fils curs engleza an 4 sem. 1 2015
TRANSCRIPT
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5. DESIGN OF STRUCTURAL MEMBERS
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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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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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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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Fig. 5.10.Recommendations for connecting components of tension members
Largely distanced components are connected either with laces (Fig. 5.11a) or
battens (Fig.5.11b). See also 5.4.3.3.
Fig. 5.11.Recommendations for connecting components of tension members
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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5. DESIGN OF STRUCTURAL MEMBERS
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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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5. DESIGN OF STRUCTURAL MEMBERS
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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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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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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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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
EIPF == corresponds to flexural buckling;
2
y2
y2L
EIPF == corresponds to flexural buckling;
+==
2
2
r2c
3L
EIGI
i
1PF corresponds to flexural-torsion buckling.
5.4.2.6. Flexural buckling
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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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5. DESIGN OF STRUCTURAL MEMBERS
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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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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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5. DESIGN OF STRUCTURAL MEMBERS
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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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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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5. DESIGN OF STRUCTURAL MEMBERS
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
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* 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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5. DESIGN OF STRUCTURAL MEMBERS
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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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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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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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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5. DESIGN OF STRUCTURAL MEMBERS
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* 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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5. DESIGN OF STRUCTURAL MEMBERS
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* 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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5. DESIGN OF STRUCTURAL MEMBERS
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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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5. DESIGN OF STRUCTURAL MEMBERS
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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5. DESIGN OF STRUCTURAL MEMBERS
174
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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5. DESIGN OF STRUCTURAL MEMBERS
175
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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176
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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177
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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5. DESIGN OF STRUCTURAL MEMBERS
178
( )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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5. DESIGN OF STRUCTURAL MEMBERS
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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5. DESIGN OF STRUCTURAL MEMBERS
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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181
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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5. DESIGN OF STRUCTURAL MEMBERS
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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5. DESIGN OF STRUCTURAL MEMBERS
184
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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5. DESIGN OF STRUCTURAL MEMBERS
185
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):