defense_thesis_ramy_gabr_06-22-2014_final-with notes - copy

Presented By: Ramy Hassan Mohamed Gabr

B.Sc. Ain Shams University, Structural DepartmentStructural Engineer, Parsons International

Ain Shams UniversityFaculty of Engineering

Structural Engineering Department

BEHAVIOUR AND STRENGTH OF SINGLY-SYMMETRIC CONTINUOUS I-BEAMS

Supervised By:Prof. Dr. Adel Helmy Salem

Professor of Steel Structures

Ain Shams University

Dr. Abdel-Rahim Badawy Abdel-RahimAssistant Professor of Steel Structures

Ain Shams University

Supervised by:

AGENDA

Introduction Literature Review Problem Statement Objectives Finite Element Model Verification Parametric Study Results Discussion Proposed Design Model Conclusions Recommendations for Future Study

Introduction

Open cross sections such as I-beams, are widely used in structural applications.

These sections can be classified as follows:1. Compact l < lp

2. Noncompactlp < l < lr

3. Slender l > lr

Introduction (cont’d)

Types of failure:1. Lateral-Torsional Buckling

• Short beams:non compact or slender section

• Long beams

2. Local Buckling

3. Distortional Buckling• Slender unstiffened webs

AGENDA


Literature Review

Trahair, (2008), presented the influence of restraints on the elastic buckling of monorails, without distortion, loaded at the bottom flange Trahair developed an economical strength design method for determining

the nominal LTB resistance (distortion was not taken into account).

Trahair, (2009), studied the influence of the elastic lateral-distortional buckling of single span steel monorail I-beams on its strength, using the parameter LD/LTB ratio. For beams with bottom flange loading, and unrestrained bottom flange,

smaller LD/LTB ratios were encountered, but they increase when rigid web stiffeners or top flange torsional restraints were provided at the supports.

Literature Review (cont’d)

Kitipornchai et. al, (1986) studied the effect of moment gradient and load position on buckling capacities of singly-symmetric beams subjected to different ratios of end moments. Traditional moment gradient factors worked reasonably well with singly-

symmetric sections subjected to single-curvature bending. For cases with reverse-curvature bending, the study found that the Cb

factors were unsafe when the maximum moment caused compression in the small flange and overly conservative when the maximum moment caused compression in the larger flange.


Andrade et. al, (2006), evaluated the elastic critical moment (Mcr) in doubly and singly-symmetric I-section cantilevers. Different parameters were introduced in the paper such as: degree of mono-symmetry (r), load type and load position with respect to the shear center. The 3-factor formula which was included in the ENV version of the

Eurocode 3 was extended to cantilevers by providing approximate analytical expression to determine C1, C2 and C3 factors.


Avik and Ashwini, (2005), studied the effect of distortional buckling of simply supported I-section of different degrees of mono-symmetry (r). For fairly long beams, Cb values obtained from the study agree with

SSRC Guide (1998) recommendations as buckling is guided by flexural-torsional buckling.

For short beams, the difference is significant since Cb values dependent not only on the degree of beam singly-symmetry (r) but on the span to beam height (L/h) ratio where the distortional buckling is predominant.

Avik and Ashwini, (2006), extended the investigation to the case of reverse-curvature bending. It was shown from the results presented that the available design

specifications provide over estimated Cb values for the two load cases (point and distributed) considered


Helwig, et. al, (1997), investigated the lateral-torsional buckling of singly-symmetric I-beams and some expressions were suggested for the moment modification factors Cb. For single-curvature bending, the finite element results showed that traditional Cb

values can be used. For reverse-curvature bending, the Cb factor was modified to agree with the FEM

results.

Mohsen, et. al, (2007)a, b, c, investigated the behavior and capacity of over-hanging singly-symmetric I-beams for various restraint conditions at the tip. The ultimate moment capacities obtained from the study are compared to those computed according to AISC specification, (2005), and BS 5950, (2000). The comparison shows that the ultimate moment capacities computed according

to the current standards and specifications vary from conservative to non-conservative, depending on the overhang length, degree of mono-symmetry and location of load with respect to the height of I-section. A design model was introduced based on the results developed from the FEM analysis

AGENDA


Problem Statement

1. The behavior of the singly-symmetric I-beams is not yet totally investigated.

2. The design of the singly-symmetric I-beams did not take the same interest in any of the standards and specifications as the double symmetric simple beam took.

3. The type of analysis in the previous work took into consideration the elastic behavior of I-beams and did not consider the geometric and material nonlinearities.

Problem Statement (cont’d)

4. Design procedures for continuous I-beams are not yet clear.

5. Different loading positions are not considered in the current standards and specifications (AISC specification considered loading and restraining conditions at centroid only).

6. Discrepancy between current standards and specifications (AISC vs. BS).

AGENDA


Objectives

1. Study the effect of different parameters on the buckling behavior and bending capacity of singly-symmetric continuous I-beams, such as the effect of:

a. Span length of the continuous beams.b. Loading position along the section height.c. Degree of mono-symmetry (r).d. Load case.e. Section height.

Span Length Span Length

Objectives (cont’d)

2. Develop and propose new design models for the beams in study.

3. Compare the proposed model results to those of the current standards and specifications.

AGENDA


Finite Element Model

A finite element model for a continuous I-beam was developed using ANSYS Program (V.12)

a. This model takes into account both material and geometry nonlinearities.

i. Non-linear stress-strain curve.ii. Initial imperfections = L/500

b. 8-Node quadrilateral thin shell element “shell 93”:i. Include initial imperfection of plates.ii. Account for plasticity, stress stiffening and large

deformations.iii. Each node has six degrees of freedom, translations (Ux,

Uy and Uz) in the nodal X, Y and Z directions, respectively, and rotations (ROT x, ROT y and ROT z) about the nodal X, Y and Z directions, respectively.

0

50

100

150

200

250

300

350

400

450

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07Strain

Str

ess

(N/m

m2 )

I

N

K

M

JL

X, UX

Z, U Z

Y, UY

r

t s

N

K

O

L

P

I

M

J

AGENDA


Verification

The Finite element model verified with past experimental work developed by Bassem, 2008.

t1

t2

1000 1000

tw

L1 L2

b1

b2

h

150 150

Verification (cont’d)

Specimen h tw b1 t1 b2 t2 L1 L2 h/tw b1/2t1 b1/2t2 Tip

OH-1 200 7.4 68 8.2 101 8.4 2000 2000 27.0 4.100 6.010 R

OH-2 200 6.0 60 8.0 100 8.0 2000 2000 33.3 3.750 6.250 T

OH-3 200 5.0 61 8.0 101 8.0 2000 1000 40.0 3.810 6.310 R

OH-4 202 5.5 61 8.0 101 8.0 2000 1000 36.7 3.810 6.310 F

OH-5 200 6.0 60 8.0 101 8.0 2000 2996 33.3 3.750 6.310 R

OH-6 201 6.0 62 8.0 101 8.0 2000 2997 33.5 3.875 6.310 F

R: Laterally and Torsionally Restrained

F: Laterally Free

T: Top Flange Laterally Restrained

b1

b2

twh

t1

t2

Verification (cont’d)

The ultimate loads from FEM are in good agreement with the experimental results with a range of deviation of + 14% to -5%, and an average of + 4%.

SPECIMENPu

Experimental(N)

PuFEM(N)

Pu FEM / Pu EXP

OH-1 36297 35218 0.97

OH-2 30019 30489 1.02

OH-3 54446 61999 1.14

OH-4 46696 47088 1.01

OH-5 19130 18207 0.95

OH-6 10104 11380 1.13

AVERAGE 1.04

AGENDA


Parametric Study

Studied parameters:a. Span Lengthb. Load Positionc. Degree of mono-

symmetry (r)

c. Load Casee. Section Height

Different Loading Positions at Mid SpanBottom Loading (BT)

Top Loading (TP) CG Loading (CG)

Span Length3000, 4000, 5000,

6000, 7000 and 8000 mm

Span Length3000, 4000, 5000,

6000, 7000 and 8000 mm

95 mm

150 mm

122 mm150 mm 185 mm 240 mm

= 0.20 = 0.35 = 0.50 = 0.65 = 0.80

Constant WebThickness = 6.0 mm

Constant FlangesThickness = 8.0 mm

Web Height = 350,500 and 650 mm

150 mm 150 mm 150 mm 150 mm StiffenersThickness = 12 mm

540 models were created to accommodate all parameters that as follows:

AGENDA


Results Discussion

Behavior of the beams: Lateral Torsional Buckling:

• Degree of mono-symmetry (r) = 0.20 and 0.35.

Results Discussion (cont’d)

Behavior of the beams: Lateral Torsional Buckling:

• Degree of mono-symmetry (r) = 0.20 and 0.35.


Behavior of the beams: Tension Flange Yielding:

• Degree of mono-symmetry (r) = 0.50 for short spans. Tension Flange Yielding + LTB:

• Degree of mono-symmetry (r) = 0.50 for long spans.


Behavior of the beams: Local Buckling:

• Degree of mono-symmetry (r) = 0.65 and 0.80 for short spans.

Compression Flange Yielding + LTB:• Degree of mono-symmetry (r) = 0.65 and 0.80 for long

spans.


1. Effect of Load Position on the Ultimate Moment Capacities

The difference between the ultimate moment capacity of a section with top loading and a section with bottom loading varies giving 40% for the cases of degree of mono-symmetry (r) equals to 0.20 and 0.35 and …..

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BOTTOMTOPCENTROID

Lb/rt

MU/M

P

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BOTTOMTTOPCENTROID

Lb/rt

MU/M

P

and 80% for (r) equals to 0. 50, 0.65 and 0.80.

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BOTTOMTOPCENTROID

Lb/rt

MU/M

P

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BOTTOMTOPCENTROID

Lb/rt

MU/M

P

The ultimate moment capacity for the top flange loading position gives the lowest values whereas the bottom flange loading position show the highest capacities due to loading at bottom flange counteract the torsion of the section.


2. Effect of Degree of Mono-Symmetry (r) on the Ultimate Moment Capacities

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ρ=0.20ρ=0.35ρ=0.50ρ=0.65ρ=0.80

Lb/rt

MU/M

P

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ρ=0.20ρ=0.35ρ=0.50ρ=0.65ρ=0.80

Lb/rt

MU/M

P

The ratio (Mu/MP) is lower for TOP than BOTTOM and CENTROID loading as loading on top of flange with the presence of geometric imperfection reach lateral-torsional buckling before its plastic capacity.


3. Effect of Load Case on the Ultimate Moment Capacities (load at bottom flange)

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OVER ONE SPANOVER TWO SPANS

Lb/rt

MU/M

P

Ultimate moment capacity of the sections with loading at the bottom flange and loading case of double (2P) is higher than loading case of a single load (1P) by 20% …

… as the compression flange will tend to freely sway and the same behavior will be recognized in the second loaded span if combined with the first span it will result in some stabilization in the section with respect to its lateral move as each span will balance the compressive and tensile stresses at the top flange with the other span.

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Lb/rt

MU/M

P


3. Effect of Load Case on the Ultimate Moment Capacities (load at centroid)

Ultimate moment capacity of the sections with loading at the centroid of the section and loading case of double (2P) is higher than loading case of a single load (1P) by 15%.

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Lu/rt

MU/M

P

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Lu/rt

MU/M

P


4. Effect of Section Height on the Ultimate Moment Capacities

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WEB DEPTH = 350 mm

WEB DEPTH = 500 mm

WEB DEPTH = 650 mm

Lb/rt

MU/M

P

Web 350mm section was the highest although the failure load for the web 650mm was the highest among all studied sections, higher heights sections fail by LTB giving a lower ultimate moment capacity with the high plastic capacity of the deeper I beams …

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0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1WEB DEPTH = 350 mm

WEB DEPTH = 500 mm

WEB DEPTH = 650 mm

Lb/rt

MU/M

P

… the ratio will end up with values lower than those values from the smaller heights that have a relatively ultimate moment capacity close to its plastic capacity giving higher (Mu/MP) ratio

AGENDA


Proposed Design Model

Compact Nonompact

Load on One Span

BTLoad

CGLoad

TPLoad

BTLoad

CGLoad

TPLoad

BTLoad

CGLoad

TPLoad

BTLoad

CGLoad

TPLoad

Load on Two Spans Load on One Span Load on Two Spans

130 ≤ Lb/rt ≤ 360

40 ≤ Lb/rt < 130

200 ≤ Lb/rt ≤ 400

80 ≤ Lb/rt < 200

Proposed Design Model (cont’d)

The general form of the equation derived for the current study is represented as follow:

22

edr

Lc

r

Lba

M

M

t

b

t

b

p

u

yxp FZM

bL is the length between points that are either braced against lateral displacement of compression flange or braced against twist of the cross section

tr is the radius of gyration of the flange components in flexural compression plus one-third of the web area in compression

is the degree of mono-symmetry which is calculated as the ratio of the moment of inertia about the minor axis of the compression flange to the moment of inertia about the minor axis of the whole cross section


Factors a, b, c, d and e are as follows:

Lb/rt a b c d e

Comapct

1P

BOTTOM 40-130 0.738 2.18E-03 -2.25E-05 0.254 -2.59E-01130-360 1.352 -5.95E-03 9.28E-06 -0.192 1.10E-01

TOP 40-130 0.912 -4.90E-03 -6.80E-07 0.077 -3.17E-02130-360 0.665 -3.75E-03 7.20E-06 0.082 -1.73E-01

CENTROID 40-130 0.859 -9.40E-04 -1.80E-05 0.105 -3.08E-02130-360 0.999 -5.13E-03 9.09E-06 0.006 -9.61E-02

2P

BOTTOM 40-130 0.900 1.97E-03 -1.82E-05 -0.274 3.18E-01130-360 1.443 -5.63E-03 8.30E-06 -0.045 -6.52E-02

TOP 40-130 1.012 -5.36E-03 4.12E-06 -0.312 2.00E-01130-360 0.584 -2.54E-03 4.11E-06 0.009 -1.56E-01

CENTROID 40-130 0.826 6.55E-03 -6.23E-05 -0.252 2.34E-01130-360 1.358 -6.81E-03 1.01E-05 0.146 -5.01E-01

Non Compact

1P

BOTTOM 80-200 1.052 -5.20E-03 8.78E-06 0.290 -3.65E-01200-400 0.598 -5.65E-04 -1.48E-07 -0.585 7.94E-01

TOP 80-200 0.878 -7.43E-03 1.83E-05 0.256 -3.59E-01200-400 0.111 5.43E-04 -1.16E-06 -0.058 -5.35E-03

CENTROID 80-200 1.105 -8.63E-03 2.06E-05 0.394 -5.17E-01200-400 0.337 1.63E-04 -9.59E-07 -0.408 4.54E-01

2P

BOTTOM 80-200 0.548 6.16E-04 -5.00E-06 0.283 -3.36E-02200-400 0.898 -2.36E-03 2.36E-06 -0.042 6.92E-02

TOP 80-200 0.909 -7.89E-03 2.07E-05 0.180 -3.01E-01200-400 0.036 1.24E-03 -2.63E-06 -0.130 9.78E-02

CENTROID 80-200 1.045 -6.67E-03 8.90E-06 1.235 -1.85E+00200-400 0.545 -1.33E-03 1.43E-06 -0.317 2.46E-01


Accuracy of the derived equationsa. The predicted moment capacity are close enough to the

corresponding values obtained from the FEM which validates the current design equations.

b. The proposed formulae produce results within the ±15% deviation lines with a maximum deviation 13%.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1Y=X Line+15% Devia-tion-15% Deviation

Mu(FE) / Mp

Mu

(pro

po

sed

mo

del

) / M

p

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1Y=X Line

+15% Deviation

-15% Deviation

Mu(FE) / Mp

Mu

(pro

po

sed

mo

del

) / M

p


Validity of the equations on other sections in range (reliability)

SpanWeb

HeightTop

FlangeLu/rt (r) Compact

nessLoad

Config.Load Pos.

Mu/Mp FEM

Mu/Mp Prop. Model

% Error

6200 520 215 113.24 0.746 Compact 1P Bottom 0.737 0.741 0.559

4500 400 110 173.77 0.283 Compact 1P Bottom 0.532 0.554 3.958

6200 390 175 138.75 0.614 Compact 1P Top 0.293 0.269 -9.135

3500 195 620 73.17 0.687 Compact 1P CG 0.700 0.751 6.818

6800 480 130 221.18 0.394 Compact 1P CG 0.257 0.297 13.354

6200 520 215 113.24 0.746 Compact 2P Bottom 0.880 0.862 -2.093

4500 400 110 173.77 0.283 Compact 2P Bottom 0.742 0.698 -6.272

3500 620 195 73.17 0.687 Compact 2P CG 0.805 0.908 11.348

3800 635 175 90.57 0.614 Noncompact 1P Bottom 0.749 0.694 -7.982

8200 550 110 334.46 0.283 Noncompact 1P Bottom 0.283 0.290 2.468

6500 530 130 214.84 0.394 Noncompact 1P Top 0.154 0.150 -2.701

7450 570 155 201.76 0.525 Noncompact 1P CG 0.265 0.241 -9.857

3800 635 175 90.57 0.614 Noncompact 2P Bottom 0.822 0.724 -13.59

3900 395 100 168.56 0.229 Noncompact 2P CG 0.405 0.360 -12.70

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1Y=X Line

+15% Error

-15% Error

Mu(FE) / Mp

Mu

(pro

po

se

d m

od

el)

/ M

p


Comparison Between Standards and Specificationsa. Case of Single Span Loading:

As the degree of mono-symmetry (r) increases, the AISC (2010) results move from being the most conservative for the case of degree of mono-symmetry (r) = 0.20 to the least conservative results at the degree of mono-symmetry (r) = 0.80, compared to the parametric study, where failure takes place at the plastic stage.

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FEM, BOTTOM FEM, TOPFEM, CENTROID AISCBS DESIGN MODEL, BOTTOMDESIGN MODEL, TOP DESIGN MODEL, CENTROID

Lb/rt

MU/

MP

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Lb/rt

MU/

MP

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Lb/rt

MU/

MP



BS (5950-1:2000) show reasonable agreement with the loading of: Bottom flange case for (r) = 0.20 and 0.35 at all span lengths studied. Top flange case for (r) = 0.50, 0.65 and 0.80 at spans 3000 and 4000 mm Centroid case for (r) = 0.50, 0.65 and 0.80 at span 5000mm. Bottom flange case for (r) = 0.50, 0.65 and 0.80 at spans 6000, 7000 and

8000mm.

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Lb/rt

MU/

MP

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Lb/rt

MU/

MP



AISC specification give lower results than the BS code for (r) = 0.20 and 0.35 and higher results for (r) = 0.80

The comparison between AISC and BS for (r) = 0.50 and 0.65 depends mainly on the ratio (Lb/rt) where AISC give results higher than the BS code for spans 3000, 4000 and 5000 and lower results for spans 6000, 7000 and 8000

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Lb/rt

MU/

MP

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Lb/rt

MU/

MP


Comparison Between Standards and Specificationsb. Case of Two-Span Loading:

The results calculated for both the AISC (2010) and the BS (5950-1:2000) for this case are identical to single span loading case, because the capacity of the beam depends on the cross section properties rather than the loading case.

BS (5950-1:2000) results lie in between the results of the bottom and top flange loading of the proposed model results unlike the case of single span loading for (r) = 0.20 and 0.35.

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FEM, BOTTOM FEM, TOPFEM, CENTROID AISCBS DESIGN MODEL, BOTTOM

Lb/rt

MU/

MP

AGENDA


Conclusions

1. 540 models were analyzed to study the behavior of mono-symmetric continues I-beams after being verified.

2. Beams with degree of mono-symmetry (r) = 0.20 and 0.35 failed by Lateral Torsional Buckling.

3. Beams with degree of mono-symmetry (r) = 0.50 for short spans failed by Tension Flange Yielding.

4. Beams with degree of mono-symmetry (r) = 0.50 for long spans failed by interaction of Tension Flange Yielding and Lateral Torsional Buckling.

5. Beams with degree of mono-symmetry (r) = 0.65 and 0.80 for short spans failed by Local Buckling.

6. Beams with degree of mono-symmetry (r) = 0.65 and 0.80 for long spans failed by interaction of Compression Flange Yielding and Lateral Torsional Buckling.

Conclusions (cont’d)

7. Handy design model, based on the parametric study, was developed using a normalized equation to cover all studied cases.

8. Span lengths and degrees of mono-symmetry (r) were incorporated in the design model.

9. The effect of load position and load case was introduced in the design model; top flange, bottom flange or centroid.

10. The design model was verified against other sections in the range with maximum deviation of 13%.

11. The difference between the ultimate moment capacity of a section with top loading and a section with bottom loading varies from 40% for (r) = 0.20 and = 0.35 to 80% for (r) = 0.50, 0.65 and 0.80.


12. The AISC results are most conservative at (r) = 0.20 and least conservative at (r) = 0.80.

13. BS (5950-1:2000) show reasonable agreement with the loading of:

Bottom flange case for (r) = 0.20 and 0.35 at all span lengths studied. Top flange case for (r) = 0.50, 0.65 and 0.80 at spans 3000 and 4000 mm Centroid case for (r) = 0.50, 0.65 and 0.80 at span 5000mm. Bottom flange case for (r) = 0.50, 0.65 and 0.80 at spans 6000, 7000 and

8000mm.

14. AISC specification give lower results than the BS code for (r) = 0.20 and 0.35 and higher results for (r) = 0.80.


15. The comparison between AISC and BS for (r) = 0.50 and 0.65 depends mainly on the ratio (Lb/rt) where AISC give results higher than the BS code for spans 3000, 4000 and 5000 and lower results for spans 6000, 7000 and 8000.

16. The effect of loading case and position is not captured in the calculation of the specifications in study as the beam is depending on the cross section properties which are the same in both loading cases.

AGENDA


Recommendations for Future Study

1. Consider different cases of loading (moving and distributed loads).

2. The effect of different mechanical properties of the material.

3. The effect of different span ratio of a continuous beam on the ultimate moment capacity of an I-section.

4. The effect of different lateral restraint on the ultimate moment capacity.

5. The effect of flange curtailment on the ultimate moment capacity.

6. The effect of web thickness on the ultimate moment capacity.

THANK YOU

defense_thesis_ramy_gabr_06-22-2014_final-with notes - copy

Documents

future study

literature review trahair

flange trahair

symmetric beams

buckling capacities

small flange

maximum moment

larger flange