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Verification Manual
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Content
1 Theoretical background ...........................................................................................................3
2 Stress analysis ..........................................................................................................................3
2.1 Geometrically linear (first order) theory .......................................................................... 4
WE-01 Compressed member ......................................................................................... 4
WE-02 Member subjected to bending ........................................................................... 6
WE-03 Member in torsion (concentrated twist moment) .............................................. 9
WE-04 Member in torsion (torsion by transverse concentrated load on mono-
symmetric I section) ..................................................................................................... 14
2.2 Geometrically nonlinear (second order) theory ............................................................. 19
WE-05 Member subjected to bending and compression ............................................. 19
WE-06 Member subjected to biaxial bending and compression ................................. 22
3 Stability analysis ....................................................................................................................26
WE-07 Lateral torsional buckling (double symmetric section & constant bending
moment) ........................................................................................................................... 26
WE-08 Lateral torsional buckling (double symmetric section & triangular bending
moment distribution) ........................................................................................................ 28
WE-09 Lateral torsional buckling (mono-symmetric section & constant moment) ....... 30
WE-10 Lateral torsional buckling (mono-symmetric section & triangular moment
distribution) ...................................................................................................................... 34
WE-11 Lateral torsional buckling (C section & equal end moments) ............................. 38
WE-12 Lateral torsional buckling (C section & equal end moments) ............................. 41
WE-13 Flexural-torsional buckling (U section) .............................................................. 44
WE-14 Interaction of flexural buckling and LTB (symmetric I section & equal end
moments and compressive force)..................................................................................... 48
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1 Theoretical background
The StabLab software uses the 14 degrees of freedom general thin-walled beam-columnfinite element (referred as Beam7 ) publ ished by Rajasekaran in the fo l lowing textbook:
CHEN, W.F. ATSUTA, T.: Theory of Beam-Columns: Space behavior and
design , Vol .2 McGraw-Hil l , 1977, pp . 539 -564
Later more researchers used and developed this element, for example:
PAPP, F.: Computer aided design of steel beam-column structures, Doctoral
thesis , Budapest Universi ty of Technology & Herio t -Wat t Universi ty of
Edinburgh, 1994-1996
The general beam-column finite element takes the effect of warping into consideration;
therefore i t is reasonable to use i t in both of the geometrically nonlinear stress analysis
and the e last ic s tabi l i ty ana lysis of spa t ia l s tee l s t ruc tures .
The verification of this analysis model is presented by comparisons with two types ofindependent resul ts :
Calculation by hand – where analytical solution is available
Calculation by triangular shell f inite element (referred as Shell3 ) – an equivalent
model is created where the plate elements of the beam member are modeled bytriangular shell f inite elements, the analysis is performed by the ConSteel software
(www.constee lsof tware .com)
2 Stress analysis
The stress analysis (computation of deflections, internal forces and reactions) of simple
structural members are verified by
Geometr ica l ly l inear ( f i rs t order) theory
Geometrica l ly non- l inear (second order) theory
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2.1 Geometrically linear (first order) theory
The analysis of simple structural members using the StabLab software (based on the
Beam7 f ini te element) are checked in the following Worked Examples (WE-01 to WE-
04) .
WE-01 Compressed member
Figure 1 shows a compressed member. The displacement of the end of themember and the compressive stress are calculated by hand, by Shell3 element and by the StabLab software.
A) Calculat ion by hand
Sectional area A 11250 m m2
Grade of material S235
E 2 10 000 N
mm2
Length of member L 4000 m m
Compressive force Fx
1000 kN
Compressive stress x
Fx
A88.889
N
mm2
End moving ex
x
L
E 1.693 mm
F ig .1 Stress analysis of compressed member
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B) Computation by Shel l3 e lement
F i g.2 Axial deflection of the compressed member – Shell3
C) Computation by StabLab
F i g.3 Axial deflection of the compressed member- Beam7
Evaluation
Table 1 shows the axial displacement of the free end of the simply supported
compressed member calculated by hand and computed by Shell3 element and by
the StabLab software . The resul ts a re accura te.
Tab.1 Stress analysis of compressed member
section property theory1 StabLab
Beam72 1 /2 Shell3
3 1 /3
HEA300
L=4000mme x [mm] 1,693 1,684 1 ,005 1,717 0 ,986
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WE-02 Member subjected to bending
Figure 4 shows a structural member which is loaded by uniformlydistributed load. The vertical displacement of the middle cross-section and themaximum bending moment of the member are calculated by hand, by Shell3
element and by the StabLab software.
A) Calculat ion by hand
Section : welded symmetric I section
flange b 200 mm tf 1 2 m m
web hw 400 mm tw 8 mm
Elastic modulus E 210000 N
mm2
Length of member L 8000 m m
Load p 30 kN
m
Inertia moment Iy 2 b tf
hw
2
tf
2
2
tw
hw3
12 246359467 mm
4
Maximum deflection ez.max5
384
p L4
E Iy 30.927 mm
Maximum bending moment My.max p L
2
8240 kN m
F i g.4 Structural member loaded by uniformly distributed load in the
vertical plane (welded I section with 200- 12 f lange and 400-8 web)
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B) Computation by Shel l3 e lement
Figure 5 shows the deflections of the member with the numerical value ofthe maximum deflection (self-weight is neglected).
F i g.5 Deflections of the member subjected to bending (with δ=50mm FE size)
C) Computation by StabLab
Figure 6 shows the deflections of the member with the numerical value of
the maximum deflection. Figure 7 shows the bending diagram with themaximum bending moment at the middle cross-section (self-weight isneglected).
F i g.6 Deflections of the member subjected to bending (with n=16 FE)
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F i g.7 Bending moment diagram of the member subjected to bending
Evaluation
Table 2 shows the maximum value of the vert ical deflections calculated by hand
and computed by Shel l3 e lement and by the StabLab software. The results are
accurate .
Tab.2 Stress analys is of member subjected to bending
section property theory1
StabLab Shel l33
Beam72
n result 1 /2 δ result 1 /3
Welded I
200-10 ;
400-8
ez .m a x [mm] 30.927
4 29,373 1 ,053 100 31,200 0 ,991
6*
30,232 1 ,023 50 31,3760 ,986
8 30,533 1 ,013 25 31,4270 ,984
16 30,823 1 ,003
My .m a x [kNm] 2404 240
1 ,0006* 240
8 240
16 240
*) given by the automatic mesh generation (default)
Notes
In the table n denotes the number of the finite element in the Beam7 model,δ denotes the size of the finite ele ments in [mm] in the Shell3 model.
The distributed load on the Beam7 model is concentrated into the FE nodes,therefore the deflections depend on the number of the fi nite elements.
The Shell3 model involves the effect of the shear deformation, therefore itshows larger deflections.
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WE-03 Member in torsion (concentrated twist moment)
Figure 8 shows a simple fork supported structural member which is loaded by a concentrated twist mome nt at the mi ddle cross -s ection. The me mber wasanalysed by hand, by Shell3 element and by the StabLab software.
A) Calculat ion by hand
Section: Welded symm etric I section
flange b 300 mm tf 1 6 m m
web hw 300 mm tw 1 0 m m
Sectional properties (by GSS m odel) It1
32 b tf
3 hw tw3 919200 mm
4
hs hw tf 316 mm
Iz 2 t f b
3
12 72000000 mm
4
I Iz
hs2
4 1797408000000mm
6
h hw 2 t f 332 mm
Elastic m odulus E 210000 N
mm2
G E
2 1 0.3( ) 80769
N
mm2
Parameter G ItE I
0.4441
m
Concentrated torsional mom ent Mx 2 5 kN m
Member length L 4000 m m
Cross-secti on positi on L2L
22000 mm
Parameters z L
22000 mm
z0 0 mm
Rotation* max
Mx
2
E I
L2
Lz
sinh L2 sinh L( )
sinh z( )
0.067 rad
max.deg max 3.852 deg
F i g.8 Simple fork supported structural member loaded by
concentrated twist moment at the middle cross-section
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B) Computation by Shel l3 e lement
Figure 9 shows the deformation of the member with the numerical value ofthe maximum rotation (self-weight is neglected). Figure 10 shows the axialstress distribution in the middle cross-section.
F ig .9 Maximum rotation of the middle cross -section
Bimoment* B Mx
sinh L2 sinh L( )
sinh z( ) 20.009kN m2
Torsinal moment* Mt Mx
L2
L
sinh L2 sinh L( )
cosh z0
3.696kN m
M Mx
sinh L2 sinh L( )
cosh z0 8.804kN m
Check equilibrium Mx.int Mt M 12.5kN m
Warping stress ef h
2
tf
2 158 mm
max ef b
2 23700 mm
2
x.maxB
Imax 263.8 N
mm2
*) Csellár, Halász, Réti: Thin-walled steel struc tures, Muszaki Könv kiadó 1965, Budapest ,
Hungary , pp. 129-131 (in hungarian)
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F i g.10 Axial stress distribution in the middle cross -section (with 25mm FE)
C) Computation by StabLab
Figure 11 shows the deflections of the member with the numerical value of
the maximum rotation (self-weight is neglected). Figure 12 shows the bimo me nt diagram with the ma ximu m bimo me nt at the mi ddle cross -section. Figure 13 shows the warping normal stress in the middle cross-section.
F i g.11 Rotation of the member due to concentrated twist moment
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Notes
In the table n denotes the number of finite element in the Beam7 model, δ denotes the size of the finit e elements in [mm] in the Shell3 model.A stiffener was applied in the shell model at mid-span in order to avoid anylocal deformation due to the introduction of the concentrated twist.
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WE-04 Member in torsion (torsion by transverse concentrated load on mono-
symmetric I section)
Figure 14 shows a simple fork supported member with mono-symmetricwelded I section which is loaded by a concentrated transverse force in the centroidof the middle cross-section. The member was analysed by hand, by Shell3 elementand by the StabLab software.
A) Calculat ion by hand
Section : Welded monsymmetric I section
top flange b1 200 mm tf1 12 m m
web hw 400 mm tw 8 mm
bottom flange b2 100 mm tf2 12 m m
Sectional properties Iz1 tf1
b13
12 8000000 mm
4 Iz2 tf2
b23
12 1000000 mm
4
Iz Iz1 Iz2 9000000 mm4
It1
3 b1 tf1
3 b2 tf2
3 hw tw
3 241067 mm
4
f
I
z1Iz1 Iz2
0.889 hs hw
t
f12
t
f22 412 mm
I f 1 f Iz hs2
1.5088 1011
mm6
ZS 248.4 mm (by GSS model of ConSteel)
zD 123.4 mm (by GSS model of ConSteel)
Elastic modul us E 210000 N
mm2
G E
2 1 0.3( ) 80769
N
mm2
Parameter G It
E I 0.784
1
m
Member length L 6000 m m
Transverse force Fy 10 kN
F i g.14 Simple fork supported member with mono-symmetricwelded I section loaded by concentrated transverse force in the
centroid
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Torsional moment Mx Fy zD 1.234kN m
Cross-secti on positi on L2
L
23000 mm
z L
23000 mm z0 0 mm
Rotation* max
Mx
2
E I
L2
Lz
sinh L2 sinh L( )
sinh z( )
3.172 deg
Bimoment* B Mx
sinh L2 sinh L( )
sinh z( ) 0.773kN m2
Torsinal moment* Mt Mx
L2
L
sinh L2 sinh L( )
cosh z0
0.501kN m
M Mx
sinh L2 sinh L( )
cosh z0 0.116kN m
Check equilibrium Mx.int Mt M 0.617kN m
Warping stress 2 18311 mm2
(by GSS model of ConSteel)
.2B
I2 93.8
N
mm2
Bending moment Mz FyL
4 15 k N m
Bending stress Mz2
Mz
Iz
b2
2 83.33
N
mm2
Axial stress in bottom flange x2
.2
Mz2
177.14 N
mm2
*) Csellár, Halász, R éti: Thin-walled s teel struc tures, Muszaki Könv kiadó 1965, Budapest,
Hungary, pp. 129-131 (in Hungarian)
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B) Computation by Shel l3 e lement
Figure 15 shows the deformation of the member with the numerical value ofthe maximum rotation (self-weight is neglected).
F i g.15 Maximum rotat ion of the middle cross -section
C) Computation by StabLab
Figure 16 shows the deformed member with the numerical value of themaximum rotation (self-weight is neglected). Figure 17 shows the bimomentdiagram with the maximum bimoment at the middle cross-section. Figure 18
shows the warping normal stress in the middle cross-section.
F i g.16 Rotation of the member due to concentrated transverse force in thecentroid of the middle cross-section (n=16)
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F i g.17 Bimoment of the member (n=16)
F i g.18 Warping normal stress in the middle cross-section (n=16)
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A) Calculat ion by hand
B) Computation by StabLab
Figure 20 shows the second order bending moment di agram of themember which was computed by the StabLab software using the Beam7 finite element model.
F i g.20 Bending moment diagram of the member (n=16)
Section:IPE 360
Sectional properties (ProfilARBED) A 7273 mm2
Iz 10430000 mm4
Elastic modulus E 210000 N
mm2
L 8000 m mLength of member
Distributed load intensity p 1 kN
m
Compressive force Fx 200 kN
Crirical foce F
cr.x
2
E Iz
L2
337.8 kN
Bending moment by first order theory Mz1 p L
2
88 kN m
Moment amplifier factor 1
1Fx
Fcr.x
2.452
Bending moment by second order theory Mz2 M z1 19.61kN m
Maximum compressive stress ymax 85 m m
c.max
Fx
A
Mz2
Iz ymax 187.3
N
mm2
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WE-06 Member subjected to biaxial bending and compression
Figure 21 shows a simple fork supported member with IPE360 equivalentwelded section (flange: 170-12,7; web: 347-8) subjected to biaxial bending aboutthe minor axis due to concentrated end moments and to compressive force.
Deflections of middle cross-section of the member are calculated by hand, byShell3 model and by the StabLab software using Beam7 model.
F i g.21 Simple fork supported member with IPE360 section
subjected to biaxial bending and compression
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A) Calculat ion by hand (using approximated method)
*) Chen, W. and Atsuta, T.: Theory of Beam-Columns, Vol. 2: Space
behavior and design, McGRAW -HILL 1977, p . 192
Section:IPE360 equivalent welded I section
Sectional properties (by EPS model ) A 6995 m m2
Iy 155238000 mm4
Iz 10413000 mm4
It 291855 mm4
I 313000000000 mm6
r 0
Iy
A
Iz
A 153. 887mm
Elastic modulus E 210000 N
mm2
G E
2 1 0.3( ) 80769
N
mm2
L 8000 m mLength of member
P 100 kNCompressive force
My 4 5 k N m Mz 7.5 kN mEnd
moments
Critical axial forces Pcr.y
2
E Iy
L2
5027 kN
Pcr.z
2
E Iz
L2
337.2 kN
Pcr.1
r 02
2
E I
L2
G It
1423.5 kN
Displacements*
C
2
8
My Mz
Pcr.y Pcr.z P
Pcr.y
Pcr.z P
Pcr.z
Pcr.y P
4
Pcr.z Pcr.y
P
My2
Pcr.z P
Mz2
Pcr.y P r 0
2Pcr. P
0.087
umax1
Pcr.z P
2
8Mz C My
55.53 mm
vmax1
Pcr.y P
2
8My C Mx
11.25mm
max C 4.991 deg
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B) Computation by Shel l3 e lement
Figure 22 shows the second order deflection of t he member which was
computed by Shell3 finite element model.
F i g.22 Deformation of the member by Shell3 FE model ( δ=43mm)
C) Computation by StabLab
Figure 23 shows the second order deflection of t he member which wascomputed by the StabLab software using the Beam7 finite element model.
F i g.23 Deformation of the member by Beam7 FE model (n=16)
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Evaluation
Table 6 shows the second order bending moment and the maximum axial
compressive stress value of the middle cross-section calculated by approximatedtheory and computed by Shell3 e lement and by the StabLab software. The accuracyof the approximated hand calculation is a bit pure, but the StabLab results of
Beam7 model comparing with the Shell3 model are accurate .
Tab.6 Second order stress analysis of member in bending and compression
sect ion displacement theory
(approximati
on)
StabLab
Beam7 Shel l3
n result δ result
IPE360equivalent
welded I
section
170-12,7
347-8
ey .m a x [mm] 55,53
2 53,00 43 51,174 53,38 25 53,03
6* 53,46
009,1)25(3
)16(7
Shell
n Beam
16 53,50
e z .m a x [mm] 11,25
2 11,10 43 10,81
4 11,10 25 10,83
6* 11,10 025 ,1
)25( csShell
)16 n( csBeam
16 11,10
φ .m a x [deg] 4,991
2 4,172 43 4,2874 4,216 25 4,433
6* 4,229 956,0
)25(3
)16(7
Shell
n Beam 16 4,239
*) given by the automatic mesh generation (default)
Notes
In the Table 6 n denotes the number of the finite elements of the Beam7 model, δ denotes the maximum size of the shell finite elements of the Shell3 model in [mm].
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3 Stability analysis
The s tabi l i ty ana lysis of s imple s t ruc tura l members using the StabLab sof tware based on
the Beam7 f ini te element models are checked by hand calculation and optionally by the
Shell3 f ini te element models in the following Worked Examples (WE-07 to WE-12 ) .
WE-07 Lateral torsional buckling (double symmetric section & constant
bending moment)
Figure 24 shows a simple fork supported member with welded section(flange: 200-12; web: 400-8) subjected to bending about the major axis due toconcentrated end moments. Critical moment of the member is calculated by
hand and by the StabLab software using the Beam7 model.
A) Calculat ion by hand
Section : welded symmetric I section
flange b 200 mm tf 1 2 m m
web hw 400 mm tw 8 mm
Sectional properties Iz 2 tf b
3
12 16000000 mm4
It1
32 b tf
3 hw tw
3 298667 mm
4
I
tf b3
24hw tf
2 678976000000 mm
6
Elastic modulus E 210000 N
mm2
G E
2 1 0.3( ) 80769
N
mm2
Member length L 6000 m m
Critical moment Mcr
2
E Iz
L2
I
Iz
L2
G It
2 E Iz
241.31kN m
F ig .24 Simple fork supported member subjected to bending aboutthe major axis (LTB)
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B) Computation by StabLab
Figure 25 shows the member subjected to lateral torsional bucklingwhich was computed by the StabLab software using the Beam7 f initeelement model.
F i g.25 LTB of simple supported structural member (n=16)
Evaluation
Table 7 shows the cri t ical moment for la teral torsional buckling of the member
which calculated by hand and computed by the StabLab software using the Beam7
model. The result is accurate .
Tab.7 Stability analysis of member in bending (LTB, L=6000mm)
section critical force theory1
Beam72
n result 1 /2
Welded I200-12 ; 400-
8
Mc r [kNm] 241,31
2 243,24 0 ,992
4 241,87 0 ,998
6* 241,79 0 ,998
16 241,77 0 ,998
*) given by the automatic mesh generation (default)
Note
In the Table 7 n denotes the number of the finite elements of the Beam7 model.
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WE-08 Lateral torsional buckling (double symmetric section & triangular
bending moment distribution)Figure 26 shows a simple fork supported member with welded section
(flange: 200-12; web: 400-8) subjected to transverse force at middle cross sectionin the main plane of the member. The critical force is calculated by hand and bythe StabLab software using Beam7 model.
A) Calculat ion by hand
Section: welded symmetric I section
flange b 200 mm tf 1 2 mm
web hw 400 mm tw 8 mm
Sectional properties Iz 2 tf b
3
12 16000000 mm
4
It1
32 b tf
3 hw tw
3 298667 mm
4
I
tf b3
24hw tf 2
678976000000 mm6
Elastic modulus E 210000 N
mm2
G E
2 1 0.3( ) 80769
N
mm2
Member length L 6000 m m
Critical force C1 1.365
Mcr C1
2
E Iz
L2
I
Iz
L2
G It
2
E Iz
329.387kN m
Fcr 4
Mcr
L 219.6 kN
F i g.26 Simple fork supported member subjected to transverse force (LTB)
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B) Computation by StabLab
Figure 27 shows the LTB of the member subjected to t ransverse force.The critical force is computed by the St abLab software using Beam7 finite element model.
F i g.27 LTB of simple supported structural member subjected to
transverse force (n=16)
Evaluation
Table 8 shows the cri t ical force for la teral torsional buckling of the member which
calculated by hand and computed by the StabLab software using Beam7 model.
The result is accurate .
Tab.8 Stability analysis of member in Bending (LTB, L=6000mm)
section critical force theory1
Beam72
n result 1 /2
Welded I200-12 ; 400-
8
Pc r [kN] 219,6
2 220,90 ,994
4 219,90 ,999
6* 219,71 ,000
16 219,71 ,000
*) given by the automatic mesh generation (default)
Note
In the Table 8 n denotes the number of the finite elements of the Beam7
model.
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WE-09 Lateral torsional buckling (mono-symmetric section & constant
moment)Figure 28 shows a simple fork supported member with welded mono-
symmetric I section (flange: 200-12 and 10 0-12; web: 400-8) subjected to equal endmoments. The critical moment is calculated by hand, by Shell3 finite element and by the StabLab software using Beam7 finite element.
F ig .28 Simple fork supported member with mono-symmetric I section subjected to equal end moments (LTB)
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A) Calculat ion by hand
Section: welded mono-symmetric I section
top flange b1 200 mm tf1 1 2 m mweb hw 400 mm tw 8 mm
bottom flange b2 100 mm tf2 1 2 m m
Sectional properties ZS 248.4 mm (by GSS model of
ConSteel)
zD 123.4 mm (by GSS model of
ConSteel)
Iz1 tf1
b13
12 8000000 mm
4 Iz2 tf2
b23
12 1000000 mm
4
Iz Iz1 Iz2 9000000 mm4
Iy 186493000 mm4 (by GSS model of
ConSteel)
It1
3 b1 tf1
3 b2 tf2
3 hw tw
3 241067 mm
4
f
Iz1
Iz1 Iz2 0.889
hs hw
tf1
2
tf2
2 412 mm
I f 1 f Iz hs2
150883555556 mm6
e hw tf2tf1
2
ZS 169.6 mm
A1 b1 tf1 2400 mm2
A2 b2 tf2 1200 mm2
qx1
Iy
zD Iz1 A1 e3
A2 hs e 3
tw
4e
4hs e
4
51.725 mm
z j zD 0.5qx 149.262 mm
Elastic modulus E 210000 N
mm2
G E
2 1 0.3( ) 80769
N
mm2
Member length L 6000 m m
Critical moment Mcr
2
E I
z
L2
I
Iz
L2
G I
t
2
E Iz z j2 z j
220.77kN m
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32
B) Computation by Shell3 element
Figure 29 shows the LTB of the mono-symmetric member subjected toequal end moments. The critical force is computed by Shell3 finiteelement model.
F i g.29 LTB of simple supported mono-symmetric structural member subjected to equal end moments (δ=50mm)
C) Computation by StabLab
Figure 30 shows the LTB of the mono-symmetric member subjected toequal end moments. The critical moment is computed by the StabLabsoftware using Beam7 finite element model.
F i g.30 LTB of simple supported mono-symmetric structural member
subjected to equal end moments (n=16)
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33
Evaluation
Table 9 shows the cri t ical moment for la teral torsional buckling of the member
which calculated by hand, by Shell3 model and computed by the StabLab softwareusing Beam7 model . The resul t i s accura te .
Tab.9 Stability analysis of mono-symmetric member subjected to equal end
moments
section critical force theory1
Beam72
Shel l33
n result 1 /2 δ result 1 /3
Welded
mono-
symmetric I200-12 ; 400-
8 ; 100-12
Mc r [kNm] 220,77
2 221,67 0 ,996 50 219,77 1 ,005
4 220,37 1 ,002 25 217,13 1 ,016
6* 220,30 1 ,002
16 220,28 1 ,002
*) given by the automatic mesh generation (default)
Note
In the Table 9 n denotes the number of the finite elements of the Beam7
model, δ denotes the maximum shell FE size.
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34
WE-10 Lateral torsional buckling (mono-symmetric section & triangular
moment distribution)
Figure 31 shows a simple fork supported member with welded mono-
symmetric I section (flange: 200-12 and 100-12; web: 400-8) subjected totransverse force at the middle cross- section of the member. The critical force iscalculated by hand, by Shell3 finite element and by the StabLab software usingBeam7 finite element.
F i g.31 Simple fork supported member with mono-symmetricwelded I section subjected to transverse force (LTB)
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35
A) Calculat ion by hand
Section: welded monsymmetric I section
top flange b1 200 mm tf1 1 2 m m
web hw 400 mm tw 8 mm
bottom flange b2 100 mm tf2 1 2 m m
Sectional properties ZS 248.4 mm (by GSS model of ConSteel)
zD 123.4 mm (by GSS model of ConSteel)
Iz1 tf1
b13
12 8000000 mm
4 Iz2 tf2
b23
12 1000000 mm
4
Iz Iz1 Iz2 9000000 mm4
Iy 186493000 mm4
(by GSS model of ConSteel)
It13
b1 tf13 b2 tf2
3 hw tw3 241067 mm4
f
Iz1
Iz1 Iz2 0.889
hs hw
tf1
2
tf2
2 412 mm
I f 1 f Iz hs2
150883555556 mm6
e hw tf2tf1
2 ZS 169.6 mm
A1 b1 tf1 2400 mm2
A2 b2 tf2 1200 mm2
qx1
Iy
zD Iz1 A1 e3
A2 hs e 3
tw
4e
4hs e 4
51.725 mm
z j zD 0.5qx 149.262 mm
Elastic m odulus E 210000 N
mm2
G E
2 1 0.3( ) 80769
N
mm2
Member length L 6000 m mCoefficients* C1 1.365 C3 0.411
Critical moment Mcr C1
2
E Iz
L
2
I
I
z
L2
G It
2
E Iz
C3 z j 2 C3 z j
213.88kN m
Fcr 4Mcr
L 142.59 kN
*) G. Sedlacek, J. Naumes: Excerpt from the Background Document to
EN 1993-1-1 Flexural buckling and lateral buckling on a common basis:
Stability assessments according to Eurocode 3 CEN / TC250 / SC3 / N1639E - rev2
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36
B) Computation by Shel l3 e lement
Figure 32 shows the LTB of the mono-symmetric member subjected totransverse force. The critical force is computed by Shell3 finite elementmodel.
F i g.32 LTB of s imple supported mono-symmetric structural member subjected to transverse force (δ=25mm)
C) Computation by StabLab
Figure 33 shows the LTB of the mono-symmetric member subjected totransverse force. The critical force is computed by the StabLab software
using Beam7 finite element model.
F i g.33 LTB of simple supported mono-symmetric structural member
subjected to transverse force (n=16)
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37
Evaluation
Table 10 shows the cri t ical force for la teral torsional buckling of the member
which calculated by hand, by Shell3 model and computed by the StabLab softwareusing Beam7 e lement. The result is accurate .
Tab.10 Stability analysis of mono-symmetric member subjected to transverse
force
section critical force theory1
Beam72
Shel l33
n result 1 /2 δ result 1 /3
Welded
mono-
symmetric I200-12 ; 400-
8 ; 100-12
Fc r [kNm] 142,59
2 143,13 0 ,996 50 141,5 1 ,008
4 142,13 1 ,003 25 139,4 1 ,023
8* 141,99 1 ,004
16 141,98 1 ,004
*) given by the automatic mesh generation (default)
Note
In the Table 10 n denotes the number of the finite elements of the Beam7
model, δ denotes the maximum shell FE size.
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38
WE-11 Lateral torsional buckling (C section & equal end moments)
Figure 34 shows a simple fork supported member with cold-formed Csection (150x100x30x2) subjected to equal end moments. The critical moment is
calculated by hand and by the St abLab software using Beam7 model.
F i g.34 Simple fork supported member with cold-formed C section subjected to equal and moments (LTB)
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40
B) Computation by StabLab
Figure 34 shows the LTB of the member with C section subjected toequal end moments. The critical moment is computed by the StabLabsoftware using Beam7 finite element model.
F i g.34 LTB of simple supported C structural member subjected toequal end moments (n=16)
Evaluation
Table 11 shows the cri t ical end moment for la teral torsional buckling of the C
member calculated by hand and computed by the StabLab software using Beam7 model. The result is accurate .
Tab.11 Stability analysis of the C member subjected to equal end moments
section critical force theory1
Beam72
n result 1 /2
Cold formed
C150x100x30x2
Mc r [kNm] 94,108
2 94,070 ,994
4 93,42 1 ,007
6* 93,381 ,008
16 93,381 ,008
*) given by the automatic mesh generation (default)
Note
In the Table 11 n denotes the number of the finite elements of the Beam7 model.
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41
WE-12 Lateral torsional buckling (C section & equal end moments)
Figure 35 shows a simple fork supported member with cold-formed Csection (150x200x30x2) subjected to equal end moments. The critical moment is
calculated by hand and by the StabLab software using Beam7 model.
F i g.35 Simple fork supported member with cold-formed C section
subjected to equal and moments (LTB)
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42
A) Calculat ion by hand
B) Computation by StabLab
Figure 36 shows the LTB of the member with C section subjected toequal end moments. The critical moment is computed by the StabLab
software using Beam7 finite element model.
Section : Cold-formed C sectionwidth of flange b 200 mmdepth d 150 mmwidth of stiffener d1 30 m m
pla te thickness t 2 mm
Cross-sectional properties (by ConSteel GSS model )
Iy 6362658 mm4
Iz 5269945 mm4
It 1734 mm4
I 35770000000 mm6
e 85.2 m m es 112.8 mm
Sectional radius* Af d t( ) t 296 mm
2
If t d t( )
3
12540299 m m
4
As d1t
2
t 58 m m2
Is
t d1t
2
3
12As
d
2
t
2
d1t
2
2
2
209399 mm4
Aw b t
2
t 398 mm2
Iw Awd
2
t
2
2
2179448 mm4
h b t
2 199 mm
qx1
Iz
e Af e2
If 2es As es2
Is 2 e h( ) Iw t
2e
4h e( )
4
30.737 mm
zD 187.8 mm
z
j
z
D
0.5q
x
203.168 mm
Length of member L 4 000 m m
Critical moment Mcr
2
E Iz
L2
I
Iz
L2
G It
2
E Iz
z j2
z j
288.68kN m
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44
WE-13 Flexural-torsional buckling (U section)
Figure 37 shows a simple fork supported member with cold-formed Usection (120x120x4) subjected to compressive force. The critical force is
calculated by hand, by Shell3 model and by the StabLab software using Beam7
element.
F i g.37 Simple fork supported member with cold-formed U section
subjected to compressive force ( f lexural-torsional buckling)
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45
A) Calculat ion by hand
Section: Cold-formed U sectionwidth of fla nge b 120 mmdepth d 120 mmplate thickness t 4 mm
Elastic modulus E 210000 N
mm2
G E
2 1 0.3( ) 80769
N
mm2
Length of member L 4 000 m m
Cross-sectional properties (by ConSteel GSS model)
A 1408 mm2
Iz 2180000 mm4
iz 39.4 mm
Iy 3699100 mm4
iy 51.3 mm
It 7927 mm4
I 5264600000 mm6
y 90.1 mm
i iy2
iz2
y2
110.915 mm
i p
Iy Iz
A64.618 mm
Critical forces Pcr.y
2
E Iy
L2
479.176 kN
P1
i2
2
E I
L2
G It
107.48 kN
Critical compressive force
Pcr
i
2
2 i p2
Pcr.y P
i
4
4 i p4
Pcr.y P
2
Pcr.y P
i
2
i p2
92.768 kN
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46
B) Computation by Shel l3 e lement
Figure 38 shows flexural torsional buckling of the member with Usection subjected to compressive force. The critical force is computed byShell3 finite element model.
F i g.38 FTB of the simple supported U structural member subjected tocompressive force (δ=25mm)
C) Computation by StabLab
Figure 39 shows the flexural torsional buckling of the member with U
section subjected to compressive force. The critical force is computed bythe StabLab software using Beam7 finite element model.
F i g.39 FTB of the simple supported U structural member subjected tocompressive force (n=16)
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47
Evaluation
Table 13 shows the cri t ical compressive force for flexural la teral buckling of the
member which calculated by hand, by Shell3 model and computed by the StabLabsoftware using Beam7 e lement . The resul ts a r e accura te .
Tab.13 Stability analysis of member subjected to compressive force
section critical force theory1
StabLab
Beam72
Shell33
n result 1 /2 δ result 1 /3
U
120x120x4cold formed
Pc r [kN] 92,77
2 93,24 0 ,995 50 94,42 0 ,983
4 92,86 0 ,999 25 93,55 0 ,992
6* 92,84 0 ,999
16 92,83 0 ,999
*) given by the automatic mesh generation (default)
Notes
In the Table 13 n denotes the number of the finite elements of the Beam7 model, δ denotes the maximum size of the shell finite elements in the Shell3 model in [mm].
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48
WE-14 Interaction of flexural buckling and LTB (symmetric I section &
equal end moments and compressive force)
Figure 40 shows a simple fork supported member with welded symmetric I
section (200-12, 400-8) subjected to compressive force and equal end moments.The critical moment with constant compressive force is calculated by hand and bythe StabLab software using Beam7 model.
F i g.40 Simple fork supported member with welded I section subjected to constant compressive force and equal end moments
(interaction)
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