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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):

EdRd ( 5.1 )

must be checked as:

1. resistance of cross-sections:

EdRd ( 5.1a )

where:

Ed 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:

EdRd,cr ( 5.1b )


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where:

Ed is the design value of an internal effort, calculated with factored loads;

Rd,cris the corresponding design buckling resistance.

There are situations, like those ones involving seismic actions, when the ductility

must also be checked as a ultimate limit state. Generally, this check is expressed in

the form:

mel

u

( 5.1c )

where:

u is the ultimate value of a deformation or of a displacement;

elis the value of the same deformation or the same displacement corresponding

to the limit of elastic behaviour;

m is the required inferior limit value.

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 yy;

some special means are to be considered with regard to the weak axis zz;

Iy>> Iz

Wy>> Wz

iy>> iz

y y y y

z z

zz


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123

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).

( a ) ( b ) ( c ) ( d )Fig. 5.2.Local buckling

5.1.4. Classification of cross-sections

Generally, given the strength of steel and aluminium alloys, failure of a metal

member subjected to loads other than tension occurs by buckling or by local

buckling. Depending on the slenderness of the element, this can happen either in the

elastic range (0 Y in figure 5.2.0) or in the plastic range (Y F in figure 5.2.0). To

manage this, EN 1993-1-1 [13] defines four classes of cross-sections of structural

members. They are best expressed for members in bending. In these definitions, the

behaviour of the material is presumed perfectly elastic up to the yielding limit and

perfectly plastic for elongations superior to the strain corresponding to the yielding

limit (Fig. 5.2.0). This model is known as the Prandtl model.


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Fig. 5.2.0.The Prandtl model for steel behaviour

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.

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.

real

Prandtlfy

0

Y F

y u


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Fig. 5.3.Possible stress distribution, depending on the cross-section class

The plastic hinge is a concept. It is a model of a cross-section where all the fibres

reached the yielding limit in tension or compression (Fig. 5.3) generated by a

bending moment, presuming a Prandtl behaviour diagram for the material, while in

the neighbour cross-sections the stress state is elastic. In reality, the stress and

strain state is more complex (Fig. 5.2.00): the material behaviour is not ideally

elasto-plastic and the plastic deformations extend on a certain length.

Fig. 5.2.00.The stresses in the region of a plastic hinge

class 4 class 3 class 2 class 1

max< fy max= fy max= fy max= fy

y y

z

z

( )

( + )

max= 0

max< y max= ymax= 0 max> y max>> y

x x

xxyy

z

z

y

fy


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Table 5.1.Limitations for the slenderness of internal walls [2]

ClassWall inbending

Wall incompression

Wall in bending and compression

Stress distribution

172

t

c 33

t

c

when > 0,5:113

396

t

c

when 0,5:

36

t

c

283

t

c 38

t

c

when > 0,5:113

456

t

c

when 0,5:5,41

tc

Stress distribution

3124

t

c 42

t

c

when > 1:33,067,0

42

t

c

+

when 1: ( ) ( )162t

c

yf

235=

fy(N/mm2) 235 275 355 420 460

1,00 0,92 0,81 0,75 0,71

Note: (+) means compression

c c c c

c cc

c

t t t t

t tt

t

Bending axis

Bending axis

cc c

c

fy

fy

fy

fy

fy

fy

c c c

c/2

fy

fy

fyfy

fy


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Table 5.2.Limitations for the slenderness of flanges [2]

ClassCompressed

flangeTension and compressed flange

Compressed edge Tension edge

Stress distribution

19

t

c

9

t

c

9

t

c

210

t

c

10

t

c

10

t

c

Stress distribution

314

t

c k21

t

c

yf

235=

fy(N/mm2) 235 275 355 420 460

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

yf

235=

fy(N/mm2) 235 275 355 420 460

1,00 0,92 0,81 0,75 0,71

2 1,00 0,85 0,66 0,56 0,51

t t t t

c c cc

c c c

c c c

c c

dt


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5.2. TORSION basic aspects

Generally, torsion is avoided in structural metal (steel or aluminium alloy)

members. There are basically two types of torsion:

St. Venant torsion (torsiunea cu deplanare liber);

warping torsion (torsiunea cu deplanare mpiedicat).

5.2.1. St. Venant torsion

It occurs when allthe following assumptions are accomplished:

the torsion moment is constant along the bar;

the area of the cross-section is constant along the bar;

there are no connections at the ends or along the bar that could prevent

cross-sections from free warping out of their planes.

It is also known as pure torsion.

Fig. 5.7.01.St. Venant torsion ([38] Fig. 2.2)

5.2.1.1. Stress and strain state

The following aspects can be noticed:

there is no increase or reduction of the length of the fibres (as there is nolongitudinal force):

x= 0 x= 0

warping (deplanarea) of the cross-section is a result of the assumption x= 0

(in order to keep the geometry);

the flanges remain rectangles


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=A

Ed dArT ( 5.21.01 )

Fig. 5.7.02.St. Venant torsion stress state

each cross-section rotates like a rigid disk(it goes out of plane but the shape

does not change);

the rotation between neighbour cross-section is the same along the bar.

.constdx

d=

= ( 5.21.02 )

5.2.2. Warping torsion

It occurs anytime when at least one of the St. Venant assumptions is not

fulfilled.

Fig. 5.7.03.Warping torsion

5.2.2.1. Stress and strain state


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When warping of the cross-section is constrained, longitudinal stresses and

additional shear stresses are developed. The following aspects can be noticed:

there are longitudinal stresses and strains:

x0 x0 w; w

the rotation between neighbour cross-section is variable along the bar.

.constdx

d

= ( 5.21.03 )

Fig. 5.7.04.Warping torsion stress state

5.2.2.2. Equilibrium equations

The following aspects can be noticed:

there is no axial force acting on the bar:

===A

wEdi,Ed 0dA0N0X ( 5.21.04 )

there are no bending moments acting on the bar:

===A

wEd,yi,Ed,y 0zdA0M0M ( 5.21.05 )

===A

wEd,zi,Ed,z 0ydA0M0M ( 5.21.06 )

Mw,Edwarping moment


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in each cross-section, the torsion moment is the sum of the St. Venant

component and the warping component (Fig. 5.7.05):

0hVdArT ewA

Ed =+= ( 5.21.07 )

Ed,wEd,tEd TTT += ( 5.21.08 )

where:

Tt,Ed the internal St. Venant torsion;

Tw,Ed the internal warping torsion.

Fig. 5.7.05.St. Venant torsion and warping torsion

As a simplification, in the case of a member with a closed hollow cross-section, such as a

structural hollow section, it may be assumed that the effects of torsional warping can be

neglected. Also as a simplification, in the case of a member with open cross section, such as Ior H, it may be assumed that the effects of St. Venant torsion can be neglected (EN 1993-1-

1 [13] 6.2.7(7)).

5.2.3. Torsion and bending

5.2.3.1. Bi-symmetrical cross-section subject to bending moment and shear force

For a I or H cross-section, the force F, acting in the plane xOz, generates only

bending moment about the y y axis (and shear force) and no torsion moment, as

the resultant forces Vwon the flanges are balanced (Fig. 5.7.06).


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Fig. 5.7.06.Shear stresses in a bisymmetrical cross-section in bending

5.2.3.2. Mono-symmetrical cross-section subject to bending moment and shear force

A force F, acting in the plane xOz in the centre of gravity of a mono-

symmetrical cross-section, generates not only bending moment about the y y axis(and shear force) but torsion moment too (Fig. 5.7.07), (Fig. 5.7.09).

Fig. 5.7.07.Shear stresses for force acting in the centre of gravity

eFhFT wefEd += ( 5.21.09 )

The shear centre (centrul de tiere, centrul de ncovoiere-rsucire) is the point

through which the applied loads must pass to produce bending without twisting.

A force F, acting in the plane xOz in the shear centre of a mono-symmetricalcross-section, generates only bending moment about the y y axis (and shear force)

and no torsion moment (Fig. 5.7.08), (Fig. 5.7.09).


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Fig. 5.7.08.Shear stresses for force acting in the shear centre

The shear centre location for different cross-sections is shown in figure 5.7.081.


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Fig. 5.7.081.Location of the shear centre ([38] Fig. 2.8)


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135

cVT EdEd = ( 5.21.10 )

eFhFcV wefEd += ( 5.21.11 )

Ed

wef

V

eFhFc

+= ( 5.21.12 )

Notations: EdwEd

f VF;V

F== ( 5.21.13 )

Ed

EdeEd

V

eVhVc

+= ( 5.21.14 )

ehc e+= ( 5.21.15 )

F acting in the centre of gravity F acting in the shear centreFig. 5.7.09.Effects of a force acting in or outside of the shear centre

5.2.4. Torsion calculation

5.2.4.1. St. Venant torsion

The case of open cross-sections

a) Rectangular cross-section

T

Edmax

I

tT = t = minimum edge ( 5.21.16 )

3

T tb3

1I = St. Venant torsional constant (noted also J) ( 5.21.17 )

t

b


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.constIG

T

dx

d

T

Ed =

==

= ( 5.21.18 )

= TEd IGT ( 5.21.19 )

b) Cross-section made of several rectangles

Rigid disk assumptions (simplifying assumptions):

1. each cross-section rotates one about the other;

2. the rotation varies from one cross-section to the other but it is constant

for all the points on the same cross-section; the cross-section does not

change its shape in plane but it can go out of plane;

3. the rotation occurs around an axis parallel to the axis of the bar.

As a result of assumption 2,

T

Ed

n

1

i,T

n

ii,Ed

n,T

n,Ed

1,T

1,Ed

IG

T

IG

T

IG

T...

IG

T

=

=

==

=

( 5.21.20 )

=n

1

3

iiT tb3

1I ( 5.21.21 )

Remark:For hot-rolled shapes,

=n

1

3

iiT tb3

I = 1,1 1,3 ( 5.21.22 )

T

maxEdmax

ItT = tmax= maximum thickness ( 5.21.23 )

=TEd IGT ( 5.21.24 )

The case of hollow sections(Fig. 5.7.10)

aVbVT baEd += ( 5.21.25 )

Fig. 5.7.10.Torsion of hollow sections

It is accepted that: (Bredt relation)

1

i

n


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2

TaVbV Edba == ( 5.21.26 )

b2

TV Eda

= ;

a2

TV Edb

= ( 5.21.27 )

a

Ed

a

aatab2

T

ta

V

== ( 5.21.28 )

b

Ed

b

bb

tba2

T

tb

V

=

= ( 5.21.29 )

min

Edmax

tA2

T

= ( 5.21.30 )

The difference in behaviour of open and hollow cross-sections in torsion is illustrated

in figure 5.7.101.

Fig. 5.7.101.St. Venant shear stresses ([38] Fig. 2.1)

5.2.4.2. Warping torsion

An exact calculation would consider the bar as a sum of shells (Fig. 5.7.11).


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Fig. 5.7.11.Shell modelling of a bar in torsion

In daily practice a simplified approach is used, based on the Vlasov theory.

The simplifying assumptionsare the following ones:

1. rigid disk behaviour:

each cross-section rotates one about the other;

the rotation varies from one cross-section to the other but it is constant

for all the points on the same cross-section;

the rotation occurs around an axis parallel to the axis of the bar (Fig.

5.7.12);

Fig. 5.7.12.Axis of rotation of the bar

2. the shear deformations are zero in the mean axis of the cross-section (Fig.

5.7.13);


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Fig. 5.7.13.Mean axis of the cross-section

3. wand ware constant on the thickness of the cross-section, because it is thin

(the mean axis is representative for the cross-section);

4. when calculating w, it is assumed that w= 0.

Based on these assumptions, the cross-section of the bar is reduced to its mean axis

(Fig. 5.7.14) and the following relations can be written between in-plane strains and

longitudinal ones (Fig. 5.7.15), considering rotation around point C:

dv'nn= ( 5.21.31 )

dx

dv

ds

du= ( 5.21.32 )

= cosnn'nn ( 5.21.33 )

== cosnn'nndv ( 5.21.34 )

Fig. 5.7.14.Mean surface of the member

= dCnnn ( 5.21.35 )

== cosdCn'nndv ( 5.21.36 )

= cosCnr ( 5.21.37 )

= drdv ( 5.21.38 )


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=

= ddsrdudx

dr

ds

du ( 5.21.39 )

By definition,

( )triangletheofarea22

dsr2ddsr

== ( 5.21.40 )

sectorial area

Notation:

[ ]2s

0

s

0

Lddsr = sectorial area (coordonatsectorial) ( 5.21.41 )

=== uddsrdu ( 5.21.42 )

==dx

du ( 5.21.43 )

Fig. 5.7.15.Geometric relations

Expressing wand w

=== EEwx ( 5.21.44 )

dAEdA2

w = ( 5.21.45 )

==A

2

A

w dAEdAB (bimoment) ( 5.21.46 )

(bimoment de ncovoiere-rsucire)

=A

2

w dAI (warping constant [L6]) ( 5.21.47 )

(moment de inerie sectorial)

Parallel between bending moment and warping torsion

zI

M

y

Ed,y

x = =w

wI

B ( 5.21.48 )

du


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y

yEd,z

zIt

SV

=

w

wEd,w

wIt

SM

= ( 5.21.49 )

=A

w dAS (first sectorial moment [L4]) (moment static sectorial) ( 5.21.47 )

The coordinates of the shear centre about the centre of gravity are:

y

AC

I

dAz

y

= ( 5.21.50 )

z

AC

I

dAy

z

= ( 5.21.51 )

5.3. TENSION MEMBERS

5.3.1. General

Tension members are largely used in truss construction, braced frames and

different other structural elements. They are also part of cable structures.

5.3.2. Types of single and built-up members

Figure 5.8 shows different 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);

* closely-spaced (slightly distanced) components (Fig. 5.8c);

* largely distanced components (Fig. 5.8d).


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Fig. 5.8.Examples of types of cross-sections used for tension members

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

Closely-spaced (slightly distanced) components are connected by welded or

bolted plates like shown in figure 5.10. See also 5.4.3.3.

( a )

( b )

( c ) ( d )

30t

24t

t1 t2

t = min(t1; t2)


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Largely distanced components are connected either with laces (Fig. 5.11a) or

battens (Fig.5.11b). See also 5.4.3.3.


5.3.3. Calculation

According to STAS 10108/078 [7], the following relation shall be satisfied:

RA

N

net

= ( 5.22 )

In (5.22) Nis the design tensile force, calculated with factored loads, Ris 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:

y y

z

z

z1

z1

z1

z1

L180iz1

L180iz1

L180iz1

L180iz1

y

y

y

y

z z

z z

z1 z1

z1 z1

z1 z1

z1 z1

( a )

( b )


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( )

( )

( )22,net11,netnet0122,net

011,net

A;AminA

td2La2A

tdbA

=

+=

=

( 5.23 )

and

dedL,net AtLA = ( 5.23.1 )

where:

=

p4

sdntA

2

0ded (EN 1993-1-1 [13] rel. (6.3)) ( 5.23.2 )

where sis the staggered pitch, the spacing of the centres of two consecutive holes

in the chain measured parallel to the member axis.

Fig. 5.12.Possible sections for establishing the net area Anet

According to EN 1993-1-1 [13], the check in the gross cross-section of the baris different from the check in the net cross-section, considering that failure in the net

cross-section is a more brittle one, so reference should be made at fu instead of fy.

When such a member is subjected to tension, limited plastic deformations occur in

the net cross-section and failure occurs when the gross cross-section reaches

yielding. Following this, the main checks for members in tension is:

0,1N

N

Rd,t

Ed (EN 1993-1-1 [13] (6.5)) ( 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,Rdis the smallest of:

* the design plastic resistance of the gross cross-section:

0M

y

Rd,pl

fAN

= (EN 1993-1-1 [13] (6.6)) ( 5.25 )

2

2

1

1

t

b

a

p

a

d0L1


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where fyis the specified minimum yield strength and the value of the safety factor

M0is given in the National Annex of the code [13]; the recommended value is:

0,10M = ( 5.26 )

* the design ultimate resistance of the net cross-section at holes for fasteners:

2M

unetRd,u

fA9,0N

= (EN 1993-1-1 [13] (6.7)) ( 5.27 )

where:

Anet the minimum 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 resistanceNpl,Rdshould 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

y

2M

unetfAfA9,0

>

( 5.30 )

In category C connections (slip resistant at ultimate in EN 1993-1-8) [14], tab.

3.2) (see table 4.4) the design tension resistance Nt,Rdof the net section at holes for

fasteners should not be taken as more than:

0M

ynet

Rd,net

fAN

= (EN 1993-1-1 [13] (6.8)) ( 5.31 )

5.4. COMPRESSION MEMBERS

5.4.1. 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


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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.

5.4.2. Buckling

5.4.2.1. Buckling and local buckling

The buckling loadis the critical force Fcrat 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 bucklingis 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,vleads 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

( a ) ( b ) ( c ) ( d )

Fcr


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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 Fcrthat the member is able to resist.

5.4.2.2. 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; = 0) (Fig. 5.14a);

torsion buckling(v = 0; 0) (Fig. 5.14b);

flexural-torsion buckling(v 0; 0) (Fig. 5.14c);

where vmeans the lateral displacement in the plane of the cross-section and is the

rotation of the cross-section in its plane.

( a ) ( b ) ( c )

Fig. 5.14.Forms of buckling

5.4.2.3. Approach methods

Fcr Fcr Fcr

v v


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Beginning with Euler, during the XVIIIthcentury, 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.

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:

0 1 0 1 0 1

L

v

F x


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1

EI

M

dx

dv1

dx

vd

232

2

2

==

+

( 5.32 )

where:vFM = ( 5.33 )

If v < L/400...L/300, then

0dx

dv ( 5.34 )

1dx

dv1

232

+ ( 5.35 )

It results:

1

EI

vF

dx

vd2

2

=

= ( 5.36 )

0vkv 2 =+ ( 5.37 )

where

EI

Fk2 = ( 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 =

the solution is the one obtained by Euler (1744):

2

2

crL

EIF

= ( 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.


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150

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.

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.

5.4.2.4. 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 < Fcrthe 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 > Fcrthe straight shape is no longer stable. After removing the transverse

load the member does not return to the straight line.

* For F = Fcrtwo positions of equilibrium are possible:

* the straight line;

* the slightly bent form.


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151

Fig. 5.17.Bifurcation and divergence of equilibrium

The divergenceof equilibrium is the approach based on the actual member,

with its imperfections, consisting of:

* physical imperfections, such as;

* variation of the mechanical properties of steel from one point to another;

* variation of residual stresses;

* variation of Youngs modulus (E);

* geometrical imperfections, like:

* initial deformation of the bar;

* eccentricity of the load with respect to the centroid line.

5.4.2.5. 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.

F

Fcr

v

bifurcation

divergence


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152

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:( ) 0LLdd intext == ( 5.42 )

Both external and internal virtual works depend on the three virtual displacements (v,

w, ). This leads to a system of differential equations [9]:

( )

=+

=++

=+

0wyFvzFiFGIEI

0yFwFwEI

0zFvFvEI

cc

2

cr

IV

cIV

y

cIV

z

( 5.43 )

where the following notations were used:

2p

2c

2c

2c izyi ++=

2

z

2

y

2

p iii +=

y

cI

zdAy =

z

y

v

w

G

G

yc

zc

C

C

z'

y'


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153

z

cI

ydAz =

G shear modulus of elasticity;

Ir torsion constant of the cross-section;

I 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 = = 0 ; v = w = = 0

x = L v = w = = 0 ; v = w = = 0

This leads to:

=

=

=

L

xsinA

L

xsinAv

L

xsinAw

3

2

1

( 5.44 )

Replacing these expressions, the following system is obtained:

( )( )

( )

=

=

=+

0AiPFAzFAyF

0AzFAPF

0AyFAPF

32c2c1c

3c2z

3c1y

( 5.45 )

where:

2

y2

yL

EIP

= ( 5.46 )

rC

ds


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154

2z

2

zL

EIP

= ( 5.47 )

+=

2

2

r2

c

L

EIGI

i

1P ( 5.48 )

Buckling means at least one of A1, A2, or A3must be different of 0. This leads to:

( )

0

iPFzFyF

zFPF0

yF0PF

2

ccc

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 (yc0; zc0), the three forces F1< F2< F3

correspond to flexural-torsion buckling.

3. For single symmetric cross-sections (yc0; zc= 0), the equation becomes:

( ) ( ) ( ) 0yFiPFPFPF 2c22

cyz =

2z

2

z1L

EIPF == corresponds to flexural buckling;

F2, F3correspond to flexural-torsion buckling.

4. For double symmetric cross-sections (yc= 0; zc= 0), the equation becomes:

( ) ( ) ( ) 0iPFPFPF 2czy =

2z

2

z1L


2

y2

y2L


+==

2

2

r2c

3L

EIGI

i

1PF corresponds to flexural-torsion buckling.

5.4.2.6. Flexural buckling


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155

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 XVIIIthcentury that:

2

2

crL

EIF

= ( 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:

2

cr

2

crL

EIF

= ( 5.52 )

where Lcr= kLis the buckling length(Fig. 5.20).

By definition, the buckling lengthis the distance between two consecutive inflection

points along the deformed shape of the bar. In the design practice, a less rigorous

definition is also accepted, as the distance between two consecutive lateral supports

along the bar.

k end fixity condition.

EN 1993-1-1 [13] defines the system length(def. 1.5.5) as the distance in a given

plane between two adjacent points at which a member is braced against lateral

displacement in this plane, or between one such point and the end of the member

and the buckling length (Lcr) (def. 1.5.6) as the system length of an otherwise

similar member with pinned ends, which has the same buckling resistance as a

given member or segment of member.

The following relations can be written:

( ) 22

2

cr

2

2

cr

22

2

cr

2

cr E

iL

E

L

iE

AL

EI

A

F

=

=

=

=

2

2

cr

E

= ( 5.53 )

where =Lcr/iis the slendernessof the bar.


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156

= 1,0 = 0,7 = 2,0 = 0,5 = 1,0

k = 1,0 k = 0,7 k = 2,0 k = 0,5 k = 1,0

Fig. 5.20.Different values of the buckling length factor

Remarks

1. The force Fcrhas a physical meaning, being the force that produces buckling of

the bar.

2. cr 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 Youngs modulus E is

constant. This happens when < p(pbeing the proportionality limit of the steel

grade), which means:

p

pp2

2

cr

E

E

=

Knowing that pis about 80% of the yielding limit, it means Eulers relation stands

only in the following ranges:

* for OL37 (S235) p104;

* for OL44 (S275) p95;

* for OL52 (S355) p85;

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 Youngs modulus which is the same.

5. As shown above, Eulers relation is no longer valid for stresses outside the

proportionality range, leading to critical forces superior to the real ones. These


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157

forces increase as the slenderness decreases. For low values of it can lead to

values of the critical stress crsuperior to the yielding limit, which is senseless.

Different researchers tried to find a more proper approach for the range where

Eulers relation no longer stands (p< p). The following ones are among

those who provided the most accurate approaches:

1. In 1889 Engesser and Considre (Fig. 5.23) proposed to use Eulers relation by

replacing Youngs modulus with the tangent modulus (Fig. 5.21):

2t

2

cr

E

= ( 5.54 )

where

d

dEt = ( 5.55 )

Fig. 5.21.The tangent modulus used by Engesser and Considre (1889)

2. In 1890 Tetmayer proposed a linear approach (Fig. 5.23):

( )1f ycr = ( 5.56 )

when = 0 cr= fyand when = pcr= p.

3. In 1910 von Krmn 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 p. The

elastic modulus in the unloaded part is E, while in the compressed part it is Et,

cr

p

fy

E

Et


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158

the tangent one (Fig. 5.21). Considering this, they propose to use Eulers relation

with a transformed elastic modulus (Fig. 5.23):

2

2

cr

T

= ( 5.57 )

where

I

IEIET ctt

+= ( 5.58 )

Itand Icbeing the moments of inertia of the tensioned and of the compressed

part of the cross-section, respectively (Fig. 5.22). Iis the moment of inertia of the

entire cross-section.

Fig. 5.22.The model proposed by von Krmn 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 Krmn. 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).

F

E

EtEt

E

< cr

< cr

= cr


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159

Fig. 5.23.Comparison among the presented models

5.4.2.7. 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;

* Youngs 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.

cr

pp

fy fy

Euler

von Krmn

Shanley

Engesser

Tetmayer


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160

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 Fcrwas measured. The purpose of these

tests was to find a connection between the critical force and the slenderness of the

bar.

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;

res= 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.

A

F crcr = ( 5.59 )

The results were represented as histograms (Fig. 5.24).


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161

Fig. 5.24.Typical distribution of tests results

2. Statistic values were calculated:

* the relative frequency of results:

==

i

iii

n

n

n

nf ( 5.60 )

* the mean value:

=

=n

1i

icri

mcr f ( 5.61 )

* the dispersion:

( )=

=n

1i

2mcr

icri

2 fs ( 5.62 )

* the standard deviation:

( )=

=n

1i

2m

cr

i

cri fs ( 5.63 )

Remark

Gausss function

( )

2m

s

xx

2

1

e2s

1xf

= ( 5.64 )

is rigorously correct. The critical stress distribution was presumed as a normal

one by introducing the computed values in Gausss function.

fi(ni)

2,28%

ks

crmcr


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162

3. A characteristic value of the critical stress was computed:

sk mcrk

cr = ( 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

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 (cr) 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 (res) 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:

cr

fy

test results

drawn curve


43/119


163

* 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 (cr) is expressed by means

of the buckling factor .

fAf

fAAN y

y

crycrcr ===

y

cr

f

= ( 5.66 )

1. In every day practice, the value of the critical stress (cr) is expressed by means

of the reduction factor .

=

== yy

cr

ycrcr fAffAAN

y

cr

f

= ( 5.66 )

2. The buckling curves are expressed [2] as function of the reduced slenderness:

1

= ( 5.67 )

where 1is the slenderness corresponding to the yielding limit in Eulers relation:

2

2

cr

E

= ( 5.68 )

y

1ycrf

Ef == ( 5.69 )

The Romanian code of practice, STAS 10108/078 [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 )

where:

E

= ( 5.71 )

i

L f= ( 5.72 )

c

E

E = ( 5.73 )

c yielding limit of the steel grade that is used;


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164

E longitudinal elasticity modulus (Youngs 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]

FactorBuckling curve

A B C

0,514 0,554 0,532

0,795 0,738 0,377

Fig. 5.26.Buckling curves according to STAS 10108/078 [7]

EN 1993-1-1 [13] uses 5 buckling curves A0, A, B, C, D. They are obtained by

means of the following relations:

22

1

+

= but 0,1 (EN 1993-1-1 [13] rel. (6.49)) ( 5.74 )

where ( ) 22,015,0 ++= ( 5.75 )

cr

y

N

Af= for Class 1, 2 and 3 cross-sections ( 5.751 )

A

B

C


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165

cr

yeff

N

fA= for Class 4 cross-sections ( 5.752 )

is an imperfection factor given in table 5.5, according to table 5.5.0

Ncr is the elastic critical force for the relevant buckling mode based on the gross

cross sectional properties.Relations (5.751) and (5.752) can also be expressed as:

A

Aeff

1

= ( 5.753 )

== 9,93f

E

y

1 ( 5.754 )

yf

235= ( 5.755 )

Table 5.5.Imperfection factors for buckling curves (EN 1993-1-1 [13] Tab. 6.1)

A0 A B C D

0,13 0,21 0,34 0,49 0,76

Reduct

ion

factor

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

1,1

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6 2,8 3,0

a0

bc

d

a

Non-dimensional slenderness Fig. 5.27.Buckling curves according to EN 1993-1-1 [13] Fig. 6.4


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166

Table 5.5.0.Selection of buckling curve for a cross-section (EN 1993-1-1 [13] Tab.

6.2)

Cross section Limits

Bucklin

g about

axis

Buckling curve

S 235

S 275

S 355S 420

S 460

Ro

lledsect

ions

b

h y y

z

z

t f

h/b>1

,2 tf40 mmy y

z z

a

b

a0

a0

40 mm < tf100y y

z z

b

c

a

a

h/b

1,2 tf100 mm

y y

z z

b

c

a

a

tf> 100 mm

y y

z z

d

d

c

c

Wel

ded

I-sect

ions tt ff

y yy y

z z

tf40 mmy y

z z

b

c

b

c

tf> 40 mmy y

z z

c

d

c

d

Ho

llow

sect

ions

hot finished any a a0

cold formed any c c

Wel

ded

box

sect

ions

t

t

f

b

h yy

z

z

w

generally (except as

below)any b b

thick welds: a > 0,5tf

b/tf< 30

h/tw


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167

EN 1993-1-1 [13] contains a table (table 5.5.0) 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 (fy430N/mm2). Curve D is generally used for some cross-

sections made of thick plates (tf40mm for welded cross-sections or tf100mm for

hot-rolled ones).

For slenderness 2,0 or for 04,0N

N

cr

Ed the buckling effects may be ignored

and only cross sectional checks apply.

5.4.3. Practical design of compressed members

5.4.3.1. 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.


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168

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) (ex. flanged cruciform section);

* closely-spaced (slightly distanced) components (Fig. 5.29c);

* largely distanced components (Fig. 5.29d).


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169

Fig. 5.29.Examples of types of cross-sections used for compression members

5.4.3.2. 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 )


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170

* 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 forfastened 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).

* The width bp(Fig. 5.33) of the plates should be 1530mm less, or greater,

than bto allow welding.

24t

t1 t2

t = min(t1; t2)

15t


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171

* 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 (zz 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 4560about the normal to the axis of the

member and they are generally made of angles not less than L40404.

* The height hp of battens should be between 0,50,8c, where c is the

distance between centres of gravity of the two components (Fig. 5.34).

b

b

bp

bp

1z1 i40L

1z1 i40L tp

y y

y y

z1z z1

z1z z1


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172

Fig. 5.34.Recommendations for connecting components of compression members

* The thickness of battens tpshould 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:

5LI

cI

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 z1z1 axis,

while Ipis 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: a120mm (Fig. 5.34).

* The distance between two consecutive battens (Fig. 5.34):

1z1 i40L ( 5.80 )

where iz1is the radius of gyration of a single component about to its axis

which is parallel to the zz axis.

1L

1z1 i40L

a c

lace

hp batten

shell shell

tp

z z

z z

z z

z1 z1

z1 z1

z1 z1

z1 z1y

yy

y y

4560


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173

* 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:member

max1 1,1 ( 5.81 )

where:

1

1z

11

i

L = ( 5.82 )

and

( )zymembermax ;max = ( 5.83 )

is the maximum of the buckling factors yand zof 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.

5.4.3.3. Checking procedure for members in compression

Practical check of compressed members consists of the following:

* check for slenderness; not anymore in EN 1993-1-1

* 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, flexural-torsional buckling or flexural buckling of

members made of distanced components is reduced to an equivalent flexural

buckling and the same type of procedure is used.


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The check for slenderness tends to become less important in modern

codes. In STAS 10108/078 [7], it consisted of the following check:

amax ( 5.84 )

where:

maxis 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 10108/078 [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.

Using these limitations was justified, as second order analysis of structures was not

a commonly used procedure at that time because of the missing calculation devices

and computer programs.

There are situation when the slenderness should be limited. One example is the

case of structural members for which the most severe loading situation contains the

seismic action. In this case, some limitations in EN 1998-1-1 [31] and in P100-1 [32]

are as follows:* for columns, in the plane where beams can form plastic hinges [32]:

e

y

max 7,0f

E7,0 == ( 5.85.01 )

* for columns, out of the plane where beams can form plastic hinges [32]:

e

y

max 3,1f

E3,1 == ( 5.85.02 )

where:

y

ef

E= ( 5.85.03 )

* for braces in V bracings [31], [32]:

emax 0,2 = ( 5.85.04 )

* for braces in X bracings [31], [32]:


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emaxe 0,23,1 ( 5.85.05 )

to avoid overloading columns in the prebuckling stage (when both

compression and tension diagonals are active).

The buckling lengthof a compressed member dependson 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 10108/078 [7], the

buckling checkmeans:

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.

The check according to EN 1993-1-1 [13] is done for flexural, torsional and flexural-

torsional buckling and it 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 (5.86.01).

0,1N

N

Rd,b

Ed (EN 1993-1-1 [13] rel. (6.46)) ( 5.86.01 )

where:

NEd the design value of the compression force;

Nb,Rd the design buckling resistance of the compression member.


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1M

y

Rd,b

fAN

= for Class 1, 2 and 3 cross-sections ( [13] rel. (6.47)) ( 5.86.02 )

1M

yeff

Rd,b

fAN

= for Class 4 cross-sections ( [13] rel. (6.48)) ( 5.86.03 )

The slenderness for flexural buckling is calculated as follows:

1

cr

cr

y 1

i

L

N

Af

== for Class 1, 2 and 3 cross-sections ( [13] rel. (6.50)) ( 5.86.04 )

1

eff

cr

cr

yeff A

A

i

L

N

fA

== for Class 4 cross-sections ( [13] rel. (6.51)) ( 5.86.05 )

where:

Lcr the buckling length in the buckling plane considered;

i the radius of gyration about the relevant axis, determined using the properties

of the gross cross-section.

== 9,93f

E

y

1 ( 5.86.06 )

yf

235= (fyin N/mm2) ( 5.86.07 )

The risk of torsional and torsional-flexural buckling must be taken into account

especially in the case of open cross-sections. The slenderness for torsional and

torsional-flexural buckling is calculated as follows:

cr

yT

N

Af= for Class 1, 2 and 3 cross-sections ( [13] rel. (6.52)) ( 5.86.08 )

cr

yeffT

N

fA= for Class 4 cross-sections ( [13] rel. (6.53)) ( 5.86.09 )

where Tcr,crTF,crcr NNbutNN


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the appropriate buckling curve may be determined from Table 5.5.0 considering the

one related to the z-axis. The following relations are suggested in [30].

2

y,cr

y

2

y,cr

L

EIN

= ( 5.86.10 )

2

z,cr

z

2

z,crL

EIN

= ( 5.86.11 )

+=

2

T,cr

w

2

T

0

T,crL

IEIG

I

AN ( 5.86.12 )

( ) ( )

+++

+= T,crz,cr

0

zy2

T,crz,crT,crz,cr

zy

0TF,cr NN

I

II4NNNN

II2

IN ( 5.86.13 )

where:2

Szy0 zAIII ++= ( 5.86.14 )

IT St. Venant torsional constant

= 3T tb3

I ( 5.86.15 )

zS the coordinate of the shear centre;

G the shear modulus (G = 70000 N/mm2);

Lcr the buckling length for torsion buckling;

Iw the warping constant;

Generally, all four buckling possibilities (flexural about the strong axis y y, flexural

about the weak axis z z, torsional and torsional-flexural) need to be checked. In the

case of bi-symmetrical cross-sections, flexural buckling is probable. In the case of

mono-symmetrical cross-sections, torsional and torsional-flexural buckling are more

probable.

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 ofcomponents in contact or slightly distanced, if the recommendations from 5.3.3.3

are fulfilled. Under these circumstances flexural bucklingwill occur about one of

the main axes.


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( )zyminz

z

zf

z

y

y

yf

y

;min

i

L

i

L

=

=

=

( 5.87 )

where yfL and zfL are respectively the buckling lengths about the main axes. The

buckling factors yand zare 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.

Fig. 5.36.Buckling of mono-symmetrical cross-sections

Under these circumstances flexural buckling may occur in the plane of

symmetry and flexural-torsion bucklingmay occur about the axis of symmetry.

( )trzzymin

trzz

trz

z

z

zf

z

y

y

yf

y

;;min

i

L

iL

=

=

=

=

( 5.88 )

y yz

z


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179

where yfL andzfL are respectively the buckling lengths about the main axes. The

buckling factors y, zandtr

z are selected from the appropriate buckling curves.

1 is a factor that takes into account the sensitivity of the cross-section to

torsion:

( )1

ic

ic411

c2

ic22

2

2

p

2

2

2

2

+

+

+= ( 5.89 )

( )z

r

2zf2

I

IL039,0Ic

+= ( 5.90 )

( ) = 3iir tb3

I ( 5.91 )

where:

= 1,0 for angles or for built-up double T cross-sections;

= 1,1 for channels;

= 1,2 for rolled double T cross-sections;

= 1,5 for built-up double T cross-sections with stiffeners;

2

z

2

y

2

p iii += ( 5.92 )

2c

2p

2

zii += ( 5.93 )

zcdefines the position of the shear centre of the cross-section.

Remark

For cross-sections made of two angles, which are commonly used for members

of lattice girders, when z60 70 = 1, which leads to flexural buckling.

3. Cross-section made of slightly distanced components (closely spaced

built-up members)(Fig. 5.33).

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).

The width bp (Fig. 5.33) of the plates should be 1530mm less, or greater,

than bto allow welding.

The thickness tpof 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.


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180

The distance (Fig. 5.33) between two consecutive connecting plates shall

comply with:

1z1 i40L ( 5.76 )

where iz1is 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 (zz 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

A special analysis must be carried out in the case of cross-sections made of several

components built-up members, which can be placed at a certain distance one

from the other (built-up membersin EN 1993-1-1 [13]), or can be very close one

to the other (closely spaced built-up members in EN 1993-1-1 [13]). In these

cases, a special attention must be paid to the shear force that arises when buckling

occurs (Fig. 5.33.01). The maximum value of this shear force is at the ends.

Presuming Eulers approach, the deformed shape of the bar can be described by a

sine function (Fig. 5.33.01). The following relations can be written:

L

xsinev 0

= ( 5.86.16 )

L

xsineFvFM 0crcr

== ( 5.86.17 )

L

xcos

L

eF

dx

dMV 0cr

== ( 5.86.18 )

L

0cos

L

eF

dx

dMVV 0cr

0x

maxEd

===

=

( 5.86.19 )

b

b

bp

bp

1z1 i40L

1z1 i40L tp

y y

y y

z1z z1

z1z z1


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0crEd eFM = ( 5.86.19 )

L

MV EdEd

= (EN 1993-1-1 [13] rel. (6.46)) ( 5.86.20 )

Fig. 5.33.01.Bending moment and shear force when buckling occurs

The checking procedure is different, depending on whether it is the case of a built-up

member, or of a closely spaced built-up member.

4. 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.Built-up members(Fig. 5.37). The components may be connected either by battens

or by laces. Irrespective of the connecting means (laces or battens), the components

must be connected with strong battens (plates) at both ends to resist the maximum

shear force.

( a ) ( b ) ( c ) ( d )

Fig. 5.37.Buckling of cross-section made of largely distanced components

Two approaches are possible:

checking the element as a whole (STAS 10108/078 [7]):

z

y y

z


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182

o the cross-section of the element is considered as a whole, neglecting the

connecting parts (laces or battens) when computing the area;

o the element is checked in compression, considering its entire length;

o buckling of a single component between two consecutive connecting

points is prevented by means of constructional recommendations;

o the connecting parts are checked;

o special requirements are given for the stiffness of the connecting parts and

for their spacing; the number of battens should be even;

checking one component (EN 1993-1-1 [13]):

o the check of the entire element is transferred to the check of a single

component between two consecutive connecting points;

o the lacings or battenings consist of equal modules with parallel chords;

o the minimum numbers of modules in a member is three;

o no other special requirements are given for the stiffness of the connecting

parts and for their spacing;

o the connecting parts are checked.

According to the recommendations of EN 1993-1-1 [13], one component of the

cross-section is checked between two consecutive connecting points as subject to a

compression force Nch,Ed:

eff

ch0Ed

EdEd,ch I2

AhM

N5,0N

+= (EN 1993-1-1 [13] rel. (6.69)) ( 5.86.21 )

where MEdis expressed considering a bow imperfectionwith a value that is given in

the code. The recommended value is:

500

Le0= ( 5.86.22 )

The elastic deformations of lacings or battenings may be considered by a continuous

(smeared) shear stiffness SVof the column. Following this, the shear force is:

500

LF

LV crEd

= ( 5.86.23 )

and

v

Ed

cr

Ed

I

Ed0EdEd

S

N

N

N1

MeNM

+= ( 5.86.24 )


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183

where:

2

eff

2

crL

EIN

= the effective critical force of the built-up member;

NEd the design value of the compression force to the built-up member;

MEd the design value of the maximum moment in the middle of the built-upmember considering second order effects;

I

EdM the design value of the maximum moment in the middle of the built-up

member without second order effects;

h0 the distance between the centroids of chords (Fig. 5.38);

Ach the cross-sectional area of one chord;

Ieff the effective second moment of area of the built-up member;

Sv the shear stiffness of the lacings or battened panel.

( )trzymintrz

21

2z

trz

y

y

yf

y

;min

i

L

=

+=

= ( 5.94 )

1depends on the type of connectors (Fig. 5.34):

* for battens:

* if 5LI

cI

11z

p , then

1z

11

iL = ( 5.95 )

* if 3LI

cI5

11z

p > , then

+=

cI

LI1

12

i

L

p

11z2

1z

11 ( 5.96 )

* for laces:

cossin

AA 2

2

D

1

= ( 5.97 )

where:

A area of the cross-section of the element;

AD area of the cross-section of diagonals (both laces);

angle of diagonals with the normal to the member axis (4560);


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For cross-sections built-up of four angles (Fig. 5.37d):

2

2

2

1

2

maxtr ++= ( 5.98 )

where: 1and 2are 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 connectingsystem must be checked at a shear force Tcwhich appears at the ends of the

member when buckling occurs:

RA012,0Tc = ( 5.100 )

Checking of the connecting parts (Fig. 5.38):

for lacings:

=

cos2

VD Ed ( 5.86.25 )

where:

D force in one diagonal (consisting of an angle usually).

The compressed lace is checked, considering its theoretical length as the

buckling length.

for battens:

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. Two allow this, the battens need

to have a certain stiffness. This stiffness requirement can be expressed by means

of the ratio:

11z

0b

LI

hI ( 5.86.26 )

where:

Ib the second moment of area of the cross-section of the batten;


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Iz1 the second moment of area of the cross-section of a single component.

This ratio is generally around 5 and it should be greater than 3.

Fig. 5.38.Behaviour of the connecting elements

* For laced system (Fig. 5.38)

cos2

TD c

= ( 5.101 )

RmA

D

dv

( 5.102 )

vC

v

vf

v i

L = ( 5.103 )

where:

D force in one diagonal (consisting of an angle);

v buckling factor about the minor axis, vv, 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;

h0

L1 L1 L1 L1

L1/2

L1/2

VEd/2 VEd/2

VEd/2 VEd/2

VEd

hp

D

M1

M2

h0 h0

h0


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* 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:

2

L

2

VM 1Ed1 = ( 5.104 )

2

LVM2M 1Ed12

== (see Fig. 5.38) ( 5.105 )

The moment on the end of a single batten (Mb) is:

4

LV

2

MM 1Ed2b

== ( 5.106 )

The shear force along a single batten (Vb) is:

c2

LV

c

M2V 1Edbb

=

= ( 5.107 )

Following this, the cross-section of the batten needs to be checked for bending

moment and shear force.

The main checks for a batten are the following ones:

RW

M

p

p

max = ( 5.108 )

f

p

pmax R

AT5,1 = ( 5.109 )

where:

Wp strength modulus of the cross-section of the batten;

Ap area of the cross-section of the batten;

6

htW

2pp

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.

For closely spaced built-up members, can be checked as a single integral member,

ignoring the effect of the shear stiffness ((SV= ) if the requirements of table 5.5.1

are fulfilled (Fig 5.38.1, Fig 5.38.2). Otherwise, it must be checked as a battened

member.


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y y

z

z

y y

z

z

y y

z

z

y y

z

z Fig. 5.38.1.Closely spaced built-up members (EN 1993-1-1 [13] Fig. 6.12)

Table 5.5.1. Maximum spacings for interconnections in closely spaced built-up or

star battened angle members (EN 1993-1-1 [13] Tab. 6.9)

Type of built-up member

Maximum spacing

between interconnections

*)

Members according to Fig. 5.38.1 connected by bolts or welds 15 imin

Members according to Fig. 5.38.2 connected by pair of battens 70 imin

*) centre-to-centre distance of interconnections

iminis the minimum radius of gyration of one chord or one angle

In the case of unequal-leg angles (Fig. 5.38.2) buckling about the y-y axis may be

verified with:

15,1

ii 0y= (EN 1993-1-1 [13] rel. (6.75)) ( 5.86.27 )

where i0is the minimum radius of gyration of the built-up member.

z z

y

y

v

v

v

v

Fig. 5.38.1.Closely spaced built-up members (EN 1993-1-1 [13] Fig. 6.13)

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


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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.

In case of using class 1 or 2 cross-sections, this check is not needed, as local

buckling would occur in the plastic range, that is after the member reach its capacity

in compression.

Local buckling may be induced either by normal compression stresses () or

by tangential ones (). 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 b

t

112

E

k

= ( 5.112 )

where:

Poissons factor (= 0,3 for steel);

t thickness of the plate;

b width of the plate;

E Youngs modulus;

k is a local buckling factor that depends on:

* the aspect ratio = 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:

3

2

cr 10b

tk8,189

= 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.

5.5. FLEXURAL MEMBERS

5.5.1. General


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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 generally 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).

5.5.2. Beams and plate girders

5.5.2.1. 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.):

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):

fils curs engleza an 4 sem. 1 2015

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