoretical Background
181
Combined flexion
The values fx is the maximum value of the bending stress in the member for the bending around
the strong axis. The value fy is the maximum value of the bending stress in the member for the bending around the weak axis.
For non-prismatic sections the values fx and fy are the local (i.e. in each intermediary section) bending stresses.
Shear buckling check
Composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
Supported sections
I Symmetric I shapes (IPE, HEA, HEB, ….)
RHS Rectangular Hollow Section (RHS)
CHS Circular Hollow Section (CHS)
L Angle section
U Channel section
T T section
PPL Asymmetric I shapes
RS Rectangular section
Cold formed section
COM Composed section in PRIMAWIN
O Solid tube
NUM Numerical section
The necessary data conditions for these sections are described in chapter "Profile conditions for code check
". The COM and NUM sections are not read out of the profile library.
I
RHS
CHS
L
U
T
PPL
RS
O
COM
NUM
Section check x x x x x x x x x x x x
Buckling check x x x x x x x x x x x x
Slender section buckling check
x x x x x x x x
LTB Check x
Shear buckling check x x x x
Scia Engineer Steel Code Check Theoretical Background
182
References
1 Règles de calcul des constrcutions en acier
ITBTP / CTICM
Régles CM Decembre 1966
Editions Eyrolles 1982
Scia Engineer Steel Code Check Theoretical Background
183
CM66 - Additif 80
CM66 - Additif 80 Code check
The beam elements are checked according to the regulations given in Additif 80
Consulted articles
The cross-section is classified according to art. 5,12. (classification 'plastic' or 'elastic').
The section is checked for tension and compression (art. 4,2), bending (art 4,3), shear force (art. 4,4), the combination of bending and axial force (art. 4,5 and art 4.6).
For the stability check, the following criteria are considered:
for lateral torsional buckling : art. 5,2.
for compression : art. 5,31.
for compression and bending : art. 5,32
A more detailed overview for the used articles is given in the following table. The chapters marked with “x” are consulted. The chapters marked with (*) have a supplementary explanation in the following chapters.
4 Resistance des sections
4,1 Règle générale (*)
4,2 Effort normale x
4,3 Moment de flexion x
4,4 Effort tranchant x
4,5 Moment de flexion et effort normal x
4,6 Momens de flexion, effort normal et effort tranchant x
5 Stabilité des éléments
5,1 Conditions de non voilement local x (*)
5,2 Résistance au déversement des poutre fléchies
5,21 Règles de contreventement latéral au voisinage des sections plastifiées
5,22 Moment ultime de déversement en flexion simple x (*)
5,23 Dimensionnement des entretoises
5,24 Résistance au déversement en flexion déviée x
5,3 Résistance au flambement
5,31 Eléments simplement comprimés x
5,32 Eléments comprimés et fléchis x
5,33 Longueur de flambement (*)
Scia Engineer Steel Code Check Theoretical Background
184
Classification of sections
For each intermediary section, the classification is determined and the proper section check is performed. The classification can change for each intermediary point.
For each load case/combination, the critical section classification over the member is used to perform the stability check. So, the stability section classification can change for each load case/combination.
However, for non-prismatic sections, the stability section classification is determined for each intermediary section.
Section check
If the sections are not according to the conditions specified in art. 5,1, the sections are checked according to the regulations given in Ref.[2].
If a torsional moment is present, the sections are checked according to the regulations given in Ref.[2].
Compression members
For the calculation of the buckling length, we refer to "Calculation of buckling ratio".
The buckling properties for a VARH element are calculated by using the critical Euler force for this member (see “Calculation of critical Euler force for VARH elements”).
Lateral-torsional buckling
For the calculation of the moment factors C1 and C2, we refer to "Calculation of moment factors for LTB", using the EC3 values.
Haunched sections (I+Ivar, Iw+Plvar, Iw+Iwvar, Iw+Ivar, I+Iwvar) and composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
Use of diaphragms
See Chapter 'Adaptation of torsional constant'.
Scia Engineer Steel Code Check Theoretical Background
185
Supported sections
I Symmetric I shapes (IPE, HEA, HEB, ….)
RHS Rectangular Hollow Section (RHS)
CHS Circular Hollow Section (CHS)
L Angle section
U Channel section
T T section
PPL Asymmetric I shapes
RS Rectangular section
Cold formed section
COM Composed section in PRIMAWIN
O Solid tube
NUM Numerical section
The necessary data conditions for these sections are described in chapter "Profile conditions for code check
". The COM and NUM sections are not read out of the profile library.
I
RHS
CHS
L
U
T
PPL
RS
O
COM
NUM
Classification Add 80
x x
Plastic section check Add 80
x x
Buck:ling check Add 80
x x
LTB check Add 80 x x
Compression + bending Add 80
x x
References
[1] Additif 80
2 Règles de calcul des constrcutions en acier
ITBTP / CTICM
Régles CM Decembre 1966
Editions Eyrolles 1982
Scia Engineer Steel Code Check Theoretical Background
186
BS5950-1:1990
BS5950-1:1990 Code Check
The beam elements are checked according to the regulations given in:
British Standard BS 5950
Structural use of steelwork in building
Part1. Code of practice for design in simple
and continuous construction:hot rolled section
British Standard distribution BS5950 Part1 1990 revised in 1992
Material properties
For standard steel grades, the yield strength py is defined according to the thickness of the element (see Table 6 Art.3.1.1.). The standard steel grades are :
Grade 43 : yield strength defined between 245 and 275 N/mm²
Grade 50 : yield strength defined between 325 and 355 N/mm²
Grade 55 : yield strength defined between 415 and 450 N/mm²
(pY in N/mm², t in mm)
Steel grade
Thickness limits
PY
Grade 43
t16 mm
275 N/Mm²
t40 mm
265 N/mm²
t63 mm
255 N/mm²
t100 mm
245 N/mm²
t16 mm
355 N/mm²
Scia Engineer Steel Code Check Theoretical Background
187
Grade 50
t40 mm
345 N/mm²
t63 mm
340 N/mm²
t100 mm
325 N/mm²
Grade 55
t16 mm
450 N/mm²
t25 mm
430 N/mm²
t40 mm
415 N/mm²
t63 mm
400 N/mm²
Remark: For cold-formed section, values for Py are not influenced by the previous table.
Remark : The reduction rules from previous table are only valid when the used material is defined as material for the selected code.
Consulted articles
According to Art. 3.5. and table 7, cross sections are classified in 4 types:
Plastic
Compact
Semi-compact
Slender
A reduction factor is applied to the design strength of the material in use for slender sections by following the rules described in Art. 3.6 and in Table 8. Partial safety factor of design strength is included in py value.
The section is checked for bending (Art.4.2.), tension (Art.4.6.), compression (Art.4.7.), shear (Art.4.2.3.), combined moment and axial force (Art. 4.8.) and biaxial moments (Art.4.9.). For the stability check, the beam element is checked for lateral torsional buckling, shear buckling, compression and bending with axial compression. Articles used for this stability check are the following:
for lateral torsional buckling : Art. 4.3.
shear buckling : Art. 4.4.5.
for compression : Art. 4.7.
for bending and axial compression : Art. 4.8.
A more detailed overview of used articles is given in the following table.
Scia Engineer Steel Code Check Theoretical Background
188
Part. 3 Section properties
3.5. Limiting proportions of cross sections Art. 3.5.1.
Art. 3.5.2.
Art. 3.5.4.
Table 7
Fig.3
3.6. Slender cross section Art. 3.6.1.
Art. 3.6.2.-3.6.3.
Art. 3.6.4.
Table 8
Part. 4 Design of structural elements
4.2. Member in bending Art. 4.2.1.3. (a) (c)
Shear capacity Art. 4.2.3.
Moment capacity with low shear Art. 4.2.5.
Moment capacity with high shear Art. 4.2.6.
4.3. Lateral torsional buckling
Member in bending Art. 4.3.7.
LTB factor
General Art. 4.3.7.1.
Equivalent uniform moment Art. 4.3.7.2.
Buckling Resistance Art. 4.3.7.3.
Bending strength pb Art. 4.3.7.4.
Equivalent slenderness LT, , , u, v
Art. 4.3.7.5.
Appendix B.
Factors m, n Art. 4.3.7.6.
Equal flanged rolled section Art. 4.3.7.7.
Buckling resistance moment for single angle Art.4.3.8.
4.4. Plate Girders
General Art. 4.4.1.
Dimensions of webs and flanges Art. 4.4.2.2. Art. 4.4.2.3.
Moment capacity Art. 4.4.4.
Section with slender webs Art. 4.4.4.2. (a)
Shear buckling resistance of thin webs Art. 4.4.5.1.
Design without using tension field action Art. 4.4.5.3. and Appendix H.1.
4.6. Axially loaded tension members
Scia Engineer Steel Code Check Theoretical Background
189
Tension capacity Art. 4.6.1.
Effective Area of simple tension members Art. 4.6.3.1. Art. 4.6.3.3.
4.7. Compression member
Slenderness Art. 4.7.3.2.
Compression resistance Art. 4.7.4.
Compressive strength Art. 4.7.5. Appendix C
4.8. Axially loaded members with moments
Tension members with moments Art. 4.8.2. + EC3 5.4.9.&Annex F
Compression members with moments Art. 4.8.3.
Local capacity check Art. 4.8.3.2.
Buckling check with exact approach Art. 4.8.3.3.2.
4.9. Members with biaxial moments
See 4.8.
Classification of sections
For each intermediary section, the classification is determined and the proper section check is performed. The classification can change for each intermediary point. For each load case/combination, the critical section classification over the member is used to perform the stability check.
So, the stability section classification can change for each load case/combination. However, for non-prismatic sections, the stability section classification is determined for each intermediary section.
Slender cross-section
Slender sections are particularly sensitive to local buckling. British Standard code (Art. 3.6.) defines stress reduction factor to prevent this phenomenon. For webs subject to moments and axial load and for circular hollow sections, the design strength py should be assumed such that the limiting proportions for semi-compact section are met. For other sections, where a slender outstand is in compression, the design strength should be reduced by the factor given in Table 8.
Section properties
The net area of a section is taken as its gross section neglecting the deduction due to fastener holes: Art. 3.3. Shear area of a cross-section is calculated by using Art. 4.2.3.
Scia Engineer Steel Code Check Theoretical Background
190
Bending moment
Before any calculation of members in bending, it's necessary to determine the shear capacity. For plastic and compact section with high shear load, moment capacity is calculated with the plastic modulus only for I and PLL sections (Art. 4.2.6. and 4.8.). For other cross-section, with plastic or compact section classification, characterised or not by a low shear load, we assumed that the moment capacity is calculated by using the same approach than for semi-compact section: the elastic modulus (elastic calculation).
Bending, shear, axial force
For plastic and compact sections, BS5950 Art. 4.8.2. & 4.8.3.2. (b) prescribes a detailed approach to determine the unity check of axially loaded members with moments. The detailed relationship allows a greater economy for plastic and compact section . In this expression, we use a reduced moment capacity Mr respectively about the major and the minor axis. Those values are determined by using EC3 Art.5.4.9. (see Ref.[5]). For semi-compact and slender section, the simplified approach is applied following Art. 4.8.2.and Art. 4.8.3.2. (a).
Lateral torsional buckling
For I sections (symmetric and asymmetric PPL), rectangular sections (solid and hollow), T sections, channel sections and angle section, the critical lateral torsional buckling moment is given by the general formula Art. 4.3.7. and Annex B2&3. For other sections, we follow conservative recommendation described in Art. 4.3.7.5. and calculation proposed in EC3 to determine the elastic critical moment Mcr EC3 Annex F1.1. Formula (F.1.) see Ref [5].
The condition to be satisfied in all the cases is that
with
Mb=Sxpb
and
(m is an equivalent uniform moment factor)
pb is the bending strength and is related to the equivalent slenderness :
in which n is an equivalent slenderness factor.
For beam without loading point between points of lateral restraint, n=1 and m depends on the ratio of the end moments at the points of restraint.
Scia Engineer Steel Code Check Theoretical Background
191
For beam loaded between point of lateral restraint, m=1 and n depend on the ratio of the end moments at the points of restraint and on the ratio of the larger moment to the mid-span free moment.
There are thus two methods for dealing with lateral torsional buckling namely:
'm approach' i.e. the 'equivalent uniform moment method' with n=1
'n approach' i.e. the 'equivalent slenderness method' with m=1
In any given situation, only one method will be admissible, taking into account that it is always conservative to use m=n=1. Since the publication of BS5950 Part 1 1990, doubt has been cast on the correctness of using n factors less than 1 in combination with an effective length LLTB less than
the length of the member L in the calculation of LTB. However, as a interim measure, pending
clarification ina future version of BS5950, it is recommended that LTB is taken as the smaller of the two following values:
By using the settings of BS5950, the user can define which method correspond to his situation or define his choice as the conservative method m=n=1.
Haunched sections (I+Ivar, Iw+Plvar, Iw+Iwvar, Iw+Ivar, I+Iwvar) and composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
Use of diaphragms
See Chapter 'Adaptation of torsional constant'.
Compression member
For member submitted to compression, we applied the recommendations given in BS 5950 and Appendix C to determine the compressive strength.
Shear buckling check
Composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
Scia Engineer Steel Code Check Theoretical Background
192
Supported sections
I Symmetric I shapes (IPE, HEA, HEB, ….)
RHS Rectangular Hollow Section (RHS)
CHS Circular Hollow Section (CHS)
L Angle section
U Channel section
T T section
PPL Asymmetric I shapes
RS Rectangular section
Cold formed section
COM Composed section in PRIMAWIN
O Solid tube
NUM Numerical section
The necessary data conditions for these sections are described in chapter "Profile conditions for code check
". The COM and NUM sections are not read out of the profile library.
I
RHS
CHS
L
U
T
PPL
RS
O
COM
NUM
Classification x x x x x x x (1) x (1) (1) (1)
Section check class 1 x x x x x x x x
Section check class 2 x x x x x x x x
Section check class 3 x x x x x x x x x x x x
Section check class 4 x x x x x x x x
Stability check class 1 x x x x x x x x
Stability check class 2 x x x x x x x x
Stability check class 3 x x x x x x x x x x x x
Stability check class 4 x x x x x x x x
Shear buckling check x x x
(1)sections are classified as class 3 cross section by default
References
[1] British Standard BS5950 Part 1 : 1990+Revised text 1992
Structural use of steel work in building
Part1 Code of practice for design in simple and continuous construction: hot rolled sections
Scia Engineer Steel Code Check Theoretical Background
193
[2] Plastic design to BS5950
J.M. Davies & B.A. Brown
The steel Construction institute
[3] Steelwork design
Guide to BS5950: Part 1: 1990
Volume 2 Worked examples (revised edition)
[4] Essentials of Eurocode 3
Design Manual for Steel Structures in Building
ECCS - N° 65, 1991
[5] Eurocode 3
Design of steel structures
Part 1 - 1 : General rules and rules for buildings
ENV 1993-1-1:1992
[6] R. Maquoi
ELEMENTS DE CONSTRUCTIONS METALLIQUE
Ulg , Faculté des Sciences Appliquées, 1988
Scia Engineer Steel Code Check Theoretical Background
194
BS5950-1:2000
BS5950-1:2000 Code Check
The background to this code check can be found within the document “BS 5950-1:2000 steel code check Theory”
Scia Engineer Steel Code Check Theoretical Background
195
SIA263
SIA263 Code check
The beam elements are checked according to the regulations given in
SIA263
Construction en acier
SIA263:2003
Material properties
The most common steel grades are used in SIA263. Their mechanical properties are described in table 1 SIA263. The following table gives the yield strength for each type of grade commonly used in function of the nominal web thickness:
t<=40 t<=40 40<t<=100 40<t<=100
fy fu fy fu
S235
S 235
235 360 215 340
S275
S 275
275 430 255 410
S355
S 355
355 510 335 490
S460
S 460
460 550 430 530
Consulted articles
The classification described in SIA263 is based on the calculation method. The calculation method in SIA263 distinguish the method used respectively to determine the internal forces and to perform the section and the stability check.
By facility, we can obviously make a parallel between the calculation method of SIA263 and the section classification proposed in EC3.
Scia Engineer Steel Code Check Theoretical Background
196
According to SIA263 Table 5a-5b , cross sections are classified in 4 types:
PP (plastic-plastic) or class 1
EP (elastic-plastic) or class 2
EE (elastic-elastic) or class 3
EER (elastic-elastic reduced) or class 4
The first letter of the classification denomination is related to the method used to calculate internal forces in the structure. The second letter indicates if we perform the section and the stability check with a elastic or a plastic approach. Finally, we must note that the steel code SIA263 is essentially oriented for symmetrical and bisymmetrical profile like I profiles. In the present modulus, others profiles are calculated by using a classic elastic approach (EE classification) and EC3 prescriptions.
The section is checked for tension, compression, shear, combination of bending and axial forces. For the stability check, the beam element is checked for lateral torsional buckling, shear buckling, compression and bending with axial compression. A more detailed overview for the used articles is given in the following table :
4 Analyse structurale et dimensionnement
4.1 Généralités x
4.2 Bases de l'analyse structurale et du dimensionnement
4.3 Modélisation
4.3.1 Classification des sections
x
4.4 Résistance des sections
4.4.1 Effort normal
x
4.4.2 Flexion x
4.4.3 Effort tranchant x
4.4.4 Flexion et effort tranchant x
4.4.5 Flexion et effort normal x
4.4.6 Sollicitations multiaxiales x
4.5 Stabilité
4.5.1 Flambage
x
4.5.2 Déversement des poutres fléchies x
4.5.3 Flexion et compression x
4.5.4 Voilement des éléments plans comprimés x
4.5.5 Voilement des éléments plans cisaillés x
4.8 Situtation de projet incendie
4.8.1 PRINCIPES
x
4.8.2 Propriétés de l'acier en cas d'incendie x
4.8.5 Méthode de calcul simplifiée x
5 Eléments de construction
5.1 POUTRES ET POTEAUX DES CLASSES DE SECTION 1 ET 2
x
5.3 Eléments comprimés à section composée
5.3.1 Barres étrésillonées ( à travers de liaison)
x
5.4 Poutres composées à âme pleine
Scia Engineer Steel Code Check Theoretical Background
197
5.4.1 Résistance à la flexion x
5.4.2 Résistance à l'effort tranchant x
5.4.3 Interaction entre flexion et effort tranchant x
Annexe B Moment critique de déversement élastique Mcr x
Annexe C Echauffement des éléments de construction en cas d'incendie x
Section classification
For each intermediary section, the classification is determined and the proper section check is performed. The classification can change for each intermediary point. For each load case/combination, the critical section classification over the member is used to perform the stability check.
So, the stability section classification can change for each load case/combination. However, for non-prismatic sections, the stability section classification is determined for each intermediary section.
Slender cross-section
The design of a section that not satisfies the table 5 of SIA263 is always performed by using a reduced area. This classification correspond to the EER method. The determination of a reduced area is based on the effective width of each compression element in the current section (Art. 4.5.4). The using of a reduced area implies the recalculation of the shear centre position, the inertia and the elastic modulus.
Sections properties
The holes due to fastener are neglected in the area of a section
Lateral torsional buckling
For double symmetric I profile, we don't have to perform any lateral torsional buckling check if
NEd/Npl,Rd0.15 and the conditions provided in Table 6 SIA263 are satisfied. For any other case, a LTB check must be perform.
Calculations described in Annex B for I,U and PPL can be applied to T sections only if the flange is subjected to compression. Otherwise, as for section not supported by SIA263 in the LTB check, we use prescriptions given in EC3 Annex F. Those rules allow us to determine a elastic critical moment for lateral torsional buckling for symmetrical (formula F.2 EC3) and non symmetrical (formula F.1. EC3) sections around the minor axis.
In the case of I, U, PPL and, T only with compression in flange, characterised by a reduced area or not, we have to determined before any calculation irc, defined as the radius of gyration of a section comprising the compression flange plus 1/3 of the compression web area, taken about an axis in the plane of the web.
Haunched sections (I+Ivar, Iw+Plvar, Iw+Iwvar, Iw+Ivar, I+Iwvar) and composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
For advanced Lateral-torsional buckling analysis, see chapter „LTBII: Lateral Torsional Buckling 2nd Order Analysis‟.
Scia Engineer Steel Code Check Theoretical Background
198
Use of diaphragms
See Chapter 'Adaptation of torsional constant'.
Shear buckling
Composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
Stability check
For double symmetric I profile PP or EP, SIA263 provides specific formula to perform the stability check of member submitted to biaxial moment. For other sections, non symmetric or from EE and EER classification, a general formula is provided to design member under mono-axial sollicitations.
Torsion check
For the cross section check inclusive torsion and warping, we refer to Chapter 'Warping check'.
Built-in beams
For built-in beam sections (IFB, SFB, THQ sections), proper section checks are performed, taking into account the local plate bending. See Chapter „Section check for built-in beams (IFB, SFB, THQ sections).
SIA263 - Fire Resistance
Fire actions effect Efi
The design effects of actions for the fire situation Efi,d,t are taken from the results of the analysis. It is recommended to use the accidental combination rules, for calculating the internal forces used in the fire resistance check.
The accidental combination is given by
Gk + Pk + Ad+ 2,iQk,i
with Gk characteristic values of permanent actions
Qk,i characteristic value of the variable action i
Ad design values of accidental action from fire exposure
2,j combination coefficients
Pk characteristic value of prestressing action
Scia Engineer Steel Code Check Theoretical Background
199
Material properties
The material properties are depending on the steel temperature.
Strength and deformation properties:
20,a,E,a
20,y,y,y
EkE
fkf
The variation in function of the steel temperature of the value for yield strength ky, and modulus of
elasticity kE, is given by tables in ref.[1], Figure 15.
In the simplified calculation method, the following default properties are considered to be constant during the analysis :
thermal elongation l/l 14 x 10-6
(a-20)
thermal conductivity a 45 W/mK
Temperature analysis - Thermal actions
In this part, the nominal temperature-time curves and the related net heat flux are described. For more info, EC3 Chapter 'Temperature analysis - Thermal actions'
Nominal temperature-time curve
See EC3 Chapter 'Nominal temperature-time curve'.
Net heat flux
See EC3 Chapter 'Net heat flux'
Scia Engineer Steel Code Check Theoretical Background
200
Steel Temperature
See Ref.[1], Annexe C.
The increase of temperature a,t in an unprotected steel member during a time interval t
thc
V/Ad,net
aa
mt,a
with Am the exposed surface area per unit length [m²/m]
V the volume of the member per unit length [m³/m]
The factor Am/V should not be taken as less than 10m-1
ca the specific heat of steel [J/kgK]
hnet,d the net heat flux per unit area [W/m²]
t the time interval [seconds]
The value should not be taken as more than 5 seconds
a the unit mass of steel [kg/m³]
The increase of temperature a,t in an insulated steel member during a time interval t
with Ap the area of fire protection material per unit length [m²/m]
V the volume of the member per unit length [m³/m]
ca the specific heat of steel [J/kgK]
cp the specific heat of fire protection material [J/kgK]
dp the thickness of the fire protection material [m]
t the time interval [seconds]
The value should not be taken as more than 30 seconds
a the unit mass of steel [kg/m³]
p the unit mass of fire protection [kg/m³]
a,t the steel temperature at time t
g,t the ambient gas temperature at time t
g,t the increase of the ambient gas temperature during the time interval
p the thermal conductivity of the fire protection material [W/mK]
The value a,t 0.0
Scia Engineer Steel Code Check Theoretical Background
201
For the increase of temperature a,t in an insulated steel member with intumescent coating, we refer to the NEN specifications, Chapter 'Steel Temperature'.
Calculation model
The calculation can be performed in 2 domains :
- strength domain
- temperature/time domain
In the strength domain, the strength Rfi,d,t(unity check) is calculated after a given time t (e.g. strength
after 45 min). In the temperature/time domain, the critical steel temperature cr,d is computed. From this critical temperature, the fire resistance time tfi,d is calculated (the time domain).
Code Check
The section and stability checks (buckling, lateral torsional buckling) are performed according to the regulations given in Ref.[1], 4.8.5.
For each member, the classification of the cross section, the section check and the stability check are performed.
The following checks are executed :
- classification of cross section : art. 4.8.5.2.
- resistance for tension members : art. 4.8.5.4.
- resistance for compression members (class 1,2 or 3) : art. 4.8.5.5..
- resistance for beams (class 1,2,3) : art. 4.8.5.6., art. 4.8.5.7., art. 4.8.5.8.
- resistance for members (class 4) : art. 4.8.5.9.
Supported sections
I Symmetric I shapes (IPE, HEA, HEB, ….)
RHS Rectangular Hollow Section
CHS Circular Hollow Section
L Angle section
U Channel section
T T section
PPL Asymmetric I shapes
Z Z section
Scia Engineer Steel Code Check Theoretical Background
202
RS Rectangular section
Cold formed section
COM Composed section
O Solid tube
NUM Numerical section
The necessary data conditions for these sections are described in chapter "Profile conditions for code check
". The COM and NUM sections are not read out of the profile library.
I
RHS
CHS
L
U
T
PPL
RS
Z
O
COM
NUM
Classification x x x x x x x x (1) x (1) (1) (1)
Section check PP x x(2) x(3)
Section check EP x x(2) x(3)
Section check EE x x x x x x x x x x x x x
Section check EER x x x x x x
Stability check PP x x x x x x x x x x x x x
Stability check EP x x x x x x x x x x x x x
Stability check EE x x x x x x x x x x x x x
Stability check EER x x x x x x
Shear buckling check x x x
LTB x x(4) x(4) x(4) x(4) x(4) x x(4) x(4) x(4) x(4) x(4) x(4)
(1) sections are classified as class 3 cross section by default.
(2) check according to EN 1993-1-1
(3) check according to ENV 1993-1-1
(4) general formula for Mcr
References [1] SIA263
Construction en acier
SIA263:2003
[2] SIA263/1
Construction en acier / Spécification complémentaires
SIA263/1:2003
Scia Engineer Steel Code Check Theoretical Background
203
GBJ 17-88
The GBJ 17-88 code check
The beam elements are checked according to the regulations given in :
National standard of the People‟s Republic of China
Code for design of steel structures
GBJ 17-88
Beijing 1995
Material properties
The used steel grades are:
Grade3
16Mn
16Mnq
15Mn
15Mnq
For Steel3, the following groups are defined according to the element thickness (in mm):
Group Diameter or thickness of bars Thickness of L-, I- and U sections
Thickness of Plates
1 <=40 <=15 <=20
2 >40-100 >15-20 >20-40
3 >20 >40-80
The design values are (in N/mm²)
Steel Group Thickness f fv fce fy
Steel3 1
2
3
215
200
190
125
115
110
320
320
320
235
235
235
16Mn
16Mnq
<=16
17-25
26-36
315
300
290
185
175
170
445
425
410
345
345
345
15Mn
15Mnq
<=16
17-25
26-36
350
335
320
205
195
185
450
435
415
390
390
390
Scia Engineer Steel Code Check Theoretical Background
204
with f the resistance design value for tension, compression, bending (N/mm²)
fv the resistance design value for shear (N/mm²)
fce the bearing resistance (N/mm²)
fy the yield strength (N/mm²)
Remark : The reduction rules from previous table are only valid when the used material is defined as material for the selected code. If they are not defined as GBJ material, the following rule is used
f = 0.91 x yield strength
fv = 0.58 x yield strength
Consulted articles
The section and elements are checked according to part 4 and 5. When plastic design is allowed, part 9 is supported.
A more detailed overview for the used articles of the relevant parts is given in the following table. The chapters marked with “x” are consulted.
4. Calculation of flexural members
4.1.Strength
4.1.1.
4.1.2.
x (*)
x
4.2.Overall stability (*)
4.2.1.
4.2.2.
4.2.3.
4.2.4.
x
x
x
x
4.3.Local stability (*)
4.3.1.
4.3.2.
4.3.3.
4.3.9.
x
x
x
x
5.Calculation of axially loaded members and members subjected to combined axial load and bending
5.1.Axially loaded members
5.1.1.
5.1.2.
x(*)
x(*)
5.2.Members subjected ot combined axial load and bending
Scia Engineer Steel Code Check Theoretical Background
205
5.2.1.
5.2.2.
5.2.5.
x(*)
x
x
5.3.Effective length and allowable slenderness ratio (*)
5.4.Local stability of compression members
5.4.1.
5.4.2.
5.4.3.
5.4.4.
5.4.5.
x
x
x
x
x (*)
9.Plastic design
9.1.General requirements
9.1.3.
9.1.4.
x
x
9.2.Calculation of members (*)
9.2.1.
9.2.2.
9.2.3.
9.2.4.
x
x
x
x
9.3.Allowable slenderness and detailing requirements
Appendix 1 Overall stability factor of beams
A1.1.Simply supported beam of uniform welded I section x
A1.2.Simply supported beam of rolled I section x
A1.3.Simply supported beam of rolled channel section x
A1.4.Cantilever beams of doubly symmetric I section x
A1.5.Approximate calculation of overall stability factors x
Appendix 2 Calculation of local stability of girder web
A2.1.Web plate strengthened with transverse stiffeners x(*)
A2.2.Web strengthened with transverse and longitudinal stiffeners
A2.2.Web strengthened with transverse, longitudinal and short stiffeners
Appendix 3 Stability factor of axially loaded compression members x
Section properties
The influence of the net section is neglected, i.e. only the gross area is used.
Scia Engineer Steel Code Check Theoretical Background
206
Shear buckling check
The local compressive stress c, is considered as 0.0.
Buckling curves
For welded I and PPL sections the default value for the buckling curve about the weak axis is “b”. This can be changed to “c” on users request.
Buckling length
For the calculation of the buckling length, we refer to "Calculation of buckling ratio".
The buckling properties for a VARH element are calculated by using the critical Euler force for this member (see "Calculation of critical Euler force for VARH elements").
Lateral torsional buckling
The LTB check is supported for the following sections : I section, U section, RHS section, T section, PPL section.
For the other section type, the factor b = 1.0.
Haunched sections (I+Ivar, Iw+Plvar, Iw+Iwvar, Iw+Ivar, I+Iwvar) and composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
Local stability of compressed members
For each intermediary section, the ratio‟s are determined. The section classification and the effective area properties are determined for each intermediary section for performing the section check.
For each load case/combination, the critical section classification and the effective area properties over the member are used to perform the stability check. However, for non-prismatic sections, the section classification and the effective area properties are determined for each intermediary section to perform the stability check.
When the web ratio ( dept /thickness) does not conform to the requirements, the web is reduced for calculating of the section check and stability check. A width of 20 tw sqrt(235/fy) on each side of the web is taken into account.
Scia Engineer Steel Code Check Theoretical Background
207
yw
f
235t20d
Shear buckling check
Composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
Supported sections
I Symmetric I shapes (IPE, HEA, HEB, ….)
RHS Rectangular Hollow Section (RHS)
CHS Circular Hollow Section (CHS)
L Angle section
U Channel section
T T section
PPL Asymmetric I shapes
RS Rectangular section
Cold formed section
COM Composed section in PRIMAWIN
O Solid tube
NUM Numerical section
The necessary data conditions for these sections are described in chapter "Profile conditions for code check
".
The COM and NUM sections are not read out of the profile library.
Scia Engineer Steel Code Check Theoretical Background
208
I
RHS
CHS
L
U
T
PPL
RS
O
COM
NUM
Plastic (single bending) x x
Compact section (with ) x x x x x x
Non-compact section x x x x x x x x x x
Slender section x x x x x x
Normal buckling x x x x x x x x x x x x
LTB x x x x x
Shear buckling x x x
Plastic stability check (single bending)
x x
References
[1] Chinese Steel Code
GBJ 17-88
(Chinese version)
.[2] National standard of the People‟s Republic of China
Code for design of steel structures
GBJ 17-88
Beijing 1995
Scia Engineer Steel Code Check Theoretical Background
209
Korean steel code check
The Korean steel code check
Material properties
The following design values are used :
Steel fy
t<=40 mm
fy
t>40 mm
SS41
SPS41
SPSR41
240 220
SS50 280 260
SS55 380 380
with fy the yield strength (N/mm²)
The following steel characteristics are valid :
modulus of elasticity 210000 N/mm²
shear modulus 81000 N/mm²
coefficient of linear thermal expansion 12 x 10-6
density 7850 kg/m³
Consulted articles
The section and elements are checked according to part 2 and 3. The shear buckling check is perfromed using article 7.5.2. The classiffication of sections is based on the rules of part 4.
A more detailed overview for the used articles of the relevant parts is given in the following table. The chapters marked with “x” are consulted.
TEXT
2.Allowable stress
2.1.Structural material x
2.1.1.Allowable tensile stress x
Scia Engineer Steel Code Check Theoretical Background
210
2.1.2.Allowable shear stress x
2.1.3.Allowable compressive stress x
2.1.4.Allowable bending stress
a)
b)
c)
(*)
x
x
x
2.1.5.Allowable bearing stress
3.Load and stresses
3.3.Combined stresses (*)
3.3.1.Compression force and bending moment x
3.3.2.Tensile force and bending moment x (*)
3.3.3.Shear force and tensile stress
4.Width-Thickness ratio of plates (*)
4.1.1.Cantilever plate x
4.1.2.Two side fixed plate x
4.1.3.Effective area x
4.2.CHS section and thickness ratio x
5. Tensile member
6.Compressive member
6.1.Slenderness ratio x
6.2.Buckling length x(*)
7.Beam element
7.5.Stiffener
7.5.2.Buckling verification of the web
a)
x
Section classification
For each intermediary section, the classification is determined..
For each load case/combination, the critical section classification and the effective area properties over the member are used to perform the code check. However, for non-prismatic sections, the section classification and the effective area properties are determined for each intermediary section.
When the element properties don‟t satisfy the limiting values for the ratios, the section is classified as slender. The section have to be reduced for the calculation of the stresses. For outstand
Scia Engineer Steel Code Check Theoretical Background
211
compression elements, the part that is situated on the fixed side, remains. The length of the part b‟ is calculated by the equation in which the ratio b‟/t is equal on the limiting ratio.
For internal compression elements, the remaining parts are symmetrically divided to the end of the elements. The length of the part d‟ is calculated by the equation in which the ratio d‟/t is equal on the limiting ratio.
The reduced section properties are calculated for I, U, PPL, RHS and Cold formed sections-types.
The slenderness ratios (for buckling and LTB) are calculated with the full section properties.
Section properties
The influence of the bore hole is neglected, i.e. only the gross area is used.
Buckling length
For the calculation of the buckling length, we refer to "Calculation of buckling ratio"
The buckling properties for a VARH element are calculated by using the critical Euler force for this member(see "Calculation of critical Euler force for VARH elements") .
Scia Engineer Steel Code Check Theoretical Background
212
Lateral torsional buckling
For I sections, PPL sections, U sections RHS and CHS sections, the formulas from 2.1.4 are used.
For the other supported sections, the elastic critical moment for LTB Mcr is given by
z
2
t
2
z
2
z
2
EI
GIL
I
Iw
L
EIMcr
with L LTB length
E modulus of elasticity
G shear modulus
Iw warping constant
It torsion constant
Iz moment of inertia about minor axis
With this moment Mcr, the critical LTB stress LTB is calculated :
y
crLTB
I
M
with Iy moment of inertia about major axis
The slenderness ratio for LTB LTB, is given by
LTB
LTB
E
The allowable LTB stress is calculated using the slenderness LTB with the formulas given in 2.1.3.
Haunched sections (I+Ivar, Iw+Plvar, Iw+Iwvar, Iw+Ivar, I+Iwvar) and composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
Scia Engineer Steel Code Check Theoretical Background
213
Combined stresses
For compression and bending, the following formulas are used:
1f
tt
1f
c
f
c
f
t
cbybx
by
by
bx
bx
c
c
For tension and bending, the following formulas are used :
1f
tt
1f
c
f
c
f
t
bybxt
by
by
bx
bx
bx
t
with c normal compression stress
t normal tension stress
cb bending compression stress
tb bending tension stress
cbx bending compression stress around x axis
tbx bending tension stress around x axis
cby bending compression stress around y axis
tby bending tension stress around y axis
ft allowable tension stress
fc allowable compression stress
fb allowable bending stress
fbx allowable bending stress around x axis
fby allowable bending stress around y axis
Shear buckling check
Composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
Scia Engineer Steel Code Check Theoretical Background
214
Supported sections
I Symmetric I shapes (IPE, HEA, HEB, ….)
RHS Rectangular Hollow Section (RHS)
CHS Circular Hollow Section (CHS)
L Angle section
U Channel section
T T section
PPL Asymmetric I shapes
RS Rectangular section
Cold formed section
COM Composed section in PRIMAWIN
O Solid tube
NUM Numerical section
The necessary data conditions for these sections are described in chapter "Profile conditions for code check
". The COM and NUM sections are not read out of the profile library.
I
RHS
CHS
L
U
T
PPL
RS
O
COM
NUM
Slender sections
x x x x x
Allowable stresses
x x x x x x x x x x x x
Shear buckling x x x
References
[1] Korean Standard
(Korean Version)
1983
[2] Extracts Korean Standard
(Internal English Version)
Translated by Karam Kim - 19.03.1998
[3] Regulations of Structural Standard of
Building Architecture
(internal english document)
Scia Engineer Steel Code Check Theoretical Background
215
Scia Engineer Steel Code Check Theoretical Background
216
BSK 99
BSK 99 Code check
The beam elements are checked according to the regulations given in
BSK 99
StalKonstruktioner
Boverket, Byggavdelningen, 2000
Material properties
For standard steel grades, the characteristic yield strength fyk and tensile strength fuk are defined
according to the thickness of the element (see Ref. 1, tab.2:21a and tab.2:21b)
The standard steel grades are:
Steel
Name
Type E-modulus (N/mm
2)
Poisson
Unit mass (kg /m
3)
Extensibility (m/m K)
Ultimate tensile strength (N/mm
2)
Yield strength (N/mm
2)
S235
S 235
Steel 210000 0.3 7850 12*10-6
340 235
S275
S 275
Steel 210000 0.3 7850 12*10-6
410 275
S355
S 355
Steel 210000 0.3 7850 12*10-6
490 355
S420
S 420
Steel 210000 0.3 7850 12*10-6
500 420
S460
S 460
Steel 210000 0.3 7850 12*10-6
530 460
S500
S 500
Steel 210000 0.3 7850 12*10-6
590 500
S550
S 550
Steel 210000 0.3 7850 12*10-6
640 550
S620
S
Steel 210000 0.3 7850 12*10-6
700 620
Scia Engineer Steel Code Check Theoretical Background
217
620
S690
S 690
Steel 210000 0.3 7850 12*10-6
770 690
(fyk, fuk in N/mm², t in mm)
Steel grade Thickness fuk fyk
S235, S 235 0 < t <= 16 340 235
16 < t <= 40 340 225
40 < t <= 63 340 215
63 < t <= 80 340 215
80 < t <=100
340 215
S275, S 275 0 < t <= 16 410 275
16 < t <= 40 410 265
40 < t <= 63 410 255
63 < t <= 80 410 245
80 < t <=100
410 235
S355, S 355 0 < t <= 16 490 355
16 < t <= 40 490 345
40 < t <= 63 490 335
63 < t <= 80 490 325
80 < t <=100
490 315
S420, S 420 0 < t <= 16 500 420
16 < t <= 40 500 400
40 < t <= 63 500 390
S460, S 460 0 < t <= 16 530 460
16 < t <= 40 530 440
40 < t <= 63 530 430
S500, S 500 0 < t <= 50 550 500
50 < t <= 100
550 480
S550, S 550 0 < t <= 50 640 550
50 < t <= 100
640 550
S620, S 620 0 < t <= 50 700 620
50 < t <= 100
700 580
S690, S 690 0 < t <= 50 770 690
50 < t <= 100
760 650
Scia Engineer Steel Code Check Theoretical Background
218
Remark : For cold formed section, the values for fy and fu are not influenced by the previous table.
Remark : The reduction rules from previous table are only valid when the used material is defined as material for the selected code.
Consulted articles
The cross-section is classified according to Table 6:211a . (class 1,2 or 3).
The section is checked for tension (art. 6:22), compression (6:23), bending (6:24), shear force (art. 6:26), torsion (art. 6:27), the combination of bending and axial force (art. 6:25).
A more detailed overview for the used articles is given for part 6:2 in the following table. The chapters marked with “x” are consulted. The chapters marked with (*) have a supplementary explanation in the following chapters.
6:2.Calculation of the capacity of construction elements
6:21.Limiting values of slenderness for cross section parts x
6:211.Classes of cross sections x (*)
6:212.Design methods for the different section classes x (*)
6:22.Tensile force x
6:23.Compression force x
6:231. Initial curvature, initial inclination and load eccentricity
6:232.Loss of restraint x (*)
6:233.Reduction factor for flexural buckling x
6:24.Bending moment x
6:241.Cross section classes x (*)
6:242.Shape factors in flexure x (*)
6:243.Bending moment x
6:244.Lateral torsional buckling x (*)
6:2441.Lateral bracing of beam x
6:2442.Reduction factor for LTB x
6:25. Bending and axial force
6:251.Section check x
6:252.Flexural buckling x
6:253.Flexural-torsional buckling x
6:26.Shear force and concentrated load
6:261.Shear force x(*)
6:262.Web crippling under concentrated force
6:263.Local compression
6:27.Torsional moment x
6:271.Pure torsion x
6:272.Warping
6:273.Torsional moment, shear force and bending moment x
Scia Engineer Steel Code Check Theoretical Background
219
Classification of sections
For each intermediary section, the classification is determined and the proper section check is performed using the actual internal forces. The classification can change for each intermediary point.
Effective cross-section properties for class 3 cross-section
The calculation of the effective area properties is performed according to the rules given in [5], part :23 and :24.
For each intermediary section, the classification (and if necessary, the effective area ) is determined and the proper section check is performed. The classification (and effective area) can change for each intermediary point. The most critical check is displayed on the screen.
Aeff is the effective area of the cross section when subject to uniform compression. Weff is the effective section modulus of the cross-section when subject only to moment about the relevant axis. With these properties, the section and stability check is performed.
Section properties
6:22 ; 6:243 ; 6:251 ; 6:261 : The net area properties are not taken into account .
Section check
- Double symmetric I sections (I) use the formula (6:251a) and (6:251b)
- Solid sections (O, RS) and hollow sections (RHS, CHS) use the formula (6:251c)
- For single bending, the sections U, PPL, T use formula (6:251a). For double bending the biaxial state of stress is consulted.
- All other cases use the biaxial state of stress.
The (bi)axial stress check is given by formula (3:412a) and (3:412c):
yd
22
x
ydx
f3
f
with =1.1
Compression members
6:232 : For the calculation of the buckling length, we refer to "Calculation of buckling ratio". The buckling properties for a VARH element are calculated by using the critical Euler force for this member (see "Calculation of critical Euler force for VARH elements") .
For class 3 sections, the rules given in [5], part :34 are used, including the calculating of Idef.
Scia Engineer Steel Code Check Theoretical Background
220
Stability check for torsional buckling and torsional-flexural buckling
See [5], part :37.
The design buckling resistance for torsional or torsional-flexural buckling shall be obtained using the
following reduction factor c and slenderness c :
with fyk the basic yield strength
cr the critical stress
cr,T the elastic critical stress for torsional buckling
cr,TF the elastic critical stress for torsional-flexural buckling
G the shear modulus
E the modulus of elasticity
IT the torsion constant of the gross section
CM the warping constant
iy the radius of gyration about yy-axis
iz the radius of gyration about zz-axis
lT the buckling length of the member for torsional buckling
y0 the position of the shear center
ly the buckling length for flexural buckling about the yy-axis
Scia Engineer Steel Code Check Theoretical Background
221
The calculation of cr based on [6], part 6.2.3.(5).
Lateral-torsional buckling
Alternatively to the regulations given in 6:2442. for bisymmetric sections, the elastic critical moment for LTB Mcr for I sections (symmetric and asymmetric), RHS (Rectangular Hollow Section) sections and CHS (Circular Hollow Section) sections, can be calculated using the formula given by the
general formula F.2. Annex F Ref. 3.
For the calculation of the moment factors C1, C2 and C3 we refer to "Calculation of moment factors for LTB".
For the other supported sections, the elastic critical moment for LTB Mcr is given by
z2
t
z2
z2
EI
L²GI
I
Iw
L
EIMcr
with E the modulus of elasticity
G the shear modulus
L the length of the beam between points which have lateral restraint (= lLTB)
Iw the warping constant
It the torsional constant
Iz the moment of inertia about the minor axis
Scia Engineer Steel Code Check Theoretical Background
222
See also Ref. 4, part 7 and in particular part 7.7. for channel sections.
For class 3 section, Izdef according to [5], part :44 is used.
Haunched sections (I+Ivar, Iw+Plvar, Iw+Iwvar, Iw+Ivar, I+Iwvar) and composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
Use of diaphragms
See Chapter 'Adaptation of torsional constant'.
Shear force ( shear buckling)
The shear buckling check is using the values for v from table 6:261 in column 2.
The value for w is (according to [5], part :26, (18:26d)) taken as below :
2
w
w
2
w
w
k
yk
w
ww
a
b34.500.4k1
b
aif
a
b00.434.5k1
b
aif
E
f
t
b
k
81.0
with Ek the modulus of elasticity
fyk the yield strength
a the field length
bw the field height
tw the web thickness
Scia Engineer Steel Code Check Theoretical Background
223
a
bw
Composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
Supported sections
I Symmetric I shapes (IPE, HEA, HEB, ….)
RHS Rectangular Hollow Section
CHS Circular Hollow Section
L Angle section
U Channel section
T T section
PPL Asymmetric I shapes
RS Rectangular section
Cold formed section
COM Composed section in PRIMAWIN
O Solid tube
NUM Numerical section
The necessary data conditions for these sections are described in chapter "Profile conditions for code check
". The COM and NUM sections are not read out of the profile library.
I
RHS
CHS
L
U
T
PPL
RS
O
COM
NUM
Classification x x x x x x x x x (1) (1) (1)
Section check
double bending
x x x x x x x x x x x x
Class 3 support x x x x x x
Buck:ling check x x x x x x x x x x x x
LTB check x x x x x x x x x x x x
Compression + x
Scia Engineer Steel Code Check Theoretical Background
224
bending
double bending
Compression + bending
single bending
x x x x x x x x
Compression + LTB
double bending
x
Shear buckling x x x x
Torsional check x
(1) sections are classified as class 2 cross section by default.
References
[1] BSK 99
StalKonstruktioner
Boverket, Byggavdelningen, 2000
[2] Swedish Regulations for Steel Structures
BSK
SBI Swedish Institute of Steel Construction, Publication 118, 1989
[3] Eurocode 3
Design of steel structures
Part 1 - 1 : General rules and rules for buildings
ENV 1993-1-1:1992, 1992
4 R. Maquoi
ELEMENTS DE CONSTRUCTIONS METALLIQUE
Ulg , Faculté des Sciences Appliquées, 1988
[5] Torsten Höglund
K18, Dimensionering av Stalkonstruktioner
Utdrag ur Handboken Bygg, kapitel K18 och K19
C E Fritzes AB, Stockholm
[6] ENV 1993-1-3:1996
Eurocode 3 : Design of steel structures
Part 1-3 : General rules
Supplementary rules for cold formed thin gauge members and sheeting
CEN 1996
Scia Engineer Steel Code Check Theoretical Background
225
IS 800
IS:800 Code check
The beam elements are checked according to the regulations given in
IS 800 Draft version (for 3rd
Revision)
Material properties
The following steel grades are supported :
Grade/ Classification Yield stress(Mpa) Ultimate tensile stress(Mpa)
A/Fe410WA 250(<20mm), 240(20mm to 40mm), 230(>40mm) 410
B/Fe410WB 250(<20mm), 240(20mm to 40mm), 230(>40mm) 410
C/Fe410WC 250(<20mm), 240(20mm to 40mm), 230(>40mm) 410
Fe440 300(<16mm), 290(16mm to 40mm), 280(>41mm to 63mm)
440
Fe440B 300(<16mm), 290(16mm to 40mm), 280(>41mm to 63mm)
440
Fe490 350(<16mm), 330(16mm to 40mm), 320(>41mm to 63mm)
490
Fe490B 350(<16mm), 330(16mm to 40mm), 320(>41mm to 63mm)
490
Fe540 410(<16mm), 390(16mm to 40mm), 380(>41mm to 63mm)
540
Fe540B 410(<16mm), 390(16mm to 40mm), 380(>41mm to 63mm)
540
The string in the column „Grade/Classification‟ is used to determine the proper yield stress reduction.
Consulted articles
The cross-section is classified according to Table 3.1.
The section is checked for tension (Section 6), compression (Section 7), bending (Section 8) and the combination of forces (Section 9).
Scia Engineer Steel Code Check Theoretical Background
226
A more detailed overview for the used articles is given in the following table. The chapters marked with “x” are consulted. The chapters marked with (*) have a supplementary explanation in the following chapters.
3.7. Classification of Cross Section x(*)
6.1. Tension members x
6.2. Design strength due to Yielding of Gross section
7.1. Design Strength x
8.2. Design strength in bending x
8.2.1. Laterally supported beam
8.2.1.1. Section with slender webs x
8.2.1.2. When factored shear force < 0.6 Vd x
8.2.1.3. When factored shear force > 0.6 Vd x
8.2.2. Laterally unsupported beam x
8.2.2.1. Elastic Lateral Torsional Buckling moment x
8.4. Shear x
8.4.1. The nominal plastic shear resistance x
8.4.2. Resistance to shear buckling x
9.1. General x
9.2. Combined Shear and bending x
9.3. Combined Axial Force and Bending Moment x
Appendix F x
Remarks
- the design of slender compression elements is outside the scope of this implementation
- the shear buckling check is only using the Simple Post Critical Method
Classification of sections
For each intermediary section, the classification is determined and the proper section check is performed. The classification can change for each intermediary point.
For each load case/combination, the critical section classification over the member is used to perform the stability check. So, the stability section classification can change for each load case/combination.
However, for non-prismatic sections, the stability section classification is determined for each intermediary section
Scia Engineer Steel Code Check Theoretical Background
227
The cross sections are classified as
- class 1 : plastic
- class 2 : compact
- class 3 : semi-compact
- class 4 : slender section
The class 4 (slender) section check is not supported. For this sections a class 3 (semi-compact) section check is performed.
Section properties
The net area properties are not taken into account .
Section check
In the case of high shear for class 3 section, the allowable normal stress is reduced with a factor (1-
). When torsional shear stress is present, the VonMisis criterium is checked.
Compression members
For the calculation of the buckling length, we refer to "Calculation of buckling ratio". The buckling properties for a VARH element are calculated by using the critical Euler force for this member (see "Calculation of critical Euler force for VARH elements") .
Stability check for torsional buckling and torsional-flexural buckling
The design buckling resistance Nb,Rd for torsional or torsional-flexural buckling shall be obtained using buckling for buckling around the weak axis, and with relative slenderness given by :
Scia Engineer Steel Code Check Theoretical Background
228
with fyb the basic yield strength
cr the critical stress
cr,T the elastic critical stress for torsional buckling
cr,TF the elastic critical stress for torsional-flexural buckling
G the shear modulus
E the modulus of elasticity
IT the torsion constant of the gross section
CM the warping constant
iy the radius of gyration about yy-axis
iz the radius of gyration about zz-axis
lT the buckling length of the member for torsional buckling
y0 the position of the shear center
ly the buckling length for flexural buckling about the yy-axis
Scia Engineer Steel Code Check Theoretical Background
229
Lateral-torsional buckling
The elastic critical moment for LTB Mcr for I sections (symmetric and asymmetric), RHS (Rectangular Hollow Section) sections and CHS (Circular Hollow Section) sections, can be calculated using the formula given by Annex F.
For the calculation of the moment factors C1, C2 and C3 we refer to "Calculation of moment factors for LTB".
For the other supported sections, the elastic critical moment for LTB Mcr is given by
z2
t
z2
z2
EI
L²GI
I
Iw
L
EIMcr
with E the modulus of elasticity
G the shear modulus
L the length of the beam between points which have lateral restraint (= lLTB)
Iw the warping constant
It the torsional constant
Iz the moment of inertia about the minor axis
Haunched sections (I+Ivar, Iw+Plvar, Iw+Iwvar, Iw+Ivar, I+Iwvar) and composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
For advanced Lateral-torsional buckling analysis, see chapter „LTBII: Lateral Torsional Buckling 2nd Order Analysis‟.
Scia Engineer Steel Code Check Theoretical Background
230
Use of diaphragms
See Chapter 'Adaptation of torsional constant'.
Supported sections
The following standard sections are defined :
I Symmetric I shapes (IPE, HEA, HEB, ….)
RHS Rectangular Hollow Section
CHS Circular Hollow Section
L Angle section
U Channel section
T T section
PPL Asymmetric I shapes
Z Z section
RS Rectangular section
Cold formed section
COM Composed section ( sheet welded, section pairs, …)
O Solid tube
NUM Numerical section
The necessary data conditions for these sections are described in chapter "Profile conditions for code check
". The COM and NUM sections are not read out of the profile library.
In the following matrix is shown which sections are supported for the different analysis parts in the Indian steel Code check :
I
RHS
CHS
L
U
T
PPL
RS
Z
O
COM
NUM
Section Classification x x x x x x x x (1) x (1) (1) (1)
Section check class 1 x x x
Section check class 2 x x x
Section check class 3 x x x x x x x x x x x x x
Section check class 4
Stability check class 1 x x x
Stability check class 2 x x x
Stability check class 3 x x x x x x x x x x x x x
Stability check class 4
Shear buckling check x x x
(1) sections are classified as class 3 cross section by default.
Scia Engineer Steel Code Check Theoretical Background
231
References
[1] IS:800
2005
Scia Engineer Steel Code Check Theoretical Background
232
EAE code check
The beam elements are checked according to the regulations given in
Instrucción EAE
Documento 0 de la Instrucción de Acero Estructural
Comisión Permanente de Estructuras de Acero
November 2004
Material properties
For standard steel grades, the yield strength fy and tensile strength fu are defined according to
Capítulo VI of Ref. 1.
Steel Grade fy (N/mm²)
fu (N/mm²)
S 235 235 360
S 275 275 430
S 355 355 510
S 275 N/NL 275 390
S 355 N/NL 355 490
S 420 N/NL 420 540
S 460 N/NL 460 570
S 275 M/ML 275 380
S 355 M/ML 355 470
S 420 M/ML 420 520
S 460 M/ML 460 550
S 460 Q/QL/QL1
460 570
S 235 W 235 360
S 355 W 355 510
S 235 H 235 360
S 275 H 275 430
S 355 H 355 510
S 275 NH/NLH 275 370
S 355 NH/NLH 355 470
S 460 NH/NLH 460 550
S 275 MH/MLH 275 360
S 355 MH/MLH 355 470
S 420 MH/MLH 420 500
S 460 MH/MLH 460 530
Scia Engineer Steel Code Check Theoretical Background
233
The name of the steel grade (e.g. 'S 355 W') is used to identify the steel grade.
Remark : For cold formed section, the values for fy and fu are not influenced by the previous table
Remark : For cold formed sections, the average yield strength fya can be used (by setting the proper data flag in the Cross Section input dialog) according to Ref.[4].
The average yield strength is determined as follows :
with fyb the tensile yield strength = fy
fu the tensile ultimate strength
t the material thickness
Ag the gross cross-sectional area
k is a coefficient depending on the type of forming :
k = 0.7 for cold rolling
k = 0.5 for other methods of forming
n the number of 90° bends in the section
Consulted articles
The beam elements are checked according to the regulations given in " Instrucción EAE, Documento 0 de la Instrucción de Acero Estructural, Comisión Permanente de Estructuras de Acero, November 2004".
The cross-sections are classified according to Artículo 20 of Capítulo V. All classes of cross-sections are included. For class 4 sections (slender sections) the effective section is calculated in each intermediary point, according to Artículo 20 of Capítulo V.
The member check is executed according to Capítulo IX. The stress check is taken from art. 34.: the section is checked for tension (art. 34.2.), compression (art. 34.3.), bending (art. 34.4.), shear (art. 34.5.), torsion (art. 34.6.) and combined bending, shear and axial force (art. 34.7.1., art. 34.7.2. and art. 34.7.3.).
The stability check is taken from art. 35.: the beam element is checked for buckling (art. 35.1.), lateral torsional buckling (art. 35.2.), and combined bending and axial compression (art. 35.3.).
The shear buckling is checked according to prEN 1993-1-5:2003, Chapter 5.
For I sections, U sections and cold formed sections warping can be considered.
A check for critical slenderness and torsion moment is also included.
For integrated beams, the local plate bending is taken into account for the plastic moment capacity and the bending stresses in the section. The out-of-balance loading is checked.
Scia Engineer Steel Code Check Theoretical Background
234
A more detailed overview for the used articles is given in the following table. The chapters marked with “x” are consulted. The chapters marked with (*) have a supplementary explanation the following chapters.
Instrucción EAE
20. Clasificación de las secciones transversales (*)
20.2. Clasificación de las secciones transversales metálicas x
20.3. Criterios de asignación de Clase en secciones metálicas no rigidizadas x
20.7. Características de la sección reducida en secciones transversales esbeltas
x
34. Estado límite de resistencia de las secciones
34.1. Principios generales del cálculo x
34.1.2. Características de las secciones transversales x(*)
34.2. Esfuerzo axil de tracción x
34.3. Esfuerzo axil de compresión x
34.4. Momento flector x
34.5. Esfuerzo cortante x
34.6. Torsión x(*)
34.7. Interacción de esfuerzos
34.7.1. Flexión y cortante x
34.7.2. Flexión y esfuerzo axil x
34.7.3. Flexión, cortante y esfuerzo axil x
35. Estado límite de inestabilidad
35.1. Elementos sometidos a compresión x(*)
35.2. Elementos sometidos a flexión x
35.3. Elementos sometidos a compresión y flexión x(*)
35.5. Abolladura del alma a cortante x
35.7. Interacción
35.7.1. Cortante, flexión y esfuerzo axil x
For cold formed sections EN 1993-1-3 is applied.
6.1.2. Axial tension x
6.1.3. Axial compression x
6.1.5. Shear force x
6.1.6. Torsional moment x
Scia Engineer Steel Code Check Theoretical Background
235
Classification of sections
For each intermediary section, the classification is determined and the proper section check is performed. The classification can change for each intermediary point.
For each load case/combination, the critical section classification over the member is used to perform the stability check. So, the stability section classification can change for each load case/combination.
However, for non-prismatic sections, the stability section classification is determined for each intermediary section.
Effective cross-section properties for class 4 cross-section
The calculation of the effective area is performed with the direct method (sigma_d = fy,k).
For each intermediary section, the classification (and if necessary, the effective area ) is determined and the proper section check is performed. The classification (and effective area) can change for each intermediary point. The most critical check is displayed on the screen.
For each load case and combination, the most critical effective area properties are saved :
Aeff is the effective area of the cross section when subject to uniform compression. Weff is the effective section modulus of the cross-section when subject only to moment about the relevant axis. eN is the shift of the relevant centroidal axis when the cross section is subject to uniform compression.
With these critical properties, the stability check is performed.
For non-prismatic elements, the effective area properties are calculated on each intermediary section, also for the stability check.
Section properties
The net area properties are not taken into account .
The shear lag effects are neglected .
Torsion check
For the cross section check inclusive torsion and warping, we refer to Chapter 'Warping check'.
Built-in beams
For built-in beam sections (IFB, SFB, THQ sections), proper section checks are performed, taking into account the local plate bending. See Chapter „Section check for built-in beams (IFB, SFB, THQ sections)‟.
Scia Engineer Steel Code Check Theoretical Background
236
Compression members
For the calculation of the buckling length, we refer to chapter "Calculation of buckling ratio"
The buckling properties for a VARH element are calculated by using the critical Euler force for this member (see chapter “Calculation of critical Euler force for VARH elements”).
Lateral-torsional buckling
For I sections (symmetric and asymmetric), RHS (Rectangular Hollow Section) sections and CHS (Circular Hollow Section) sections, the elastic critical moment for LTB Mcr is given by the general
formula F.2. Annex F Ref. 5. For the calculation of the moment factors C1, C2 and C3 we refer to "Calculation of moment factors for LTB".
For the other supported sections, the elastic critical moment for LTB Mcr is given by
z2
t
z2
z2
EI
L²GI
I
Iw
L
EIMcr
with E the modulus of elasticity
G the shear modulus
L the length of the beam between points which have lateral restraint (= lLTB)
Iw the warping constant
It the torsional constant
Iz the moment of inertia about the minor axis
See also Ref. 3, part 7 and in particular part 7.7. for channel sections.
Haunched sections (I+Ivar, Iw+Plvar, Iw+Iwvar, Iw+Ivar, I+Iwvar) and composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
For advanced Lateral-torsional buckling analysis, see chapter „LTBII: Lateral Torsional Buckling 2nd Order Analysis‟.
Use of diaphragms
See Chapter 'Adaptation of torsional constant'.
Scia Engineer Steel Code Check Theoretical Background
237
Combined bending and axial compression
For prismatic members the value My,Ed is the maximum value of the bending moment around the strong axis in the member. The value Mz,Ed is the maximum value of the bending moment around the weak axis in the member.
For non-prismatic sections, the values My,Ed and Mz,Ed are the concurrent bending moments for each intermediary section.
Interaction Method Calculation of Czz
By default for Czz the formula given in Ref.[1] is used:
In this formula however the position of the factor eLT is incorrect. For exact analysis the formula according to Ref.[9] can be used:
Shear buckling check
Composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.
Supported sections
I Symmetric I shapes (IPE, HEA, HEB, ….)
RHS Rectangular Hollow Section
CHS Circular Hollow Section
L Angle section
U Channel section
T T section
PPL Asymmetric I shapes
Z Z section
RS Rectangular section
Cold formed section
COM Composed section in PRIMAWIN
O Solid tube
NUM Numerical section
The necessary data conditions for these sections are described in chapter "Profile conditions for code check
Scia Engineer Steel Code Check Theoretical Background
238
". The COM and NUM sections are not read out of the profile library.
I
RHS
CHS
L
U
T
PPL
RS
Z
O
COM
NUM
Classification x x x x x x x x (1) x (1) (1) (1)
Section check class 1 x x x
Section check class 2 x x x
Section check class 3 x x x x x x x x x x x x x
Section check class 4 x x x x x x
Stability check class 1
x x x
Stability check class 2
x x x
Stability check class 3
x x x x x x x x x x x x x
Stability check class 4
x x x x x x
Shear buckling check x x x x
(1) Sections are classified as class 3 cross section by default.
References
1 Instrucción EAE
Documento 0 de la Instrucción de Acero Estructural
Comisión Permanente de Estructuras de Acero
November 2004
2 Essentials of Eurocode 3
Design Manual for Steel Structures in Building
ECCS - N° 65, 1991
3 R. Maquoi
ELEMENTS DE CONSTRUCTIONS METALLIQUE
Ulg , Faculté des Sciences Appliquées, 1988
[4] ENV 1993-1-3:1996
Eurocode 3 : Design of steel structures
Part 1-3 : General rules
Supplementary rules for cold formed thin gauge members and sheeting
CEN 1996
[5] Eurocode 3
Design of steel structures
Part 1 - 1/ A1 : General rules and rules for buildings
ENV 1993-1-1:1992/A1, 1994
Scia Engineer Steel Code Check Theoretical Background
239
[6] Eurocode 3
Design of steel structures
Part 1 - 2 : General rules - Structural fire design
ENV 1993-1-2:1995, 1995
[7] Model Code on Fire Engineering
ECCS - N° 111
May 2001
[8] Eurocode 1
Basis of design and actions on structures
Part 2-2 : Actions on structures - Actions on structures exposed to fire
ENV 1991-2-2:1995
Scia Engineer Steel Code Check Theoretical Background
240
Calculation of buckling ratio
Introduction to the calculation of buckling ratio
For the calculation of buckling ratio, several methods can be applied.
The general method is described in chapter "Calculation buckling ratio – general formula".
For crossing diagonals, the buckling ratio is explained in chapter "Calculation buckling ratios for crossing diagonals".
For VARH elements, the critical Euler force is calculated according to the method given in chapter "Calculation of critical Euler force for VARH elements".
For lattice tower members, see the chapter "Calculation buckling ratio for lattice tower members".
When using member buckling data the buckling ratio can be calculated from a stability analysis. See chapter Calculation of buckling ratio – From Stability Analysis.
Calculation buckling ratio – general formula
For the calculation of the buckling ratios, some approximate formulas are used. These formulas are treated in reference [1], [2] and [3].
The following formulas are used for the buckling ratios (Ref[1],pp.21) :
for a non sway structure :
24)+11+5+24)(2+5+11+(2
12)2+4+4+24)(+5+5+(=l/L
21212121
21212121
for a sway structure :
4+x
x=l/L1
2
with L the system length
E the modulus of Young
I the moment of inertia
Ci the stiffness in node i
Mi the moment in node i
Fi the rotation in node i
Scia Engineer Steel Code Check Theoretical Background
241
21212
12
21
8+)+(
+4=x
EI
LC= i
i
i
ii
M=C
The values for Mi and i are approximately determined by the internal forces and the deformations, calculated by load cases which generate deformation forms, having an affinity with the buckling form. (See also Ref.[5], pp.113 and Ref.[6],pp.112).
The following load cases are considered:
load case 1 : on the beams, the local distributed loads qy=1 N/m and qz=-100 N/m are used, on the columns the global distributed loads Qx = 10000 N/m and Qy =10000 N/m are used.
load case 2 : on the beams, the local distributed loads qy=-1 N/m and qz=-100 N/m are used, on the columns the global distributed loads Qx = -10000 N/m and Qy= -10000 N/m are used.
The used approach gives good results for frame structures with perpendicular rigid or semi-rigid beam connections. For other cases, the user has to evaluate the presented bucking ratios. In such cases a more refined approach (from stability analysis) can be applied.
The following rule applies specifically to ky: in case both the calculation for load case 1 and load case 2 return ky = 1,00 then ky is taken as kz. This rule is used to account for possible rotations of the cross-section.
Scia Engineer Steel Code Check Theoretical Background
242
Calculation buckling ratios for crossing diagonals
For crossing diagonal elements, the buckling length perpendicular to the diagonal plane, is calculated according to Ref.[4], DIN18800 Teil 2, table 15. This means that the buckling length sK is dependent on the load distribution in the element, and it is not a purely geometrical data anymore.
In the following chapters, the buckling length sK is defined,
with sK buckling length
l member length
l1 length of supporting diagonal
I moment of inertia (in the buckling plane) of the member
I1 moment of inertia (in the buckling plane) of the supporting diagonal
N compression force in member
N1 compression force in supporting diagonal
Z tension force in supporting diagonal
E elastic modulus
Continuous compression diagonal, supported by continuous tension diagonal
NN
Z
Z
l/2
l1/2
l5.0s
lI
l1I1
lN4
lZ31
ls
K
3
1
3
1
K
See Ref.[4], Tab. 15, case 1.
Scia Engineer Steel Code Check Theoretical Background
243
Continuous compression diagonal, supported by pinned tension diagonal
NN
Z
Z
l/2
l1/2
l5.0s
lN
lZ75.01ls
K
1
K
See Ref.[4], Tab. 15, case 4.
Pinned compression diagonal, supported by continuous tension diagonal
NN
Z
Z
l/2
l1/2
)1lZ
lN(
4
lZ3)IE(
1lZ
lN
l5.0s
1
2
2
1
d1
1
K
See Ref.[4], Tab. 15, case 5.
Scia Engineer Steel Code Check Theoretical Background
244
Continuous compression diagonal, supported by continuous compression diagonal
NN
N1
N1
l/2
l1/2
l5.0s
lI
l1I1
lN
lN1
ls
K
3
1
3
1
1
K
See Ref.[4], Tab. 15, case 2.
Continuous compression diagonal, supported by pinned compression diagonal
NN
N1
N1
l/2
l1/2
1
1
2
KlN
lN
121ls
See Ref.[4], Tab. 15, case 3 (2).
Scia Engineer Steel Code Check Theoretical Background
245
Pinned compression diagonal, supported by continuous compression diagonal
NN
N1
N1
l/2
l1/2
)N
lN
12(
l
lN)IE(
l5.0s
1
1
2
1
2
3
d
K
See Ref.[4], Tab. 15, case 3 (3).
Scia Engineer Steel Code Check Theoretical Background
246
Calculation of critical Euler force for VARH elements
Definitions
A VARH element is defined as follows :
The member has the properties of a symmetric I secion (formcode=1), where only the height is linear variable along the member. The system length for buckling around the local yy axis (strong axis), is equal to member length.
For this non-prismatic section, the critical Euler force is given in Ref[7].
Calculation of the critical Euler force
For a VARH element (form node i to node j), we can define:
L beam length
Ii, Ij moment of inertia at end i and j
Ai, Aj
sectional area at end i and j
E modulus of Young
Ncr critical Euler force
Ri, Rj
beam stiffness at end i and j
The stiffness R and R' is given by:
EI
LR=R
EI
LR=R
M=R
ijj
iii
I
I=
i
j
The critical Euler force is given by
L
EI=N 2
i2cr
To calculate , the next steps are followed :
Scia Engineer Steel Code Check Theoretical Background
247
1. Calculate L, Ii, Ij, Ri, Rj, R'i, R'j, ξ
2. We suppose that
2
1>
1-
3. Calculate a, b, c and d as follows
)]lncotg(+2
1(
1)-(+[1
1=d
]1-
)ln(sin-[1
1=c=b
)]lncotg(-2
11)(-(+[1
1=a
4
1-
)1-(=
2
2
2
2
2
4. For a beam in non-sway system, we solve
0=RRbc)-(ad+Rd+Ra+1 jiji
For a beam in sway system, we solve
0=bc))-(ad-d+c-b-(aRR+-)d-(1R+)a-(1R2
ji22
j2
i
5. When a solution is found, we check if
2
1>
1-
6. If not, then recalculate a,b,c en d as follows :
Scia Engineer Steel Code Check Theoretical Background
248
])-(
))+2
1(-)-
2
11)((-(
+[11
=d
])-(
1)-(2-[1
1=c=b
]-
))+2
1(-)-
2
11)((-(
+[11
=a
-
-
2
-2
-
2
and resolve the proper equation of 4.
Calculation buckling ratio for lattice tower members When the national code EC-ENV is selected, the following buckling configuration can be selected.
For each configuration, the critical slenderness to be considered, is defined.
The values are taken from Ref.[8].
y
y
z z
v
v
We define :
iyy radius of gyration around yy axis
izz radius of gyration around zz axis
ivv radius of gyration around vv axis
With the option 'Bracing members are sufficiently supported', the effective slenderness may be reduced as follows:
- for vv-axis : vv7.035.0
- for yy-axis : yy7.050.0
The buckling curve 'b' is used.
Scia Engineer Steel Code Check Theoretical Background
249
Leg with symmetrical bracing
vvi
L
Leg with intermediate transverse support
yyi
L
Scia Engineer Steel Code Check Theoretical Background
250
Leg with staggered bracing
vv
yy
i
52.1)2a,1amax(
i
L
Single Bracing
vvi
L
Scia Engineer Steel Code Check Theoretical Background
251
Single Bracing with SBS (Secondary Bracing System)
yy
2
vv
1
i
L
i
L
Cross bracing
yy
com
com
y
E
E
com
com
com
com
1
1
1b
1
com
sup
1b
2bcomb
'
2
zz
'
2
yy
'
2
vv
1
i
L
f
E
58.070.0K
L
L
K
1125.0K
0.15.0K
1125.0
F
F
K
1138.075.0K
LKLKL
i
L,
i
L
i
L
Scia Engineer Steel Code Check Theoretical Background
252
with Lcom Length of compressed member (L2 from figure)
Fcom Force in compressed member (L2 from figure)
Fsup Force in supporting member (member crossing member L2)
E Modulus of Young
fy Yield strength
Cross bracing with SBS
3bcomb
'
3
zz
'
3
yy
'
3
zz
2
yy
2
vv
1
LKLKL
i
L,
i
L
i
L,
i
L
i
L
with Lcom Length of compressed member (L3 from figure)
Fcom Force in compressed member (L3 from figure)
Fsup Force in supporting member (member crossing member L3)
Kb See Chapter 'Cross bracing'
Scia Engineer Steel Code Check Theoretical Background
253
K Bracing
zz
3
yy
3
zz
2
yy
2
vv
1
i
L,
i
L
i
L,
i
L
i
L
Horizontal Bracing
L
1R0
P
PR
73.0R316.0R085.0k
i
Lk
1
2
2
vv
with P1 Compression load
P2 Tensile load
Scia Engineer Steel Code Check Theoretical Background
254
Horizontal Bracing with SBS
L
1R0
P
PR
73.0R316.0R085.0k
i
Lk
1
2
2
yy
with P1 Compression load
P2 Tensile load
Scia Engineer Steel Code Check Theoretical Background
255
Discontinuous Cross bracing with horizontal member
N1 N2
N1N2
F F
a
a
cos)2N1N(,FmaxF
i
a,
i
a2
Sd
vvyy
with F normal force to check
FSd actual compression force in horizontal member
N1 tensile force in diagonal
N2 compression force in diagonal
Scia Engineer Steel Code Check Theoretical Background
256
Calculation of buckling ratio – From Stability Analysis When member buckling data from stability are defined, the critical buckling load Ncr for a prismatic member is calculated as follows:
Edcr NN
Using Euler‟s formula, the buckling ratio k can then be determined:
With: Critical load factor for the selected stability combination
NEd Design loading in the member
E Modulus of Young
I Moment of inertia
s Member length
In case of a non-prismatic member, the moment of inertia is taken in the middle of the element.
References
[1] Handleiding moduul STACO VGI
Staalbouwkundig Genootschap
Staalcentrum Nederland
5684/82
[2] Newmark N.M. A simple approximate formula for effective end-fixity of columns
J.Aero.Sc. Vol.16 Feb.1949 pp.116
[3] Stabiliteit voor de staalconstructeur
uitgave Staalbouwkundig Genootschap
[4] DIN18800 Teil 2
Stahlbauten : Stabilitätsfälle, Knicken von Stäben und Stabwerken
November 1990
[5] Rapportnr. BI-87-20/63.4.3360
Controleregels voor lijnvormige constructie-elementen
IBBC Maart 1987
Scia Engineer Steel Code Check Theoretical Background
257
[6] Staalconstructies TGB 1990
Basiseisen en basisrekenregels voor overwegend statisch belaste constructies
NEN 6770, december 1991
[7] Y. Galéa
Flambement des poteaux à inertie variable
Construction Métallique 1-1981
[8] NEN-EN 50341-3-15
Overhead electrical lines exceeding AC 45 kV - Part 3: Set of National Normative Aspects
Number 15: National Normative Aspects (NNA) for The Netherlands
Scia Engineer Steel Code Check Theoretical Background
258
Calculation of moment factors for LTB
Introduction to the calculation of moment factors
For determining the moment factors C1 and C2 for lateral torsional buckling (LTB), we use the standard tables which are defined in Ref.[1] Art.12.25.3 table 9.1.,10 and 11.
The current moment distribution is compared with several standard moment distributions. These standard moment distributions are moment lines generated by a distributed q load, a nodal F load, or where the moment line is maximum at the start or at the end of the beam.
The standard moment distribution which is closest to the current moment distribution, is taken for the calculation of the factors C1 and C2.
The factor C3 is taken out of the tables F.1.1. and F.1.2. from Ref.[2] - Annex F.
Calculation moment factors
Moment distribution generated by q load
For ENV 1993, IS800 and CM66
if M2 < 0
C1 = A* (1.45 B
* + 1) 1.13 + B
* (-0.71 A
* + 1) E
*
C2 = 0.45 A* [1 + C* eD*
(½ + ½)]
if M2 > 0
C1 = 1.13 A* + B
* E
*
C2 = 0.45A*
Scia Engineer Steel Code Check Theoretical Background
259
For DIN18800 and ONORM4300
if M2 < 0
C1 = A* (1.45 B
* + 1) 1.12 + B
* (-0.71 A
* + 1) E
*
C2 = 0.45 A* [1 + C
* e
D* (½ + ½)]
if M2 > 0
C1 = 1.12 A* + B
* E
*
C2 = 0.45A*
with : l+q|M2|8
lq=A
2
2*
ql
|M2|94=C
2
*
l+q|M2|8
|M2|8=B
2
* )
ql
|M2|-72(=D
2
2
*
For DIN18800 / ONORM 4300
0.77-1.77=E*
For ENV 1993 and IS800
2.70<E*
0.52+1.40-1.88=E*2
For NEN6770/6771, SIA263
E*=1.75-1.05*+0.30*² and E*<2.3
For CM66
2.70<E*
0.52+1.40-1.88=E*2
Scia Engineer Steel Code Check Theoretical Background
260
Moment distribution generated by F load
FM2 M1 = Beta M2
l
M2 < 0
C1 = A** (2.75 B
** + 1) 1.35 + B
** (-1.62 A
** + 1) E
**
C2 = 0.55 A** [1 + C
** e
D** (½ + ½)]
M2 > 0
C1 = 1.35 A** + B
** E
**
C2 = 0.55 A**
with : +Fl|M2|4
Fl=A **
+Fl|M2|4
|M2|4=**B
Fl
|M2|38=C **
)Fl
|M2|-32(=D
2**
The values for E** can be taken as E
* from chapter "Moment distribution generated by q load".
Scia Engineer Steel Code Check Theoretical Background
261
Moment line with maximum at the start or at the end of the beam
M2 M1 = Beta M2
l
C2 = 0.0
For DIN18800 / ONORM 4300
0.77-1.77=1C
For ENV 1993 / IS800
2.70<1C and
0.52+1.40-1.88=1C2
For CM66
For NEN6770/6771, SIA263 Code
E*=1.75-1.05*+0.30*² and E*<2.3
References
[1] Staalconstructies TGB 1990
Stabiliteit
NEN 6771 - 1991
[2] Eurocode 3 : Design of steel structures
Part 1-1 : General rules and rules for buildings
ENV 1993-1-1:1992
Scia Engineer Steel Code Check Theoretical Background
262
LTBII: Lateral Torsional Buckling 2nd Order Analysis
Introduction to LTBII
For a detailed Lateral Torsional Buckling analysis, a link was made to the Friedrich + Lochner LTBII application Ref.[1].
The FriLo LTBII solver can be used in 2 separate ways:
1) Calculation of Mcr through eigenvalue solution
2) 2nd
Order calculation including torsional and warping effects
For both methods, the member under consideration is sent to the FriLo LTBII solver and the respective results are sent back to Scia Engineer.
A detailed overview of both methods is given in the following chapters.
Eigenvalue solution Mcr
The single element is taken out of the structure and considered as a single beam, with:
- Appropriate end conditions for torsion and warping
- End and begin forces
- Loadings
- Intermediate restraints (diaphragms, LTB restraints)
The end conditions for warping and torsion are defined as follows:
Cw_i Warping condition at end i (beginning of the member)
Cw_j Warping condition at end j (end of the member)
Ct_i Torsion condition at end i (beginning of the member)
Ct_j Torsion condition at end j (end of the member)
To take into account loading and stiffness of linked beams, see chapter “Linked Beams”.
For this system, the elastic critical moment Mcr for lateral torsional buckling can be analyzed as the solution of an eigenvalue problem:
Scia Engineer Steel Code Check Theoretical Background
263
With
Critical load factor
Ke Elastic linear stiffness matrix
Kg Geometrical stiffness matrix
For members with arbitrary sections, the critical moment can be obtained in each section, with: (See Ref.[3],pp.176)
With
Critical load factor
Myy Bending moment around the strong axis
Myy(x) Bending moment around the strong axis at position x
Mcr(x) Critical moment at position x
The calculated Mcr is then used in the Lateral Torsional Buckling check of Scia Engineer.
For more background information, reference is made to Ref[2].
0KK ge
)x(MxM
MmaxM
yycr
yycr
Scia Engineer Steel Code Check Theoretical Background
264
2nd
Order analysis
The single element is taken out of the structure and considered as a single beam, with:
o Appropriate end conditions for torsion and warping
o End and begin forces
o Loadings
o Intermediate restraints (diaphragms, LTB restraints)
o Imperfections
To take into account loading and stiffness of linked beams, see chapter “Linked Beams”.
For this system, the internal forces are calculated using a 2nd
Order 7 degrees of freedom calculation.
The calculated torsional and warping moments (St Venant torque Mxp, Warping torque Mxs and Bimoment Mw) are then used in the Stress check of Scia Engineer (See chapter “Warping Check – Stress Check”).
Specifically for this stress check, the following internal forces are used:
o Normal force from Scia Engineer
o Maximal shear forces from Scia Engineer / FriLo LTBII
o Maximal bending moments from Scia Engineer / FriLo LTBII
Since Lateral Torsional Buckling has been taken into account in this 2nd
Order stress check, it is no more required to execute a Lateral Torsional Buckling Check.
For more background information, reference is made to Ref[2].
Supported National Codes The following codes are supported for the analysis of Mcr.
- EC3 - ENV
- EC3 - EN
- DIN18800
- ONORM
- NEN
- SIA
- IS
- EAE
Scia Engineer Steel Code Check Theoretical Background
265
For the following national codes, the 2nd Order analysis approach is supported.
- EC3 - ENV
- EC3 - EN
- DIN18800
- ONORM
- NEN
- SIA
- EAE
Supported Sections
The following table shows which cross-section types are supported for which type of analysis:
FRILO LTBII CSS Scia Engineer CSS Eigenvalue
analysis
2nd
Order
analysis
Double T I section from library x x
Thin walled geometric I x x
Sheet welded Iw x x
Double T unequal IPY from library x x
Thin walled geometric asymmetric I x x
Haunched sections x x
Welded I+Tl x x
Sheet welded Iwn x x
HAT Section IFBA, IFBB x x
U cross section U section from library x x
Thin walled geometric U x x
Thin walled Cold formed from library x x
Cold formed from graphical input x x
Double T with top flange angle
Welded I+2L x
Sheet welded Iw+2L x
Rectangle Full rectangular from library x
Full rectangular from thin walled geometric
x
Static values double symmetric
all other double symmetric CSS x
Static values single symmetric
all other single symmetric CSS x
Remark: Haunched sections are replaced by equivalent asymmetric I sections, by ignoring the middle flanges.
Scia Engineer Steel Code Check Theoretical Background
266
The following picture illustrates the relation between the local coordinate system of Scia Engineer and FriLo LTBII. Special attention is required for U sections due to the inversion of the y and z-axis.
For more information, reference is made to Ref[2]
Scia Engineer Steel Code Check Theoretical Background
267
Loadings
The following load impulses are supported:
- Point force in node (if the node is part of the exported beam)
- Point force on beam
- Line force in beam
- Moment in node (if the node is part of the exported beam)
- Moment on beam
- Line moment in beam (only for Mx in LCS)
The supported load impulses and their eccentricities are transformed into the local LCS of the exported member.
The dead load is replaced by an equivalent line force on the beam.
Load eccentricities are replaced by torsional moments.
The forces in local x-direction are ignored, except for the torsional moments.
In Frilo LTBII a distinction is made between the centroid and the shear center of a cross-section. Load impulses which do not pass through the shear center will cause additional torsional moments.
Imperfections
In the 2nd
Order LTB analysis the bow imperfections v0 (in local y direction) and w0 (in local z direction) can be taken into account.
v0
y, v0
z
y
Scia Engineer Steel Code Check Theoretical Background
268
For DIN, ONORM, EC-EN and EAE the imperfections can be calculated according to the code. The codes indicate that for a 2
nd Order calculation which takes into account LTB, only the imperfection v0
needs to be considered.
The sign of the imperfection according to code depends on the sign of Mz in Scia Engineer.
Initial bow imperfection v0 for DIN and ONORM
The imperfection is calculated according to Ref.[6] article 2.2
For prismatic uniform members:
Resistance check Section Bucking curve v0
EE
(Elastic)
any a0 L/1050
any a L/900
any b L/750
any c L/600
any d L/450
EP
PP
(Plastic)
I section a0 L/700
I section a L/600
I section b L/500
I section c L/400
I section d L/300
For non-uniform members, the bow imperfection is considered at the centre of the buckling system length L.
Initial bow imperfection v0 for EC-EN and EAE
The imperfection is calculated according to Ref.[4] article 5.3.4(3)
00 ekv
With k Factor taken from the National Annex of EC-EN
Factor taken as 0,5 for EAE
e0 Bow imperfection of the weak axis
Scia Engineer Steel Code Check Theoretical Background
269
The value of e0 is taken from following table:
Buckling curve
eo /L – elastic analysis
eo/L – plastic analysis
a0 1/350 1/300
a 1/300 1/250
b 1/250 1/200
c 1/200 1/150
d 1/150 1/100
With L Member system length
Initial bow imperfections v0 and w0 for other supported codes
For all other supported codes (EC-ENV, NEN and SIA) as well as DIN, ONORM, EC-EN and EAE the user can manually input the imperfections v0 and w0.
Scia Engineer Steel Code Check Theoretical Background
270
LTB Restraints LTB restraints are transformed into 'Supports' (Ref.[2] p22), with horizontal elastic restraint Cy:
Cy = 1e15 kN/m
The position of the restraint z(Cy) is depending on the position of the LTB restraint (top/bottom).
The use of an elastic restraint allows the positioning of the restraint since this is not possible for a fixed restraint. (Ref.[2] p23)
Specifically for U-sections, an elastic restraint Cz is used with position y(Cz) due to the rotation of U-sections in the FriLo LTBII solver. (see Chapter “Supported Sections”)
Scia Engineer Steel Code Check Theoretical Background
271
Diaphragms Diaphragms are transformed into 'Elastic Foundations' of type „elastic restraint‟ (Ref.[2] p25). Both a
horizontal restraint Cy and a rotational restraint C are used.
The elastic restraint Cy [kN/m^2] is calculated as follows (Ref.[2] p52 and Ref.5 p40):
2
LSCy
With
S Shear stiffness of the diaphragm
L Diaphragm length along the member
The above formula for Cy is valid in case the bolt pitch of the diaphragm is set as „br‟. For a bolt
pitch of „2br‟ the shear stiffness S is replaced by 0,2 S (Ref.5 p22).
The shear stiffness S for a diaphragm is calculated as follows (Ref.7,3.5 and Ref.8,3.3.4.):
L
K+K
10a.=S
s
21
4
With a Frame distance
Ls Length of the diaphragm
K1 Factor K1 of the diaphragm
K2 Factor K2 of the diaphragm
The position of the restraint z(Cy) is depending on the position of the diaphragm.
Specifically for U-sections, an elastic restraint Cz is used with position y(Cz) due to the rotation of U-sections in the FriLo LTBII solver. (see Chapter “Supported sections”)
Scia Engineer Steel Code Check Theoretical Background
272
The rotational restraint C [kNm/m] is taken as vorhC (see Chapter “Adaptation of Torsional Constant”)
Linked Beams
Linked beams are transformed into 'Supports' (Ref.[2] p22), with elastic restraint.
The direction of the restraint is dependent on the direction of the linked beam:
If the linked beam has an angle less then 45° with the local y-axis of the beam under consideration, the restraint is set as Cy. In all other cases the restraint is set as Cz.
The position of the restraint z(Cy) or y(Cz) is depending on the application point of the linked beam (top/bottom).
The position is only taken into account in case of a flexible restraint (Ref.[2] p23).
The end forces of the linked beam are transformed to point loads on the considered 1D member,
- in z -direction for linked beams considered as y-restraint
- in y- direction for linked beams considered as z-restraint
Specifically for U-sections, if the linked beam has an angle less then 45° with the local y-axis of the beam under consideration, the restraint is set as Cz. In all other cases the restraint is set as Cy. This is due to the rotation of U-sections in the FriLo LTBII solver. (see Chapter “Supported Sections”)
Scia Engineer Steel Code Check Theoretical Background
273
Limitations and Warnings
The FRILO LTB solver is used with following limitations
o Only straight members are supported
o LTBII analysis is done for the whole 1D member, not for a part of the member, not for more members together
o When a LTB system length is inputted which differs from the member length, a warning will be given. Intermediate lateral restraints should be defined through LTB restraints, diaphragms and linked beams.
During the analysis, the FriLo LTBII solver may return a warning message. The most important causes of the warning message are listed here.
Eigenvalue solution Mcr
- Lateral Torsional Buckling is not governing – relative slenderness < 0,4
Due to the low relative slenderness, no LTB check needs to be performed. In this case it is not required to use the FriLo LTBII solver.
- Design Torsion! Simplified analysis of lateral torsional buckling is not possible.
Due to the torsion in the member it is advised to execute a 2nd
order analysis instead of an eigenvalue calculation.
- Bending of U-section about y-axis!
The program calculates the minimum bifurcation load only.
2nd
Order Analysis
- Load is greater then minimum bifurcation load (Error at elastic calculation – system is instable in II.Order )
The loading on the member is too big, a 2nd
order calculation cannot be executed.
- You want to calculate the structural safety with Elastic-Plastic method. This analytical procedure cannot be used for this cross-section. It is recommended to use the Elastic-Elastic method.
Plastic calculation is not possible, use imperfection according to code elastic instead of plastic.
For more information, reference is made to Ref[1] and [2].
Scia Engineer Steel Code Check Theoretical Background
274
References
[1] FriLo LTBII software Friedrich + Lochner Lateral Torsional Buckling 2
nd Order Analysis
Biegetorsionstheorie II.Ordnung (BTII)
http://www.frilo.de
[2] Friedrich + Lochner LTBII Manual BTII Handbuch Revision 1/2006
[3] J. Meister
Nachweispraxis Biegeknicken und Biegedrillknicken
Ernst & Sohn, 2002
[4] Eurocode 3
Design of steel structures
Part 1 - 1 : General rules and rules for buildings
EN 1993-1-1:2005
[5] J. Schikowski
Stabilisierung von Hallenbauten unter besonderer Berücksichtigung der Scheibenwirkung von Trapez- und Sandwichelementdeckungen, 1999 http://www.jschik.de/
[6] DIN 18800 Teil 2
Stahlbauten
Stabilitätsfälle, Knicken von Stäben und Stabwerken
November 1990
[7] E. Kahlmeyer
Stahlbau nach DIN 18 800 (11.90)
Werner-Verlag, Düsseldorf
[8] Beuth-Kommentare
Stahlbauten
Erläuterungen zu DIN 18 800 Teil 1 bis Teil 4, 1.Auflage
Beuth Verlag, Berlin-Köln 1993
Scia Engineer Steel Code Check Theoretical Background
275
Profile conditions for code check
Introduction to profile characteristics The standard profile sections have fixed sections properties and dimensions, which have to be present in the profile library.
The section properties are described in chapter "Data for general section stability check".
The required dimension properties are described in chapter "Data depending On the profile shape".
Data for general section stability check
The following properties have to be present in the profile library for the execution of the section and the stability check :
Description Property number
Iy moment of inertia yy 8
Wy elastic section modulus yy 10
Sy statical moment of area yy 6
Iz moment of inertia zz 9
Wz elastic section modulus zz 11
Sz statical moment of area zz 7
It* torsional constant 14
Wt* torsional resistance 13
A0 sectional area 1
Iyz centrifugal moment 12
iy radius of gyration yy 2
iz radius of gyration zz 3
Mpy plastic moment yy 30
Mpz plastic moment zz 31
fab fabrication code
0=rolled section (default value)
1=welded section
2=cold formed section
105
The fabrication code is not obligatory.
When the section is made out of 1 plate, the properties marked with (*) can be calculated by the calculation routine in the profile library. When this is not the case, these properties have to be input by the user in the profile library.
The plastic moments are calculated with a yield strength of 240 N/mm².
Scia Engineer Steel Code Check Theoretical Background
276
Data depending On the profile shape
I section
Formcode 1
PSS Type .I.
Property Description
49 H
48 B
44 t
47 s
66 R
74 W
140 wm1
61 R1
146
109 1
B
s
w
t
R
R1
a
H
Scia Engineer Steel Code Check Theoretical Background
277
RHS
Formcode 2
PSS Type .M.
Property Description
49 H
48 B
67 s
66 R
109 2
B
s
H
R
Scia Engineer Steel Code Check Theoretical Background
278
CHS
Formcode 3
PSS Type .RO.
Property Description
64 D
65 s
109 3
D
w
Scia Engineer Steel Code Check Theoretical Background
279
Angle section
Formcode 4
PSS Type .L.
Property Description
49 H
48 B
44 t
61 R1
66 R
74 W1
75 W2
76 W3
109 4
B
R
R1
w1
w2
t
w3
w1
w2
Scia Engineer Steel Code Check Theoretical Background
280
Channel section
Formcode 5
PSS Type .U.
Property Description
49 H
48 B
44 t
47 s
66 R
68
41
61 R1
146
109 5
B
s
H
t
R
R1
a
Scia Engineer Steel Code Check Theoretical Background
281
T section
Formcode 6
PSS Type .T.
Property Description
49 H
48 B
44 t
47 s
66 R
61 R1
62 R2
146 1
147 2
109 6
B
s
t
R
a1
H
a2
R1
R2
Scia Engineer Steel Code Check Theoretical Background
282
Full rectangular section
Formcode 7
PSS Type .B.
Property Description
48 B
67 H
109 7
B
H
Scia Engineer Steel Code Check Theoretical Background
283
Full circular section
Formcode 11
PSS Type .RU.
Property Description
64 D
109 11
D
Scia Engineer Steel Code Check Theoretical Background
284
Asymmetric I section
Formcode 101
PSS Type
Property Description
49 H
48
44
47 s
42 Bt
43 Bb
45 tt
46 tb
66 R
109 101
R
H
Bt
Bb
tt
tb
Scia Engineer Steel Code Check Theoretical Background
285
Z section
Formcode 102
PSS Type .Z.
Property Description
49 H
48 B
44 t
47 s
67 R
61 R1
109 102
B
s
t
H
RR1
Scia Engineer Steel Code Check Theoretical Background
286
General cold formed section
Each section is considered as a composition of rectangular parts. Each part represents a plate unit which is considered as element for defining the effective width. The start and end parts are considered as unstiffened elements, the intermediate parts are considered as stiffened parts.
This way of definition of the section assumes that the area is concentrated at its centre line. The rounding in the corners is ignored.
Description Property number Value
form code 109 110
Dy* 22
Dz* 23
CM* 26
buckling curve around yy axis 106 (1)
buckling curve around zz axis 107 (1)
buckling curve for LTB 108 (1)
(1) The values for the buckling curves are defined as follows:
1 = buckling curve a
2 = buckling curve b
3 = buckling curve c
4 = buckling curve d
The conditions are that the section is an open profile. Only the geometry commands O, L, N, A may be used in the geometry description.
When the section is made out of 1 plate, the properties marked with (*) can be calculated by the calculation routine in the profile library. The properties from the reduced section can be calculated by the code check.
When the section is made out of more than 1 plate, the properties marked with (*) can NOT be calculated by the calculation routine in the profile library. The properties from the reduced section can be calculated, except for the marked properties. These properties have to be input by the user in the profile library.
Scia Engineer Steel Code Check Theoretical Background
287
Formcode 110
PSS Type
Property Description
44 s
61 r
48 B
142 sp
143 e2
68 H
109 110
Remark:
r is rounding, special for KLS section (Voest Alpine)
sp is number of shear planes
B
H
e2
s
Scia Engineer Steel Code Check Theoretical Background
288
Cold formed Angle section
Formcode 111
PSS Type
Property Description
44 s
61 r
48 B
68 H
109 111
B
s
H
r
Scia Engineer Steel Code Check Theoretical Background
289
Cold formed Channel section
Formcode 112
PSS Type
Property Description
44 s
61 r
48 B
49 H
109 112
B
s
H
r
Scia Engineer Steel Code Check Theoretical Background
290
Cold formed Z section
Formcode 113
PSS Type
Property Description
44 s
61 r
48 B
49 H
109 113
B
s
H
R
Scia Engineer Steel Code Check Theoretical Background
291
Cold formed C section
Formcode 114
PSS Type
Property Description
44 s
61 r
48 B
49 H
68 c
109 114
B
s
H
r
c
Scia Engineer Steel Code Check Theoretical Background
292
Cold formed Omega section
Formcode 115
PSS Type
Property Description
44 s
61 r
48 B
49 H
42 c
109 115
B
s
H
c
R
Scia Engineer Steel Code Check Theoretical Background
293
Cold formed C section eaves beam
Formcode 116
PSS Type
Property Description
49 H
44 t
48 B
61 r1
68 R
163 A
109 116
Scia Engineer Steel Code Check Theoretical Background
294
Cold formed C Plus section
Formcode 117
PSS Type
Property Description
49 H
44 t
48 B
61 r1
68 R
164 PL
167 APL
109 117
Scia Engineer Steel Code Check Theoretical Background
295
Cold formed ZED section
Formcode 118
PSS Type
Property Description
49 H
44 t
42 Bt
43 Bb
61 r1
68 R
109 118
Scia Engineer Steel Code Check Theoretical Background
296
Cold formed ZED section asymmetric lips
Formcode 119
PSS Type
Property Description
49 H
44 t
42 Bt
43 Bb
61 r1
68 R
164 PL
109 119
Scia Engineer Steel Code Check Theoretical Background
297
Cold formed ZED section inclined lip
Formcode 120
PSS Type
Property Description
49 H
44 t
42 Bt
43 Bb
61 r1
68 R
164 PL
165 AL
109 120
Scia Engineer Steel Code Check Theoretical Background
298
Cold formed Sigma section
Formcode 121
PSS Type
Property Description
48 B
166 B1
49 H
50 H1
51 H2
68 R
44 t
61 r1
109 121
Scia Engineer Steel Code Check Theoretical Background
299
Cold formed Sigma section stiffened
Formcode 122
PSS Type
Property Description
48 B
166 B1
49 H
50 H1
51 H2
68 R
44 t
61 r1
164 PL
109 122
Scia Engineer Steel Code Check Theoretical Background
300
Cold formed Sigma Plus section
Formcode 123
PSS Type
Property Description
48 B
166 B1
49 H
50 H1
51 H2
68 R
44 t
61 r1
164 PL
167 APL
109 123
Scia Engineer Steel Code Check Theoretical Background
301
Cold formed Sigma section eaves beam
Formcode 124
PSS Type
Property Description
48 B
166 B1
49 H
50 H1
51 H2
68 R
44 t
61 r1
163 A
109 124
Scia Engineer Steel Code Check Theoretical Background
302
Cold formed Sigma Plus section eaves beam
Formcode 125
PSS Type
Property Description
48 B
166 B1
49 H
50 H1
51 H2
68 R
44 t
61 r1
163 A
164 PL
167 APL
109 125
Scia Engineer Steel Code Check Theoretical Background
303
Cold formed ZED section both lips inclined
Formcode 126
PSS Type
Property Description
49 H
44 t
42 Bt
43 Bb
61 r1
68 R
164 PL
165 AL
109 126
Scia Engineer Steel Code Check Theoretical Background
304
Cold formed I-Plus section
Formcode 127
PSS Type
Property Description
49 H
44 t
48 B
61 r1
68 R
164 PL
167 APL
168 a
109 127
Scia Engineer Steel Code Check Theoretical Background
305
Cold formed IS-Plus section
Formcode 128
PSS Type
Property Description
48 B
166 B1
49 H
50 H1
51 H2
68 R
44 t
61 r1
164 PL
167 APL
168 a
109 128
Scia Engineer Steel Code Check Theoretical Background
306
Cold formed Sigma section asymmetric
Formcode 129
PSS Type
Property Description
42 Bt
43 Bb
166 B1
49 H
50 H1
51 H2
68 R
44 t
61 r1
164 PL
109 129
Scia Engineer Steel Code Check Theoretical Background
307
Rail type KA
Formcode 150
PSS Type .KA.
Property Description
148 h1
149 h2
150 h3
151 b1
152 b2
153 b3
154 k
155 f1
156 f2
157 f3
61 r1
62 r2
63 r3
158 r4
159 r5
160 a
109 150
r1
r2
r4
r3
r5
b3
k
b2
b1
f3
f2
f1
h1
h3h2
Scia Engineer Steel Code Check Theoretical Background
308
Rail type KF
Formcode 151
PSS Type .KF.
Property Description
48 b
154 k
49 h
153 b3
155 f1
157 f3
148 h1
149 h2
61 r1
62 r2
63 r3
109 151
r1
r2r2
r2 r2
r3
k
bb3
f3
f1
h
h1 h2
Scia Engineer Steel Code Check Theoretical Background
309
Rail type KQ
Formcode 152
PSS Type .KQ.
Property Description
48 b
154 k
49 h
153 b3
155 f1
149 h2
150 h3
61 r1
109 152
b
k
b3
r1
h3
h2
f1
Scia Engineer Steel Code Check Theoretical Background
310
Warping check
Stress check
In cross sections subject to torsion, the following is checked:
Ed,wEd,tEd,VzEd,VyEd,tot
Ed,wEd,MzEd,MyEd,NEd,tot
M
y2
Ed,tot
2
Ed,tot
0M
y
Ed,tot
M
y
Ed,tot
f1.13
3
f
f
with fy the yield strength
tot,Ed the total direct stress
tot,Ed the total shear stress
M = M0 (class 1,2 and 3 section)
= M1 (class 4 section)
M0 the partial safety factor for resistance of cross-sections where failure is caused by yielding
M1 the partial safety factor for resistance of cross-sections where failure is caused by buckling
N,Ed the direct stress due to the axial force on the relevant effective cross-section
My,Ed the direct stress due to the bending moment around y axis on the relevant effective cross-section
Mz,Ed the direct stress due to the bending moment around z axis on the relevant effective cross-section
w,Ed the direct stress due to warping on the gross cross-section
Vy,Ed the shear stress due to shear force in y direction on the gross cross-section
Vz,Ed the shear stress due to shear force in z direction on the gross cross-section
t,Ed the shear stress due to uniform (St. Venant) torsion on the gross cross-section
w,Ed the shear stress due to warping on the gross cross-section
Scia Engineer Steel Code Check Theoretical Background
311
The warping effect is considered for standard I sections and U sections, and for (= “cold formed
sections”) sections. The definition of I sections and U sections, and sections are described in „Profile conditions for code check‟.
The other standard sections ( RHS, CHS, Angle section, T section and rectangular sections) are considered as warping free. See also Ref.[2], Bild 7.4.40.
Calculation of the direct stress due to warping
The direct stress due to warping is given by (Ref.[2] 7.4.3.2.3, Ref.[3])
m
MwEd,w
C
wM
with Mw the bimoment
wM the unit warping
Cm the warping constant
Scia Engineer Steel Code Check Theoretical Background
312
I sections
For I sections, the value of wM is given in the tables (Ref. [2], Tafel 7.87, 7.88). This value is added to the profile library. The diagram of wM is given in the following figure:
The direct stress due to warping is calculated in the critical points (see circles in figure).
The value for wM can be calculated by (Ref.[5] pp.135) :
mM hb4
1w
with b the section width
hm the section height (see figure)
Scia Engineer Steel Code Check Theoretical Background
313
U sections
For U sections, the value of wM is given in the tables as wM1 and wM2 (Ref. [2], Tafel 7.89). These values are added to the profile library. The diagram of wM is given in the following figure :
The direct stress due to warping is calculated in the critical points (see circles in figure).
sections
The values for wM are calculated for the critical points according to the general approach given in Ref.[2] 7.4.3.2.3 and Ref.[8] Part 27.
The critical points for each part are shown as circles in the figure.
Scia Engineer Steel Code Check Theoretical Background
314
Calculation of the shear stress due to warping
The shear stress due to warping is given by (Ref.[2] 7.4.3.2.3, Ref.[3])
s
0
M
m
xsEd,w tdsw
tC
M
with Mxs the warping torque (see "Standard diagrams for warping torque, bimoment and the St.Venant torsion")
wM the unit warping
Cm the warping constant
t the element thickness
I sections
The shear stress due to warping is calculated in the critical points (see circles in figure)
For I sections, we have the following :
A4
wtbtdsw M
2/b
0
M
Scia Engineer Steel Code Check Theoretical Background
315
U sections, sections
Starting from the wM diagram, we calculate the value
s
0
M tdsw
for the critical points.
The shear stress due to warping is calculated in these critical points (see circles in figures)
Scia Engineer Steel Code Check Theoretical Background
316
Plastic Check For doubly symmetric I sections of class 1 and class 2 (plastic check), the interaction formula given in Ref.[10] is used.
b
tw
tf
h Hy y
z
z
Used variables
Section Properties
A sectional area
b width
H heigth of section
tf flange thickness
tw web thickness
h = H - tf
Aw = 1.05 (h+tf) tw for rolled section
Aw = h tw for welded sections
ff tb2A
A
A ff
fw 1
Wz,pl plastic section modulus around z axis
Wy,pl plastic section modulus around y axis
Material Properties
Scia Engineer Steel Code Check Theoretical Background
317
fy,d yield strength
y,d shear strength
Internal forces
NSd normal force
My,Sd bending moment around y axis
Mz,Sd bending moment around z axis
Mw,Sd bimoment
Vy,Sd shear force in y direction
Vz,Sd shear force in z direction
Mxp,Sd torque due to St. Venant
Mxs,Sd warping torque
Plastic capacities
Npl,Rd = A fy,d
Mz,pl,Rd = Wz,pl fy,d
Vz,pl,Rd = Aw y,d
d,y
2
w2
fRd,pl,xp2
thbtM
My,pl,Rd = Wy,pl fy,d
2
hMM Rd,pl,zRd,pl,w
Vy,pl,Rd = Af y,d
2
hVM Rd,pl,yRd,pl,xs
Rd,pl
Sd
N
Nn
Rd,pl,y
Sd,y
yM
Mm
Rd,pl,z
Sd,z
zM
Mm
Rd,pl,w
Sd,w
wM
Mm
Scia Engineer Steel Code Check Theoretical Background
318
Rd,pl,y
Sd,y
yV
Vv
Rd,pl,z
Sd,z
zV
Vv
Rd,pl,xp
Sd,xp
xpM
Mm
Rd,pl,xs
Sd,xs
xsM
Mm
Shear force reduction
Sign
p=sign ( Mz,Sd x Mw,Sd)
ww
swwss
np1s4
Scia Engineer Steel Code Check Theoretical Background
319
Unity checks:
Remark: the values between must be > 0.
Standard diagrams for warping torque, bimoment and the St.Venant torsion
The following 6 standard situations are given in the literature (Ref.[2], Ref.[3]).
The value is defined as follows :
m
t
CE
IG
with Mx the total torque
= Mxp + Mxs
Mxp the torque due to St. Venant
Mxs the warping torque
Mw the bimoment
IT the torsional constant
CM the warping constant
E the modulus of elasticity
G the shear modulus
Scia Engineer Steel Code Check Theoretical Background
320
Torsion fixed ends, warping free ends, local torsional loading Mt
Mx
L
aMM
L
bMM
txb
txa
Mxp for a side
)xcosh(
)Lsinh(
)bsinh(
L
bMM txp
Mxp for b side
)'xcosh(
)Lsinh(
)asinh(
L
aMM txp
Mxs for a side
)xcosh(
)Lsinh(
)bsinh(MM txs
Mxs for b side
)'xcosh(
)Lsinh(
)asinh(MM txs
Mw for a side
)xsinh(
)Lsinh(
)bsinh(MM t
w
Mw for b side
)'xsinh(
)Lsinh(
)asinh(MM t
w
Scia Engineer Steel Code Check Theoretical Background
321
Torsion fixed ends, warping fixed ends, local torsional loading Mt
Mx
L
aMM
L
bMM
txb
txa
Mxp for a side
3D
L
1k2kbMM txp
Mxp for b side
4D
L
1ka2kMM txp
Mxs for a side 3DMM txs
Mxs for b side 4DMM txs
Mw for a side 1D
MM t
w
Mw for b side 2D
MM t
w
Scia Engineer Steel Code Check Theoretical Background
322
Scia Engineer Steel Code Check Theoretical Background
323
Torsion fixed ends, warping free ends, distributed torsional loading mt
Mx
2
LmM
2
LmM
txb
txa
Mxp
)Lsinh(
)'xcosh()xcosh()x
2
L(
mM t
xp
Mxs
)Lsinh(
)'xcosh()xcosh(mM t
xs
Mw
)Lsinh(
)'xsinh()xsinh(1
mM
2
tw
Scia Engineer Steel Code Check Theoretical Background
324
Torsion fixed ends, warping fixed ends, distributed torsional loading mt
Mx
2
LmM
2
LmM
txb
txa
Mxp
)Lsinh(
)'xcosh()xcosh()k1()x
2
L(
mM t
xp
Mxs
)Lsinh(
)'xcosh()xcosh()k1(
mM t
xs
Mw
)Lsinh(
)'xsinh()xsinh()k1(1
mM
2
tw
)2
Ltanh(
2
L
1k
Scia Engineer Steel Code Check Theoretical Background
325
One end free, other end torsion and warping fixed, local torsional loading Mt
Mx
txa MM
Mxp
)Lcosh(
)'xcosh(1MM txp
Mxs
)Lcosh(
)'xcosh(MM txs
Mw
)Lcosh(
)'xsinh(MM t
w
Scia Engineer Steel Code Check Theoretical Background
326
One end free, other end torsion and warping fixed, distributed torsional loading mt
Mx
LmM txa
Mxp
)Lcosh(
)xsinh())Lsinh(L1()xcosh(L'x
mM t
xp
Mxs
)Lcosh(
)xsinh())Lsinh(L1()xcosh(L
mM t
xs
Mw
)Lcosh(
)xcosh())Lsinh(L1()xsinh(L1
²
mM t
w
Scia Engineer Steel Code Check Theoretical Background
327
Decomposition of arbitrary torsion line Since the Scia Engineer solver does not take into account the extra DOF for warping, the determination of the warping torque and the related bimoment, is based on some standard situations.
The following end conditions are considered:
warping free
warping fixed
This results in the following 3 beam situations :
situation 1 : warping free / warping free
situation 2 : warping free / warping fixed
situation 3 : warping fixed / warping fixed
Scia Engineer Steel Code Check Theoretical Background
328
Decomposition for situation 1 and situation 3
The arbitrary total torque line is decomposed into the following standard situations:
n number of torsion lines generated by a local torsional loading Mtn
one torsion line generated by a distributed torsional loading mt
one torsion line with constant torque Mt0
The values for Mxp, Mxs and Mw are taken from the previous tables for the local torsional loadings Mtn and the distributed loading mt. The value Mt0 is added to the Mxp value.
Decomposition for situation 2
The arbitrary total torque line is decomposed into the following standard situations:
one torsion line generated by a local torsional loading Mtn
one torsion line generated by a distributed torsional loading mt
The values for Mxp, Mxs and Mw are taken from the previous tables for the local torsional loading Mt and the distributed loading mt.
References
[1] ENV 1993-1-3:1996
Eurocode 3 : Design of steel structures
Part 1-3 : General rules – Supplementary rules for cold formed thin gauge members and sheeting
CEN 1996
[2] Stahl im Hochbau
14. Auglage Band I/ Teil 2
Verlag Stahleisen mbH, Düsseldorf 1986
[3] Kaltprofile
3. Auflage
Verlag Stahleisen mbH, Düsseldorf 1982
[4] Roik, Carl, Lindner
Biegetorsionsprobleme gerader dünnwandiger Stäbe
Verlag von Wilhem ernst & Sohn, Berlin 1972
[5] Dietrich von Berg
Krane und Kranbahnen – Berechnung Konstruktion Ausführung
B.G. Teubner, Stuttgart 1988
Scia Engineer Steel Code Check Theoretical Background
329
[6] DASt-Richtlinie 016
Bemessung und konstruktive Gestaltung von Tragwerken aus dünnwandigen kaltgeformten Bauteilen
Stahlbau-Verlagsgesellschaft, Köln 1992
[7] Esa Prima Win
Steel Code Check Manual
SCIA
EPW 3.10
[8] C. Petersen
Stahlbau : Grundlagen der Berechnung und baulichen Ausbildung von Stahlbauten
Friedr. Vieweg & Sohn, Braunschweig 1988
[9] Eurocode 3
Design of steel structures
Part 1 - 1 : General rules and rules for buildings
ENV 1993-1-1:1992, 1992
[10] I. Vayas,
Interaktion der plastischen Grenzschnittgrössen doppelsymmetrischer I-Querschnitte
Stahlbau 69 (2000), Heft 9
Scia Engineer Steel Code Check Theoretical Background
330
Check of numerical sections
Stress check
The stress calculation for a numerical section is as follows:
with vm the VonMises stress, the composed stress
tot the total normal stress
tot the total shear stress
N the normal stress due to the normal force N
My the normal stress due to the bending moment Myy around y axis
Mz the norma stress due to the bending moment Mzz around z axis
Vy the shear stress due to shear force Vy in y direction
Vz the shear stress due to shear force Vz in z direction
Ax the sectional area
Ay the shear area in y direction
Az the shear area in z direction
Wy the elastic section modulus around y axis
Wz the elastic section modulus around z axis
z
zVz
y
y
Vy
z
zzMz
y
yy
My
x
N
VzVytot
MzMyNtot
2tot
2totvm
A
V
A
V
W
M
W
M
A
N
3
Scia Engineer Steel Code Check Theoretical Background
331
Use of diaphragms
Adaptation of torsional constant
See Ref.[1], Chapter 10.1.5., Ref.2,3.5 and Ref.3,3.3.4..
When diaphragms (steel sheeting) are used, the torsional constant It is adapted for symmetric/asymmetric I sections, channel sections, Z sections, cold formed U, C , Z sections.
The torsional constant It is adapted with the stiffness of the diaphragms:
12
³sI
)th(
IE3C
200b125if100
bC25.1C
125bif100
bCC
s
EIkC
C
1
C
1
C
1
vorhC
1
G
lvorhCII
s
sk,P
aa
100k,A
a
2
a100k,A
effk,M
k,Pk,Ak,M
2
2
tid,t
with l the LTB length
G the shear modulus
vorhC the actual rotational stiffness of diaphragm
CM,k the rotational stiffness of the diaphragm
CA,k the rotational stiffness of the connection between the diaphragm and the beam
CP,k the rotational stiffness due to the distortion of the beam
k numerical coefficient
= 2 for single or two spans of the diaphragm
= 4 for 3 or more spans of the diaphragm
EIeff bending stiffness of per unit width of the diaphragm
s spacing of the beam
ba the width of the beam flange (in mm)
Scia Engineer Steel Code Check Theoretical Background
332
C100 rotation coefficient - see table
h beam height
t thickness beam flange
s thickness beam web
References
[1] ENV 1993-1-3:1996
Eurocode 3 : Design of steel structures
Part 1-3 : General rules
Supplementary rules for cold formed thin gauge members and sheeting
CEN 1996
[2] E. Kahlmeyer
Stahlbau nach DIN 18 800 (11.90)
Werner-Verlag, Düsseldorf
[3] Beuth-Kommentare
Stahlbauten
Scia Engineer Steel Code Check Theoretical Background
333
Erläuterungen zu DIN 18 800 Teil 1 bis Teil 4, 1.Auflage
Beuth Verlag, Berlin-Köln 1993
Scia Engineer Steel Code Check Theoretical Background
334
Section check for built-in beams (IFB, SFB, THQ sections)
Introduction
For the national codes EC-ENV, EC-EN, NEN6770/6771, DIN18800 and SIA263, special checks are performed for built-in beams, according to Ref.[1].
Reduction of plastic moment capacity due to plate bending
bu
e1
e2=bo
bo
tu
0.5 q0.5 q
Scia Engineer Steel Code Check Theoretical Background
335
bu
e1
e2=bo
bo
tu
0.5 q0.5 q
to
bu
e1
bo
tu
0.5 q0.5 q
e2=0
to
When the lower plate is loaded by q-load (uniform distributed load), the effective area of the loaded plate (flange) for the calculation of the plastic capacity is reduced as follows :
Scia Engineer Steel Code Check Theoretical Background
336
for THQ and IFB beams :
for SFB beam :
oouueff AAA
with e1, e2, tu, bu see the figures above
q load on flange, plate (as N/m)
fy yield strength
M partial safety factor
see formula
u =
o analog to u, but with
bu=bo
e1=bo
tu=to
e2=tw
Scia Engineer Steel Code Check Theoretical Background
337
Plastic interaction formula for single bending and shear force
The following plastic interaction formula can be used, when single bending around yy-axis My,Sd, in combination with shear force Vz,Sd, is acting :
y,pl
fm
Rd,z,pl
Sd,z
m
v
Rd,y,pl
Sd,y
W2
hA
0.1V
V
A
A
M
M
with My,Sd, Vz,Sd internal forces
Mpl,y,Rd plastic bending capacity around yy axis
Vpl,z,Rd plastic shear capacity in z direction
Av shear area (see figure)
Am = A - | Ao,x - Au,x | (see figure)
hf = h+tu/2-to/2 (see figure)
Wpl,y plastic section modulus around yy axis - reduced if necessary
Scia Engineer Steel Code Check Theoretical Background
338
Plastic check for plate in bending
The following condition for the plate in bending must be verified:
with e1, e2, tu see figures
q load on flange, plate (as N/m)
= qmax+qmin
(Ksi)
q
qq minmax
fy yield strength
M partial safety factor
0.5 q (1+Ksi)0.5 q (1-Ksi)
Scia Engineer Steel Code Check Theoretical Background
339
Stress check for slim floor beams
Normal stress check
At the edges of the bottom plate, the following composed stress check is performed:
Shear stress check in plate
In the middle of the bottom plate, transverse shear stress is checked:
u
minmax
M
y2
x
t
)q,q(
2
3
f²3
Scia Engineer Steel Code Check Theoretical Background
340
Torsion check due to unbalanced loading
for IFB and SFB beams:
12
bEtEI
GI
EIh2L
L
Ltanh
2
QeLM
htb
M
2
3
L
L
L
Ltanh
12
QeLM
I
tM
3
f
3
ooo
t
ofk
k
kmax,w
foo
max,w
max,w
k
k
max,t
t
omax,t
max,t
M
y
max,wmax,t
with to, bo see figures
hf = h+tu/2-to/2 (see figure)
It torsional constant for complete section
E modulus of Young
G shear modulus
L system length for Lyz
Q,e see figure
Scia Engineer Steel Code Check Theoretical Background
341
Q
e
for THQ beams :
2
V
b
e1
4
qL Rd,z,pl
f
with e, bf see figure
hf = h+tu/2-to/2 (see figure)
q load on flanges, plate (as N/m)
= qmax+qmin
(Ksi)
q
qq minmax
Scia Engineer Steel Code Check Theoretical Background
342
q maxq min
bf
ee
References
[1] Multi-Storey Buildings in Steel
Design Guide for Slim Floors with Built-in Beams
ECCS N° 83 - 1995
Scia Engineer Steel Code Check Theoretical Background
343
Effective cross-section properties for lattice tower angle members
Effective cross-section properties for compressed lattice tower angle members The effective cross-section properties shall be based on the effective width beff of the leg. See Ref.[1], Chapter J.2.3.
b
The effective width shall be obtained from the nominal width of the leg, assuming uniform stress distribution:
bb
f
235
43.0K
K4.28
t
b
eff
y
c
c
p
p
p
Scia Engineer Steel Code Check Theoretical Background
344
For a rolled angle:
2
p
p
p
p
p
98.0213.1
91.02213.191.0
0.191.0
For a cold formed angle:
2
p
p
p
p
p
98.0213.1
3
404.05
213.1809.0
0.1809.0
with t the thickness
b the nominal width
fy the yield strength in Mpa
References
[1] EN 50341-1:2001
Overhead electrical lines exceeding AC 45 kV Part 1: General requirements
top related