steel structures

Serban

DIMA

Bogdan

STEFANESCU

STEEL STRUCTURES

– basic elements –

DEDICATION

This book is a tribute to the memory of Professor Dragos GEORGESCU, the author

of the curricula of STEEL STRUCTURES courses at the French and English

Department of the Technical University of Civil Engineering of Bucharest.

STEEL STRUCTURES – basic elements

5

CONTENTS

Chapter 1 : STEEL STRUCTURES ……………………………....…. 11

1.1. TYPES OF CONSTRUCTION WORKS WITH STEEL STRUCTURES ..… 11

1.2. DESIGN. FABRICATION. ERECTION ………………...…..………………… 13

1.3. BASIS OF DESIGN ………………………………...……………...…………… 14

1.4. STRUCTURAL MEMBERS ………………...…………………...…………….. 16

1.5. STRUCTURAL SYSTEMS ………………...…………………...……………... 22

1.5.1. Structural philosophy …………………………….………………….…….. 22

1.5.2. Structures with a single column …………………………….…..……….. 23

1.5.2.1. Structural philosophy …………………………..…………………...……… 23

1.5.2.2. Structural systems ……………………………..…………………………… 26

1.5.3. Structures with a number of columns in a line ………………….…….. 27

1.5.4. Structures with a number of orthogonal column lines ……..………... 28

1.5.4.1. Structural philosophy ………………….…………………………………… 28

1.5.4.2. Single storey buildings …………………………..………….……………… 29

1.5.4.3. Multi-storey buildings …………………………..………….…..…………… 30

Chapter 2 : RELIABILITY OF STEEL STRUCTURES ………....…. 33

2.1. GENERAL ASPECTS ..………………………………………………………… 33

2.2. ALLOWABLE STRESS METHOD (DETERMINISTIC METHOD) ….…….. 33

2.3. PROBABILISTIC ANALYSIS OF RELIABILITY ….………………………... 38

2.3.1. Probabilistic bases ……………………………….………………..……….. 38

2.3.2. Resistance randomness ……………….……….…….…….…...…..…….. 38

2.3.3. Force randomness …………………….………….………………..……….. 41

2.3.4. Safety analysis ……………………..………….…….…….……...…..…….. 42


6

2.3.5. Probabilistic methods …………………….………….…………....……….. 43

2.3.6. The semi-probabilistic limit states method (level 1) ………………….. 43

2.3.6.1. Limit states …………………………..……….…….…………..…………… 43

2.3.6.2. Actions …………………………..……………..……………….…………… 44

2.3.6.3. Design values of actions …………………………..………….…………… 45

2.3.6.4. Load combinations (combinations of actions) ……..…………….…....… 45

2.3.6.5. Material design properties ………….….……………..…….…...………… 48

2.3.6.6. Ultimate limit state …………………………..………..……….…….……… 49

2.3.6.7. Serviceability limit state …………………………..………….……….….… 50

2.3.6.8. Conclusive remarks ………….………………..….……….….….………… 51

2.3.7. The reliability index ββββ method (level 2) ………………………………….. 51

2.3.8. The probabilistic method (level 3) ……….………….….……….……….. 55

Chapter 3 : STRUCTURAL STEEL ………....……………………….. 57

3.1. MATERIALS ..…………………………………………………………………… 57

3.2. CHEMICAL COMPOSITION. CRYSTALLINE STRUCTURE ...…………… 58

3.3. STEEL MAKING ...……………………………………………………………… 63

3.4. ROLLING PROCESS INFLUENCE ...………………………………………… 64

3.5. RESISTANCE AGAINST BRITTLE FRACTURE ...………………………… 66

3.6. INFLUENCES OF TEMPERATURE ON THE PROPERTIES OF STEEL .. 67

3.7. FATIGUE BEHAVIOUR ...……………………………………………………… 67

3.8. CORROSION ..…………………….…………....………….…………………… 69

3.9. SHAPES ..……………………………………...………………………………… 70

3.9.1. General ……………………………………………………………….……….. 70

3.9.2. Structural imperfections ……………….………………………….……….. 71

3.9.2.1. Residual stresses …………………………..…...…………….….………… 71

3.9.2.2. Non-homogeneity of mechanical properties …….……….…..…..……… 72

3.9.3. Geometrical imperfections ….…………….….…………………..……….. 73

3.10. STRUCTURAL STEEL REQUIREMENTS …….…………………………… 73

3.11. STRUCTURAL STEEL GRADES …….….………………....….…………… 79


7

Chapter 4 : CONNECTING DEVICES ………....……………...…….. 81

4.1. GENERAL ..……………………………………………………………………… 81

4.2. WELDING ..………………….………….…….……….………………………… 81

4.2.1. General ……………………………………………………………….……….. 81

4.2.2. Weldability ……………………………………..…………………….……….. 82

4.2.3. Structural welding process and materials ……………….…….……….. 83

4.2.4. Metallurgic phenomena in the welding process ……………….…..….. 85

4.2.5. Thermal phenomena in welding process …….….…….…….….…..….. 86

4.2.6. Welding positions ……………………………………………..…….…..….. 87

4.2.7. Weld details …………………...………………………………..…….…..….. 88

4.2.8. Welding defects …………….……...….…………….…….…..…….…..….. 89

4.2.9. Weld inspection methods …………….……....……...………..…..…..….. 90

4.2.10. Strength of welded joints …………………...……...………..…...…..….. 91

4.2.10.1. Butt welds ………………..….…………………………………………...… 92

4.2.10.2. Fillet welds ………………..….……...……...…………………………...… 95

4.3. BOLTS ..……………………………….…….…………………………………… 102

4.3.1. General ……………………………………………………………….……….. 102

4.3.2. Classification of bolts ……………………………………….…….……….. 103

4.3.3. Behaviour and design resistance of bolts ……………………...…..….. 104

4.3.3.1. Loading and tightening ………………..….……………………………...… 104

4.3.3.2. Behaviour of normal bolts in shear connections …….…….….……....… 105

4.3.3.3. Behaviour of high strength bolts in slip connections …………….……… 109

4.3.3.4. Behaviour of bolts in tension …………….………………………………… 111

4.3.3.5. Design resistance of bolts according to STAS 10108/0–78, C133–82 .. 112

4.3.4. Spacing of holes ……………………...…..…………………………………. 113

4.3.5. Categories of bolted connections according to EUROCODE 3 …….. 114

4.3.6. Examples of calculation ………………………………….………...…..….. 115

4.3.6.1. General aspects ………………...…..……….…………………………...… 115

4.3.6.2. Connection loaded only in its plane …….………...……….……….......… 116

4.3.6.3. Connection loaded normally on its plane ……….………..…….……...… 118


8

Chapter 5 : DESIGN OF STRUCTURAL MEMBERS ……….....….. 121

5.1. BASIS OF DESIGN ..…………………………………………………………… 121

5.1.1. Design method …………….………………………….…………….……….. 121

5.1.2. Stability of steel structures …………….………………………………….. 121

5.1.3. Cross-section particularities …………….……………………….……….. 122

5.1.4. Classification of cross-sections …………..….….………….……….…... 123

5.1.5. Elastic and plastic design ………….….…..…..……….…….……..…….. 126

5.2. TENSION MEMBERS ..………………………………………………………… 129

5.2.1. General …………….…………….…………….……….…………….……….. 129

5.2.2. Types of single and built-up members …….………..………….……….. 129

5.2.3. Calculation ……….….………….…………….……….…………….……….. 131

5.3. COMPRESSION MEMBERS ..………………………………………………… 132

5.3.1. General …………….…………….…………….……….…………….……….. 132

5.3.2. Buckling ………………….……………….…….….……….………...…..….. 133

5.3.2.1. Buckling and local buckling ………………....…..……….……………...… 133

5.3.2.2. Forms of buckling …………………………....…..……….……………...… 134

5.3.2.3. Approach methods ………….………………....…..……..……………...… 134

5.3.2.4. Bifurcation and divergence of equilibrium ………….…………..……...… 137

5.3.2.5. The general equation of stability ………….……………………..……...… 138

5.3.2.6. Flexural buckling ………….………………………..……………..……...… 141

5.3.2.7. Buckling curves ………….………….……………...……………..……...… 145

5.3.3. Practical design of compressed members …….…….….……...…..….. 151

5.3.3.1. Cross-section philosophy ………………....…..…...…….……………...… 151

5.3.3.2. Types of members in compression …….………….....…..….....……...… 152

5.3.3.3. Connecting elements of a compressed member ………………..........… 153

5.3.3.4. Checking procedure for members in compression ……………….......… 156

5.4. FLEXURAL MEMBERS ..…………….…………….….……….……………… 164

5.4.1. General …………….…………….…………….……….…………….……….. 164

5.4.2. Beams and plate girders …….…….….……...………………………...….. 164

5.4.2.1. Cross-section philosophy ………………....…..…...…….……………...… 164

5.4.2.2. Behaviour of beams ………………............…..…...…….……………...… 168

5.4.2.3. Main checks for a member in bending ………….….…......…………...… 171


9

5.4.2.4. Procedure for sizing of the cross-section ………………....…………...… 178

5.4.3. Lattice girders …….…….….……………….....………………………...….. 180

5.4.3.1. General ………………....…..…...………………………...……………...… 180

5.4.3.2. Geometric schemes ………………....…………………...……………...… 182

5.4.3.3. Cross-sections of bars ………………....………………...……………...… 183

5.4.3.4. Joint details ……………………………...………………...……………...… 183

5.4.3.5. Calculation of efforts in bars ……………....……….…....……………...… 184

5.4.3.6. Main checks ……………...…………………..…………...……………...… 184

BIBLIOGRAPHY ……….....…………………………………………..... 185


10

1. STEEL STRUCTURES

11

Chapter 1

STEEL STRUCTURES

1.1. TYPES OF CONSTRUCTION WORKS WITH STEEL STRUCTURES

Construction works is the general term including both buildings (apartment

houses, offices, schools, etc.) and civil engineering works (TV towers, tanks, etc.).

Structure (Structural system) is an assemblage of load carrying structural

members joined to provide the required strength, stiffness and ductility of a

construction work.

Cladding is the exterior covering of the structure.

By cladding (roof + side wall) a certain volume is separated from the atmosphere.

This separation is made to create in the interior all the conditions required by a

human activity that can not be developed in open air.

Construction works with steel structures can be classified in three types, depending

on the presence or role of cladding:

1. Type S.C. (Construction work = Structure + Cladding)

This is the most general type (Fig. 1.1).

Fig. 1.1. Type S.C. construction work

This type (S.C.) of construction works is largely represented by all kind of buildings:

• one storey industrial buildings (Fig. 1.2a);

• apartment houses, offices, hotels, schools, colleges etc. (Fig. 1.2b);

• sport halls, theatres etc.

Side wall Roof Structure Cladding

1. STEEL STRUCTURES

12

( a ) ( b )

Fig. 1.2. Examples of type S.C. construction works

2. Type S. (Construction work = Structure only)

This type (Fig. 1.3) is represented by all kind of civil engineering works when

cladding is not necessary, like:

• transmission towers (Fig. 1.3a);

• pipe-lines (Fig. 1.3b) etc.

( a ) ( b )

Fig. 1.3. Examples of type S. construction works

Cladding

Structure

Cladding

Structure

1. STEEL STRUCTURES

13

3. Type C. (Construction work = Structural cladding)

This type (Fig. 1.4) is represented by all kind of civil engineering works when

cladding is structural, like:

• tanks (Fig. 1.4a);

• spherical vessels (Fig. 1.4b);

• chimneys (Fig. 1.4c);

• silos etc.

( a ) ( b ) ( c )

Fig. 1.4. Examples of type C. construction works

1.2. DESIGN. FABRICATION. ERECTION

A steel structure results by assembling on site a number of various structural

members, like beams, columns etc. (Fig. 1.5) prefabricated in fabrication shops.

Fig. 1.5. Examples of structural members

Truss Beam

Beam-column Column

H

p p

p

N N

M

M M

N H

Q Q Q M

1. STEEL STRUCTURES

14

The main steps to realise a steel structure are:

• design of the structure;

• fabrication of structural members in fabrication shops (using plates and

profiles which are produced in steel works);

• transport of structural members on site;

• erection of the structure by assembling structural members on site.

All the technical activities involved, meaning design, production of shapes and

plates, fabrication of the structural members and erection must comply with

requirements contained in principles and application rules provided by the codes.

1.3. BASIS OF DESIGN

A structure shall satisfy the following requirements during its intended lifetime:

1. It must sustain with appropriate degrees of reliability all actions to occur during its

construction and intended use.

2. It must remain fit for its required use.

This usually leads to two types of requirements to be checked:

• strength requirement – in order to resist all actions to occur during its intended

lifetime;

• stiffness requirement – in order to remain fit for its required use (allowable

displacements).

Fig. 1.6. Main steps to create and analyse the model of a structure

Actual configuration

Calculation scheme

Actions

Effects of actions

h h

L L

IC IC

IB

p H x

y y

z

z

x

Q+ M+

N+

1. STEEL STRUCTURES

15

The strength requirement is expressed by

dd CE ≤ ( 1.1 )

In eq. (1.1) and in figure 1.6:

Ed is the design value of that effect of actions:

N – axial force (+ tension; - compression);

M – bending moment;

Q – shear force;

Mt – torsion moment.

N, M, Q, Mt are efforts and they are effects of external forces.

Cd is the design capacity of the structural member, for the considered effort N, M,

Q or Mt.

The stiffness requirement is expressed by:

a∆≤∆ ( 1.2 )

where:

∆∆∆∆ – the calculated deformation;

∆∆∆∆a – the allowable deformation.

Example

Fig. 1.7. Example

Strength requirement

8Lp

ME2

Sdd⋅== (calculated)

RWMC Rdd ⋅== (calculated)

p

L

f

MSd

1. STEEL STRUCTURES

16

dd CE ≤ RdSd MM ≤ RW8Lp 2

⋅≤⋅

Stiffness requirement

EILp

3845

f4⋅⋅==∆ (calculated)

300L

faa ==∆ (allowable)

a∆≤∆ aff ≤ 300L

EILp

3845 4

≤⋅⋅

In the above relations:

W – section modulus of the cross-section;

R – design strength of the steel grade that is used;

EI – stiffness of the cross-section of the member.

The strength requirements and the stiffness ones can be found in codes of

practice as principles and application rules.

Principles comprise:

• general statements and definitions for which there is no alternative;

• requirements and analytical models for which no alternative is permitted.

Application rules, usually called recommendations in the codes, are recognised

rules that follow the principles and satisfy their requirements. It is allowed to use

alternative rules, different from the recommendations (application rules) given in the

codes, provided that it is proved that the alternative rules comply with the principles

and provide at least the same reliability.

1.4. STRUCTURAL MEMBERS

Structural members are prefabricated in fabrication shops using a large range

of products for steel construction produced in steel works:

• standard profiles (shapes)

angle I shape (W shape) channel steel pipe etc.

1. STEEL STRUCTURES

17

• rolled plates.

Some built-up elements like plate girders or box sections are fabricated in

fabrication shops, usually by welding.

The main structural members can be classified with respect to the dominant efforts N

(axial force), M (bending moment), Q (shear force), as follows:

1. Beam is a structural member whose primary function is to carry loads transverse

to its longitudinal axis (Fig. 1.8). The dominant effort is M (bending moment).

Fig. 1.8. Beam

Equilibrium relations

Fig. 1.9. Typical stress distribution for a beam

NSd = 0 T – C = 0 T = C ( 1.3 )

MSd ≠ 0 MSd = T ⋅ z MRd ( 1.4 )

where:

MSd – action (bending moment produced by external forces);

MRd – capacity (resistant bending moment);

C – resultant of compression normal stresses on the cross-section;

T – resultant of tension normal stresses on the cross-section.

M

L

p

C

T z

z

x

z

z

y y

MSd0

NSd=0

QSd0

1. STEEL STRUCTURES

18

Remark: The cross-section must be developed (Fig. 1.10) in the plane of the acting

bending moment M in order to increase the resistant bending moment MRd, i.e. in the

plane of the acting forces (greater h → greater z → greater MRd = T ⋅ z).

Fig. 1.10. Typical development of the cross-section

Typical problem: The risk of lateral instability (lateral buckling) (Fig. 1.11a) or local

instability (local buckling) (Fig. 1.11b) is typical for metal (steel or aluminium alloy)

members subjected to bending moment.

( a ) ( b )

Fig. 1.11. Typical instability problems for metal members in bending

Depending on the practical solution adopted for a beam, the following ones are the

most commonly used cross-sections:

1.a. Rolled beam is a structural beam produced by rolling (hot rolling). The most

commonly used shapes (Fig. 1.12) for beams are the following ones:

IPE, HE, HL, HD, HP, W, UB, UC IPN UAP UPN

Fig. 1.12. The most commonly used hot rolled shapes for beams

1. STEEL STRUCTURES

19

1.b. Plate girder (Fig. 1.13) is a built-up structural beam, usually made of welded

rolled plates (sometimes they may be bolted or riveted, especially in the case of

aluminium alloy).

Fig. 1.13. Typical plate girder cross-section

1.c. Lattice girder (Fig. 1.14) is a built-up structural beam made of a triangulated

system of bars subjected to axial forces. It is able to resist forces acting in it’s plane.

Fig. 1.14. Example of lattice girder

MSd ≠ 0 MSd = MRd = C ⋅ h (or MRd = T ⋅ h) ( 1.5 )

NSd = 0 T + D ⋅cosα – C = 0 α

−=cos

TCD ( 1.6 )

h

h

Top chord

Web members

Bottom chord

L

M

C

T

D

1. STEEL STRUCTURES

20

Truss (Fig. 1.15) is a lattice girder used in the roof framing.

Fig. 1.15. Example of truss

1.d. Cold-formed shape (Fig. 1.16) is a cross-section obtained from plates by

bending or by rolling at normal temperature. They are especially used for purlins

(secondary beams of the roof structure).

Fig. 1.16. Examples of cold-formed cross-sections used for beams

2. Column (Fig. 1.17) is a structural member whose primary function is to carry

loads acting in its longitudinal axis. The dominant effort is N.

( a ) ( b )

Fig. 1.17. Examples of columns

Remark: The fact that practically all the compressed structural members are sized by

the buckling resistance of the member is typical for steel structures. In the concrete

structures the loss of stability is an uncommon phenomenon.

For the column in fig 1.17a the strength requirement (1.1) turns into:

( )2e

2

RdSd h2EI

PP⋅

⋅π=≤ ( 1.7 )

External force Critical force

P P

buckling buckling he

1. STEEL STRUCTURES

21

As a result, in order to avoid buckling in any vertical plane, the cross-section

must be developed in its plane, like shown in figure 1.18.

Fig. 1.18. Examples of cross-sections for columns

3. Beam-column (Fig. 1.19) is a structural member whose primary function is to

carry both transverse to longitudinal axis and acting in its longitudinal axis forces.

The dominant efforts are M and N.

Fig. 1.19. Example of beam-column

Remark: The following are typical for the cross-sections used in metal structures:

• the cross-section is preferentially developed in the plane of the acting bending

moment with regard to the strong axis y-y (Fig. 1.20a);

• in the situations when it is necessary, the moment of inertia (second moment of

the area) with regard to the weak axis z-z is improved (Fig. 1.20c).

Beam Column Beam-column

Iy >> Iz Iy Iz Iz is improved by lips

( a ) ( b ) ( c ) Fig. 1.20. Examples of cross-sections for beams, columns and beam-columns

P

P

N M H

h

M = H × h

y y y y y y

z

z

z

z

z

z

lip

1. STEEL STRUCTURES

22

4. Structural wall (Fig. 1.21) is a structural member whose primary function is to

carry both vertical and horizontal forces acting in the plane of the wall.

Fig. 1.21. Example of structural wall

4.a. Vertical bracing (Fig. 1.22) is a structural wall made of a triangulated system of

bars subjected to axial forces.

Fig. 1.22. Example of vertical bracing

1.5. STRUCTURAL SYSTEMS

1.5.1. Structural philosophy

The concept of steel structural system is largely influenced by some

particularities of structural steel as a material and of the behaviour of the structural

members. As a result, steel design is based on its own structural philosophy, which

presents some particularities in comparison with the concept of structural systems in

reinforced concrete, brick or timber.

H

P

H

P P P P P P

H H

1. STEEL STRUCTURES

23

1.5.2. Structures with a single column

1.5.2.1. Structural philosophy

Problem 1 (Fig. 1.23)

Lead to ground (Fig. 1.23a) a vertical force P (gravitational) acting at the level

h from the ground in the plane xOy.

( a ) ( b ) Fig. 1.23. Leading a vertical force to the ground

Solution

Use a vertical bar on the acting line of the force P to connect the point A to the

point B on the ground (Fig. 1.23b).

Remarks

1. This solution is the most economical, thanks to the following:

• the path AB is the shortest one to carry the force P to the ground;

• only the force P is to carry on the load path AB (according to a principle of

structural mechanics, a force translates on its acting line by its value).

2. This solution, corresponding to the case of a vertical force, can also be applied in

the case of an inclined force P.


Lead to the ground a horizontal force H (wind, seismic action, etc.) parallel to

the ground, acting at the level h.

x

h

P point A

y

h

P

A

B

N = P

O

1. STEEL STRUCTURES

24

Fig. 1.24. Leading a horizontal force to the ground

General remark

In accordance with a principle of structural mechanics, a force H displaces

parallel to itself by its value H and a bending moment M. As a result, it is much more

expensive to carry a horizontal force to the ground than to carry a vertical one.

Solution a (Fig. 1.25)

Use a bar transverse to the acting line of the force H to connect the point A to

the point B on the ground.

Fig. 1.25. Solution a for leading a horizontal force to the ground

Remark a

Using this solution, the required area of material to carry a horizontal force H

could be 5 to 10 times (in some cases even more) greater than the required area to

carry the same force acting vertically P = H.

x

O y

h

H

point A

h

H A

B

Q = H

M = H × h

1. STEEL STRUCTURES

25

Solution b (Fig. 1.26)

Use a vertical bracing; the simplest one is a triangulated system.

Fig. 1.26. Solution b for leading a horizontal force to the ground

Remark b

This solution is more economical, because the force H is carried to the ground

by axial forces. For instance, if the force H = P the steel consumption is 2 to 3 times

greater than for the same force P acting vertically, depending on the distance a

between the supports. The greater the distance a is, the arm lever increases and, as

a result, the forces diminish.


Lead to the ground a vertical force P and a horizontal force H parallel to the

ground, acting at the level h from the ground, in the plane xOy.

Fig. 1.27. Leading a horizontal force and a vertical force to the ground

H

h

T C

a

α⋅==

=α⋅+α⋅=

cos2H

TC

HcosTcosCTC

x

y O

h

H

P

point A

1. STEEL STRUCTURES

26

Solutions (Fig. 1.28)

Four possible solutions are presented, based on the previously discussed ones:

• (a) cantilever;

• (b) structural wall (solved as a vertical bracing);

• (c) a triangulated system;

• (d) guyed tower.

The solution (d) represents a combination between (a) and (c). The cables must be

in tension in any loading case so they need to be pretensioned. As a result, the initial

tension in the cables Tinit must be greater than the highest compression CH produced

by the force H. This solution is generally required by high rise TV towers.

( a ) ( b ) ( c ) ( d ) Fig. 1.28. Solutions for leading a horizontal force and a vertical force to the ground

1.5.2.2. Structural systems

Some structural systems based on the solutions presented in figure 1.28 are

shown in figure 1.29. These solutions are developed in order to realise spatial

structures, required both by stability requirements and by the effects of horizontal

forces H acting on any direction.

Fig. 1.29. Structural systems with a single column

H H H H P P P P

T C Compressed bar

Pretensioned cables Tinit > CH

1 1

2 2

3 3 4 4

1 – 1

2 – 2

3 – 3

4 – 4

1. STEEL STRUCTURES

27

1.5.3. Structures with a number of columns in a line

Figure 1.30 shows a steel structure designed to support a pipe-line.

Fig. 1.30. Steel structure for sustaining a pipe-line

This solution is typical for steel structures and is characterized by:

• cantilever columns (C) (Fig. 1.30), sized to resist the vertical forces P and the

horizontal forces H transverse to the line of columns; they also provide the

required stiffness in the transverse plane (each column resists its own P and H

forces); for this reason, their cross-sections are developed in the plane of the

acting bending moment produced by the transverse forces H;

• a vertical bracing (VB) (Fig. 1.30), sized to resist all the horizontal forces ΣΣΣΣL

acting in the longitudinal direction and to provide the required strength and

stiffness in the longitudinal direction;

• two continuous beams (B) (Fig. 1.30), sized to resist the vertical loads P acting

between columns and to transmit them to the columns; at the same time, the

beams connect the columns in the longitudinal direction.

Remarks

The vertical bracing is typical for a steel structure. It is located in the middle of the

structure, to allow a good behaviour of the structure to the effects of temperature

variations. Built-up cross-sections able to resist bending moments in two planes like

those ones in figure 1.31 are to be avoided due to their high cost of fabrication.

Fig. 1.31. Cross-sections that are not very common for steel columns

C C

A

A

ΣL

VB

B H

P

B

A – A

1. STEEL STRUCTURES

28

1.5.4. Structures with a number of orthogonal column lines

1.5.4.1. Structural philosophy

Problem 4

Lead to the ground vertical (P), horizontal (H) and inclined (I) forces acting on

the roof or on the floor of a building (Fig. 1.32).

Fig. 1.32. Leading to the ground forces acting on the roof

Solutions

Figure 1.33 shows three possible solutions, which are compared in table 1.1

from the point of view of their strength, stiffness and ductility properties.

Strength is the resistance to the forces S (N, Q, M, Mt) produced by the loads.

Stiffness is the resistance to the deformations ∆, γ, θ produced by the loads.

Ductility is the capacity to dissipate energy by large plastic deformations.

Fig. 1.33. Possible solutions for leading forces acting on the roof

Solution 1: M.R.F. = Moment Resisting Frame

Solution 2: C.B.F. = Concentrically Braced Frame

Solution 3: E.B.F. = Eccentrically Braced Frame

1

1

plastic hinge

buckling

plastic zone

H I P

1. STEEL STRUCTURES

29

Table 1.1. Comparison among possible solutions

Strength Stiffness Ductility

M.R.F. good poor very good

C.B.F. good very good poor

E.B.F. good good good

1.5.4.2. Single storey buildings

Figure 1.34 shows a typical structure of a single storey industrial building,

based on the structural philosophy discussed above.

Fig. 1.34. A typical steel structure for a single storey industrial building

The structure is composed of:

• transverse MRF, sized to resist vertical (P) and horizontal (H) forces and to

provide the required strength and stiffness in the transverse plane; each MRF

resists its own P and H forces and their cross-sections are developed in the plane

of the acting bending moment M produced by the transverse forces H;

• vertical bracing VB, sized to resist all longitudinal forces ΣΣΣΣL acting in the

longitudinal direction and to provide the required strength and stiffness in the

longitudinal direction;

TRANSVERSE SECTION SIDE VIEW PLAN VIEW

H P

P H

HLB

crane CRG

Pr HTB VB CRG

L

Pr VB

L

RHB

MRF

HTB

1. STEEL STRUCTURES

30

• roof framing, consisting of roof horizontal bracing RHB, composed of horizontal

transverse bracing HTB and horizontal longitudinal bracing HLB, in order to

provide torsional rigidity of the structure and purlins Pr to resist vertical forces

acting on the roof and to transmit them to the MRF;

• crane runway girders CRG, to resist the forces produced by cranes and to

transmit their P and H forces to the MRF and L forces to the VB.

Remark:

Trusses are often used instead of girders for long span buildings. In this case

MRF is composed of columns and trusses, usually pin connected, like in figure 1.35.

Fig. 1.35. A steel structure for a single storey industrial building using trusses

1.5.4.3. Multi-storey buildings

Figure 1.36 shows a modern concept of a multi-storey steel structure

composed of two systems:

• a frame system (F), resisting both vertical (P) and horizontal (H and L) forces; this

could be a moment resisting frame (MRF), a concentrically braced frame (CBF)

or an eccentrically braced frame (EBF);

• a gravitational system, resisting only vertical forces (P).

Rigid diaphragm floors and side frame systems provide the torsional rigidity of the

whole building, which is fundamental for the good behaviour of the structure when

subjected to horizontal loads.

Figure 1.37 shows three very well known present-day performances in high-

rise skyscrapers construction.

Truss (T)

Crane runway girder (CRG)

Purlin (Pr)

Column (C)

1. STEEL STRUCTURES

31

Fig. 1.36. A modern concept of a multi-storey steel structure

Petronas Towers Sears Tower Empire State

452m – 88 floors – 1998 442m – 108 floors – 1974 381m – 1931

Fig. 1.37. Present-day performances in skyscrapers

PLAN 1 1

Frame system (F)

Gravitational system (G)

SECTION 1 – 1

MRF CBF EBF

1. STEEL STRUCTURES

32

Figure 1.38 shows the tallest building in the world, Taipei 101, situated in

Taipei, Taiwan.

Taipei 101

509m – 101 floors – 2004

Fig. 1.38. Present-day tallest building in the world

2. RELIABILITY OF STEEL STRUCTURES

33

Chapter 2

RELIABILITY OF STEEL STRUCTURES

2.1. GENERAL ASPECTS

In order to check the safety of a structure it is necessary to assess whether a

dangerous situation, able to make the structure unusable, might be reached due to

some extreme events. There are three types of methods to make the analysis of

steel structure reliability:

• deterministic methods, which consider all parameters with their deterministic

values;

• probabilistic methods, which consider all parameters and the relations among

them as random variables; they are difficult to carry on and they need a very

sophisticated mathematical procedure; they also need a great amount of data

about loads, material properties etc.;

• semi-probabilistic methods, which use probabilistic models to establish the

values for actions and capacities but they compare them using deterministic

models; most of present day design codes for steel structures use such methods.

Generally, when checking the safety of a structural element or of a whole

structure, the following requirements are to be satisfied:

• strength requirement;

• stiffness requirement.

In some cases, like seismic design, ductility requirements need also to be fulfilled.

2.2. ALLOWABLE STRESS METHOD (DETERMINISTIC METHOD)

In this method the strength requirement is expressed by the following relation:

allσ≤σ ( 2.1 )

In this equation (2.1) the allowable stress σall is given by:


34

c

call

σ=σ ( 2.2 )

where c is a global safety coefficient taking into account the following possibilities:

• actual nominal loads considered in calculating the effective stress σ in equation

(2.1) could be greater than assumed;

• actual nominal yielding stress σc in equation (2.2) could be lower than presumed;

• fabrication and/or erection may produce unfavourable effects.

The stiffness requirement is expressed by the following equation (same as

(1.2)):

a∆≤∆ ( 2.3 )

where ∆ and ∆a are the calculated and the allowable deformation respectively.

Critical remark

The method considers only a simultaneous increase of the loads that can

unfavourably affect a correct analysis of the reliability, especially when permanent

loads (dead loads) are significantly smaller than the imposed ones (live loads).

The following two examples point out the facts that:

• a snowfall is always more dangerous for the structural members of a roof when

the cover is in steel sheeting than for a concrete slab (example 2.1);

• on the same roof structure, the effect of wind suction is always more dangerous

when the cover is very light, like in steel sheeting construction (example 2.2).

Example 2.1.

For the structure shown in figure 2.1 the following are given:

• snow load: s = 1.2 kN/m2

• steel OL37: c = 240N/m2; c = 1,5; 2call mmN160

c=σ=σ

the following are required:

• size the bottom chord of the truss considering two possible solutions:

1. concrete slab cover: 3 kN/m2

2. steel sheeting cover: 0.1 kN/m2

• for the chosen bottom chord shapes, examine the behaviour under the action of a

snow-load increased from 1,2kN/m2 (a) to 2,0kN/m2 (b).


35

Fig. 2.1. Example 2.1

a. Bottom chord sizing →→→→ snow = 1,2kN/m2

Concrete slab cover Steel sheeting cover

Dead load (D):

• concrete slab cover: 3,0kN/m2

• steel structure weight: 0,4kN/m2

Dead load (D):

• steel sheeting cover: 0,1kN/m2


D = 3,4kN/m2 D = 0,4kN/m2

Live load (L):

• snow load: 1,2kN/m2

Live load (L):


Total load

p = D + L = 3,4 + 1,2 = 4,6 kN/m2

Total load

p = D + L = 0,4 + 1,2 = 1,6 kN/m2

Total load on truss

q = p ⋅ Lt = 4,6 × 10 = 46 kN/m

Total load on truss

q = p ⋅ Lt = 1,6 × 10 = 16 kN/m

Bending moments (M)

kNm331282446

8Lq

M22

=×=⋅= kNm115282416

8Lq

M22

=×=⋅=

Axial efforts (N)

kN11043

3312hM

N === kN3843

1152hM

N ===

Cross-section shapes

150 ×××× 150 ×××× 12 A = 69,6cm2 80 ×××× 80 ×××× 8 A = 24,6cm2

Strength check

a2

2

3

mmN159106,69101104

AN σ<=

××==σ a

22

3

mmN156106,2410384

AN σ<=

××==σ

roof cover

3m

side wall

L = 24m

q (kN/m)

Lt = 10m

8Lq

M2⋅=


36

Security level

51,1159240

c c ==σσ= 54,1

156240

c c ==σσ=

b. Bottom chord check →→→→ snow = 2,0kN/m2


p = D + L = 3,4 + 20 = 5,4 kN/m2

q = p ⋅ Lt = 5,4 × 10 = 54 kN/m

kNm388882454

8Lq

M22

=×=⋅=

kN12963

3888hM

N ===

22

3

mmN2,186106,69101296

AN =

××==σ

29,12,186

240c c ==

σσ=

p = D + L = 0,4 + 20 = 2,4 kN/m2

q = p ⋅ Lt = 2,4 × 10 = 24 kN/m

kNm172882424

8Lq

M22

=×=⋅=

kN5763

1728hM

N ===

22

3

mmN1,234106,2410576

AN =

××==σ

0,1025,11,234

240c c ≅==

σσ=

Remark

Under the action of a snow load increased from 1,2kN/m2 to 2,0kN/m2, which

is to be expected to occur in the intended life of the building:

• in the case of a concrete slab cover the safety coefficient decreases from c=1,51,

but a value of 1,29 still remains;

• in the case of steel sheeting cover, practically there is no more load carrying

capacity, as the safety factor decreases from c=1,54 to c=1,0.

This remark underlines a weakness of the method, that the cross-section of the steel

solution was not well sized. A possible increase of the snow load from the value

considered in design to an accidental one has a more unfavourable effect on a light

roof than on a heavy one. This cannot be outlined by the allowable stress method.

Example 2.2.

Check the bottom chord of the truss sized in example 2.1 for the same

structure without side walls (Fig. 2.2) under the action of a wind suction

gw=0,6kN/m2. The check is to be made considering summer time, i.e. without the

effect of snow load.


37



Dead load (D): D = 3,4kN.m2 Dead load (D): D = 0,4kN.m2

Live load (L):

• wind suction: gw = –0,6kN/m2

Live load (L):

• wind suction: gw = –0,6kN/m2

Total load

p = D + L = 3,4 – 0,6 = 2,8 kN/m2

Total load

p = D + L = 0,4 – 0,6 = –0,2 kN/m2

Total load on truss

q = p ⋅ Lt = 2,8 × 10 = 28 kN/m

Total load on truss

q = p ⋅ Lt = –0,2 × 10 = –2,0 kN/m

kNm201682428

8Lq

M22

=×=⋅= kNm1448

240,28Lq

M22

−=×−=⋅=

kN6723

2016hM

N === kN483144

hM

N −=−==

150 ×××× 150 ×××× 12 A = 69,6cm2 80 ×××× 80 ×××× 8 A = 24,6cm2

a2

2

3

mmN6,96106,6910672

AN σ<=

××==σ

Iy = 144,4cm4

Iz = 318cm4

( ) ( )2

452

2yf

y2

y,cr 6000104,144101,2

L

IEN

××××π=⋅⋅π

=

( ) ( )2

452

2yf

y2

z,cr 2400010318101,2

L

IEN

××××π=⋅⋅π

=

Ncr,y = 83kN; Ncr,z = 11,4kN

Ncr = min. (Ncr,y; Ncr,z)= 11,4kN

48,26,96

240Rc c ==

σ= 24,0

484,11

NN

c cr ===

roof cover qg

qw 6m

Lt = 10m

3m


38

Remarks

1. When the wind suction reaches gw = – 0,4kN/m2, the total load for steel sheeting

cover becomes:

p = D + L = 0,4 – 0,4 = 0 kN/m2 N = 0 kN σ = 0 N/mm2;

2. When the wind suction reaches gw = – 0,6kN/m2:

• in the case of concrete slab cover the safety factor increases from 1,51 to 2,48;

• in the case of steel sheeting cover the safety coefficient decreases to 0,24 (<

1,0); a collapse is to be expected.

Conclusions

1. In the allowable stress method any progress in information is difficult to be

considered because all parameters affecting the reliability of the structure are

included in the unique safety coefficient c.

2. Taking into account the above remarks, at present most of the codes replaced

the allowable stress method by the limit state method, which is a semi-

probabilistic method.

2.3. PROBABILISTIC ANALYSIS OF RELIABILITY

2.3.1. Probabilistic bases

A more rational approach to analyse the problem of structural safety is a

probabilistic one. In such a model of analysis, all the parameters whose uncertainty

can influence the reliability of structures, especially those ones concerning

resistance and loads, are considered as random variables.

2.3.2. Resistance randomness

The strength capacity R(s) of a structural member with respect to a certain

internal force S (N, M, Q) may be expressed in a general form by:

( ) ( )cR,fsR Ω= ( 2.4 )


39

where ΩΩΩΩ is the cross-sectional characteristic corresponding to the internal force S,

i.e.:

Ω = A for members in tension;

Ω = W for members in bending.

For industrially fabricated steel structural members, the cross sectional

characteristic Ω may be considered as a deterministic value. The yield stress c

must be considered as a random variable.

The following steps are to be followed to define the random variable x = c:

• consider the results on a sample of n = Σni tensile specimen tests (i.e. n values of

yield stress c);

• according to the values given in table 2.1, draw the histogram in figure 2.3,

noticing that the normalized area of any rectangle on the histogram represents

the ratio:

==i

iii n

nnn

f ( 2.5 )

where ni is the number of samples satisfying the condition:

c,i < x ≤ c,i + ∆c ( 2.6 )

where ∆c = 20 N/mm2 as shown in figure 2.3.

Table 2.1. Example of values of the yielding limit c

Results association

Frequency of

results

Calculation • mean value xm (N/mm2) • dispersion D (N2/mm4)

Interval of

association

Interval central

values xi

Absolute ni

Relative fi

fi xi

(xi – xm)2

fi (xi – xm)2

220 ÷ 240 230 20 0.05 11.5 4140.923 207.0461 240 ÷ 260 250 19 0.0475 11.875 1966.923 93.42882

260 ÷ 280 270 59 0.1475 39.825 592.9225 87.45607

280 ÷ 300 290 140 0.35 101.5 18.9225 6.622875 300 ÷ 320 310 101 0.2525 78.275 244.9225 61.84293

320 ÷ 340 330 40 0.1 33 1270.923 127.0923 340 ÷ 360 350 21 0.0525 18.375 3096.923 162.5884

n = 400 Sfi = 1,0 xm= 294.35 D=746.0775 s = (D)0,5 = 27.31442


40

0 5% 5%15%

35%25%

10% 5%

00.10.20.30.4

220 240 260 280 300 320 340 360 Fig. 2.3. Histograms corresponding to the values in table 2.1

It is to observe that any rectangle fi represents the relative frequency of the

results (simple probability) and in this case the normalized area of the whole

histogram is:

1fi = ( 2.7 )

• calculate the mean value:

=

⋅=n

1iiim xfx ( 2.8 )

(for the case in table 2.1, xm = 294N/mm2)

• calculate the dispersion:

( )=

−⋅==n

1i

2mii

2 xxfsD ( 2.9 )

(for the case in table 2.1, D = 746N2/mm4)

• calculate the standard deviation:

( )=

−⋅=n

1i

2mii xxfs ( 2.10 )

(for the case in table 2.1, s = 27,3N/mm2)

The values xm and s define the random variable.

The histogram in figure 2.3 may be represented by the normal (Gaussian)

function of probability density described by (Fig. 2.4):

( )2

m

sxx

21

e2s

1xf

−⋅−

⋅π⋅

= ( 2.11 )

The characteristic value of the yield stress c may be defined in a probabilistic

manner by the following relation:

skmk ⋅−σ=σ ( 2.12 )


41

Codes usually accept k = 2, which represents a probability of 2,28% (inferior

fractil p) that the yield stress will not be inferior to k. It means:

s2mk −σ=σ ( 2.13 )

The fractil p is defined as that value of the yield stress for which there is a probability

p for the yield stress to be inferior to that value.

By noting:

mxs

v = ( 2.14 )

where v is the coefficient of variation, equation (2.13) becomes:

( )v21mk −σ=σ ( 2.15 )

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

Fig. 2.4. Gaussian function of probability density for the yielding limit randomness

2.3.3. Force randomness

The internal force S(Fk) in a certain cross-section of a structural member, with

regard to the type of load and the structural model of calculation, may be written as:

S(Fk) = Ψ(L) ( 2.16 )

where:

L represents the acting loads;

Ψ are formulas derived from accepted principles of structural model of calculation.

Example:

For a simply supported beam, the maximum bending moment is:

f(x)

inferior fractil ( p = 2.28% )

x = c

k ks m


42

8Dq

MF2

max⋅==

In this case, the load L = q is considered to be the random variable:

x = L = q

( ) x8

Dx

2

⋅=Ψ

A histogram may be drawn in the same way as described for steel randomness,

determining the mean value Fm and the standard deviation s for loads (Fig. 2.5).

0

0.005

0.01

0.015

0.02

Fig. 2.5. Gaussian function of probability density for force randomness

Accepting the formula Ψ as deterministic, equation (2.16) becomes:

S(Fk) = S(L) ( 2.17 )

The characteristic value Fk, depending on the loads, may be written as:

skFF mk ⋅+= ( 2.18 )

Codes usually accept k = 1,645, corresponding to a 5% probability for the value Fk to

be exceeded (superior fractil p).

2.3.4. Safety analysis

Basically, to assess the safety of a structure in the probabilistic concept

means to check that the probability p of exceeding a given limit state is not greater

than an a priori chosen probability pu, depending on the consequences of reaching

that limit state (Fig. 2.6).

p ≤ pu ( 2.19 )

f(F)

superior fractil

F Fm

Fk ks


43

0

0.005

0.01

0.015

0.02

Fig. 2.6. Example of safety analysis

2.3.5. Probabilistic methods

Basically, three methods are to be considered:

• the semi-probabilistic method (level 1);

• the reliability index ββββ method (level 2);

• the exact probabilistic method (level 3).

2.3.6. The semi-probabilistic limit states method (level 1)

2.3.6.1. Limit states

A limit state can be defined as the situation when one of the criteria

governing the design performance of the structure is no more fulfilled.

There are two categories of limit states:

1. ultimate limit states, which are related to the maximum carrying capacity of the

structure (or part of the structure). It is to consider here:

• strength capacity;

• loss of equilibrium;

• formation of plastic mechanism;

• general and/or local stability.

f(S) f(R)

S, R

f(S)

f(R)

P


44

2. serviceability limit states, which refer to the normal use of the structure. The

following are to be considered:

• deformations affecting the use, appearance or efficiency of the structure;

• local damage (local buckling in some cases, cracks in weld seams, etc.);

• vibrations able to lead to resonance phenomena.

2.3.6.2. Actions

An action (F) is any cause producing internal efforts S (meaning N, M or Q)

and deformations ∆∆∆∆ (f, θ or γ). Actions (F) can be classified:

1. by their origin:

• direct actions – forces (loads) applied on the structure;

• indirect actions – imposed deformations (temperature effects, support

settlements, etc.)

2. by their time variation (according to the Romanian code of actions):

• permanent (Pi) – they act continuously, practically with the same intensity, during

the entire lifetime of the structure (e.g. self-weight of the structure (dead loads));

• quasi-permanent (Ci) – they act at high intensity for long periods, or they act very

frequently (e.g. some quasi-fixed fittings or equipment);

• variable (Vi) – their intensity varies with time or they can miss for long periods

(e.g. imposed (live) loads, wind action, snow action, etc.);

• exceptional (Ei) – they act very rarely during the lifetime of the structure (e.g.

seismic action, explosions, etc.).

3. by their spatial variation:

• fixed actions – position and direction never change (e.g. self-weight);

• free actions – changes may occur in value, position or direction (e.g.

imposed loads, wind load, snow load, seismic action, etc.).

4. by their nature and/or the structural response:

• static actions – action that does not cause significant acceleration of

the structure or structural members;

• dynamic actions – action that causes significant acceleration of the

structure or structural members.


45

2.3.6.3. Design values of actions

The design value Fd of an action is expressed by:

iiid FnF ⋅= ( 2.20 )

where:

Fi is the characteristic value of that action (2.18);

ni is the partial safety factor for the action Fi, being Fi = (Pi, Ci, Vi, Ei).

2.3.6.4. Load combinations (combinations of actions)

1. According to the Romanian code STAS 10101/0A-77, two design situations are

considered:

• Fundamental combination

⋅⋅+⋅+⋅ iigiiii VnnCnPn ( 2.21 )

• Special combination

1idiii EVnCP +⋅++ ( 2.22 )

In equations (2.21) and (2.22):

ng is a factor taking into account the probability of simultaneous action of a

number of variable actions (Vi) at their highest intensity:

• ng = 1 for one Vi;

• ng = 0,9 for two or three Vi;

• ng = 0,8 for four or more Vi.

nid is a factor representing the long lasting part of a variable action; ni

d < 1.

The ultimate limit states are usually examined considering the effects of the

design values of actions, while for serviceability limit states the characteristic

values of actions are generally used.

2. Structural EUROCODES is a programme for establishing a set of harmonised

technical rules for the design of construction works in Europe. In a first stage,

they are intended to be an alternative to the national design codes and in the

end, they will replace the national rules. The Structural Eurocode programme

comprises the following standards, each one consisting of a number of Parts:


46

EN 1990 Eurocode 0: Basis of structural design

EN 1991 Eurocode 1: Actions on structures

EN 1992 Eurocode 2: Design of concrete structures

EN 1993 Eurocode 3: Design of steel structures

EN 1994 Eurocode 4: Design of composite steel and concrete structures

EN 1995 Eurocode 5: Design of timber structures

EN 1996 Eurocode 6: Design of masonry structures

EN 1997 Eurocode 7: Geotechnical design

EN 1998 Eurocode 8: Design of structures for earthquake resistance

EN 1999 Eurocode 9: Design of aluminium structures

The design situations to be considered according to EN 1990 [10] are:

• persistent situations, related to the normal use of the structure;

• transient situations, which can appear during construction or repair;

• accidental situations, like storm, fire action etc.;

• seismic situations (earthquakes).

According to the European code [10], three types of combinations of actions are

to be considered when designing steel members:

• For persistent and transient design situations, the most unfavourable of:

>≥

⋅⋅⊕⋅⋅⊕⋅⊕⋅1i

i,ki,0i,Q1,k1,01,QP1j

j,kj,G QQPG ( 2.23a )

>≥

⋅⋅⊕⋅⊕⋅⊕⋅⋅1i

i,ki,0i,Q1,k1,QP1j

j,kj,Gj QQPG ( 2.23b )

• for accidental design situations

( ) >≥

⋅⊕⋅⊕⊕⊕1i

i,ki,21,k1,21,1d1j

j,k QQorAPG ( 2.24 )

• for seismic design situations

>≥

⋅⊕⊕⊕1i

i,ki,2Ed1j

j,k QAPG ( 2.25 )

In relations (2.23), (2.24), (2.25) the meanings are as follows:

= “the combined effect of”;

⊕ = “combined with”;

Gk,j = characteristic value of permanent action j;

P = relevant representative value of a prestressing action;

Qk,1 = characteristic value of the leading variable action 1;


47

Qk,i = characteristic value of the accompanying variable action i;

Ad = design value of an accidental action;

AEd = design value of seismic action EkIEd AA ⋅= ;

AEk = characteristic value of seismic action;

I = importance factor, given in EUROCODE 8 [11];

G,j = partial factor for permanent action j;

P = partial factor for prestressing actions;

Q,i = partial factor for the variable action i;

0 = factor for combination value of a variable action;

1 = factor for frequent value of a variable action;

2 = factor for quasi-permanent value of a variable action;

= a reduction factor for unfavourable permanent actions G.

The value for and factors may be set by the National annex. Some examples

of recommended values of factors for buildings are given in table 2.2.

Table 2.2. Examples of recommended values of factors for buildings [10]

Action 0 1 2

Imposed loads in buildings:

Category A: domestic, residential areas 0,7 0,5 0,3

Category B: office areas 0,7 0,5 0,3

Category D: shopping areas 0,7 0,7 0,6

Category E: storage areas 1,0 0,9 0,8

Category H: roofs 0,0 0,0 0,0

Snow loads on buildings (Finland, Iceland, Norway, Sweden)

0,7 0,5 0,2

Wind loads on buildings 0,6 0,2 0,0

Temperature (non-fire) in buildings 0,6 0,5 0,0

3. According to the American codes ASCE 7–98 [3] (the latest version is from 2002)

and LRFD [4], the following combinations shall be investigated:


48

( )( ) ( ) ( )

( ) ( )( )

H6,1E0,1D9,0H6,1W6,1D9,0

S2,0L5,0E0,1D2,1R or S or L5,0L5,0W6,1D2,1

W8,0 or L5,0R or S or L6,1D2,1R or S or L5,0HL6,1TFD2,1

FD4,1

r

r

r

⋅+⋅+⋅⋅+⋅+⋅

⋅+⋅+⋅+⋅⋅+⋅+⋅+⋅

⋅⋅+⋅+⋅⋅++⋅+++⋅

+⋅

( 2.26 )

being:

D = dead load (Pi + Ci)

F = load due to fluids with well-defined pressures and maximum heights

Fa = flood load

H = load due to lateral earth pressure, ground water pressure or pressure

of bulk materials

L = live load (Vi imposed loads)

Lr = roof live load

W = wind load

S = snow load

T = self-straining force

E = earthquake load

R = rain water or ice

2.3.6.5. Material design properties

The design value d of a material property is generally defined as:

M

kd

= ( 2.27 )

where:

k = characteristic value of the considered material property;

γM = partial safety factor for the considered material property.

For the design strength R of a structural steel, equation (2.27) becomes:

M

kRγσ= ( 2.28 )

being ( )v21mk ⋅−⋅σ=σ (see equation (2.15)).


49

2.3.6.6. Ultimate limit state

In the limit state method (also called the method of extreme values), the

probabilistic condition in equation (2.19) p < pu is replaced by:

Sd ≤ Rd ( 2.29 )

which means that the maximum probable internal design effort Sd does not exceed

the minimum probable design resistance capacity Rd. In equation (2.29):

Sd = S(niFi) is the internal design effort, calculated using design values of actions

and taking into account respectively the load combinations in eqs.

(2.21) and (2.22) or (2.23), (2.24) and (2.25) or (2.26), depending on

the code;

d = (Rk/γM) is the corresponding design resistance, calculated using the

design strength of steel.

Example 2.3.

Size the bottom chord of the truss in example 2.1 according to the Romanian

codes, using OL37 with R = 220 N/mm2 (Fig. 2.7).



Dead load (D):

• cover concrete slab: 3,0kN/m2


Dead load (D):

• steel sheeting cover: 0,1kN/m2


D = 3,4kN/m2 D = 0,4kN/m2 Live load (L):


07,12,14,3

4,02,2SD

4,02,2ns =×−=⋅−=

Live load (L):


07,22,14,0

4,02,2SD

4,02,2ns =×−=⋅−=

q

h = 3m N

L = 24m

Lt = 10m


50

Total load

p = ndD + nsS =

= 1,1 × 3,4 + 1,07 × 1,2 = 5,02 kN/m2

Total load

p = D + L = 0,4 + 1,2 =

= 1,1 × 0,4 + 2,07 × 1,2 = 2,92 kN/m2

Total load on truss

q = p ⋅ Lt = 5,02 × 10 = 50,2 kN/m

Total load on truss

q = p ⋅ Lt = 2,92 × 10 = 29,2 kN/m

Bending moments (M)

kNm4,36148

242,508Lq

M22

=×=⋅= kNm4,21028

242,298Lq

M22

=×=⋅=

Axial efforts (N)

kN8,12043

4,3614hM

N === kN8,7003

4,2102hM

N ===

Cross-section shapes

120 ×××× 120 ×××× 12 A = 55 cm2 100 ×××× 100 ×××× 10 A = 38,4 cm2

Strength check

RmmN2191055

108,1204AN 2

2

3

<=×

×==σ RmmN5,182104,38108,700

AN 2

2

3

<=××==σ

Remarks

1. The 150×150×12 shape sized by the allowable stress method could be

reduced to 120×120×12.

2. The 80×80×8 shape sized by the allowable stress method must be increased to

100×100×10.

2.3.6.7. Serviceability limit state

The most common serviceability limit state to be checked is the deformation

check. It will be verified that:

∆d ≤ ∆a ( 2.30 )

where:

∆d = ∆(Fi) is the design deformation, calculated using the characteristic (nominal)

values of actions;

∆a is an allowable deformation given in codes or requested by the owner.


51

2.3.6.8. Conclusive remarks

1. At present, the limit state method is the design method provided in most of the

important codes.

2. It represents a more accurate model compared to the allowable stress method

because it separates the material randomness from the load randomness and it

accepts different approaches for different types of loads.

2.3.7. The reliability index ββββ method (level 2)

In a general form, equation (2.29) becomes at limit:

Sd = Rd ( 2.31 )

Equation (2.31) may be written:

• in the subtract model Rjanitin – Cornell as:

E = Sd – Rd = 0 ( 2.32 )

• in the logarithmic model Freudenthal – Rosenblueth as:

0RS

lnEd

d == ( 2.33 )

In equations (2.32) and (2.32) E = 0 is the reliability function, expressing (Fig. 2.8):

• E < 0 : safety range;

• E > 0 : unsafe range;

• E = 0 : the border between safety and unsafe range.

Fig. 2.8. The reliability index β method (level 2)

Xi

Xj

Xn

Unsafe range E > 0

Safety range E < 0

space E

limit hypersurface E = 0

mE

fE

E×sE


52

In the case of a simple internal effort S (= N, M or Q), the reliability index ββββE is

defined as the reverse of the coefficient of variation vE of the function E:

E

E

EE s

mv1 −=−=β ( 2.34 )

Equation (2.34) may be written as:

0sm EEE =⋅β+ ( 2.35 )

In equations (2.34) and (2.35) mE and sE are the mean value and, respectively, the

standard deviation of the function E.

Figure 2.8 shows the physical significance of the reliability index ββββE which

represents in hyper-space E the distance calculated in standard deviations sE

between the point with the abscissa mE and the point with the abscissa E = 0,

located on the random hyper-surface which defines the border between safe and

unsafe behaviour, corresponding to a certain probability pu = p(βE).

The properties of the main statistic characteristics for two variables, X1 and

X2, are given in table 2.3.

Table 2.3. Main statistic characteristics

Y mY DY vY

X1 mX1 DX1 vX1

C C 0 0

C⋅X1 C⋅mX1 C2⋅ DX1 vX1

X1 ± C mX1 ± C DX1 Cmvm

1X

!X1X

±⋅

X1 + X2 mX1 + mX2 DX1 + DX2 2X1X

22X

22X

21X

21X

mmvmvm

+⋅+⋅

X1 – X2 mX1 – mX2 DX1 + DX2 2X1X

22X

22X

21X

21X

mmvmvm

−⋅+⋅

X1 ⋅ X2 mX1 ⋅ mX2 2X2

2X1X2

1X DmDm ⋅+⋅ 22X

21X vv +

X1 / X2 mX1 / mX2 2X2

1X1X2

2X22X

DmDmm

1 ⋅+⋅⋅ 22X

21X vv +


53

For the two models presented above, the reliability index ββββ, taking into

account the relations in table 2.3, becomes:

SR

SRRS DD

mm+

−=β − ( 2.36 )

2S

2R

S

R

RS

ln vv

mm

ln

+

=β ( 2.37 )

Table 2.4 shows a correspondence between the index ββββ and the probability pu of

losing the safety for ββββS–R (S and R – normal distributions) and ββββlnS/R respectively (S

and R – lognormal distributions).

The American code provides the ββββlnS/R index (2.37) and the following targets

were selected:

loading earthquake + live + dead under 75,1loading wind+ live + dead under 5,2

loadingsnow and/or live + dead under sconnection for 5,4loadingsnow and/or live + dead under members for 3

=β=β=β=β

( 2.38 )

Table 2.4. Correspondence between the index ββββ and the probability pu

pu ββββS–R; ββββlnS/R ββββS–R; ββββlnS/R pu

10-1 1,29 1,0 1,59 × 10-1

10-2 2,33 1,5 6,68 × 10-2

10-3 3,09 2,0 2,27 × 10-2

10-4 3,72 2,5 6,21 × 10-3

10-5 4,27 3,0 1,35 × 10-3

10-6 4,75 3,5 2,33 × 10-4

10-7 5,20 4,0 3,17 × 10-5

10-8 5,61 4,5 3,40 × 10-6

10-9 6,00 5,0 2,90 × 10-7

10-10 6,35 5,5 1,90 × 10-8


54

Example 2.4.

Calculate the index βS–R and βlnS/R for the beam in figure 2.9:


Given:

• for the loading:

• mean value: mq = qm = 20kN/m

• variation factor: vq = 0,1

• for the steel in use:

• mean value: mRc = Rm = 294N/mm2

• dispersion: DRc = 744N2/mm4

Calculate for the loading q (S):

23

22qM mmN5,169

1035412600020

W12Lm

Wm

m =××

×=⋅⋅

==σ

42222q

2qq mmN41,020vmD =×=⋅=

42622

4

q

222

mmN3,28741035412

6000D

W12L

qW12

LDD =×

××=⋅

⋅=

⋅

⋅=σ

1,05,1693,287

mD

v ===σ

σσ

Calculations for the material (R):

mRc = 294N/mm2

DRc = 744N2/mm4

093,0294744

mD

vRc

Rc ===σ

Calculate the index βS–R (2.36):

q

L = 6m

12Lq

M2⋅=

I24; Wy = 354cm3


55

0,3877,33,287744

5,169294DD

mmDD

mm

Rc

Rc

SR

SRRS >=

+−=

+−=

+−=β

σ

σ−

Calculate the index βlnS/R (2.37):

0,3033,41,0093,0

5,169294

ln

vv

mm

ln

vv

mm

ln

2222Rc

Rc

2S

2R

S

R

RS

ln>=

+=

+

=+

=βσ

σ

Remarks

1. In this method, the general condition p ≤ pu (2.19) is replaced by:

β ≥ βu ( 2.39 )

which expresses the condition E > 0 (S > R); βu is a risk a priori accepted.

2. At present, this method is used especially to calibrate the partial safety factors in

the limit state method and the coefficients ni in the load combinations; in the

future it is to be expected that the index ββββ method will replace the limit state

method.

3. In order to improve the index ββββ method two tendencies are to be observed in

scientific works:

• a more adequate location of points on the hyper-surface E = 0;

• an extension of the method to various non-normal distributions.

2.3.8. The probabilistic method (level 3)

In this method the reliability analysis is based on the general condition p ≤ pu

(2.19), where p is the probability of E > 0, being:

( ) 0R,,R,R;S,,S,SE n21n21 = ( 2.40 )

a function of random variables Si and Ri and pu an accepted risk, depending on the

consequences.

At present this method is used only in scientific works.


56

3. STRUCTURAL STEEL

57

Chapter 3

STRUCTURAL STEEL

3.1. MATERIALS

Figure 3.1 shows, in term of diagram σ–ε (stress–strain), the behaviour of

various materials in the process of loading and unloading.

Fig. 3.1. Models of behaviour diagrams

Figure 3.1a shows a typical linear elastic behaviour. Loading and unloading follow

the same straight line O – A – O. The elastic deformation disappears just after

unloading. The diagram in figure 3.1b is a non-linear elastic behaviour. The elastic

deformation disappears just after unloading, but the loading and unloading line is no

longer straight, even it remains the same for both processes. Figure 3.1c shows a

viscous behaviour. Loading follows a curve (O – A), while unloading goes on a

different path (A – B – O). The plastic deformation disappears in time (B – O). The

–

–

O O O

O O

O

A A A

A A A’ A A’

A’’ A’’’

B

B B

B B’

( a ) ( b ) ( c )

( d ) ( e )

( f )

3. STRUCTURAL STEEL

58

diagram in figure 3.1d is a typical elasto-plastic behaviour. Loading follows a path

(O – A) which is different from the unloading one (A – B). Unloading goes on a

straight line, the elastic deformation is removed, but the plastic deformation is kept.

Figure 3.1e shows a bilinear elasto-plastic behaviour. Loading (O – A – A’) and

unloading (A’ – B) follow different straight lines. The elastic deformation is removed,

but the plastic deformation is kept. Structural steel is generally modelled as bilinear

elasto-plastic symmetrical material, having a large yielding plateau (Fig. 3.1f).

3.2. CHEMICAL COMPOSITION. CRYSTALLINE STRUCTURE

More than 70 elements in the Mendeleyev Periodic System are metals. Metals

form, together with their alloys, a large multitude of substances, having very diverse

properties. In spite of this diversity, the energy spectrum of electrons in metals

represents the common characteristic that allows all metals to be described from a

single standpoint. Metals and their alloys possess a crystalline structure formed by a

crystalline lattice (Fig. 3.2), where the atoms are placed in the knots of the lattice.

Fig. 3.2. Types of crystalline lattice of steel

The crystal structure of a metal is not strictly periodical in each given instant of time,

because of oscillating motion of some quasi-free valence electrons (Paulus model) in

the lattice. The polarization of the surrounding atoms occurs in the electric field of

these electrons. Atoms become dipoles (i.e. a system with two electric charges +q,

–q) and the interaction of dipoles creates the metallic bond, expressed by attraction

forces between sufficiently distant atoms in the lattice. The greater the number of

valence electrons is, the greater the strength of the metallic bond is.

When molten metal solidifies, the crystallization process begins around some

centres of crystallization (Fig. 3.3). Every centre grows up and so, when the

solidifying process comes to the end, a lot of crystallites called grains appear.

3. STRUCTURAL STEEL

59

beginning advanced close to final

Fig. 3.3. Scheme of the crystallization process

The crystal structure of iron, as a theoretic pure metal, consists of ferrite

grains (αFe). Iron is a mild and very plastic material (c = 120 N/mm2; u = 250 ÷ 330

N/mm2, εu = 50%), with a non-linear diagram σ–ε (Fig. 3.4).

Fig. 3.4. Behaviour diagrams

Carbon steel is an iron-carbon alloy, with a carbon percentage limited at 0,15

÷ 0,25%.

Remarks concerning the chemical composition:

1. Each percentage of 0,1% increases the ultimate stress (u) by 80 ÷ 90 N/mm2

and the yield stress (fy) by 40 ÷ 50 N/mm2.

2. The carbon percentage must be limited at 0,25%, since a percentage superior to

that dramatically diminishes the strain at rupture εu (= εr) and unfavourably affects

weldability.

U

high alloy steel

low alloy steel

carbon steel

iron

F (failure)

perlite

ferrite

Fe

Y E

P

arctg E

y 0,2% ~18% ~30% ~50%

p e

fy u

3. STRUCTURAL STEEL

60

3. The carbon steel grade (0,15% < %C < 0,25%) called mild steel, which is a very

largely used structural steel, is characterised by the following:

• the crystalline structure is composed of ferrite-perlite grains;

• the ferrite grains give a very good plasticity εu (= εr) ≅ 25%;

• the perlite grains (perlite = a mechanical mixture of six parts of ferrite and

one part of cementite – Fe3C), containing an average percentage of

0,90%C, give a good resistance;

• the behaviour is perfectly linear elastic up to the limit of proportionality σp

(point P on σ–ε diagrams) and it is defined by the linear law of Hooke

σ = E⋅ε ( 3.1 )

perlite grains restrain the tendency of ferrite grains to deform plastically;

• the behaviour is still elastic in the range P – E on σ–ε diagrams up to the

elastic limit (point P), but it is no longer linear;

• in the elastic range, the form of the lattice is modified under loading but the

distances between atoms remain in the limit of the full active metallic bond

and thus, after unloading, the lattice regains its initial form (Fig. 3.5);

Fig. 3.5. Elastic deformation of the crystalline structure

• at point Y mild steel undergoes yielding i.e. a large elongation horizontal

plateau, without any increase of the stress;

• plastic deformation is a shear irreversible one (Fig. 3.6); a complete tensile

test puts in evidence these phenomena (Fig. 3.7);

Fig. 3.6. Shear deformation of the crystalline structure

a

a a – 2a

a + 1a

P P

3. STRUCTURAL STEEL

61

Fig. 3.7. Tensile test of a standard specimen

A0 = initial cross-section area of the specimen;

d0 = initial diameter of the cross-section of the specimen;

L0 = initial distance between gauge points;

Au = minimum (ultimate) cross-section area of the specimen after failure;

Lu = final distance between gauge points;

%100L

L100

LLL

00

0uur ×∆=×−=ε=ε ( 3.2 )

• after yielding, a new internal equilibrium is to be made up in a typical strain

hardening (line Y – U) i.e. for a deformation ∆2 > ∆1 a force F2 > F1 is

necessary;

• at point U, the σ–ε diagram falls away and stops at the elongation value εu

(= εr), called the ultimate elongation at rupture, which corresponds to the

specimen breaking;

4. The σ–ε diagram outlines the typical qualities of a structural steel:

• Plasticity = property to keep a total or partial plastic deformation;

• Ductility = plastic quality appreciated by the ratio plastic

ultimate

εε

;

• Tenacity = property to keep large deformation before failure, under great

forces;

A0

Au

d0

L0

Lu = L0 + L

F F

3. STRUCTURAL STEEL

62

5. The Prandtl model is usually accepted for the stress–strain diagram (σ–ε) in order

to simplify calculation (Fig. 3.8).

Fig. 3.8. The Prandtl model for steel behaviour

Low alloy structural steel is steel alloyed with manganese, which has the

quality to increase the ultimate stress and the yield stress, without an important

diminution of plasticity. The percentage of Mn is limited for structural steel to

1,7÷1,8%, because it favours fragility.

Remarks

1. A percentage of 0,22÷0,25%C increases the yield stress from fy = 120 N/mm2 (of

iron) to fy =240 N/mm2 and it diminishes the strain at rupture εεεεu from 50% to 25%.

2. A percentage of 0,20÷0,22%C together with a percentage of 1,3÷1,5Mn increase

the yield stress of a high strength low alloy steel to fy = 360÷400 N/mm2 and they

diminish the strain at rupture εεεεu to the limit allowed for structural steel, i.e.

15÷20%.

3. Generally, an increase of steel strength is associated with decrease of ductility.

4. Quite recently, the Luxemburg steel manufacturer ARBED succeeded in

producing a structural steel that increases the values of the yielding limit fy. It is a

new technology, called QST (Quenching and Self Tempering):

• after the last rolling pass, an intense water cooling is applied to the whole

surface of the shape, so that the skin is quenched (Fig. 3.9);

real

Prandtl fy

3. STRUCTURAL STEEL

63

Entry QST Bank Quenching Self Tempering

1600°F ≅ 871°C 1100°F ≅ 593°C

Fig. 3.9. The idea of the QST process

• cooling is interrupted before the core is affected by quenching and the outer

layers are tempered by the flow of heat from the core to the surface, during

the temperature homogenization phase.

Due to its metallurgical principle, without increasing the percentage of alloying

elements, it results:

• a high yield strength (fy ≅ 500 N/mm2) without a decrease of εu strain;

• an excellent weldability.

For some necessities, as well as for high strength bolts, high alloy steels are

produced with great ultimate (tensile) strength (fu = 800÷1000 N/mm2) and high yield

stress (0,2 = 640÷900 N/mm2) but with a poor ductility.

3.3. STEEL MAKING

Steel making is a hearth-process, based on the refining principle. The molten

blast-furnace iron is saturated with carbon, about 4% by weight. By heating, this

molten metal (the melt temperature of steel is superior to that of iron) and, by

introducing oxygen (Fig. 3.10), carbon in excess is reduced in the refining process

and furnace iron is transformed in steel. In most of the steel-making procedures, the

primary reaction is the combination of carbon and oxygen to form a gas. If oxygen in

excess is not removed (by adding ferrosilicon, aluminium, etc.) the gaseous product

water

3. STRUCTURAL STEEL

64

FeO continues to evolve during solidification. This oxide (FeO) is very dangerous,

because it makes steel fragile. In the Romanian codes there are three qualities:

• not killed steel – poorly deoxidized (n);

• killed steel – deoxidized (k);

• strongly killed steel – strongly deoxidized (with aluminium or silicon) (kf).

Killed steels are characterised by a relatively high degree of uniformity in

composition and properties. Low alloy steels are always killed.

Fig. 3.10. Simplified scheme of the steel making process

Silicon increases the strength of steel and favours the formation of a fine grain

structure. Aluminium is a good deoxidiser.

Sulphur and phosphorus, impurities that result in the steelmaking process,

must be limited in structural steel at about 0,05% each, since they unfavourably

affect steel fragility.

3.4. ROLLING PROCESS INFLUENCE

Most of the products for steel construction are obtained by hot rolling process.

Basically, the hot rolling process consists of passing, in several steps, of a steel bar

through a certain space determined by two shaped rolling cylinders which rotate in

contrary directions (Fig. 3.11) realizing both the advance of the bar and the changing

of its shape from one step to another (Fig. 3.12).

Fig. 3.11. Simplified scheme of the hot-rolling process

Burned gases

Oxygen lance Burner Gas or liquid fuel

Molten furnace iron

t1 t2 < t1

3. STRUCTURAL STEEL

65

1st step 2nd step 3rd step final step

Fig. 3.12. Simplified scheme of the steps to realise a hot-rolled product

The temperature and the timing of the process are controlled. Every passing

between rolling cylinders produces the self-strain hardening (H) of steel (grains are

broken and pressed) and every period between two passings allows the tendency

(R) of recrystallization (grains tend to return to their initial form). Finally, a much more

compact structure with fine grains is obtained and, as a result, both the yield stress

and the ultimate strength increase (Fig. 3.13).

Fig. 3.13. Simplified scheme of grains evolution during hot-rolling

It is important to note that the increase in strength depends on the thickness

of the material, i.e. the less the thickness is the greater the yield stress and the

ultimate strength are.

The rolling process is very important in increasing both the ultimate and the

yield strengths. Two steels with the same chemical composition have different

strengths, depending on the metallurgical process.

T 1000÷1200°C H1 (hardening) R1 (recristallization) 1st step

hardening

Hi

Ri ith step

Hfinal

Rfinal

Gfinal Ginitial G (magnitude of grains)

600÷800°C

3. STRUCTURAL STEEL

66

3.5. RESISTANCE AGAINST BRITTLE FRACTURE

Under some circumstances, plastic qualities of a structural steel can be

unfavourably affected. That is especially the case of:

• dynamic loading;

• low temperature.

Under such circumstances, plastic deformations do not develop and brittle fractures

occur. A brittle fracture is characterised by a sharp unexpected fracture, without

previous plastic deformation. The tendency to brittle fracture, called toughness, is

appreciated by a pendulum test (Fig. 3.14). A special hammer breaks a typical

specimen under a mechanical work L equal to:

L = G⋅(h1 – h2) ( 3.3 )

where G is the weight of the hammer which drops from the height h1 and, after

producing the rupture of the specimen, rises to the height h2. The mechanical work L

(3.3), measured in Joules, characterizes the resistance to brittle fracture. Codes

require a minimum value of 27J in order to provide a good behaviour.

Fig. 3.14. Simplified scheme of the toughness test

Toughness is strongly unfavourably influenced by low temperatures (Fig. 3.15).

0

20

40

60

80

100

120

140

-25

-23

-21

-19

-17

-15

-13

-11 -9 -7 -5 -3 -1 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Fig. 3.15. Variation of the mechanical work L with temperature

55

2 45°

Charpy specimen (V notch)

10

10

h2

h1

Temperature (°C)

Mechanical work (J)

27J Transition temperature

3. STRUCTURAL STEEL

67

Codes define the transition temperature (T) (Fig. 3.15) as the temperature

below which toughness decays to an unacceptable value.

3.6. INFLUENCES OF TEMPERATURE ON THE PROPERTIES OF STEEL

Mechanical properties (E, c (fy), u (fu), δδδδu) vary with temperature (Fig.3.16).

It is to notice that:

• up to T = 100°C there are no significant changes;

• for T > 200°C u (fu) and c (fy) decrease dramatically while δδδδu increases;

Young’s modulus (E) also decreases; as a result, the resistance of steel

structures to fire is an important problem and special measures are necessary;

• for T < 0°C u (fu) and c (fy) increase but, as it was showed before, the fragility

of steel increases.

0.000

0.200

0.400

0.600

0.800

1.000

20 100 200 300 400 500 600 700 800 900 1000 1100 1200 Fig. 3.16. Variation of some mechanical properties of steel with temperature

3.7. FATIGUE BEHAVIOUR

Fatigue tests show the lowering of mechanical strength after a cycle of stress

having an oscillating intensity in time (Fig. 3.17).

y

,y,fy f

fk θ

θ =

EE

k ,Eθ

θ =

p

,p,pk

σσ

= θθσ

(ºC)

3. STRUCTURAL STEEL

68

Fig. 3.17. Types of loading cycles

In the elastic range, the variation of fatigue stress σ0 depends on (Fig. 3.18):

• the type of loading cycle (Fig. 3.17);

• the designed details.

Fig. 3.18. Wöhler diagram

In modern codes, the fatigue stress is defined with regard to (Fig.3.19):

• the range of stresses: ∆σ = σmax – σmin, where compression is (–);

• the designed details.

Fig. 3.19. ∆σ–n diagrams depending on the designed details

constant

alternate

time

max max

max max

min min

min min

n (number of cycles)

O

6 × 106 3 × 106 106

parent metal

butt weld

(log())

n

n

n (log(n)) 104 105 106 107 108

3. STRUCTURAL STEEL

69

Mf

R

γσ∆≤σ∆ ( 3.4 )

where:

∆σR = fatigue strength;

γMf = 1,25,..., 1,35 partial safety factor.

In the plastic range the fatigue stress largely depends on the magnitude of the

plastic deformations. The phenomenon is called low cycle fatigue.

The number n of critical cycles may be determined according to:

( ) ( ) 83,1log4,2nlog −=⋅+ ( 3.5 )

It results:

ε = ±5% n = 20

ε = ±1% n = 933

ε = ±0,5% n = 4926

ε = ±0,3% n = 16785

The following experimental values were obtained:

• fracture at 650 cycles for ε = 1%;

• fracture at 16 cycles for ε = 2,5%.

The fatigue behaviour in the plastic range is a very important phenomenon for

buildings in seismic zones.

3.8. CORROSION

In contact with atmosphere, structural steel corrodes. Rust FeO (iron-oxide) is

formed, whose volume is twice grater than the corroded pattern material, and so the

material deteriorates.

Protection against corrosion is typical for steel structural members and may

be realised:

1. by painting, containing:

• one or more primer coatings;

• two or more coloured paint coatings.

Painting is the most common rust-preventing method.

2. by galvanizing and electro-zinkage (zinc plating), typical for corrugated sheets;

3. STRUCTURAL STEEL

70

3. using non-corrosive steel;

4. using stainless steel.

3.9. SHAPES

3.9.1. General

Steel structures design is based on the use of:

1. standard profiles, obtained by hot rolling process (Fig. 3.20):

Fig. 3.20. Standard profiles

2. built-up sections, obtained by welding hot rolled plates (Fig. 3.21):

Fig. 3.21. Built-up cross-sections

IPE, W, UB HEA, HEB, HEM, UC IPN UPN UAP

L, LNP L, LNP IPET, HEAT, HEBT

ROR RHS, MSH, TPS, RAUTA, VHP

hollow sections

3. STRUCTURAL STEEL

71

3. thin-walled cold-formed shapes, obtained by bending cold rolled thin plates or

sheets (Fig. 3.22):

Fig. 3.22. Cold-formed shapes

The most recent works tend to consider in calculation the actual structural

elements, with their structural and geometrical imperfections, that is a tendency to

abandon the ideal perfect bars.

The following aspects are to be considered:

• structural imperfections of the material;

• geometrical imperfections of the structure.

3.9.2. Structural imperfections

The most important structural imperfections of steel structural members are:

• the presence of residual stresses;

• the non-homogeneity of mechanical properties over the cross-section.

3.9.2.1. Residual stresses

At the end of the rolling process the temperature is equal to approximately

600°C. The exposed parts (the flange edges and mid web) tend to cool faster than

the area around the flange to web joints. As a result, the complete cooling of the

most exposed parts precedes the cooling of the flange to web joint areas, which

remain hot for a longer period. This produces self-balanced residual stresses, whose

distribution on the cross-section depends on the shape of the section (Fig. 3.23).

Tests showed that residual stresses can be great enough, especially in jumbo

sections (res = 0,5c ... c).

3. STRUCTURAL STEEL

72

Fig. 3.23. Example of distribution of residual stresses

Remarks

1. Great residual stresses also appear in welded built-up sections.

2. Residual stresses are less important in cold-formed sections.

3.9.2.2. Non-homogeneity of mechanical properties

Non-homogeneity of mechanical properties is a result of many factors but the

most important one is the difference in thickness between web and flange, in hot

rolled shapes as well as in welded ones.

In cold-formed sections an important increase of the yielding limit (fy) and of

the ultimate strength (fu) occurs in the corner areas as a result of the hardening due

to the bending (or rolling) process (Fig. 3.24).

Fig. 3.24. Increase of mechanical properties in the corner areas

1

1

1

2 3 4 5 6 7 8

2

3 4 5 6

78

2 3 4 5 6 7 8

fu

fy

rolling bending initial plate

increase due to cut down

strength

fibre

( – )

( – ) ( – )

( + )

( + )

( + )

3. STRUCTURAL STEEL

73

3.9.3. Geometrical imperfections

The most important problems to be considered are:

• variation of cross-section properties (A, I, W) along the member;

• mid-span initial deflection (f0);

• load eccentricity (e0).

Tests on different shapes proved that the variation of cross-section properties

along the member is very poor, so they may practically be considered as

deterministic variables.

Generally, the mid-span deflection f0 is described by the ratio f0/L (L being the

span). The statistical distribution of f0/L looks like the one in figure 3.25.

Fig. 3.25. Statistical distribution of the mid-span deflection f0

Codes usually accept a value f0 equal to:

1000

Lf0 = ( 3.6 )

The load eccentricity e0 can be designed or accidental and, depending on the

values, it may be neglected or not in calculation.

3.10. STRUCTURAL STEEL REQUIREMENTS

In order to provide good elasto-plastic behaviour and for technological

reasons, structural steels shall comply with the following requirements (Fig. 3.26):

L f0

actual

n

f0/L ×10 -3

0,2 0,7 1,0

3. STRUCTURAL STEEL

74

• a high strength, appreciated by a yield stress fy ≥≥≥≥ 235 N/mm2 and fu/fy > 1,2;

• a quite large yielding plateau;

• an adequate tenacity ( = property to keep large plastic deformation before failure

under great forces), appreciated by εεεεu ≥≥≥≥ 15% and εεεεu ≥≥≥≥ 20εεεεy;

• a good resistance against brittle fracture, appreciated by the fracture energy

KV(+20°°°°C) ≥≥≥≥ 27J;

• a good weldability, i.e. the property of steel to be welded in normal conditions.

Fig. 3.26. Typical behaviour diagram of structural steel

Remarks

1. The high strength is the major advantage of steel, relative to the strength of other

common structural materials: wood, masonry, concrete, etc. Unlike masonry and

concrete, which are weak in tension, steel is strong both in tension and

compression.

An image of the strength of steel is given by the ratio:

γ

= RcL ( 3.7 )

A physical meaning of this factor is the greatest length of a bar whose cross-

section is able to support its self-weight.

R

LRLRA

VR

AN

==⋅=⋅==

In these relations:

R – design strength of steel;

fu

fy

u y

3. STRUCTURAL STEEL

75

– weight per unit volume of steel;

N – axial force generated by the self-weight of the bar;

A – area of the cross-section of the bar;

V – the volume of the bar;

– normal stress on the most loaded cross-section of the bar.

Here are some values of this factor (cL) for different materials:

• ordinary structural steel: 2800 ÷ 4000 m;

• aluminium alloys: 2500 ÷ 10000 m;

• reinforced concrete in compression: 100 ÷ 1300 m;

• glass fibre: 96000 ÷ 184000 m;

• carbon fibre: 150000 ÷ 210000 m.

2. The capacity of steel to yield is a fundamental requirement for structural steel:

a) Residual stresses result from the fabrication process both in rolled profiles

and in welded built-up structural members (Fig. 3.27). The tendency to

shorten of the warmer flange to web joint is braked by the other fibres of the

cross-section. Consequently, once cooling completed, the flange to web joint

remains in tension, while the other parts are in compression. Residual

stresses, in tension and in compression, are self-balanced on the cross-

section.

Fig. 3.27. Example of residual stress distribution

b) Figures 3.28 and 3.29 show the effects of residual stresses on a loaded

structural member in tension and in bending, respectively.

3. STRUCTURAL STEEL

76

Fig. 3.28. Influence of residual stresses on a tensioned member

Fig. 3.29. Influence of residual stresses on a member in bending

It is to note that the stress distribution according to the theory is strongly

affected by the existing residual stresses.

c) For the members in figures 3.28 and 3.29 the evolution of the total stress

σ = σres + σN (or σM) ( 3.8 )

in the bottom flange is examined in figure 3.30 (the phenomena in the bottom

flange are the same for the member in tension or in bending).

It is to observe (σF = σN or σM):

• when σF ≤ fy – σres all fibres behave in the elastic range (Fig. 3.30a, b);

• when fy – σres < σF ≤ fy some fibres behave elastically and some plastically

(Fig. 3.30c);

• when σF = fy all fibres behave plastically (Fig. 3.30d).

= res + N

= res + M

3. STRUCTURAL STEEL

77

Fig. 3.30. Evolution of the total stress in the bottom flange

As a result:

• the uniform stress distribution in the bottom flange, usually accepted in

current design, is a conventional one (it is correct only in the final state,

when σF = fy);

• the elastic behaviour in the meaning no plastic deformation after unloading

is real only for the entire section as a whole, or when σF is insignificant;

• in order to allow all fibres to reach the yield stress, steel used for structural

members must possess a large plastic yielding plateau. When this

fundamental requirement is not satisfied, the material has a fragile

behaviour and a fragile rupture is to be expected in the moment when in a

single fibre, the most stressed one, the stress reaches its ultimate strength

value (like cast iron, glass, etc.).

d) The good plastic behaviour is also fundamental in the situations of a structural

member in tension with an important hole (Fig.3.31) or in the case of a

structural member in tension with a non-homogeneous cross-section.

σ = σres + σF σF

σres

σ

σ

σ

σ

a)

b)

c)

d)

3. STRUCTURAL STEEL

78

Fig. 3.31. Stress distribution around a hole in a member in tension

3. An adequate tenacity and the avoidance of brittle fracture are fundamental

requirements for a structural steel in order to create the possibility to transform

the structure into a plastic mechanism (Fig.3.32), able to dissipate energy. The

capacity to dissipate energy is a very important requirement for a structure

situated in seismic regions when it is subjected to severe earthquake motions.

Fig. 3.32. Example of plastic mechanism for a structure during earthquake

4. A good weldability in normal conditions is an important requirement in order to

avoid an uneconomical cost of fabrication.

The above qualities of a structural steel result from:

• chemical composition and crystalline structure;

• the process of steel making;

• the rolling process influence.

σmax

σmin

σ

σaverage

σ

σ σ

3. STRUCTURAL STEEL

79

3.11. STRUCTURAL STEEL GRADES

Table 3.1 shows the main characteristics for the most common Romanian

structural steel grades, according to STAS 500/2–80.

Table 3.1. Main characteristics for some Romanian steel grades

Yield strength σc

Design strength R (N/mm2)

thickness t (mm)

Steel type Nominal steel grade

Ultimate tensile

strength σr

(N/mm2)

Ultimate elongation

δ%

t ≤ 16 16 < t≤ 40

Material factor

γm = σc/R

240 230 Carbon steel

OL37 360 ÷ 440 25...26

220 210

1,09

280 270 OL44 430 ÷ 540 22...25

260 250

1,08

350 340

High strength

low alloyed OL52 510 ÷ 630 21...22

315 300

1,11

Remarks

1. Steel grades with fine grains (OCS) are fabricated for special welded structures.

2. Corrosion resistant structural steels (RCA, RCB) and stainless steels (ORC) are

fabricated for special use.

The unified European pre-standard EUROCODE 3 [2] uses the steel grades

defined in the European standard EN 10025:1990+A1:1993 [5]. This standard was

adopted in Romania as SR EN 10025:1990+A1:1994 [6]. Although there is a good

correspondence between the steel grades defined by STAS 500/2–80 and the ones

described in [5] and [6], an exact equivalence is not possible.

Steels used in [2] are designated as follows (EN 10027–1).

Table 3.2. Example of steel grade designation according to EN 10027–1

Letter Figure Symbol 1 Symbol 2

example S 355 J2

Letter S for structural steel

Figure Minimum value of the yielding limit in N/mm2 for the lowest range of

thickness

Symbol 1 Fracture energy in Joules for a given temperature, defined as follows:

3. STRUCTURAL STEEL

80

Table 3.3. Definition of the fracture energies

°C 20 0 –20 –30 –40 –50 –60

27 J JR J0 J2 J3 J4 J5 J6

40 J KR K0 K2 K3 K4 K5 K6

60 J LR L0 L2 L3 L4 L5 L6

Symbol 2 Obtaining mean of steel:

M : thermo mechanical

N : normalised or by normalising rolling

Q : quenched and tempered

G : other characteristics followed by 1 or 2 digits if necessary.

Table 3.4. shows examples of values of some mechanical characteristics for the

most common structural steel grades contained in [5].

Table 3.4. Mechanical characteristics for some steel grades contained in [5]

Thickness (mm)

t ≤ 40mm 40mm < t ≤ 80mm

Steel grade

Ultimate strength fu

(N/mm2)

Yielding strength fy

(N/mm2)

Ultimate strength fu

(N/mm2)

Yielding strength fy

(N/mm2)

S235 360 235 340 215

S275 430 275 410 255

S355 510 355 490 335

The partial factors γγγγM are applied to the various characteristic values of resistance as

follows:

• resistance of cross-sections to excessive yielding, including local buckling γγγγM0;

• resistance of members to instability assessed by member checks γγγγM1;

• resistance of cross-sections in tension to fracture γγγγM2;

• resistance of joints special provisions.

The values of the partial factors γγγγM may be defined in the National Annex. The

following numerical values are recommended for buildings:

γM1 = 1,0

γM2 = 1,25

4. CONNECTING DEVICES

81

Chapter 4

CONNECTING DEVICES

4.1. GENERAL

Connecting devices for steel structures are:

• Welds – are largely used in fabrication of structural members in shops;

• Bolts – are largely used in assembling structural members on the field;

• Rivets – at present they are practically abandoned due to their complicate

technology and high cost.

4.2. WELDING

4.2.1. General

Welding is a technological process that realizes the junction of the members

of a structure into a monolithic elastic network.

To execute a weld, one needs:

• a heat source;

• some adequate additional material.

The weld seam results after local melting in the area of welding (Fig. 4.1). A

number of welding passes, called weld layers, are necessary.

Fig. 4.1. Scheme of a welding process

heat source

parent metal

solidified weld

additional material

weld layer (seam)

molten pool


82

The integrity of the welded structure depends on its ability to deform

plastically during fabrication, erection and service. The ability of the welded structure

to deform plastically, avoiding brittle failure primary depends upon:

1. weldability of steel;

2. welding procedure selection;

3. avoidance of notches both in design and fabrication;

4. adequate quality control and inspection.

4.2.2. Weldability

Weldability is defined as “the capacity of a metal to be welded under

fabrication conditions imposed into a specific suitably designed structure and to

perform satisfactorily in the intended service life”.

Weldability is largely depending on the reaction of steel to the drastic heating

and cooling cycle of arc welding. Three of the most important steel properties that

influence weldability are:

• the chemical composition;

• the structural grain size;

• the thickness of the material.

Chemical composition. The brittleness that steel may reach after rapid cooling from

high temperature is directly proportional to the carbon content. In order to avoid

brittle failure of the welded structure it is necessary:

• to limit the content in carbon to 0,20 ÷ 0,22%;

• to limit the content in carbon of the additional material to 0,08,..., 0,12%.

Structural grain size. There is a linear relationship between the ferrite grain size and

the Charpy transition temperature between ductile and brittle behaviour; the greater

the grain size is the greater the transition temperature is. Weldability also varies with

grain size meaning it is favoured by a reduced grain size.

High heat input welds show a larger grain size than the same process at a

lower heat input, because they provide a slower cooling rate. That is why

recommendations usually limit the thickness of a weld layer at about 6mm. A

subsequent pass will refine the grains of a previous pass.


83

Thickness. Because of their greater mass, thick plates extract heat from the weld

area and cool the weld more rapidly than the same weld on thin plates. As a result,

weldability is affected. There are two possibilities to avoid a tendency to brittle

fracture:

• to limit the thickness of plates;

• to pre-heat the pieces and to hold them at a temperature of a few hundred

degrees before the welding operation.

Conclusions:

Weldability is increased by:

• low carbon content;

• fine grain size;

• restricted low thickness;

and, conversely, is reduced by:

• high carbon content;

• coarse grain;

• big thickness.

4.2.3. Structural welding process and materials

Fusion welding processes vary largely, according to the applied heat source

and to how the molten pool is protected against atmosphere. The most common

welding processes used in commercial structural steel fabrication are:

1. Manual shielded metal arc process (Fig.4.2)

The heat source is the electric arc formed between the electrode and the

parent metal. The developed heat produces a quick melting of the external

coatings of electrodes containing aluminium, silicon and other deoxidizers, which

protect the area surrounding the arc and the weld pool. This process is widely

applicable to any kind of welds.

2. Submerged arc process (Fig.4.3)

The heat source is the electric arc formed between the electrode and the

parent metal. The protection of the weld pool, better as in the shielded arc

process, is provided by a granulated deoxidizer flux automatically thrown in


84

advance and at the same speed of the welding process. This procedure is highly

productive for long weld seams.

Fig. 4.2. Scheme of the manual shielded metal arc process

Fig. 4.3. Scheme of the submerged arc process

3. Gas shielded metal arc process (GMAW - Gas Metal Arc Welding) with

consumable electrode (MIG and MAG). The arc protection is provided by an

inert gas (MIG) or by a chemically active gas (MAG). This procedure is used in

welding mild steel and low alloy steel.

4. Gas shielded metal arc process with non-consumable electrode. The arc is

produced between a tungsten element and the parent metal. The protection is

provided by argon. This procedure is used especially for welding stainless steel

or aluminium alloys.

additional material

coating

direction of travel

metal arc

weld pool (molten pool)

electrode

protective gas

protecting slag

solidified weld (weld deposit)

parent metal

metal arc

molten pool

flux feed line

granular flux

parent metal

recovered flux

direction of travel

solidified weld (weld deposit) slag

bar electrode (continuous wire)


85

5. Electro-slag welding is a special procedure to weld very thick steel parts with

only one pass in a vertical position.

4.2.4. Metallurgic phenomena in the welding process

Essentially, there are three metallurgic phenomena:

1. A hard zone appears in the parent metal near the weld seam, which can lead to

so-called cold cracking (Fig. 4.4). The origin of this phenomenon is assigned to

the hydrogen absorbed by the weld material in the molten state. The tendency to

brittle cracks may be moderated by pre-heating the part to be welded and by

using electrodes with basic coating.

Fig. 4.4. Scheme of the material structure near a weld seam

2. Lamellar tearing is a separation or a crack in the base metal, caused by

through-thickness weld shrinkage stairs (Fig. 4.5). It is a result of the reducing of

ductility in the through-thickness direction, which can be lower than in the

conventional longitudinal tests.

Fig. 4.5. Lamellar tearing

hardness

cracks

2 … 6mm

lamellar tearing


86

3. Hot cracking can occur in the molten area. These cracks form during the

solidification process and they are explained by the presence of some impurities

solidifying at a lower temperature than steel (Fig. 4.6).

Fig. 4.6. Hot cracks

4.2.5. Thermal phenomena in welding process

The heating-cooling cycles during welding produce (Fig. 4.7):

• internal stresses (residual stresses);

• deformations (Fig. 4.8).

The greater deformations are the lower stresses are.

Fig. 4.7. Residual stresses and residual deformations

Fig. 4.8. Example of residual deformations after welding

steel plate

longitudinal shrinkage weld seam

res = (0,5 … 1,0) × fy


87

4.2.6. Welding positions

The most common welding positions are shown in figure 4.9.

1. Flat position

butt welds fillet welds

2. Horizontal position

3. Vertical position

4. Overhead position

Fig. 4.9. Welding positions

Flat position requires the simplest technology. The overhead position is the

most complicated one.


88

4.2.7. Weld details

In order to avoid unfavourable weld details, the following are recommended:

1. Avoid overwelding (Fig. 4.10). This requires the use of an appropriate weld

size, not larger than the one given by calculation.

Fig. 4.10. Example of oversized weld seam

2. Avoid asymmetry (Fig. 4.11).

Fig. 4.11. Example of asymmetric weld seams

3. Avoid lamellar tearing (Fig. 4.12). Lamellar tearing means failure of a hot rolled

plate or of a hot rolled shape because of cracks formed along the rolling direction.

These cracks create separation plans among longitudinal fibres.

Fig. 4.12. Example of details that may favour lamellar tearing

OK NO oversized weld (too much heating)

desirable

desirable

notch effect

lamellar tearing


89

4. Avoid susceptible details (Fig. 4.13). Some details might favour lamellar tearing

or brittle fractures.

Fig. 4.13. Examples of susceptible details and improved ones

5. Avoid weld fatigue (Fig. 4.14). Any change in section should be “stream-lined”.

Fig. 4.14. Example of “stream-lined” details to avoid fatigue and brittle fractures

4.2.8. Welding defects

Welding defects are:

• cracks – the worse defect;

• blow holes – metallurgic defect;

• lack of penetration;

• porosity;

• slag inclusions.

susceptible details improved details

NO YES

“stream line”

“stream line”


90

4.2.9. Weld inspection methods

1. Visual Test (VT)

It is the most economical test. The magnifying glass detects surface

imperfections, porosity, slag, cracks, irregularities, etc.

2. Dye (Liquid) Penetrant Test (DPT) (Fig. 4.15)

This test uses a red dye penetrant applied to the work from a pressure spray can.

Fig. 4.15. Dye penetrant test

3. Magnetic Particle Test (MPT) (Fig. 4.16)

A magnetizing current is introduced over a dry red magnetic powder. This

induces a magnetic field in the work that will be distorted by any cracks or

inclusions, located on or near the surface.

Fig. 4.16. Magnetic particle test

This method will indicate surface defects, like fine cracks not to be observed by

liquid penetration (cracks filled with slag, difficult for liquid to penetrate).

4. Radiographic Test (RT.)

Radiographic testing is basically an X-ray film process. Internal defects may be

put in evidence (porosity, blow holes, slag inclusions, cracks appear as darker

stains (spots) on the film).

subvisible crack

red penetrant applied in excess

excess removed

visible indication

white developer applied

current

red dry powder


91

5. Ultrasonic Test (UT)

The ultrasonic inspection process is analogous to radar. The method is based on

the variations in reflections due to differences in acoustic properties (pulse echo)

caused by defects (at the boundary).

4.2.10. Strength of welded joints

In the Romanian code STAS 10108/0–78 [7] there are two important types of

weld seams, with respect to their behaviour and to their design models:

• butt welds;

• fillet welds.

Fig. 4.17. Classification of weld seams according to STAS 10108/0–78 [7]

The main difference is that in this model butt welds behave like parent material, while

fillet welds resist always by shear stresses τ.

Checking a welded connection generally consists of the following steps:

1. Establishing the design cross-section and its geometrical characteristics;

2. Reducing loads in the centre of gravity of the cross-section;

3. Establishing the stress distribution on the cross-section;

4. Checking the seam in the most loaded points.

The beginning and the end of a weld seam are generally weak zones. That is

why they are neglected when establishing the strength of the joint. In order to avoid

end lap weld seams corner weld seams overlapping weld seams

butt weld seams fillet weld seams


92

“losing” a part of the seam, it is possible to use some additional pieces from where to

start and to end welding. These pieces are made of copper (Fig. 4.18). In the end

they are cut down and the entire seam is reliable. The use of additional pieces is

compulsory for butt welds.

Fig. 4.18. Example of using additional plates

Weld seams are noted on drawings according to SR EN 22553 (ISO 2553).

4.2.10.1. Butt welds

The design cross-section of the weld seam must be established before any

design procedure.

Fig. 4.19. Dimensions of a butt weld seam

ds LaA ⋅= ( 4.1 )

)a2(LLd ⋅−= ( 4.2 )

a – the throat (effective thickness); it is equal to the thickness of the thinner

joined member (Fig. 4.19);

Ld – the design length of the seam; it is obtained by deducing the bad parts of the

seam from the actual length L (4.2); if additional plates are used it is equal to

the actual length of the seam (Fig. 4.19).

additional plates cutting line

a

a

a

a

L


93

1. Butt weld subjected to axial force (N) (Fig. 4.20)

Fig. 4.20. Butt weld seam subjected to axial force

The stress distribution is constant on the cross-section:

sA

N=σ ( 4.3 )

2. Butt weld subjected to shear force (T) (Fig. 4.21)

Fig. 4.21. Butt weld seam subjected to shear force

Generally, the stress distribution is a parabola described by Juravski’s relation:

yIw

ST⋅⋅=τ ( 4.4 )

where:

S – static moment of the area of the part of the cross-section that tends to

slide in the point where is calculated;

w – width of the cross-section in the point where is calculated;

Iy – moment of inertia (second moment of the area) of the cross-section about

y-axis (axis normal to the shear force).

The maximum shear stress is obtained in the neutral axis (Fig. 4.21a), where the

static moment S has the maximum value:

N N Ld

a

( a ) ( b )

Ld

a

t

t

hw tw

b

T

T

T

T

y y

z

z

y y y y

z

z

z

z


94

y

maxmax Iw

ST⋅

⋅=τ ( 4.5 )

In cases where there is an important variation in the value of the width w of the

cross-section, Juravski’s relation describes a leap in the diagram and the

parabola is flattened. In these cases, a simplified distribution is accepted (Fig.

4.21b), considering that the entire shear force is resisted only by the web.

swA

T=τ ( 4.6 )

wwsw haA ⋅= ( 4.7 )

3. Butt weld subjected to bending moment (M) (Fig. 4.22)

Fig. 4.22. Butt weld seam subjected to bending moment

Generally, the linear stress distribution is described by Navier’s relation:

zIM

y

⋅=σ ( 4.8 )

where:

Iy – moment of inertia (second moment of the area) of the cross-section about

y-axis (axis normal to the plane of the bending moment).

z – the distance from the considered point to the neutral axis (in the plane of

the bending moment).

The maximum stress is obtained when z takes the greatest value:

y

maxy

max WM

zIM =⋅=σ ( 4.9 )

where:

Wy – cross-section modulus about y-axis (axis normal to the plane of the

bending moment).

M M Ld

a

y y

z

z


95

4. Butt weld connection subjected to axial force, shear force and bending moment

(N, T, M) (Fig. 4.23)

Fig. 4.23. Butt weld seam subjected to axial force, shear force and bending moment

Solving the general problem given in figure 4.23 means using linear superposition

of relations (4.3) – (4.9) and checking the stress state in the most loaded points by

means of relations (4.10) – (4.12).

s

ysMNmax Rz

IM

AN ≤⋅±=σ±σ=σ ( 4.10 )

sfT R≤τ ( 4.11 )

( ) s2T

2MNeq R3 ≤⋅+±= ; NN = ; z

IM

yM ⋅= ( 4.12 )

When using relation (4.12), and must be calculated in the same point (z*) and in

the same loading situation.

The values of the normal design strength sR and of the shear design strength sfR

according to the Romanian code STAS 10108/0–78 [7] may be found in table 4.1.

4.2.10.2. Fillet welds

The profile of a fillet weld can have different shapes:

flat convex concave concave with

Fig. 4.24. Possible profiles of a fillet weld unequal legs

T

M

N

t

hw

t

tw

b

y y

z

z N M T

z*

M

N


96

In the model used in the Romanian code STAS 10108/0–78 [7] the design thickness

of the cross-section of the seam is defined by the height of the greatest isosceles

triangle that can be inscribed in the cross-section of the weld seam (Fig. 4.25):

Fig. 4.25. Design cross-section of a fillet weld seam

Once the thickness of the design cross-section (throat) established, the design

section of the weld seam is obtained by bringing the rectangles defined by relations

(4.13) and (4.14) in the plane of the connection.

ds LaA ⋅= ( 4.13 )

)a2(LLd ⋅−= ( 4.14 )

a – the effective throat thickness (Fig. 4.25) (design thickness of the cross-

section of the seam);

Ld – the design length of the seam; it is obtained by deducing the bad parts of the

seam from the actual length L (4.14); these parts are situated at each end.

The effective throat thickness a can be 25, 3, 35, 4, 5, 6, 7 ... mm and it generally

shall satisfy the following requirements (Fig. 4.25), (Fig.4.26a):

minmax t7,0at3,0 ⋅≤≤⋅ ( 4.15 )

For shapes like angles (Fig.4.26b) or channels (Fig.4.26c):

min2max t7,0at3,0 ⋅≤≤⋅ ( 4.16 )

( )pg1max t85,0 ;t7,0minat3,0 ⋅⋅≤≤⋅ ( 4.17 )

( a ) ( b ) ( c )

Fig. 4.26. Geometric requirements for the effective throat thickness of fillet welds

a a a a

a t1

t2

a1

tg

tp tp

tg

a2

a1

a2


97

where:

tg – thickness of the gusset;

tp – thickness of the shape (profile);

tmin – the minimum thickness of the connected elements (min ti).

There are also limitations for the length Ld of the weld seam (Fig. 4.27):

( )a60L

mm40b

U , L shapes rolledhot fora15plates fora6

d ⋅≤≤

−⋅⋅

(STAS 10108/0–78) ( 4.18 )

Fig. 4.27. Geometric requirements for the length of fillet weld seams

Depending on their position with respect to the main force, fillet weld seams

can be classified as:

• side (longitudinal) weld (Fig.4.28a);

• end (transverse) weld (Fig.4.28b);

• combined weld (Fig.4.28c).

( a ) ( b ) ( c )

Fig. 4.28. Types of fillet weld seams

Combined welds are not recommended because of the different stiffness of side and

end welds, which generates a non-uniform behaviour of the connection. Tests

showed that fillet welds generally fail due to tangential stresses that are developed in

inclined planes at 45°. Following this, the design relations are as follows.

N N b

L a L

model stress distribution

real stress distribution


98

1. Fillet weld subjected to axial force

• when the force acts in the centroid line of the connection (Fig. 4.29)

Fig. 4.29. Axial force acting in the centroid line of a fillet weld connection

)a2(LLd ⋅−= ( 4.19 )

ds La2A ⋅⋅= ( 4.20 )

s

N AN=τ ( 4.21 )

• when the force acts with an eccentricity from the centroid line of the

connection (e.g. angles, channels, etc.) (Fig. 4.30)

Fig. 4.30. Axial force acting with an eccentricity by the centroid line of a fillet weld

)a2(LL 11d ⋅−= ( 4.21 )

)a2(LL 22d ⋅−= ( 4.22 )

1d11s LaA ⋅= ( 4.23 )

2d22s LaA ⋅= ( 4.24 )

b

ebNN1

−⋅= ( 4.25 )

be

NN2 ⋅= ( 4.26 )

N N

L a L

a

a

Ld

N N N1

N2

L

a1 L

a2 L

a1

a2

e

b

Ld1

Ld2


99

1s

11N A

N=τ ( 4.27 )

2s

22N A

N=τ ( 4.28 )

2. Fillet weld subjected to shear force

• when the shear force acts together with a bending moment, Juravski’s relation

is used

y

T IwST

⋅⋅=τ ( 4.29 )

or in cases where there is an important variation in the value of the width w of

the cross-section, the simplified relation (4.30) may be used, where Asw is the

area of the cross-section that resists the shear force (area of the web for I and

H shapes)

sw

T AT=τ ( 4.30 )

• when the shear force does not act together with a bending moment (a

“scissors-like” force or a force acting in the plane of the connection, in the

centre of gravity of the connection, on any direction), relation (4.31) is used,

where As is the total area of connection

s

T AT=τ ( 4.31 )

3. Fillet weld subjected to axial force, shear force and bending moment acting

normally to the plane of the connection (Fig. 4.31)

Fig. 4.31. Fillet weld connection subjected to moment acting normally on the plane

T

M

N

element cross-section

connection design cross-section

y y

z N T M 1

2


100

Solving the general problem given in figure 4.31 means using linear superposition

of the previously presented relations and checking the stress state in the most

loaded points, always keeping in mind that all stresses that are developed in a

fillet weld connection are shear ones.

s

N AN=τ ( 4.32 )

sw

T AT=τ ( 4.33 )

or, by using the general relation (not a common situation)

y

T IwST

⋅⋅=τ ( 4.33’ )

zIM

yM ⋅=τ ( 4.34 )

maxy

Mmax, zIM ⋅= ( 4.34’ )

The checks to be done are:

• in the farthest points away from the centre of gravity welded connection (point

1 in figure 4.31)

sfMN R≤τ±τ ( 4.35 )

• theoretically, in any point on the cross-section and especially at the edge of

the web for I cross-section, the geometric sum of stresses (point 2 in figure

4.31)

( ) ( ) sf

2T

2MN R≤τ+τ±τ ( 4.36 )

The values of the shear design strength sfR for fillet weld seams according to the

Romanian code STAS 10108/0–78 [7] may be found in table 4.1.

4. Fillet weld subjected to axial force, shear force and bending moment acting in the

plane of the connection (Fig. 4.32)

According to the previously presented relations,

s

N AN=τ ( 4.37 )

s

T AT=τ ( 4.38 )


101

zII

M

zxxM ⋅

+=τ ( 4.39 )

xII

M

zxzM ⋅

+=τ ( 4.40 )

Fig. 4.32. Fillet weld connection subjected to in-plane moment

Considering sfR given in table 4.1 for fillet weld seams, the check to be made in

the farthest point away from the centre of gravity (point 3 in figure 4.32) is:

( ) ( ) sf

2zMT

2xMN R≤τ±τ+τ±τ ( 4.41 )

In all the previously presented fillet weld connections whenever the seams are

doubled (they are situated on both sides of a plate), the areas and the moments of

inertia are doubled on the same geometric configuration.

Table 4.1. Strength of weld seams according to STAS 10108/0–78 [7]

Weld type Compression Tension Shear

Butt weld RRsc = RRs

i = for automatic welding, followed by non-destructive tests

R8,0Rsi ⋅= for manual welding

R6,0Rsf ⋅=

Fillet weld – – R7,0Rsf ⋅=

R = design strength of the parent material Whenever a connection contains in the same cross-section butt welds

and fillet welds, it is treated as a whole and only the checks differ, depending

on whether the checked point is situated on butt weld or on fillet weld.

T

M N

a design cross-section

x

z

x x

z

z

N xM

T

zM

3


102

4.3. BOLTS

4.3.1. General

The more general term “fasteners” includes bolts and rivets. The behaviour of

rivets is very much alike the behaviour of bolts and they are very rarely used today.

Bolts are connecting elements largely used on field at the erection stage when

structural members are to be assembled in order to realise a steel structure. Figure

4.33 shows a steel frame built on field using bolted connections.

Fig. 4.33. Example of steel frame built on field using bolted connections

Bolts used for structures generally consist of the following components:

• a metal cylindrical shank, partially threaded and having a head, usually

hexagonal (Fig. 4.34a);

• a nut, usually hexagonal (Fig. 4.34b);

• one or two washers, usually round (Fig. 4.34c).

( a ) ( b ) ( c )

Fig. 4.34. Components of a bolt

A bolted connection results by twisting the nut until a firm contact is obtained

between the plates to be assembled (Fig. 4.35a). In bolted connections subjected to


103

vibration, spring washers (Grower) (Fig. 4.35b) or lock nuts (Fig. 4.35c) should be

used in order to avoid any loosening of the nuts.

( a ) ( b ) ( c )

Fig. 4.35. Possible components of a bolted connection

4.3.2. Classification of bolts

Bolts can be classified as:

• normal bolts;

• high strength bolts.

Table 4.2 shows the mechanical properties of the most common bolts used in steel

structures.

Table 4.2. Main mechanical properties of the most common bolts [2]

Type Grade fub (N/mm2) fyb (N/mm2) εεεεu (%) fkb (N/mm2)

4.6 400 240 22 240

Normal bolts 5.6 500 300 20 300

6.8 600 480 8 420

High strength 8.8 800 640 12 560

bolts 10.9 1000 900 9 700

fub is the minimum tensile strength determined on the entire bolt

fyb is the minimum yield stress determined on the entire bolt

εu is the ultimate strain

fkb is the characteristic strength value, equal to the lower between fyb and 0,7fub

The diameters in mm of the bolts usually used in steel structures are: 10, 12, 14, 16,

18, 20, 22, 24, 27, 30, 33, 36.


104

4.3.3. Behaviour and design resistance of bolts

4.3.3.1. Loading and tightening

The behaviour and the design resistance of bolts substantially depend on:

• loading type;

• tightening type.

Loading type. From the loading type point of view, bolts can be classified as:

• bolts loaded perpendicular to their axis (shear connections) (Fig.4.36a);

• bolts axially loaded (tension connections) (Fig.4.36b).

( a ) ( b )

Fig. 4.36. Loading types of bolts

Tightening type. Tightening can be:

• normal tight;

• controlled tight.

In both types of tightening, the bolt is introduced in a 2...3mm larger diameter hole.

Normal tight is defined as the tightness that exists when members to be connected

are in firm contact. This may usually be realised by the full effort of a man using an

ordinary wrench. The tightening produces a self-stress loading consisting of:

• tension in the bolt, balanced by compression in the plates (a certain friction also

results between plates in contact);

• a twisting moment in the bolt balanced by friction between the plate and the

washer and between this one and the nut.

Controlled tight is defined as the tightness corresponding to a fully pre-tensioned

bolt. The control of tightening refers to the preload force Nt to be induced in the

shank of the bolt by a twisting moment Mt applied to the nut (by using a calibrated

impact wrench or by using “turn-off the nut” method).

F/2

F/2

F/2 F/2

F/2 F/2

F


105

4.3.3.2. Behaviour of normal bolts in shear connections

Figure 4.37 shows the behaviour of a normal bolt in a shear connection.

Fig. 4.37. Stress distribution in a “bearing type” connection

The following states can be noticed when loading a bolted connection normally on

the axis of the bolt (Fig. 4.38):

• Phase 1 The bolt is generally introduced in a 2...3 mm larger hole and it is

normally tightened. A friction force Ff results between plates in contact. In this

phase, when loading, no relative displacement is noticed until the load F reaches

the friction limit Ff (Fig. 4.38).

Fig. 4.38. Typical load – deformation curve for a usual “bearing type” connection

F

F/2

F/2

bearing pressure

model used for the stress distribution

shear force

real stress distribution

Phase 1

Phase 2

Phase 3

Phase 4

F Fu

Ff

L = L – L0


106

• Phase 2 When F = Ff, slipping of the joint begins under a force F practically

constant. Slipping stops when the contact shank – plates is realised.

• Phase 3 is characterized by an elastic behaviour, meaning that the

displacement ∆∆∆∆L is proportional to force F.

• Phase 4 is characterized by a plastic behaviour, i.e. large deformations occur for

a slight load increase and the joint fails at an ultimate value Fu.

Failure at the ultimate load can be one of the following:

1. collapse due to hole failure in bearing (Fig.4.39a);

2. collapse due to bolt failure in shear (Fig.4.39b);

3. collapse by shear failure of the connected plates (Fig.4.39c);

4. collapse by failure of plates in tension (Fig.4.39d).

( a ) ( b ) ( c ) ( d )

Fig. 4.39. Typical failure modes for a usual “bearing type” connection

1. Bearing failure of plate (Fig.4.39a). Plate failure is a result of the bearing force

produced at the contact between the bolt and the plates in connection. The

bearing resistance of a bolt is:

bg,p

ming,p

bg,pg,p

RtdN

RtdN

⋅⋅=

⋅⋅=

( 4.42 )

where:

d is the nominal diameter of the bolt;

t is the smallest thickness of plates in contact;

min

t is the minimum value of the sum of the thickness of the plates which tend to

displace in the same direction;

F F F F

d e1

b F/2 F/2 F/2 F/2

Bearing failure of plate

Shear failure of bolt

Longitudinal shear

failure of plate

Plate failure in tension


107

m

kbg,p

RR

γ⋅β= is the design strength calculated with:

• Rk – the characteristic strength of plates (= fy);

• γm = 1,25 – partial safety factor of the material;

• β = 2,0 usually.

2. Shear failure of bolt (Fig.4.39b). The bolt fails in shear under a force per shear

plane (Nf,p) equal to:

bf

2bfbp,f R

4d

RAN ⋅⋅π=⋅= ( 4.43 )

where:

bfR is the shear design resistance of the bolt

m

kbf

R6,0R

γ⋅=

• Rk – the characteristic resistance of the bolt;

• γm = 1,25 – partial safety factor of the material;

Ab is the cross-section area of the bolt equal to:

• 4d

A2

b⋅π= when the shear plane passes through the unthreaded part of

the bolt (d is the nominal diameter of the bolt);

• 4d

A20

b

⋅= when the shear plane passes through the threaded part of the

bolt.

d89,02

dddd mn

0res ⋅≅+== (Fig. 4.40)

dn = diameter of the core of the shank;

dm = average diameter;

d = nominal diameter;

dres = resistant diameter.

Fig. 4.40. Cross-section of the bolt and the resistant area [12]

d dn dm dres


108

The design resistance in shear of a bolt is:

bfbfp,fff RAnNnN ⋅⋅=⋅= ( 4.44 )

where:

nf is the number of shear planes.

3. Longitudinal shear failure of plate (Fig.4.39c). In order to avoid shear failure of

plates, the following requirement should be satisfied:

ff1 NRt2d

e ≥⋅⋅

− ( 4.45 )

The minimum required edge distance e1 (Fig.4.39c) results from relation (4.45),

where Rf is the shear design strength of the material of the plate. The minimum

required edge distance e1 is generally given in codes (if eactual > e1 there is no

need to check the condition (4.45)). Usually, it is greater than two times the

diameter of the hole.

4. Plate failure in tension (Fig.4.39d). Generally, the elastic stress distribution

around a hole is the one given in figure 4.41a.

Fig. 4.41. Stress distribution around a hole

If the hole is assumed to be an ellipse it can be proved that the maximum stress

is given by the following relation:

+⋅σ=σca2

1avmax ( 4.46 )

where:

F F F

F/2 F/2 ( a ) ( b )

2c

2a

t

d

b

1 1

real distribution model distribution


109

av – average stress in the plate;

a – half of the axis normal to the stress (Fig. 4.41a);

c – half of the axis along the stress (Fig. 4.41a).

In the special case of a circular hole, it results:

yavmax f3 ≤⋅= (for structural steel) ( 4.47 )

Based on the good plastic properties of structural steel, which is a fundamental

requirement in this case, the simplified distribution given in figure 4.41b is

accepted. This leads to the following condition, according to the Romanian code

STAS 10108/0–78 [7]:

( ) FRtdb ≥⋅⋅− ( 4.48 )

where:

b – width of the plate that is being checked (Fig. 4.41b);

d – diameter of the hole (Fig. 4.41b);

t – thickness of the plate that is being checked (Fig. 4.41b);

R – design strength of the material of the plate;

F – axial force in the checked cross-section (1-1).

Remark The uniform stresses distribution which is assumed in calculus when

checking an element is unfavourably affected by the presence of the hole.

4.3.3.3. Behaviour of high strength bolts in slip connections

Tightening control refers to the pre-load force Nt to be induced in the shank of

the bolt by the twisting moment Mt applied to the nut. Codes generally accept an

empirical relation like the following one:

dN2,0M tt ⋅⋅= ( 4.49 )

between the pre-load force Nt and the applied twisting moment Mt, where d is the

diameter of the bolt.

Based on the fact that the greater pressure is the greater the friction force is,

in order to obtain a maximum capacity of the connection, a maximum pre-load force

Nt needs to be applied. According to the Romanian code C133–82 [8], the pre-load

force should be:

cbt RAkN ⋅⋅= ( 4.50 )


110

where:

k – behaviour factor;

k = 0,8 for 8.8 bolt grade;

k = 0,7 for 10.9 bolt grade;

Ab – area of the cross-section of the bolt in the threaded zone; it may be taken

from tables or it may be calculated using the approximate formulae:

4d

A2s

b⋅π= ( 4.51 )

d89,0ds ⋅≅ ( 4.52 )

d – nominal diameter of the bolt;

Rc – yield strength of the bolt (fyb in table 4.2);

The pre-load force Nt may be practically obtained by:

• using a dynamometric wrench calibrating the required Mt;

• turning-off the nut tightening (after the first snug tight, an additional turning is

applied, representing an amount of a complete turn i.e. 0,25 to 0,75 turn).

An important friction appears between plates (Fig. 4.42) as a result of the tightening.

Under these circumstances, the slip resistance of a pre-loaded bolt is [8]:

tff NfnmN ⋅⋅⋅= ( 4.53 )

where:

m – working condition factor (it has the meaning of a partial safety factor);

m = 0,95 for static loading;

m = 0,85 for dynamic loading;

nf – number of friction (slip) interfaces;

f – slip factor; according to [8] it generally may be considered as:

f = 0,25 for cleaned surfaces without any brushing;

f = 0,35 for brushed surfaces using wire brushes or for burnt surfaces;

f = 0,50 for blasted surfaces;

Nt – the pre-load force.

The equation (4.53) shows that the slip resistance of a bolt increases when the pre-

load force Nt increases. Following this, a higher strength bolt allows a higher slip

resistance. It may be also noticed that the greater the slip factor f is the greater the

slip resistance is. A treatment of the surfaces in contact improves friction.


111

Fig. 4.42. The basic principles of a slip connection

Figure 4.43 shows the general behaviour of a shear connection. It can be

noticed that the ultimate load Fu is the same for a given bolt and it corresponds to the

failure of a bearing type connection (which is produced by the lowest value between

the force that causes failure of the plates and the force that causes shear failure of

the bolt). The presence of the pre-load force Nt only increases the range of elastic

behaviour and it delays slipping but it has no practical influence on the ultimate

capacity of the connection.

Fig. 4.43. General behaviour of a shear connection

4.3.3.4. Behaviour of bolts in tension

Tension is applied on the bolt (Fig. 4.44) at the contact between one plate and

the head of the bolt (or the washer which is under the head) at one end and at the

Nf Nf

Nf/2

Nf/2

Nf/2

Nf/2

Nt

Nt

friction forces

normally tightened connection

partially pre-loaded slip connection

pre-loaded slip connection

F

Fu

Nf1

Nf2

L


112

contact between the other plate and the washer which is under the nut at the other

end. A bolt in tension fails in the most reduced cross-section, in the threaded zone of

the shank. The area of the cross-section of the bolt in this zone can be taken from

tables or it may be calculated using relations (4.51) and (4.52).

Fig. 4.44. Bolt in tension

4.3.3.5. Design resistance of bolts according to STAS 10108/0–78 [7], C133–82 [8]

1. Bolts in tension connections bibi,cap RAN ⋅= ( 4.54 )

where:

Ab – area of the cross-section of the bolt (from table or using rel. (4.51));

biR – tension design strength of the bolt, as given in table 4.3.

2. Ordinary bolts in shear connections

( )p,fg,pf,cap N;NminN = ( 4.55 )

bg,p

ming,p RtdN ⋅⋅= ( 4.56 )

bfbp,f RAN ⋅= ( 4.57 )

where the terms are explained at relations (4.42) and (4.43) and values of the

design strength are given in table 4.3.

3. High-strength bolts in slip connections

tff NfnmN ⋅⋅⋅= ( 4.58 )

where the terms are explained at relation (4.53).


113

4. Bolts used in tension and shear connections

• Shear connections

Apart from checks using relations (4.54) and (4.55) for the capable forces, an

interaction check is needed. This check is based on the von Mises criterion.

ANL

=σ ( 4.59 )

ANT

=τ ( 4.60 )

R3 22 ≤τ⋅+σ ( 4.61 )

where:

NL – the force acting along the axis of the bolt;

NT – the force acting normal to the axis of the bolt;

A – area of the cross-section of the bolt; if shear occurs in the threaded zone of

the shank the reduced area given by relation (4.51) shall be used.

R – design strength of the steel grade of the bolt;

• Slip connections

The force NL reduces the pre-load Nt and it unfavourably affects the capacity of

the connection. The capable force is in this case:

( )Ltff NNfnmN −⋅⋅⋅= ( 4.62 )

Table 4.3. Design strength for bolts according to STAS 10108/0–78 [7]

Bolt grade Steel grade of plates Design strength [N/mm2]

m 4.6 5.6 6.6*) OL37 OL44 OL52

Shear bfR 0,6 130 160 180 – – –

Bearing bg,pR 1,6 – – – 350 415 500

Tension biR 0,8 170 210 240 – – –

*) They are no longer in fabrication

4.3.4. Spacing of holes

In order to avoid failure of plates between neighbour holes and to prevent

corrosion between connected elements, codes usually give some limitations


114

concerning the spacing of holes for bolts and rivets. In the Romanian code STAS

10108/0–78 [7], they are as follows (Fig. 4.45):

( )t12;d8mined3 00 ≤≤ ( 4.63 )

( )t8;d4mined2 010 ≤≤ ( 4.64 )

( )t8;d4mined5,1 020 ≤≤ ( 4.65 )

( )21 t;tmint = ( 4.66 )

where:

d0 – diameter of the hole;

e – spacing between centres of fasteners on any direction;

e1 – end distance from the centre of a hole to the adjacent end of any part,

measured parallel to the loading direction;

e2 – edge distance from the centre of a fastener hole to the adjacent edge of any

part, measured normally to the loading direction;

t – minimum thickness of exterior plates.

Fig. 4.45. Spacing of holes

4.3.5. Categories of bolted connections according to EUROCODE 3

Table 4.4 presents a classification of bolted connections given in EC3 [2]:

e e e

e

e1 e1

t1

t2

e2

e2


115

Table 4.4. Categories of bolted connections according to EUROCODE 3 [2]

Shear connections

Category Criteria Remarks

A bearing type

N ≤ Nf N ≤ Np,g

• No pre-loading required • No special provisions for surfaces treatment • All grades 4.6 to 10.9

B slip resistant at

serviceability state

Nserv ≤ Nf, serv N ≤ Nf

N ≤ Np,g

• Pre-loaded high strength bolts • No slip at serviceability limit state • Surfaces treatment

C slip resistant at ultimate state

N ≤ Nf N ≤ Np,g

• Pre-loaded high strength bolts • No slip at ultimate limit state • Surfaces treatment

Tension connections

D non-preloaded

N ≤ Ni

• No pre-loading required • No special provisions for surfaces treatment • All grades 4.6 to 10.9

E preloaded

N ≤ Ni • Pre-loaded high strength bolts • No slip at ultimate limit state • Surfaces treatment

4.3.6. Examples of calculation

4.3.6.1. General aspects

Checking a fastened connection generally consists of the following steps:

1. Establishing the design cross-section of the connection, that consists of points;

2. Reducing loads in the centre of gravity of the cross-section;

3. Establishing the load distribution on the cross-section;

4. Checking the most loaded fastener.

A force acting on any direction in the centre of gravity of the connection

uniformly distributes its effects on all fasteners in the connection. A moment acting in

the centre of gravity of the connection distributes its effects on each fastener

proportionally to the distance from that fastener to the centre of rotation. The first

three steps of the checking procedure are the same for all types of fastened


116

connections (rivets, bolted connections, slip connections). The influence of the type

of fastener appears only in the final step, when establishing the capable force.

4.3.6.2. Connection loaded only in its plane (Fig. 4.46)

Fig. 4.46. Fastener connection loaded only in its plane

The force produced in a fastener i by the moment M (Fig. 4.46) is proportional

to the displacement i. This displacement is normal to the radius of the point, ri, and

it is proportional to that radius, considering a rotation .

iri ⋅= ( 4.67 )

As all fasteners are identical, they have the same stiffness K. The force Ni produced

by the moment in a fastener can be expressed as:

ii rKKNi ⋅⋅=⋅= ( 4.68 )

The moment is resisted by all the fasteners in the connection:

=

⋅=n

1jjj rNM ( 4.69 )

where n is the number of fasteners in the connection.

Using relation (4.68) in relation (4.69), the following relations can be written:

=

⋅⋅=n

1j

2jrKM ( 4.70 )

N

T M

z

x x x

z

design cross-section

ri

M,iN xN,iN x

M,iN

zT,iN

zM,iN


117

=

=⋅ n

1j

2jr

MK ( 4.71 )

Following this, the force Ni produced by the moment in the fastener i is:

in

1j

2j

i rr

MN ⋅=

=

( 4.72 )

Based on the following notations:

2i

2i

2i zxr += ( 4.73 )

i

ii

xM,i r

zNN ⋅= ( 4.74 )

i

ii

zM,i r

xNN ⋅= ( 4.75 )

it can easily be proved that:

( )

=

+⋅= n

1j

2j

2j

ixM,i

zx

zMN ( 4.76 )

( )

=

+⋅= n

1j

2j

2j

izM,i

zx

xMN ( 4.77 )

( ) ( )2zM,i

2xM,ii NNN += ( 4.78 )

It is obvious that the most loaded fastener is the one situated at the greatest distance

from the centre of gravity of the connection.

For the problem in figure 4.46:

nN

NxN,i = ( 4.79 )

nT

NzT,i = ( 4.80 )

Based on relations (4.76), (4.77), (4.79) and (4.80), the resultant force in the most

loaded fastener is obtained for the maximum value of ri:

( ) ( )2zM,i

zT,i

2xM,i

xN,imax,i NNNNN +++= ( 4.81 )

This force must be less than the capable force of the fastener:

capmax,i NN ≤ ( 4.82 )


118

Depending on the type of fastener, Ncap may be calculated using relation (4.55) for

rivets and bolts in ordinary shear connections or relation (4.58) for high-strength

bolts in slip connections.

4.3.6.3. Connection loaded normally on its plane (Fig. 4.47)

The model accepted by the Romanian code STAS 10108/0–78 [7] assumes

the end-plate as infinitely rigid. A force acting on any direction in the centre of gravity

of the connection uniformly distributes its effects to all fasteners in the connection.

The model used for calculating the efforts produced by a bending moment M

resembles to the one used for a reinforced concrete cross-section. A moment

equation should be written by the centre of compressions (Fig. 4.47b):

=

⋅=n

1jjj rNM ( 4.83 )

Based on the infinite rigidity of the end plate assumption, efforts in each fastener are

proportional to the distance ei from that fastener to the neutral axis (Fig. 4.47b).

ieKNi ⋅= ( 4.84 )

where K is a constant.

Fig. 4.47. Fastener connection loaded normally on its plane

( a ) ( b ) ( c ) ( d ) ( e )

N

M T

ei ri

hi

x

z

xN,iN

zT,iN

xM,iN


119

These equations are hard to be handled, so a simplified approach is used: the

compression centre is on the same line with the rotation axis, which is situated on

the last line of fasteners (Fig. 4.47e). In this case:

iii her == ( 4.85 )

where hi is the distance from fastener i to the line of least tensioned fasteners (Fig.

4.47e). Under these circumstances, the force produced in a fastener i by the

moment M (Fig. 4.47e) is proportional to the fastener elongation li. This elongation

is proportional to the distance hi, considering a rigid body rotation .

ii hl ⋅=∆ ( 4.86 )

As all fasteners are identical, they have the same stiffness K. The tension force Ni

produced by the moment in a fastener can be expressed as:

iii hKlKN ⋅⋅=∆⋅= ( 4.87 )

The moment is resisted by all the fasteners in the connection:

=

⋅=n

1jjj hNM ( 4.88 )

where n is the number of fasteners in the connection. Replacing (4.87) in (4.88), it

can easily be proved that:

=

⋅⋅=n

1j

2j

hKM ( 4.89 )

=

=⋅ n

1j

2jh

MK ( 4.90 )

Following this, the force Ni,M produced by the moment in the fastener i is (Fig. 4.47e):

in

1j

2j

x hh

MN

M,i⋅=

=

( 4.91 )

and it has the maximum value for the maximum distance hi. The forces produced by

the axial force N (Fig. 4.47c) and by the shear force T (Fig. 4.47d) are:

nN

NxN,i = ( 4.92 )

nT

NzT,i = ( 4.93 )

When solving the problem in figure 4.47a, there are basically three groups of

checks that need to be done:


120

A. Check in the longitudinal direction of the fastener (Fig. 4.47c), (Fig. 4.47e):

xmaxM,i

xN,imax,i NNN += ( 4.94 )

capmax,i NN ≤ ( 4.95 )

where Ncap is calculated using relation (4.54).

B. Check in the plane of the connection (Fig. 4.47d):

capzT,i NN ≤ ( 4.96 )

where the transverse capable force Ncap is calculated using relation (4.55) for

rivets and bolts in ordinary shear connections. For high-strength bolts in slip

connections the following interaction checks apply.

C. Interaction check, depending on the type of fastener:

• Shear connections

A check based on the von Mises criterion is used. The normal stress and

the tangential stress are calculated in the shared cross-section of the

fastener. Relations (4.59), (4.60) and (4.61) are used.

• Slip connections

The longitudinal force in the bolt reduces the pre-load Nt and it unfavourably

affects the capable force. The capable force is in this case:

( )[ ]xM,i

xN,itff NNNfnmN +−⋅⋅⋅= ( 4.97 )

If the end-plate stands on a support that is welded on the column, it is

considered that the shear force is directly transferred to this support and in-plane

checks are no longer necessary, as the fasteners do not carry this force.

5. DESIGN OF STRUCTURAL MEMBERS

121

Chapter 5

DESIGN OF STRUCTURAL MEMBERS

5.1. BASIS OF DESIGN

5.1.1. Design method

Generally, in most of the present day codes the design of structural steel

members is based on the limit states design method, as shown at title 2.3.6.1, taking

into account:

• ultimate limit states;

• serviceability limit states.

The application of this design method to steel structures presents some

particularities due to the particular behaviour of steel structures.

5.1.2. Stability of steel structures

Due to the high strength of structural steels, structural steel members are

slender ones. As a result, typically for steel structures, the ultimate limit state of

resistance, expressed by relation (2.29):

Sd ≤ Rd ( 5.1 )

must be checked as:

1. resistance of cross-sections:

Sd ≤ Rd ( 5.1a )

where:

Sd is the design value of an internal effort, calculated with factored loads;

Rd is the corresponding design resistance, calculated with the design strength.

2. buckling resistance of members:

Sd ≤ Rd,cr ( 5.1b )


122

where:

Sd is the design value of an internal effort, calculated with factored loads;

Rd,cr is the corresponding design buckling resistance.

5.1.3. Cross-section particularities

The most common cross-sections of steel structural members are developed

in the plane of the acting bending moment (Fig. 5.1). This is typical for metal

structural members and they are generally characterized by:

Fig. 5.1. Typical metal cross-section

As a result:

• all the strength, stiffness and stability requirements are to be satisfied by the

cross-section itself with regard to the strong axis y–y;

• some special means are to be considered with regard to the weak axis z–z;

• torsion rigidity is very poor, Ir ≅ 0; generally, metal structures are designed to

avoid torsion in such structural members;

• the slenderness of the web and the stresses in the compressed flange can lead

to local buckling (typical for metal members) affecting the load carrying capacity

of structural members; generally, local buckling can be:

• local buckling of flanges of members in compression (Fig. 5.2a);

• local buckling of the web of members in compression (Fig. 5.2b);

• local buckling of the compressed flange of members in bending (Fig. 5.2c);

• local buckling of the web of members in bending (Fig. 5.2d).

Iy >> Iz

Wy >> Wz

iy >> iz

y y y y

z z

z z


123

( a ) ( b ) ( c ) ( d ) Fig. 5.2. Local buckling

5.1.4. Classification of cross-sections

Depending on the stress state that causes local buckling, cross-sections of

structural members are classified as [2] (Fig. 5.3):

Class 1 – cross-sections that can form a plastic hinge with sufficient rotation

capacity to allow redistribution of bending moments. Only class 1 cross-

sections may be used for plastic design.

Class 2 – cross-sections that can reach their plastic moment resistance but local

buckling may prevent development of a plastic hinge with sufficient

rotation capacity to permit plastic design (redistribution of bending

moments).

Class 3 – cross-sections in which the calculated stress in the extreme

compression fibre can reach the yield strength but local buckling may

prevent development of the full plastic bending moment.

Class 4 – cross-sections in which it is necessary to take into account the effects of

local buckling when determining their bending moment resistance or

compression resistance.

Fig. 5.3. Possible stress distribution, depending on the cross-section class

class 4 class 3 class 2 class 1

max < fy max = fy max = fy max = fy

y y

z

z

( – )

( + )


124

The class of a cross-section is the maximum among its components. Tables 5.1, 5.2,

5.3 show the requirements for different cross-sectional classes.

Table 5.1. Limitations for the slenderness of internal walls [2]

Class Wall in bending

Wall in compression

Wall in bending and compression

Stress distribution

1

72tc ≤

33tc ≤

when > 0,5: 113396

tc

−≤

when 0,5:

36tc ≤

2

83tc ≤

38tc ≤

when > 0,5: 113456

tc

−≤

when 0,5:

5,41tc ≤

Stress distribution

3

124tc ≤

42tc ≤

when > –1: 33,067,0

42tc

+≤

when –1: ( ) ( )162tc −−≤

fy (N/mm2) 235 275 355 420 460

yf235

= 1,00 0,92 0,81 0,75 0,71

Note: (+) means compression

c c c c

c c c

c

t t t t

t t t

t

Bending axis

Bending axis

c c c c

fy

fy

fy

fy

fy

fy

c c c c/2

fy

fy

fy

fy

fy


125

Table 5.2. Limitations for the slenderness of flanges [2]

Tension and compressed flange Class

Compressed flange Compressed edge Tension edge

Stress distribution

1

9tc ≤

9tc ≤

9tc ≤

2 10

tc ≤

10tc ≤

10tc ≤

Stress distribution

3

14tc ≤ k21

tc ≤

fy (N/mm2) 235 275 355 420 460

yf235

= 1,00 0,92 0,81 0,75 0,71

Note: (+) means compression

Table 5.3. Limitations for the slenderness of the walls of round tubes [2]

Class Cross-section in bending and/or compression

1 d/t 502

2 d/t 702

3 d/t 902

fy (N/mm2) 235 275 355 420 460

1,00 0,92 0,81 0,75 0,71 yf

235 =

2 1,00 0,85 0,66 0,56 0,51

t t t t

c c c c

c c c

c c c

c c

d t


126

5.1.5. Elastic and plastic design

Depending on the general stability of the structural member and on the cross-

section class, different types of analysis can be used. This is illustrated on simple

example of a fixed beam subjected to a uniformly distributed load. It is presumed that

the cross-section of the beam is the same on its entire length. In relation (5.1) Sd is

the bending moment produced by exterior forces in the most loaded cross-section,

while Rd is the resisting bending moment of the cross-section.

1. Elastic–critical design: class 4 cross-sections

Efforts in structural members are calculated using an elastic model and the cross-

section is checked using a critical stress distribution (local buckling occurs before

reaching the yielding limit in the most compressed fibre).

Fig. 5.4. Elastic–critical design

12

LqMS

2

maxd⋅== ( 5.2 )

crelcapcr,dd WMRR ⋅=== ( 5.3 )

ycr f < ( 5.4 )

ymax < ( 5.5 )

where cr and max are respectively the stress and the elongation in the most

compressed fibre, while y is the yielding elongation. The safety check (5.1) is:

dcapcrel

2

maxd RMW12

LqMS ==⋅≤⋅== ( 5.6 )

2. Elastic–elastic design: class 3 cross-sections

Efforts are calculated using an elastic model and the cross-section is checked

using the elastic stress distribution.

q max < fy/2

max = cr < fy

L 24Lq

M2

2⋅=

12Lq

M2

1⋅=


127

12

LqMS

2

maxd⋅== ( 5.7 )

yelcapel,dd fWMRR ⋅=== ( 5.8 )

ycr f ≥ ( 5.9 )

ymax ≥ ( 5.10 )

where fy is the stress in the most compressed fibre. Local buckling occurs after

reaching the yielding limit in the most compressed fibre. The safety check (5.1) is:

dcapyel

2

maxd RMfW12

LqMS ==⋅≤⋅== ( 5.11 )

Fig. 5.5. Elastic–elastic design

3. Elastic–plastic design: class 2 cross-sections

Efforts are calculated using an elastic model and the cross-section is checked

using the plastic stress distribution.

12

LqMS

2

maxd⋅== ( 5.12 )

yplcappl,dd fWMRR ⋅=== ( 5.13 )

Fig. 5.6. Elastic–plastic design

q

q

L

L

max fy/2

max = fy

max = fy

max < fy

24Lq

M2

2⋅=

24Lq

M2

2⋅=

12Lq

M2

1⋅=

12Lq

M2

1⋅=


128

ycr f > ( 5.14 )

ymax > ( 5.15 )

Plastic hinges appear at the two ends of the beam, but local buckling that occurs

after the formation of the plastic hinges does not allow plastic redistribution of

efforts and the development of a third plastic hinge in the middle of the span. The

safety check (5.1) is:

dcapypl

2

maxd RMfW12

LqMS ==⋅≤⋅== ( 5.16 )

4. Plastic–plastic design: class 1 cross-sections

Efforts are calculated using a plastic model and the cross-section is checked

using the plastic stress distribution. As the load increases, plastic hinges appear

at the two ends of the beam. After that, these cross-sections rotate freely in the

plastic range under constant bending moment and the bending moment in the

middle of the span increases till the formation of a third plastic hinge. No local

buckling occurs until the development of this plastic mechanism.

Fig. 5.7. Plastic–plastic design

16

LqMMMS

2

21maxd⋅==== ( 5.17 )

yplcappl,dd fWMRR ⋅=== ( 5.18 )

ycr f > ( 5.19 )

ymax >> ( 5.20 )

dcapypl

2

maxd RMfW16

LqMS ==⋅≤⋅== ( 5.21 )

q

L

max = fy

max = fy

16Lq

M2

1⋅=

16Lq

M2

2⋅=


129

5.2. TENSION MEMBERS

5.2.1. General

Tension members are largely used in truss construction, braced frames and

different other structural elements. They are also part of cable structures.

5.2.2. Types of single and built-up members

Figure 5.8 shows different types of cross-sections used for tension members:

Fig. 5.8. Examples of types of cross-sections used for tension members

In built-up members, consisting of two or more main components (Fig. 5.8b, c,

d), the parts are connected in order to behave like a single shape (Fig. 5.8a).

Built-up members can be realised using:

• components in contact (Fig. 5.8b);

• slightly distanced components (Fig. 5.8c);

• largely distanced components (Fig. 5.8d).

( a )

( b )

( c ) ( d )


130

Components in contact are usually connected by intermittent or continuous

weld seams as shown in figure 5.9. The continuous ones are preferred.

Fig. 5.9. Recommendations for connecting components of tension members

Slightly distanced components are connected by welded or bolted plates like

shown in figure 5.10. See also 5.3.3.3.


Largely distanced components are connected either with laces (Fig. 5.11a) or

battens (Fig.5.11b). See also 5.3.3.3.


30t

24t

t1 t2

t = min(t1; t2)

y y

z

z

z1

z1

z1

z1

L1 80iz1

L1 80iz1

L1 80iz1

L1 80iz1

y

y y

y

z z

z z

z1 z1

z1 z1

z1 z1

z1 z1

( a )

( b )


131

5.2.3. Calculation

According to STAS 10108/0–78 [7], the following relation shall be satisfied:

RAN

net

≤= ( 5.22 )

In (5.22) N is the design tensile force, calculated with factored loads, R is the design

strength of the steel grade and Anet is the minimum net area in a cross-section

perpendicular to the axis of the tension member, or any diagonal or zigzag section.

For the case in figure 5.12 it is to be considered the minimum of:

( )( )( )22,net11,netnet

0122,net

011,net

A;AminA

td2La2A

tdbA

−−

−

−

=⋅⋅−+⋅=

⋅−= ( 5.23 )

Fig. 5.12. Possible sections for establishing the net area Anet

According to EUROCODE 3 [2], the main check for members in tension is:

0,1NN

Rd,t

Ed ≤ ( 5.24 )

In (5.24) NEd is the design tensile force, calculated with factored loads, while the

design tension resistance of the cross-section Nt,Rd is the smallest of:

• the design plastic resistance of the gross cross-section:

0M

yRd,pl

fAN

⋅= ( 5.25 )

where fy is the specified minimum yield strength and the value of the safety factor

M0 is given in the National Annex of the code [2]; the recommended value is:

0,10M = ( 5.26 )

• the design ultimate resistance of the net cross-section at holes for fasteners:

2

2

1

1

t

b

a

p

a

d0 L1


132

2M

unetRd,u

fA9,0N

⋅⋅= ( 5.27 )

where:

Anet– the net area of the cross-section, as shown in figure 5.12;

fu – the ultimate strength of the steel grade;

M2 – safety factor given in the National Annex; the recommended value is:

25,12M = ( 5.28 )

Where ductile behaviour is required (in case of capacity design, requested for

a good seismic behaviour), the design plastic resistance Npl,Rd should be less than

the design ultimate resistance of the net cross-section at holes for fasteners Nu,Rd, so

the following condition shall be satisfied:

Rd,plRd,u NN > ( 5.29 )

which leads to:

0M

2M

u

ynet

ff

AA

9,0

⋅>⋅ ( 5.30 )

In category C connections (see table 4.4) the design tension resistance Nt,Rd

of the net section at holes for fasteners should not be taken as more than:

0M

ynetRd,net

fAN

⋅= ( 5.31 )

5.3. COMPRESSION MEMBERS

5.3.1. General

Compression members may be found in structures as columns, components

of truss constructions, elements of braced frames and as different other structural

elements. Purely axially loaded members (either in tension or in compression) are

not frequent among structural elements but, for some members, the other loads like

the torsion moment, the bending moment and the shear force may be neglected.

Compression in a structural member is frequently associated with bending moment

and shear force but in order to be able to analyze such an element, axially

compressed members need to be studied first.


133

5.3.2. Buckling

5.3.2.1. Buckling and local buckling

The buckling load is the critical force Fcr at which a perfectly straight member

in compression assumes a deflected position (Fig. 5.13a). Buckling is a limit state, in

the meaning that once the force Fcr is reached the deflection increases until the

collapse of the bar is reached. The member should be subjected only to loads

inferior to the critical force (F < Fcr).

Local buckling is the loss of local stability of a part of a member, produced

by in-plane stresses. Stresses that lead to local buckling can be either normal

compression stresses (), or shear stresses (). In case of compression members,

this means that a certain value of the force Fcr,v leads to the local buckling of the web

(Fig. 5.13b), of the flanges (Fig. 5.13c) or of both of them (Fig. 5.13d).

Fig. 5.13. Buckling

Remarks:

1. Local buckling is not necessarily a limit state of a compression member. The

member is often able to resist compression loads superior to Fcr,v, the force that

produced local buckling.

2. Local buckling reduces the critical force Fcr that the member is able to resist.

( a ) ( b ) ( c ) ( d )

Fcr


134

5.3.2.2. Forms of buckling

When subjected to an axial compression force, a straight member may lose

its stability in one of the following forms (Fig. 5.14):

• flexural buckling (v ≠ 0; ϕ = 0) (Fig. 5.14a);

• torsion buckling (v = 0; ϕ ≠ 0) (Fig. 5.14b);

• flexural-torsion buckling (v ≠ 0; ϕ ≠ 0) (Fig. 5.14c);

where v means the lateral displacement in the plane of the cross-section and ϕϕϕϕ is the

rotation of the cross-section in it’s plane.

( a ) ( b ) ( c )

Fig. 5.14. Forms of buckling

5.3.2.3. Approach methods

Beginning with Euler, during the XVIIIth century, different researchers tried to

express the equilibrium and the failure mode of a perfectly straight member

subjected to axial compression. The most common approach methods used for

studying buckling of elements in compression are the following ones:

• the static method;

• the design methods of Statics;

• the energetic method.

Fcr Fcr Fcr

v v

ϕ ϕ


135

1. The static method

A static criterion is established to express equilibrium. It is based on the analogy

with the balance of a ball on a surface (Fig. 5.15). Based on this, three different

situations can be illustrated:

• stable;

• limit;

• unstable.

stable limit unstable

Fig. 5.15. A static criterion for expressing equilibrium

In figure 5.15 the initial state is (0) and the final one is (1). In the limit case there

are more positions that allow equilibrium. The use of this method is illustrated

with the following example of pin connected bar in axial compression (Fig. 5.16).

Fig. 5.16. The balance of a pin connected bar in compression

Two positions of equilibrium are possible:

• the straight line;

• the slightly curved line.

The following relations can be written:

1EIM

dxdv

1

dxvd

232

2

2

=−=

+

( 5.32 )

where:

vFM ⋅= ( 5.33 )

If v < L/400...L/300, then

0 1 0 1 0 1

L

v

F x


136

0dxdv ≅ ( 5.34 )

1dxdv

1232

≅

+ ( 5.35 )

It results:

1EI

vFdx

vd2

2

=⋅−= ( 5.36 )

0vkv 2 =⋅+′′ ( 5.37 )

where

EIF

k2 = ( 5.38 )

kxcosCkxsinCv 21 ⋅+⋅= ( 5.39 )

Considering the limit conditions,

x = 0 v = 0 C2 = 0

x = L v = 0 kLsinC0 1 ⋅= sin kL = 0 kL = π

the solution is the one obtained by Euler (1744):

2

2

cr LEI

F⋅= ( 5.40 )

2. The design methods of Statics

This means the use of the two known methods:

• the method of efforts;

• the method of displacements.

They are used mainly in computer programs. It generally means solving a

problem of eigenvalues.

3. The energetic method

It is based on the laws of energy conservation:

intext LL = ( 5.41 )

where:

Lext – work produced by exterior actions;

Lint – work produced by internal efforts.


137

Remarks

1. The energetic method generally leads to values of the critical forces which are

superior to the real ones. This is because the chosen deflected shape is not the

real one. This method can be used in complicated cases.

2. Classic problems and those ones that are found in codes are usually solved

using the static method.

3. The design methods of Statics are generally used for structures.

5.3.2.4. Bifurcation and divergence of equilibrium

The bifurcation of equilibrium is the approach based on the theoretical

member; the axis is perfectly straight and the load acts rigorously in the centre of

gravity of the cross-section of the element. The behaviour in this model is as follows:

• For F < Fcr the straight form of the bar is stable. If a force acts transversely to its

axis the bar is bent. After removing the transverse load the member returns to the

straight line.

• For F > Fcr the straight shape is no longer stable. After removing the transverse

load the member does not return to the straight line.

• For F = Fcr two positions of equilibrium are possible:

• the straight line;

• the slightly bent form.

Fig. 5.17. Bifurcation and divergence of equilibrium

The divergence of equilibrium is the approach based on the actual member,

with its imperfections, consisting of:

F

Fcr

v

bifurcation

divergence


138

• physical imperfections, such as;

• variation of the mechanical properties of steel from one point to another;

• variation of residual stresses;

• variation of Young’s modulus (E);

• geometrical imperfections, like:

• initial deformation of the bar;

• eccentricity of the load with respect to the centroid line.

5.3.2.5. The general equation of stability

We consider the general case of a member in compression. The cross-section

has no axis of symmetry (Fig. 5.18). The bar is pin connected at both ends.

Fig. 5.18. Virtual displacement of the cross-section of the bar

In figure 5.18 letters have the following meanings:

G – centre of gravity of the cross-section;

C – torsion centre of the cross-section;

v – displacement along y axis;

w – displacement along z axis;

– rotation of the cross-section in the plane yOz;

A virtual displacement is considered.

The energetic equation has the form:

z

y

v

w

G

G’

yc

zc

C

C’

z'

y'


139

( ) 0LLdd intext =−= ( 5.42 )

Both external and internal virtual works depend on the three virtual displacements (v,

w, ϕ). This leads to a system of differential equations [9]:

( )

=′′⋅⋅+′′⋅⋅−′′⋅⋅−−⋅

=′′⋅⋅+′′⋅+⋅

=′′⋅⋅−′′⋅+⋅

0wyFvzFiFGIEI

0yFwFwEI

0zFvFvEI

cc2cr

IV

cIV

y

cIV

z

( 5.43 )

where the following notations were used: 2p

2c

2c

2c izyi ++=

2z

2y

2p iii +=

yc I

zdAy ⋅

=

zc I

ydAz ⋅

=

G – shear modulus of elasticity;

Ir – torsion constant of the cross-section;

Iω – warping constant of the cross-section;

=A

2 dAI

dsrd ⋅= (Fig. 5.19).

Fig. 5.19. Diagram for coordinates

The constraints at limits are:

x = 0 v = w = ϕ = 0 ; v” = w” = ϕ“ = 0

x = L v = w = ϕ = 0 ; v” = w” = ϕ“ = 0

This leads to:

r C

ds


140

⋅=

⋅=

⋅=

Lx

sinA

Lx

sinAv

Lx

sinAw

3

2

1

( 5.44 )

Replacing these expressions, the following system is obtained:

( )

( )( )

=⋅⋅−−⋅⋅−⋅⋅=⋅⋅−⋅−=⋅⋅+⋅−

0AiPFAzFAyF0AzFAPF0AyFAPF

32c2c1c

3c2z

3c1y

( 5.45 )

where:

2y

2

y LEI

P⋅

= ( 5.46 )

2z

2

z LEI

P⋅= ( 5.47 )

⋅+= 2

2

r2c

L

EIGIi1

P ( 5.48 )

Buckling means at least one of A1, A2, or A3 must be different of 0. This leads to:

( )

0iPFzFyF

zFPF0yF0PF

2ccc

cz

cy

=⋅−⋅−⋅

⋅−−⋅−

( 5.49 )

which is the general equation of stability:

( ) ( ) ( ) ( ) ( ) 0PFzFPFyFiPFPFPF y2c

2z

2c

22czy =−⋅⋅−−⋅⋅−⋅−⋅−⋅− ( 5.50 )

Remarks:

1. This equation has three solutions.

2. For non-symmetric cross-sections (yc ≠ 0; zc ≠ 0), the three forces F1 < F2 < F3

correspond to flexural-torsion buckling.

3. For single symmetric cross-sections (yc ≠ 0; zc = 0), the equation becomes:

( ) ( ) ( )[ ] 0yFiPFPFPF 2c

22cyz =⋅−⋅−⋅−⋅−

2z

2

z1 LEI

PF⋅== corresponds to flexural buckling;

F2, F3 correspond to flexural-torsion buckling.

4. For double symmetric cross-sections (yc = 0; zc = 0), the equation becomes:

( ) ( ) ( ) 0iPFPFPF 2czy =⋅−⋅−⋅−


141

2z

2

z1 LEI

PF⋅== corresponds to flexural buckling;

2y

2

y2 LEI

PF⋅

== corresponds to flexural buckling;

⋅+== 2

2

r2c

3 L

EIGIi1

PF corresponds to flexural-torsion buckling.

5.3.2.6. Flexural buckling

The theoretical study of buckling began with the pin connected bar (Fig. 5.16),

under the following circumstances:

• the axis of the member is rigorously straight;

• the compression load acts strictly in the centre of gravity of the cross-section;

• the cross-section is bi-symmetrical;

• the material is homogenous and has a perfectly elastic behaviour (E=constant).

Considering this, Euler proved in the XVIIIth century that:

2

2

cr LEI

F⋅= ( 5.51 )

This is rigorously exact if:

• the deflected shape is a sinusoid;

• the elastic modulus E is constant;

• the moment of inertia of the cross-section is constant all along the bar.

This relation was then extended to other types of restraints at the ends:

2f

2

cr LEI

F⋅= ( 5.52 )

where Lf = µµµµL is the effective length (buckling length) (Fig. 5.20).

The following relations can be written:

( ) 2

2

2

f

2

2f

22

2f

2cr

E

iL

E

LiE

ALEI

AF ⋅=⋅=⋅⋅=

⋅⋅= 2

2

cr

E

⋅= ( 5.53 )

where λλλλ=Lf/i is the slenderness of the bar.


142

µ = 1,0 µ = 0,7 µ = 2,0 µ = 0,5 µ = 1,0

Fig. 5.20. Different values of the buckling length factor µµµµ

Remarks

1. The force Fcr has a physical meaning, being the force that produces buckling of

the bar.

2. σcr is not a real stress, it is a conventional one; during buckling of the bar, the

stress distribution on the cross-section is no longer constant.

3. The relation (5.53) stands only in the range where Young’s modulus E is

constant. This happens when σ < σp (σσσσp being the proportionality limit of the steel

grade), which means:

p

pp2

2

cr

E

E ⋅≥ ≤⋅=

Knowing that σp is about 80% of the yielding limit, it means Euler’s relation stands

only in the following ranges:

• for OL37 → λp ≥ 104;

• for OL44 → λp ≥ 95;

• for OL52 → λp ≥ 85;

4. For values of the slenderness superior to those ones above, the use of superior

quality steels is not rational, as the critical load is the same for all kinds of steel,

depending only on Young’s modulus which is the same.

5. As shown above, Euler’s relation is no longer valid for stresses outside the

proportionality range, leading to critical forces superior to the real ones. These

forces increase as the slenderness λλλλ decreases. For low values of λλλλ it can lead to

values of the critical stress σσσσcr superior to the yielding limit, which is senseless.


143

Different researchers tried to find a more proper approach for the range where

Euler’s relation no longer stands (σ σp λ < λp). The following ones are among

those who provided the most accurate approaches:

1. In 1889 Engesser and Considère (Fig. 5.23) proposed to use Euler’s relation by

replacing Young’s modulus with the tangent modulus (Fig. 5.21):

2t

2

cr

E

⋅= ( 5.54 )

where

dd

Et = ( 5.55 )

Fig. 5.21. The tangent modulus used by Engesser and Considère (1889)

2. In 1890 Tetmayer proposed a linear approach (Fig. 5.23):

( )1f ycr ⋅−⋅= ( 5.56 )

when = 0 → σcr = fy and when = p → σcr = σp.

3. In 1910 von Kàrmàn and Iassinski, at the same time with Engesser, proposed a

new approach. They presumed that bending associated to buckling elastically

unloads the tensioned part of the cross-section, while in the rest of the cross-

section compression increases in the elasto-plastic range (Fig. 5.22). This

happens when the average stress on the cross section is greater than σσσσp. The

elastic modulus in the unloaded part is E, while in the compressed part it is Et,

the tangent one (Fig. 5.21). Considering this, they propose to use Euler’s relation

with a transformed elastic modulus (Fig. 5.23):

cr

p

fy

E

Et


144

2

2

cr

T

⋅= ( 5.57 )

where

I

IEIET ctt ⋅+⋅= ( 5.58 )

It and Ic being the moments of inertia of the “tensioned” and of the compressed

part of the cross-section, respectively (Fig. 5.22). I is the moment of inertia of the

entire cross-section.

Fig. 5.22. The model proposed by von Kàrmàn and Iassinski (1910)

4. In 1946 Shanley showed that none of the previous theories was rigorously

correct. He proved that the values of critical average stresses are between the

values given by Engesser and those ones given by von Kàrmàn. He accepted

that bending associated to buckling does not change the direction of strains, so it

does not unload a part of the cross-section. The behaviour of the entire cross-

section is in the elasto-plastic range (Fig. 5.23).

Fig. 5.23. Comparison among the presented models

F

E

Et

Et E

< cr

< cr

= cr

cr

p p

fy fy

Euler

von Kàrmàn

Shanley

Engesser

Tetmayer


145

5.3.2.7. Buckling curves

All the above theories were developed for the ideal straight bar made of a

perfect elastic and isotropic material, loaded in the centre of gravity of the cross-

section along the axis of the member. In everyday practice, we have to deal with the

actual industrial bar, which has a lot of imperfections:

• structural (physical) imperfections:

• steel is not homogenous and isotropic (the ideal material does not exist);

• the yielding limit varies:

• from one point to another on the cross-section;

• from one cross-section to another along the bar;

• from one bar to another;

• Young’s modulus E is not a constant;

• residual stresses of different origins:

• thermal (rolling procedure, welding procedure, cutting procedure,

etc.)

• mechanical (cold forming, straightening, etc.);

• geometrical imperfections:

• initial deflections of the bar;

• allowed variations of the cross-section along the bar;

• eccentricity of the load with respect to the axis of the bar.

Tests showed that a bar in compression has deflections starting from the

beginning of loading. These deflections increase step-by-step as the load increases.

These were among the main reasons that led to the idea of studying buckling

on the actual bar. Such a work was done by ECCS (European Convention for Steel

Structures) which conducted an experimental analysis. More than 1000 (1067)

specimens of real bars were tested in seven countries (Belgium, France, Germany,

Great Britain, Italy, Netherlands and Yugoslavia) in about ten years during the

decade 1960 – 1970.

Tested bars were either rolled or built-up by welding and their slenderness

was between 40 and 170. The critical force Fcr was measured. The purpose of these

tests was to find a connection between the critical force and the slenderness of the

bar.


146

The following were considered as random variables:

f0 = initial eccentricity of the load;

e0 = initial deflection of the bar;

A = initial area of the cross-section of the bar;

t = thickness of the flanges and of the web of the cross-section;

fy = yield stress;

σres = residual stress.

Tests showed that the values of critical forces for series of 8 to 20 identical bars

have a distribution close to a Gauss normal type one. Following this, the analysis

consisted basically of the following steps:

1. For a series of identical bars the critical forces were measured and a critical

stress was calculated by dividing the force to the initial area of the cross-section.

AF

crcr = ( 5.59 )

The results were represented as histograms (Fig. 5.24).

Fig. 5.24. Typical distribution of tests results

2. Statistic values were calculated:

• the relative frequency of results:

==i

iii n

nnn

f ( 5.60 )

• the mean value:

fi (ni)

2,28%

ks

cr mcr


147

=

⋅=n

1i

icri

mcr f ( 5.61 )

• the dispersion:

( )=

−⋅=n

1i

2mcr

icri

2 fs ( 5.62 )

• the standard deviation:

( )=

−⋅=n

1i

2mcr

icri fs ( 5.63 )

Remark

Gauss’s function

( )2

m

sxx

21

e2s

1xf

−⋅−⋅

⋅= ( 5.64 )

is rigorously correct. The critical stress distribution was presumed as a normal

one by introducing the computed values in Gauss’s function.

3. A characteristic value of the critical stress was computed:

sk mcr

kcr ⋅−= ( 5.65 )

Codes usually accept k = 2 (Fig. 5.24), which corresponds to a probability of

2,28%. The same value was used in this case.

4. The same procedure was used for different values of the slenderness of the bar.

The results were put on a diagram.

5. A curve was drawn to connect all these points using the MONTE CARLO

procedure.

Fig. 5.25. Example of drawing a buckling curve

cr

fy

test results

drawn curve


148

The results of tests led to the following conclusions:

1. The variation of the area of the cross-section, and of the thickness does not have

an important influence on the critical stress.

2. The critical stress (σσσσcr) is influenced by the initial deflection of the bar (e0) and by

the eccentricity of the load (f0).

3. The yield limit (fy) and the residual stresses (σσσσres) have a very important influence

on the critical stress.

4. Residual stresses have different influences on the resisting capacity of the cross-

section with respect to one of the two main axes; this means that critical stresses

depend on the buckling axis; the same cross-section has different values of the

critical stress, depending on the plane of buckling.

All these prove that a single curve is not enough. Following this, every

important code of practice uses three, four, five or six buckling curves, depending on:

• the shape of the cross-section;

• the axis of the cross-section (the plane of buckling);

• the yield limit of the steel grade.

Remarks

1. In every day practice, the value of the critical stress (σσσσcr) is expressed by means

of the buckling factor .

fAf

fAAN yy

crycrcr ⋅⋅=⋅⋅=⋅=

y

cr

f

= ( 5.66 )

2. The buckling curves are expressed [2] as function of the reduced slenderness:

1

= ( 5.67 )

where 1 is the slenderness corresponding to the yielding limit in Euler’s relation:

2

2

cr

E

⋅= ( 5.68 )

y

1ycr fE

f ⋅= = ( 5.69 )

The Romanian code of practice, STAS 10108/0–78 [7], uses three buckling

curves A, B and C. The relations defining the three curves are as follows:

( ) ( ) 2222 5,05,0

1

⋅−⋅++⋅+= ( 5.70 )


149

where:

E

= ( 5.71 )

i

L f= ( 5.72 )

c

E

E ⋅= ( 5.73 )

c – yielding limit of the steel grade that is used;

E – longitudinal elasticity modulus (Young’s modulus) of steel;

i – radius of gyration of the cross-section about the axis of buckling (normal to

the plane of buckling);

Lf – buckling length in the plane of buckling;

The values of factors and are given in table 5.4.

Table 5.4. Values of factors and [7]

Buckling curve Factor

A B C

0,514 0,554 0,532

0,795 0,738 0,377

Fig. 5.26. Buckling curves according to STAS 10108/0–78 [7]

A

B

C


150

EUROCODE 3 [2] uses 5 buckling curves A0, A, B, C, D. They are obtained

by means of the following relations:

1

122

≤−Φ+Φ

= ( 5.74 )

( )[ ]22,015,0 +−⋅+⋅=Φ ( 5.75 )

where is an imperfection factor whose values are given in table 5.5.

Table 5.5. Values for factor [2]

A0 A B C D

0,13 0,21 0,34 0,49 0,76

EUROCODE 3 [2] contains a table (table 6.2 [2]) that recommends the use of

the proper buckling curve, depending on the shape of the cross-section, on the

buckling axis, on the steel grade and on the thickness of the parts of the cross-

section. Curve A0 is recommended for some cross-sections made of S460, which

has a high yielding limit (fy ≥ 430N/mm2). Curve D is generally used for some cross-

sections made of thick plates (tf ≥ 40mm for welded cross-sections or tf ≥ 100mm for

hot-rolled ones).

0

0.2

0.4

0.6

0.8

1

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9 Fig. 5.27. Buckling curves according to EUROCODE 3 [2]

A0

A

B

C

D

=0,2


151

5.3.3. Practical design of compressed members

5.3.3.1. Cross-section philosophy

The cross-section of a compressed bar depends on the following:

• the value of the compression force;

• the type of structural member;

• the buckling length of the member;

• the presence of other loads (bending moments, shear forces, etc.);

• the type of connecting detail at the ends of the member.

The capable load of a tensioned member depends on the area of the cross-

section and does not depend on the shape of the cross-section. On the contrary, for

a member in compression, the capable load fundamentally depends on the shape of

the cross-section. It is very important to have the material away from the centroid line

of the member (Fig. 5.28) in order to get a greater radius of gyration. Two cross-

sections having the same area but different shapes will have different critical forces.

NO YES NO YES

Fig. 5.28. Cross-section philosophy for members in compression

Same as for tensioned members, members in compression can be made of:

• a single hot rolled or a single cold formed shape (Fig. 5.29a);

• built-up cross-sections:

• components in contact (Fig. 5.29b);

• slightly distanced components (Fig. 5.29c);

• largely distanced components (Fig. 5.29d).


152

Fig. 5.29. Examples of types of cross-sections used for compression members

5.3.3.2. Types of members in compression

The most common types of members in compression are:

• members of braced systems (Fig. 5.30);

Fig. 5.30. Examples of types of cross-sections used for members of braced systems

( a )

( b )

( c ) ( d )


153

• columns (Fig. 5.31);

Fig. 5.31. Examples of types of cross-sections used for columns

5.3.3.3. Connecting elements of a compressed member

In order to act like a whole, a built up cross-section must comply with the

following rules and recommendations:

1. For components in contact

• The weld seams should be continuous.

• If welds are not continuous, the gap among seams should be less than 15t

along the force and 24t transverse to the force (Fig. 5.32), where t is the

minimum thickness of the connected elements.

• If they are connected with fasteners they shall comply with the rules for

fastened connections.

Fig. 5.32. Recommendations for connecting components of compression members

2. For slightly distanced components

• Connecting plates are usually square (Fig. 5.33). Their length and width bp

shall be greater than 0,8b, where b is the width of the connected (back to

back) components (flanges or web).

24t

t1 t2

t = min(t1; t2)

15t


154

• The width bp (Fig. 5.33) of the plates should be 15–30mm less, or greater,

than b to allow welding.

• The thickness tp of connecting plates (Fig. 5.33) should be greater than

b/10 to allow protection against corrosion. In strong aggressive

environments it should be greater than b/6.

• The distance (Fig. 5.33) between two consecutive connecting plates shall

comply with:

1z1 i40L ⋅≤ ( 5.76 )

where iz1 is the radius of gyration of a single component about its axis

which is parallel to b (parallel to the plane which does not meet the cross-

section material (z–z plane)).

• There will be at least two connecting plates along a member, even if its

length would not demand it. For tension members there should be at least

one connecting plate along a member.

Fig. 5.33. Recommendations for connecting elements of compression members

3. For largely distanced components

• Components are connected with battens, laces and shells (Fig. 5.34).

• Battened solutions are simpler to realise and therefore they are more often

used. They should not be used when the member is subjected to bending

moment associated to compression.

• Laced compressed members have a greater stiffness but they are more

difficult to realise. They are recommended especially when the member is

subjected to bending moment associated to compression.

• Shells (Fig. 5.34) increase the torsion stiffness of the member.

• Laces should be inclined at 45°–60° about the normal to the axis of the

member and they are generally made of angles not less than L40×40×4.

b

b

bp

bp

1z1 i40L ⋅≤

1z1 i40L ⋅≤ tp

y y

y y

z1 z z1

z1 z z1


155

• The height hp of battens should be between 0,5–0,8c, where c is the

distance between centres of gravity of the two components (Fig. 5.34).

Fig. 5.34. Recommendations for connecting components of compression members

• The thickness of battens tp should be greater than c/50 and than 8mm.

• Battens should have greater stiffness than the components of the cross-

section. It is desirable to satisfy the following recommendation:

5LIcI

11z

p ≥ ( 5.77 )

If this is not possible, for the accuracy of the calculation model it is

necessary that:

3LIcI

11z

p ≥ ( 5.78 )

where:

Iz1 is the moment of inertia of a single component about the z1–z1 axis,

while Ip is the moment of inertia of the cross-section of the batten:

12

htI

3pp

p

⋅= ( 5.79 )

• The distance between the two components of the cross-section of the

member must allow protection against corrosion: a ≥ 120mm (Fig. 5.34).

• The distance between two consecutive battens (Fig. 5.34):

1z1 i40L ⋅≤ ( 5.80 )

1L

1z1 i40L ⋅≤

a c

lace

hp batten

shell shell

tp

z z

z z

z z

z1 z1

z1 z1

z1 z1

z1 z1 y

y y

y y

45°–60°


156

where iz1 is the radius of gyration of a single component about to its axis

which is parallel to the z–z axis.

• There will be at least two pairs of connecting plates (battens) along a

battened member, even if its length would not demand it. For tension

members, at least a pair of battens is necessary.

• The slenderness of one component of the member between two

consecutive joints should be so that:

membermax1 1,1 ⋅≥ ( 5.81 )

where:

11z

11

iL

→= ( 5.82 )

and

( )zymembermax ;max = ( 5.83 )

is the maximum of the buckling factors y and z of the element about its

two main axes.

• For laced members, as well as for battened ones, components shall be

connected with strong battens at both ends of the bar. The height of these

end battens should be at least equal to c (Fig. 5.34).

• It is allowed to have the intersections between the axes of the laces at the

exterior edges (Fig. 5.34) of the element components.

5.3.3.4. Checking procedure for members in compression

Practical check of compressed members consists of the following:

• check for slenderness;

• check for buckling (main check);

• check for local buckling;

• check of connecting elements in case of built-up members.

This checking procedure is established for flexural buckling of bars having the cross-

section made of a single shape or of in contact components. Any other type of

buckling, like torsion buckling or flexural buckling of members made of distanced


157

components is reduced to an equivalent flexural buckling and the same type of

procedure is used.

The check for slenderness tends to become less important in modern

codes. In STAS 10108/0–78 [7], it consists of the following check:

amax ≤ ( 5.84 )

where:

max is the maximum slenderness of the member about the two main axes;

( )zymax ;max = ( 5.85 )

a is the allowable slenderness for that type of structural member; the values are

given in codes; in STAS 10108/0–78 [7] they are as follows:

a = 120 for important members, such as main columns, compressed chord of lattice

girders, or web members (of lattice girders) near supports;

a = 150 for secondary columns, web members of lattice girders, members of

vertical bracing between columns etc.;

a = 250 for members of the horizontal bracing of roofs.

The buckling length of a compressed member depends on the following:

• the supporting systems at the ends – it depends whether the member is pin-

connected or it is fixed;

• the distance among any connections along the member – these connections

might oppose to deflections on their direction;

• the variation of the load along the member – behaviour is different for a member

loaded with the same compression force in any cross-section and for one with a

variable load.

According to the Romanian code of practice STAS 10108/0–78 [7], the

buckling check means:

RA

N

min

≤⋅

( 5.86 )

where:

N – the axial compression load produced by factored loads;

A – the area of the cross-section (it is the gross area, not the net one, as the

check is on the member and not on a cross-section);

R – design strength of the steel grade;

min – minimum of the buckling factors.


158

The check generally consists of the following steps:

1. calculate each slenderness about the main axes, corresponding to each

buckling mode;

2. extract the buckling factors depending on the steel grade, on the cross-section

shape, on the buckling axis and on the slenderness of the member;

3. buckling check, using relation (5.86).

Depending basically on the type of cross-section, some particular aspects of the

checking procedure need to be pointed out:

1. Bi-symmetrical cross-sections (Fig. 5.35) made of a single shape or built-up of

components in contact or slightly distanced, if the recommendations from 5.3.3.3

are fulfilled. Under these circumstances flexural buckling will occur about one of

the main axes.

( )zymin

zz

zf

z

yy

yf

y

;min

iL

iL

=

→=

→= ( 5.87 )

where yfL and z

fL are respectively the buckling lengths about the main axes. The

buckling factors y and z are selected from the appropriate buckling curves.

Fig. 5.35. Flexural buckling of bi-symmetrical cross-sections

2. Mono-symmetrical cross-sections (Fig. 5.36) made of a single shape or built-

up of components in contact or slightly distanced, if the recommendations from

5.3.3.3 are fulfilled.


159

Fig. 5.36. Buckling of mono-symmetrical cross-sections

Under these circumstances flexural buckling may occur in the plane of

symmetry and flexural-torsion buckling may occur about the axis of symmetry.

( )trzzymin

trzz

trz

zz

zf

z

yy

yf

y

;;min

iL

iL

=

→⋅=

→=

→=

( 5.88 )

where yfL and z

fL are respectively the buckling lengths about the main axes. The

buckling factors y, z and trz are selected from the appropriate buckling curves.

γγγγ ≥ 1 is a factor that takes into account the sensitivity of the cross-section to

torsion:

( ) 1ic

ic411

c2ic

22

2

2p

2

2

2

2

≥

+

⋅⋅−+⋅

⋅+= ( 5.89 )

( )

z

r

2zf2

IIL039,0I

c⋅⋅+= ( 5.90 )

( ) ⋅⋅= 3iir tb

3

I ( 5.91 )

where:

α = 1,0 for angles or for built-up double T cross-sections;

α = 1,1 for channels;

α = 1,2 for rolled double T cross-sections;

α = 1,5 for built-up double T cross-sections with stiffeners;

2z

2y

2p iii += ( 5.92 )

2c

2p

2 zii += ( 5.93 )

zc defines the position of the shear centre of the cross-section.

y y

z

z


160

Remark

For cross-sections made of two angles, which are commonly used for members

of lattice girders, when λz ≥ 60 ÷ 70 γ = 1, which leads to flexural buckling.

3. Cross-section made of largely distanced components (Fig. 5.37). The

components may be connected either by battens or by laces and the

recommendations from 5.3.3.3 must be fulfilled.

( )trzymin

trz

21

2z

trz

yy

yf

y;min

iL

=

→+=

→= ( 5.94 )

( a ) ( b ) ( c ) ( d )

Fig. 5.37. Buckling of cross-section made of largely distanced components

1 depends on the type of connectors (Fig. 5.34):

• for battens:

• if 5LIcI

11z

p ≥ , then

1z

11 i

L = ( 5.95 )

• if 3LIcI

511z

p ≥> , then

+⋅⋅=

cILI

112

iL

p

11z2

1z

11 ( 5.96 )

• for laces:

cossin

AA

2

2

D1 ⋅

⋅= ( 5.97 )

where:

A – area of the cross-section of the element;

AD – area of the cross-section of diagonals (both laces);

y y

z

z


161

– angle of diagonals with the normal to the member axis (45°–60°);

For cross-sections built-up of four angles (Fig. 5.37d):

22

21

2maxtr ++= ( 5.98 )

where: 1 and 2 are computed for each couple of faces

max = max ( y; z)

In all these cases of largely distanced components, the following restriction must

be fulfilled:

( )trzy1 ;max > ( 5.99 )

where 1 is the buckling factor for a single component of the member, on the

length between two consecutive battens or two consecutive joints of laces. This

requirement is generally fulfilled if relation (5.81) is fulfilled.

Remarks

A. In both cases, either laced or battened compressed members, the connecting

system must be checked at a shear force Tc which appears at the ends of the

member when buckling occurs:

RA012,0Tc ⋅⋅= ( 5.100 )

Fig. 5.38. Behaviour of the connecting elements

• For laced system (Fig. 5.38)

c c

c c

L1 L1 L1 L1

L1/2

L1/2

Tc/2 Tc/2

Tc/2 Tc/2

Tc

hp

D

M1

Mb


162

cos2

TD c

⋅= ( 5.101 )

RmA

D

dv

⋅≤⋅

( 5.102 )

vC

v

vf

v iL

→= ( 5.103 )

where:

D – force in one diagonal (consisting of an angle);

v – buckling factor about the minor axis, v–v, of the lace (angle);

vfL – buckling length of the lace (its theoretical length);

Ad – area of the cross-section of a lace;

iv – radius of gyration of the cross-section of a lace about the minor axis;

m – behaviour factor depending on the type of lace;

m = 0,9 angle with uneven legs, when the greater one is welded;

m = 0,75 angle with even legs;

• For battened system (Fig. 5.38)

Based on the assumption that the static scheme is a frame with rigid

beams, it may be considered that the inflexion point on the vertical

elements is situated at the middle of the distance between two “beams”.

As a result, the following relations may be written:

2L

2T

M 1c1 ⋅= ( 5.104 )

2LT

M2M 1c1b

⋅=⋅= (see Fig. 5.38) ( 5.105 )

The moment on the end of a single batten (Mp) is:

4LT

2M

M 1cbp

⋅== ( 5.106 )

The shear force along a single batten (Tp) is:

c2LT

cM2

T 1cpp ⋅

⋅=⋅

= ( 5.107 )

The main checks for a batten are the following ones:

RWM

p

pmax ≤= ( 5.108 )


163

fp

pmax R

AT

5,1 ≤⋅= ( 5.109 )

where:

Wp – strength modulus of the cross-section of the batten;

Ap – area of the cross-section of the batten;

6ht

W2pp

p

⋅= ( 5.110 )

ppp htA ⋅= ( 5.111 )

B. In both cases the welded connection between laces or battens and

components must be checked. They are fillet welds.

Checking for local buckling of a compressed member means to prove that

local buckling does not occur before buckling of the member (the critical stress that

causes local buckling is greater than the one that causes buckling of the member) or,

if this happens, to prove that its influence on the resistance of the member has been

taken into account. This problem generally appears when using thin-walled cold-

formed shapes.

Local buckling may be induced either by normal compression stresses () or

by tangential ones (ττττ). For purely compressed members tangential stresses have

small values and do not cause troubles from this point of view.

The general form of the local buckling critical stress in the elastic range is:

( )2

2

2

cr bt

112Ek

⋅−⋅⋅⋅= ( 5.112 )

where:

– Poisson’s factor ( = 0,3 for steel);

t – thickness of the plate;

b – width of the plate;

E – Young’s modulus;

kσ is a local buckling factor that depends on:

• the aspect ratio α = a/b, a, b, being the dimensions of the plate;

• the supports on the borders of the plate (pinned, fixed);

• the loading type.

This relation can be found as:


164

32

cr 10btk8,189 ⋅

⋅⋅= N/mm2 ( 5.113 )

Remark

The slenderness of the parts of hot-rolled shapes generally respects restrictions to

avoid local buckling in the elastic range. These limits can be found in codes.

5.4. FLEXURAL MEMBERS

5.4.1. General

Flexural members may be classified as:

• beams and plate girders;

• lattice girders.

Beams and plate girders are flexural members with a solid cross-section:

• beams are hot-rolled shapes;

• girders are built-up elements, usually by welding.

Lattice girders are structural systems able to carry a bending moment. They are

composed of axially loaded members.

The most common flexural members are:

• purlins (secondary beams of the roof structure);

• main beams and secondary beams of floors;

• travelling crane runway girders etc;

• trusses (lattice girders used as main flexural members of roofs structures).

5.4.2. Beams and plate girders

5.4.2.1. Cross-section philosophy

Generally, when designing a member subjected to bending moment, there are

five types of limit states to be checked:

• ultimate limit states (U.L.S.):


165

1. strength limit state Wy,nec;

2. lateral-torsional buckling limit state Wz,nec;

3. local buckling limit state;

4. fatigue limit state;

• serviceability limit state (S.L.S.):

5. deflection limit state Iy,nec.

The cross-section of a flexural member may be found between two extreme

virtual solutions (Fig. 5.39a and b), which outline the importance of each

component part of a typical beam:

a. Rectangular cross-section (Fig. 5.39a):

hbA ⋅= ( 5.114 )

12hA

12hb

I23

y⋅=⋅= ( 5.115 )

6hA

6hb

zI

W2

max

yy

⋅=⋅== ( 5.116 )

6RhA

RWM ycapy

⋅⋅=⋅= ( 5.117 )

b. A fictitious cross-section made only of flanges, where the whole area is

concentrated (Fig. 5.39b):

hb2A

2A ⋅=⋅= ( 5.118 )

4hA

2h

2A2I

22

y⋅=

⋅⋅= ( 5.119 )

2hA

h2

4hA

zI

W2

max

yy

⋅=⋅⋅== ( 5.120 )

2RhA

RWM ycapy

⋅⋅=⋅= ( 5.121 )

It is to notice that both cross-sections have the same area (A) (Fig. 5.39).

However, the fictitious one has a much superior resisting bending moment (three

times greater) than the first one. This cross-section cannot be realised.

Metal cross-sections tend to have as much material as possible in the flanges

(50%–60% of the total area of the cross-section). The main reasons for the material

on the web are to make the connection between the two flanges (one in tension and

one in compression), to restraint their relative slipping and to resist shear forces. For


166

a properly designed metal cross-section, about 75–85% of the bending moment is

resisted by the flanges and about 15–25% by the web.

( a ) ( b )

Fig. 5.39. Extreme solutions for the cross-section of a metal beam

The most common cross-section solutions are:

• beams, channels (Fig. 5.40):

IPN IPE HE UPN UAP

Fig. 5.40. Beams and channels

• cellular beams (Fig. 5.41):

Fig. 5.41. Cellular beams

h h

b

y y y y

A/2

z z

z z

A


167

• thin-walled cold-formed cross-sections (Fig. 5.42):

Fig. 5.42. Cold-formed shapes

• built-up cross-sections (Fig. 5.43):

Fig. 5.43. Built-up cross-sections

• plate girders, box cross-sections (Fig. 5.44):

Fig. 5.44. Plate girders, box cross-sections

Choosing the cross-section type involves taking into account the ratio

between the material cost and the labour cost. Rolled cross-sections mean low

amount of labour cost and a higher amount of material, while built-up cross-section

involve less material and more labour cost.

Generally, the height of the beam depends on the following:


168

• the type of loading (static, dynamic etc.);

• the amount of loading (great forces, large deformations etc.);

• the function of the beam (main beam, secondary beam etc.);

• the static scheme of the beam.

It generally results between the following limits:

• D/15 and D/8 for simply supported beams;

• D/30 and D/10 for continuous beams;

where D is the span of the beam.

5.4.2.2. Behaviour of beams

Assumptions

1. The behaviour of steel follows Prandtl diagram (Fig. 5.45):

Fig. 5.45. Prandtl diagram

Remarks

• The assumption of elastic behaviour up to reaching the yielding limit is a

simplifying one.

• All codes demand a certain elongation at failure. Some codes demand an

important yielding plateau. Special provisions are given for structures in

seismic areas.

2. Bernoulli’s assumption of plane cross-sections is accepted in the elastic range as

well as in the plastic range.

Prandtl

real

fy

y


169

3. The value of the equivalent stress according to the von Mises criterion:

22eq 3 ⋅+= ( 5.122 )

established for the elastic range is accepted for the plastic range too.

Elastic and plastic behaviour of the cross-section

The normal stresses that are developed in a fibre on the cross-section are a

result of the elongation of that fibre. As Bernoulli’s assumption is accepted both in

the elastic and in the plastic range, the elongation distribution on the cross-section is

linear until failure. Presuming a monotonically increasing static loading, failure of the

cross-section may occur either by local buckling or by yielding (lateral buckling is

neglected here, as it deals with the entire beam and not with the cross-section).

Several stages may be noticed on the cross-section when the load increases:

A. Elongations are less than y on the entire cross-section (max < y). The entire

cross-section behaves elastically, according to Hooke’s law (Fig. 5.46(A)):

E ⋅= ( 5.123 )

B. Elongations reach y in the extreme fibres (max = y). In the case of a class 4

cross-section, failure has already occurred by local buckling before reaching this

stage. In the case of a class 3 cross-section, failure may occur by local buckling

at any time after reaching this stage. The entire cross-section is still in the elastic

range and it follows Hooke’s law (Fig. 5.46(B)).

C. Elongations reach y in a certain fibre (at a certain level zy) (Fig. 5.46(C)). In the

fibres that are closer to the neutral axis (z < zy) the stress is less than the

yielding limit fy and the behaviour is still elastic, according to Hooke’s law (5.123).

In the other fibres (z zy) the material has plastified ( = fy) and it is no longer

able to resist any increase of the stress. The increase of the bending moment is

resisted by the neighbour fibres that have not yet reached the yielding limit. This

stage can be reached only by class 2 and class 1 cross-sections. Class 3 cross-

sections may fail at any time during this stage.

D. Elongations are much greater than y on most of the cross-section (max >> y).

Practically, the entire cross-section behaves plastically and the plastic hinge

appears. The central part of the cross-section, where the material is still in the

elastic range ( < y), is very reduced and its influence on the resisting bending

moment of the cross-section may be neglected, so the entire cross-section is


170

considered to be plastified ( = fy) (Fig. 5.46(D)). This concept is obviously a

model; in reality, one cannot have yielding in tension and yielding in compression

in neighbour fibres. Both class 2 and class 1 cross-sections can reach this stage

but after that class 2 cross-sections may fail at any time, while class 1 cross-

sections are able to continue deforming by large rotations. Considering the

Prandtl behaviour diagram, this is the maximum capable bending moment that a

cross-section may have:

⋅⋅⋅+

+⋅⋅⋅⋅=4

hf

2h

t2t

2h

ftb2M wy

ww

wypl ( 5.124 )

y

2w

ww

pl f8

ht2t

2htb2M ⋅

⋅+

+⋅⋅⋅= ( 5.125 )

yypl fS2M ⋅⋅= ( 5.126 )

where Sy is the static moment of half of the cross-section about the y–y axis.

yplpl fWM ⋅= ( 5.127 )

E. Elongations are much greater than y on most of the cross-section (max < y) and

they still increase. The cross-section is no longer able to resist any increase of

the bending moment and it rotates freely, in the same way as an articulation, but

keeping the bending moment that generated the plastic hinge. Deformations

increase until the structure becomes a plastic mechanism and it fails. It is the

structure that fails and not the cross-section; only class 1 cross-sections can

reach this stage.

Remarks

1. The plastic hinge resembles to an articulation, as any increase of the bending

moment produces free rotation of the cross-section.

2. However, it is different, because the plastic hinge has a bending moment which is

kept. Free rotations appear from that moment on.

3. Both class 2 and class 1 cross-sections are able to develop plastic hinges but

only class 1 cross-sections are able to assure enough rotation of the cross-

section to allow plastic redistribution of bending moments.

4. The difference between bi-rectangular stress distribution on the cross-section and

the one involved by a fully correct geometric assumption may be neglected.

5. Wel has a physical meaning; Wpl (5.127) does not have a physical meaning.


171

Fig. 5.46. Elastic and plastic behaviour of the cross-section of a beam

5.4.2.3. Main checks for a member in bending

Generally, there are five groups of checks that need to be taken into account

when designing a beam. The limit states that are checked may be ultimate limit state

(ULS) or serviceability limit state. They are as follows:

1. Strength checks (ULS);

2. Lateral buckling checks (ULS);

3. Local buckling checks (ULS);

y y

z

z

(A)

(B)

(C)

(D)

max < y

max = y

max > y

max >> y

y

max < fy

max = fy

max = fy

max = fy

fy

ymaxy

ymax

y

y

ymaxmax

fzIM

zIM

z

z

<⋅=

⋅=

<⋅=⋅=

ymaxy

ymax

y

y

ymaxmax

fzIM

zIM

z

z

=⋅=

⋅=

=⋅=⋅=

yy

yyy

y

ymaxmax

iff

iffzIM

z

z

≥=

<<⋅=

>⋅=⋅=

ymax

ymaxmax

f

z

z

==

>>⋅=⋅=


172

4. Fatigue checks (ULS);

5. Deflection checks (SLS);

1. Strength checks

Considering the stress distribution in figure 5.47, according to the Romanian code

STAS 10108/0–78 [7], the most important strength checks are the following ones:

• in the cross-section where the bending moment has the greatest value:

RWM

zIM

y

ymax

y

ymax ≤=⋅= ( 5.128 )

where R is the design strength of the steel grade.

• in the cross-section where the shear force has the greatest value:

R6,0RId

STf

y

maxymax ⋅≅≤

⋅⋅

= ( 5.129 )

where:

Symax – the maximum static moment of the part of the cross-section that tends to

slip in the point where is calculated (in the neutral axis);

d – the width of the cross-section in the point where is calculated.

For cross-sections with an important change of the width of the cross-section,

that are very common for metal members (ex. for I and H cross-sections,

channels, box cross-sections), like the one in figure 5.47, a simplified shear

stress distribution is accepted:

R6,0Rht

Tf

ww

⋅≅≤⋅

= ( 5.130 )

• in the cross-sections where both the bending moment and the shear force have

important values, the von Mises criterion needs to be checked:

( ) R1,13 22*eq ⋅≤⋅+= ( 5.131 )

where * and are stresses in the same point of the same cross-section.

• in the case where an important local force F acts transversely to the beam in the

plane of the web, a check for the local stress may be needed; the problem

generally appears at the contact between the web and the flange; it generally has

the following form:

RLt

F

wL ≤

⋅= ( 5.132 )


173

where L is the stiff bearing length of the local load (F). It is obtained accepting a

45º slope distribution of stresses (stream line) in the web under the transverse

force (F).

• in the case of bi-axial bending, the next relation is accepted:

R1,1WM

WM

z

z

y

ymax ⋅≤+= ( 5.133 )

Fig. 5.47. Stress distribution in the cross-section of a beam

2. Lateral buckling checks

In the same way as for any member in compression, the buckling problem appears

for the compressed flange. Generally, buckling (Fig. 5.48) may not occur in the plane

of the web, as the compressed flange is continuously connected through the web

material to the tensioned part of the cross-section, the tension flange. The stabilizing

effect of the tension zone transforms free transverse buckling into lateral-torsional

buckling, causing lateral bending and twisting of the beam.

Fig. 5.48. Lateral-torsional buckling of a beam

t

t

hw

b

y y

z

z

tw

max

*

span

Lateral buckling of the flange

Torsion (twisting of the beam)


174

According to the Romanian code STAS 10108/0–78 [7], the check for lateral

buckling uses the following relation:

RW

M

yg

y ≤⋅

( 5.134 )

where:

My – the maximum value of the bending moment along the beam;

g – lateral buckling factor (instability factor) that is function of a transformed

slenderness coefficient tr;

For checking the beam between two consecutive lateral supports of the compressed

flange:

i

L

zfl

fztr ⋅

= ( 5.135 )

For checking a beam with a bi-symmetrical cross-section without any lateral support

of the compressed flange:

i

L

z

fztr ⋅

⋅= ( 5.136 )

For checking a beam with a mono-symmetrical cross-section without any lateral

support of the compressed flange:

i

L

z

fz1tr ⋅⋅= ( 5.137 )

In relations (5.135), (5.136) and (5.137) the terms have the following meanings:

Lfz – the buckling length of the compressed flange of the beam;

izfl – radius of gyration of the compressed flange about the z–z axis (Fig. 5.47);

32

bbt12bt

AI

i3

zfl

zflzfl ⋅

=⋅

⋅== ( 5.138 )

iz – radius of gyration of the entire cross-section about the z–z axis;

,1 – factors that take into account the torsion stiffness of the cross-section;

– factor that takes into account the bending moment diagram along the beam;

3,2MM

30,0MM

05,175,1

2

1

2

1

2 ≤

⋅+⋅−= ( 5.139 )

where M1 and M2 are the values of the bending moment (taken with their algebraic

signs from the diagram) at the two ends of the checked part of the beam, 21 MM ≥ .


175

3. Local buckling checks

Local buckling may occur as a result of the action of normal compression stresses

(), of tangential ones (ττττ), or of their combination. The problem appears in the case

of the compressed flange and in the case of the web. Generally, local buckling of the

compressed flange (in the elastic range) is prevented from the beginning, when

sizing the cross-section. If in figure 5.47 we make the notation:

( ) 2tbb w−=′ ( 5.140 )

then, according to the Romanian code STAS 10108/0–78 [7], if

≤′

52OLfor1344OLfor1437OLfor15

tb

( 5.141 )

there is no need to check for the local buckling of the compressed flange, as it will

occur in the plastic range.

Generally, checking for the local buckling of the web involves using an interaction

relation containing actual values of stresses and critical values of stresses:

( )crLcrcrL ,,,,,F ( 5.142 )

For example, such a relation, given in the Romanian code STAS 10108/0–78 [7], is:

m

2

cr

2

Lcr

L

cr

*

≤

+

+

( 5.143 )

The check refers to a web panel which is limited by two consecutive stiffeners and by

the two flanges (Fig. 5.49):

Fig. 5.49. Local buckling check of the web of a beam

a a a d

a/2

e hw

y y

z

z *

M1 M2

T1 T2

M

T


176

The terms contained in relation (5.143) have the following meanings:

m – is 1,0 for static loading or 0,9 for dynamic loading;

cr – the critical shear stress (Fig. 5.49):

2

2

w2cr mmN

td100

95125

⋅

+= ( 5.144 )

( )wh;amind = ( 5.145 )

( )d

h;amax w= ( 5.146 )

Lcr – the critical value for the local compression stress, in the case were a local

force acts transversely to the beam in the plane of the web, in the panel that

is being checked:

cr – the critical value for compression stress; for example, for static loading:

2

2

wcr mmN

te2100

700

⋅= ( 5.147 )

e – the height of the compression zone on the web (Fig. 5.49):

2

he w≤ ( 5.148 )

* – compression stress at the web to flange contact; it is calculated with the

average bending moment M for a panel zone having the length d (Fig. 5.49);

This zone will be considered adjacent to a stiffener, in the most unfavourably

loaded zone of the panel:

eIM

y

* ⋅= ( 5.149 )

2

MMM 21 += ( 5.150 )

L – the local compression stress, in the case were a concentrated force F acts on

the beam in the plane of the web:

Lt

F

wL ⋅

= ( 5.151 )

L – stiff bearing length for the local force F at the web to flange contact (see

explanations at relation (5.132);

– shear stress; it is calculated with the average shear force T on the checked

panel zone (Fig. 5.49):


177

ww ht

T⋅

= ( 5.152 )

2

TTT 21 += ( 5.153 )

4. Fatigue checks

Generally, fatigue checks refer to the fatigue phenomenon in the elastic range,

meaning when the element is subjected to a great number (500.000÷2.000.000) of

cycles of loading and unloading. Check for fatigue in the plastic range is not very

common. The relation recommended by the Romanian code STAS 10108/0–78 [7]

for fatigue check is:

R ⋅≤ ( 5.154 )

where:

ba

c⋅−

= ( 5.155 )

when the maximum stress is tension and:

ab

c⋅−

= ( 5.156 )

when the maximum stress is compression,

where a, b and c are factors that depend on the type of load concentrator and on the

number of loading cycles during the intended lifetime of the element.

max

min

= ( )maxmin ≤ ( 5.157 )

where min and max are considered with their signs (( – ) compression, ( + ) tension).

5. Deflection checks

Generally, deflection requirements are generated either by technological or by

architectural reasons. Deflections of beams should be calculated using static

nominal loads, without considering possible dynamic effects. In order to provide a

normal serviceability, the ratio between the beam deflection (f) and the span (L) is

limited, or sometimes even the deflection value (f) is limited. The limit values of the

ratio between the beam deflection (f) and the span (L) are usually between 1/200 …

1/800, depending on the beam type and on its importance (role) in the structure. The

values of the allowed deflection (fa) are given in codes. From the general Maxwell-

Mohr relation, the term in bending moment has the greatest influence, while the

other ones are generally neglected:


178

a

L

0 y

fdxIEmMf ≤

⋅⋅= ( 5.158 )

where:

L – span of the beam;

f – calculated deflection;

fa – allowable deflection;

M – bending moment diagram along the beam;

m – bending moment diagram along the beam generated by a dimensionless unity

force acting in the point and on the direction where the deflection is calculated

E – Young’s modulus;

Depending on the type of beam and on the loading conditions, some of these

checks may not be necessary and other ones may need special attention.

5.4.2.4. Procedure for sizing of the cross-section

Generally, sizing relations are aimed to help the designer to propose a good

cross-section, so that the most important checks are satisfied close to the limit. As

the strength check is the most important one for a member in bending, most of the

sizing procedures take it as the major requirement. Following this, a required

strength modulus for the proposed cross-section can be calculated:

R

MW max

ynec = ( 5.159 )

where R is the design strength of the chosen steel grade. Based on the value of

Wynec a shape may be selected from tables. It is obvious that this shape must have a

strength modulus superior to the required one.

If a plate girder is desired then, using the notations from figure 5.47, there are

four unknown dimensions to be established: tw, hw, t, b. The procedure starts with

establishing the web dimensions, by proposing a thickness of the web among the

available plates: tw = 4, 5, 6, 7, 8, 9, 10, (12, 14, 15, 16, 18, 20, 22, 25, 28, 30 …)

mm. The height of the web, hw, should be around the following value:

w

ynecw t

W15,1h ⋅≅ ( 5.160 )


179

After that, based on economic or cross-section class requirements, the slenderness

of the proposed web is checked:

maxw

wmin s

th

s ≤≤ ( 5.161 )

Sometimes it is required for the height of the web, hw, to be a round value:

• for hw 500mm rounding off 10mm;

• for 500mm < hw 1000mm rounding off 50mm;

• for hw > 1000mm rounding off 100mm;

Once the web established, the area of a single flange, Af, is estimated:

6ht

hW

A ww

w

ynecf

⋅−= ( 5.162 )

The shape of the flange is proposed, based on the following recommendations:

btA f ⋅= ( 5.163 )

( ) wt5,22,1t = ( 5.164 )

≤′

52OLfor1344OLfor1437OLfor15

tb

( 5.165 )

The following recommendation is also desirable to be fulfilled whenever possible:

5b

h3 w ≤≤ ( 5.166 )

This procedure aims to fulfil the strength check, once the steel grade checked.

It is possible to choose the steel grade, taking into account the deflection

requirement:

a

L

0 y

fdxIEmMf ≤

⋅⋅= ( 5.167 )

From this, a required moment of inertia can be extracted:

yneca Iff = ( 5.168 )

The idea is that Iynec does not depend on the steel grade, while Wynec does. So:

37OLmax37OL

ynec RM

W = ( 5.169 )

44OLmax44OL

ynec RM

W = ( 5.170 )


180

52OLmax52OL

ynec RM

W = ( 5.171 )

Based on relation (5.160), the height of the web may be estimated:

37OLw

37OLynec hW ( 5.172 )

44OLw

44OLynec hW ( 5.173 )

52OLw

52OLynec hW ( 5.174 )

It is known that, by definition:

t2h

IzI

Ww

y

max

yy +

== ( 5.175 )

+⋅= t2

hWI w

yy ( 5.176 )

Only for sizing purposes, relation (5.176) may be approximated with:

2

hWI w

yy ⋅≅ ( 5.177 )

Relation (5.177) allows to estimate a probable moment of inertia Iy, presuming that

the cross-section was sized properly, using the previously described procedure.

2

hWI

37OLw37OL

y37OL

y ⋅≅ ( 5.178 )

2

hWI

44OLw44OL

y44OL

y ⋅≅ ( 5.179 )

2

hWI

52OLw52OL

y52OL

y ⋅≅ ( 5.180 )

The chosen steel grade will be the one corresponding to the smallest value of Iy that

is superior to the required moment of inertia Iynec resulted from the deflection

requirement. Once the steel grade established, the sizing procedure is identical.

5.4.3. Lattice girders

5.4.3.1. General

Lattice girders are a step forward following the philosophy which is illustrated

in figure 5.39b, that is sending as much of the material away from the neutral axis.


181

Fig. 5.50. General philosophy of lattice girders

Following this idea, the bending moment that must be carried out by the whole

structural element is transformed in a couple of axial forces that are concentrated in

the top and in the bottom chord (Fig. 5.50). The web elements (diagonals and struts)

are designed to resist the shear force that is associated to the bending moment. This

happens under two circumstances:

1. loads act only at the joints of the lattice girder;

2. bars are pin-connected at the joints.

If anyone of these two requirements is not fulfilled, bending moments and shear

forces also appear on each bar, and the whole element must be calculated like any

frame structure. In many cases the connection between bars is realised by welding

and it does not allow free rotations. However, if the slenderness of the bars is not too

small, it is accepted to consider them as pin-connected, as the values of the bending

moment on the bar may be neglected, given their reduced flexural stiffness.

Otherwise, if the bar is short and it has a big cross-section, the flexural stiffness of

the bar needs to be taken into account, as it may cause troubles when checking

strength at its ends. The two simplifying assumptions allow a manual calculus of the

loading state in the bars. However, it is recommended to use a suitable computer

program to determine the loading state. The model that is used should be as close

as possible to the real element.

F

h

h

M

top chord

C D

T

D web members

bottom chord


182

5.4.3.2. Geometric schemes

Lattice girders are generally efficient solutions for spans starting from 15–20m

up to 60–100m. Their efficiency comes from the reduced material cost, although the

labour cost increases compared to beams or to plate girders. They can be either

plane systems (Fig. 5.51a) or spatial systems (Fig. 5.51b). When the spatial system

is developed on two directions, a reticulated structure is obtained (Fig. 5.51c).

( a ) ( b ) ( c )

Fig. 5.51. Typical geometric schemes

The geometry of the lattice girder may be established as needed. The most

important requirement when establishing the geometric scheme is that the system

must be statically determinated or statically indeterminated. Mechanisms are not

accepted for structures. This requirement can be easily fulfilled if the base module is

an indeformable shape or body, like the triangle or the tetrahedron. Critical forms

(Fig. 5.52) (systems that have the required number of constraints but they are not

properly distributed) or systems that are close to critical forms must be avoided.

Fig. 5.52. Example of bad constraint distribution

Remark

In the case of deployable structures, the system is a mechanism, as it

needs to pass quickly from the packed situation to the deployed one. However, in the

deployed state it has some blocking devices that make it act like a structure.


183

5.4.3.3. Cross-sections of bars

Generally, the cross-sections of bars are chosen among the ones available for

tension bars or for compression bars, depending on the connecting details used at

the joints. The most commonly used cross-sections are presented in figure 5.53.

Fig. 5.53. Cross-sections of bars

5.4.3.4. Joint details

Many solutions are available for joint details, depending basically on the types

of cross-sections of bars. Most of the currently used details are welded ones. The

bolted solutions are generally used for field connections. Some of the most important

requirements that need to be taken into account when choosing a joint detail are the

following ones:

• the chosen detail should be able to assure a good stream line of efforts;

• failure of an element by its connections is not a rational one;

• the chosen detail should be as close as possible to the model that was used in

calculation;

• the chosen detail should be able to transfer all efforts that arise, accidentally or

not, in the joint;

• the chosen detail should be easy to realise and safe.

Top chord

Web bars

Bottom chord


184

5.4.3.5. Calculation of efforts in bars

The most correct method to determine efforts in the bars of a lattice girder is

to use a certified and specialised computer program. In the absence of such a

program, a “manual” calculation could be accepted. In this case, bars need to be

taken into account as pin-connected at the joints. Some times, if the bars are short

and the cross-sections are big, this could place you on the unsafe side. Two

methods are available for “manual” calculation:

• the method of equilibrium of joints;

• the method of cross-sections;

The method of equilibrium of joints could be easier to handle but it is more exposed

to errors. An error affects all results obtained after the calculus that was wrong. If the

equilibrium equations are intelligently written, the method of cross-sections is simple

enough and an error once made does not affect the other results.

5.4.3.6. Main checks

The most common situation is when, based on the two assumptions:

• loads act only at the joints of the lattice girder;

• bars are pin-connected at the joints,

the bars are subjected only to axial loads. In this case, they are checked like any

member in tension or in compression, with some remarks:

1. Given the triangulated system, when establishing the in-plane buckling length of

the bar, the joints may be considered as fixed.

2. When establishing the out-of-plane buckling length of the bar, only the joints that

are blocked by a horizontal bracing may be considered as fixed.

3. When the structure is subjected to dynamic loads, the tension bars need to be

checked against vibrations. The vibration length is considered between two

consecutive fixed points.

4. The deflection check takes into account basically the axial force component from

Maxwell-Mohr’s relation.

5. Special attention is needed when checking connections.

BIBLIOGRAPHY

185

Bibliography

1. EUROCODE 1 – Actions on Structures, Final Draft 2002

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Buildings and Other Structures, 1998

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Building, 1999

5. EN 10025:1990+A1:1993 – Hot rolled products of non-alloy structural steels.

Technical delivery conditions

6. SR EN 10025:1990+A1:1994 – Produse laminate la cald din oteluri de constructie

nealiate. Conditii tehnice de livrare

7. STAS 10108/0–78 – Constructii civile, industriale si agricole. Calculul elementelor

din otel

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11. prEN 1998:2003 – Eurocode 8 – Design of structures for earthquake resistance

12. Balio G., Mazzolani F. M. – Theory and Design of Steel Structures, Chapman and

Hall, London, 1983

steel structures

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