On Development of Numerical Resistance Models of Thin-web Steel Girders
Abstract
Thin-web steel girders are attractive to be used because of their efficiency in bending. For such girders, the issue of ensuring local buckling becomes very relevant. Calculation formulas in most cases are complex and have a limited scope of application. The calculations based on numerical models make it possible to consider all the specifics of the designed element more universally. The article deals with the calculation of the buckling of the web girder under the combination of patch and shear loading by finite element modelling. Numerical models have been created, and a comparative analysis with experimental results has been carried out. The presented principles for constructing FE models (mesh size, material model, etc.) are recommended to follow when analysing the resistance and behaviour of beams with a thin web. Sensitivity analysis of the FE model with respect to the input parameters revealed the most important parameters (yield strength of steel, web thickness, geometry), the uncertainty of which needs to be taken into account when creating FE models. The convergence of the results supports the use of the finite element method in the design of steel beams for a qualitative and quantitative assessment of resistance. However, further development of unified principles for creating FE models and their verification on a larger amount of experimental data is required, as well as the determination of partial factors considering the variability and uncertainties in the obtained results and specified reliability level.
Full text
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ON DEVELOPMENT OF NUMERICAL RESISTANCE MODELS
OF THIN-WEB STEEL GIRDERS
Vi ali NADOLSKI 1, Jana MARKOVÁ 1, Vladisla PODYMAKO 2, Mi osla SÝKORA 1
1Depa men o S uc u al Reliabili y, Klokne Ins i u e, Czech Technical Uni e si y in P ague,
P ague, Czech Republic
2 Depa men o Building S uc u es, Bela usian Na ional Technical Uni e si y, Minsk, Bela us
Vi ali.Nado[email p o ec ed]; Jana.Ma ko a@c u .cz; V[email p o ec ed]m; Mi osla [email p o ec ed]
DOI: 10.35181/ ces-2023-0003
Abs ac . Thin-web s eel gi de s a e a ac i e o be used
because o hei e iciency in bending. Fo such gi de s,
he issue o ensu ing local buckling becomes e y ele an .
Calcula ion o mulas in mos cases a e complex and ha e
a limi ed scope o applica ion. The calcula ions based on
nume ical models make i possible o conside all he
speci ics o he designed elemen mo e uni e sally. The
a icle deals wi h he calcula ion o he buckling o he web
gi de unde he combina ion o pa ch and shea loading
by ini e elemen modelling. Nume ical models ha e been
c ea ed, and a compa a i e analysis wi h expe imen al
esul s has been ca ied ou . The p esen ed p inciples o
cons uc ing FE models (mesh size, ma e ial model, e c.)
a e ecommended o ollow when analysing he esis ance
and beha iou o beams wi h a hin web. Sensi i i y
analysis o he FE model wi h espec o he inpu
pa ame e s e ealed he mos impo an pa ame e s (yield
s eng h o s eel, web hickness, geome y), he unce ain y
o which needs o be aken in o accoun when c ea ing FE
models. The con e gence o he esul s suppo s he use o
he ini e elemen me hod in he design o s eel beams o
a quali a i e and quan i a i e assessmen o esis ance.
Howe e , u he de elopmen o uni ied p inciples o
c ea ing FE models and hei e i ica ion on a la ge
amoun o expe imen al da a is equi ed, as well as he
de e mina ion o pa ial ac o s conside ing he a iabili y
and unce ain ies in he ob ained esul s and speci ied
eliabili y le el.
Keywo ds
Web panel buckling, c i ical o ce, disc e iza ion, ini e
elemen me hod, impe ec ions, modelling, nume ical
model.
1. In oduc ion
Thin-web s eel beams a ac wi h hei e iciency when
subjec ed o bending. Howe e , he issue o ensu ing local
buckling becomes e y ele an o such beams. Many
heo e ical and expe imen al wo ks s udy he esis ance o
such beams, bu in mos cases, hey a e de o ed o he
in es iga ion o he beha iou o such elemen s unde he
in luence o indi idual o ce ac o s (load e ec s), such as
bending, shea , and pa ch loading. Ne e heless, in
p ac ice, he applica ion o a combined shea and pa ch
loading o beams wi h a hin web is o en encoun e ed. An
example o such a esis ance model is a s eel b idge gi de
in he p ocess o sliding (launch) on o suppo s o a c ane
gi de o an o e head c ane, in which he web o he c ane
gi de is no only locally loaded bu also subjec ed o a
shea o ce.
This combina ion o ac ions on welded I-beams wi h a
hin web was expe imen ally in es iga ed by Robe s T. M.
and Shahabian F. [1, 2], ollowed by B aun B. [3]. These
expe imen s showed a signi ican in e ac ion be ween
shea o ce and pa ch loading. As a esul , he in e ac ion
o mula and he modi ica ion o he educed s ess me hod
and he e ec i e wid h me hod desc ibed in EN 1993-1-5
unde combined loading we e p oposed [4, 5]. Howe e ,
he use o analy ical o mulas is limi ed by he complexi y
and a ea o which hey ha e been expe imen ally
con i med. As a ule, due o he complexi y o he
de o ma ion p ocess o a s eel beam aking in o accoun
he pos c i ical beha iou in he case o loss o local
buckling o he web, he models o ul ima e esis ance a e
conse a i e [6-8]. The use o nume ical me hods, in
pa icula he ini e elemen me hod, is e ec i e o
complex esis ance models. Acco ding o [4], nume ical
me hods may lead o a mo e accu a e desc ip ion o
beha iou and es ima es o ul ima e esis ance bu manuals
o speci ica ion o he pa ame e s o he models and
in e p e a ion o he esul s a e missing. The las decade
e ealed a g owing in e es in he use o compu e
(nume ical) models o he analysis o s uc u al
esis ance. While se e al s udies [9-14] showed a good
co espondence be ween expe imen s and FE models,
widely accep ed p inciples o c ea ing FE models a e
missing; his is an obs acle o hei p ac ical applica ions.
As a consequence, i is o in e es o analyse he beha iou
o s eel beams based on nume ical modelling and compa e
he esul s wi h expe imen s. The applica ion o FE models
in ol es h ee basic s eps:
de elopmen and uni ica ion o p inciples and
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pa ame e s o FE models – he main ocus o his
s udy,
e alua ion o he accu acy o esul s based on
compa ison wi h expe imen al da a – discussed
b ie ly [15],
de e mina ion o eliabili y pa ame e s such as
pa ial and sensi i i y ac o s [16].
The a icle p esen s an o e iew o he p inciples and
main pa ame e s o cons uc ing FE models (such as FE
mesh size, ma e ial model, e c.), as well as an analysis o
hei in luence on he esul o he FE model, based on
which ecommenda ions we e made on he me hods o
se ing and pa ame e alues o FE models o hin-web
s eel s uc u es. Sensi i i y analysis o he FE model o he
inpu pa ame e s (mechanical p ope ies o s eel,
geome ic dimensions) de e mined he se o pa ame e s
ha mos signi ican ly a ec he esul o he calcula ion
and o which u he esea ch is needed. Compa ison wi h
expe imen al da a shows he e iciency and accu acy o
calcula ions based on FE models.
2. FE Models
A p esen , nume ical modelling is o en used because o
he lack o he possibili y o ca ying ou a ull-scale
expe imen o o a pa ame ic s udy o he e ec s o
a ious pa ame e s. This makes i possible o e alua e he
ul ima e esis ance o comple ely new s uc u al solu ions
o which he e a e no calcula ion me hods (models). In
his wo k, nume ical modelling is done by he ini e
elemen me hod using he so wa e Abaqus. The c ea ion
and s udy o he accu acy o he applica ion o FE models
o ul ima e esis ance o hin-web welded beams is based
on he esul s o expe imen al s udies, which we e
published by Robe s T. M. and Shahabian F. [1] and
B aun B. [3]. The beams ma ked PG1-2 and PG4-2
om [1] and he beams SP600 and SP1200 om [3] a e
adop ed ( he o iginal designa ions a e kep in his s udy).
Schemes o he modelled ypes o beams a e p o ided in
Fig. 1.
In o de o elimina e possible in luence o expe imen al
e o , he es cases we e aken om di e en sou ces.
Since spacing o s i ene s is assumed o ha e a dominan
e ec on he beha io o he beams and in luences a ailu e
mode, he beams a e selec ed so as o conside di e en
cases o he a io o he s ep o he s i ene s а o he heigh
o he web, hw. Fo beams SP600 and SP1200 he dis ance
be ween he s i ene s is mo e han he heigh o he web,
and o beams PG1-2 and PG4-2 - less han hw.
FE models o he beams we e c ea ed om
measu emen s o he dimensions and mechanical
cha ac e is ics o he es ed beams. Table 1 p o ides he
geome ic pa ame e s and he mechanical p ope ies o he
s eel o he beam elemen s. The hickness o he s i ene s
and base pla es o he beams PG1-2 and PG4-2 a e 10
mm. Yield s eng h alues a e ob ained om a e aged
uniaxial ensile es da a o each beam ( h ee samples o
he web and wo o he langes). Ul ima e s eng h o s eel
was no measu ed in [1]. This alue is aken as equal o
u = 1.35 y acco ding o [17]. Due o he lack o da a,
gene ally accep ed alues a e conside ed o elas ic
modulus, E = 210 GPa, and Poisson's a io,
= 0.3.
(a)
(b)
Fig. 1: (a) Scheme o beams PG1-2 and PG4-2, (b) Scheme o beams
SP600 and SP1200.
S i ene s and base pla es o beams SP600 and SP1200
ha e he same wid h and hickness as he langes. Two es s
a e made om he same ba ch o s eel. Mechanical
p ope ies o s eel we e ob ained om he a e aged da a o
uniaxial ensile es s o specimens ( h ee es s each o
langes and webs). The elas ic modulus o he web is
equal o Ew = 177 GPa, and o he langes and s i ene s
is equal o E = 186 GPa. Poisson's a io is
= 0.3.
Tab.1: Geome ical pa ame e s and mechanical p ope ies o s eel
elemen s.
Beams SP600 SP1200 PG1-2 PG4-2
а, mm 2390 2390 600 500
hw, mm 600 1200 600 1000
w, mm 6 6 4.1 1.9
b
, mm 450 450 200 200
, mm 20 20 12.3 9.8
s
s, mm 200 200 50 50
y
w, MPa 383 383 339 247
uw, MPa 543 543 - -
y
, MPa 354 354 250 313
u
, MPa 519 519 - -
E
w, GPa 177 177 210 210
E
, GPa 186 186 210 210
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The s eng h and de o ma ion p ope ies o s eel
aken in o accoun by a s ess-s ain cu e ha e a dominan
e ec on he ul ima e esis ance o s eel s uc u es. The
esis ance o hin-web elemen s can be a ec ed by he
s ain ha dening o s eel as in he pla e, seconda y bending
s esses occu in addi ion o he p ima y s esses in he
memb ane. Conside ing he linea pa o ma e ial model
only may lead o he loss o bending s i ness o he pla es.
The ollowing ma e ial models we e conside ed in he
analysis o he obse ed e ec s: elas ic-plas ic wi hou
s ain ha dening, elas ic-plas ic wi h linea s ain
ha dening, and he quad ilinea ela ionship acco ding o
Swedish s anda d BSK07 [18] (Fig. 2).
Fig. 2: S ess-s ain cu e acco ding o BSK07.
Figu e 3 shows a compa ison o he load-displacemen
cu es o he SP600 and PG1-2 beams depending on he
selec ed ma e ial model. I can be seen ha a lowe alue
o he ul ima e load is eached in he s ess-s ain cu e
wi hou s ain ha dening. Howe e , he di e ence be ween
he esul s o he FE models is abou 1%, which
co esponds o he ac ha all ma e ial models apply o
sol e his p oblem. In he ollowing analysis, he
quad ilinea ela ionship acco ding o BSK07 is applied.
Fig. 3: Compa ison o load-displacemen cu es depending on selec ed
ma e ial model (а) Elas ic-plas ic wi hou s ain ha dening, (b)
Elas ic-plas ic wi h linea s ain ha dening, (c) Quad ilinea
ela ionship acco ding o BSK07, (d) Expe imen al da a.
The loading pla es a e modelled by he " igid" elemen .
The in e ac ion be ween he loading pla es and he op
lange is made h ough a «su ace- o-su ace» ype o
con ac wi h he p esence o ic ion. Loads a e applied as
s a ic loads, loading is pe o med by a concen a ed load
h ough he e e ence poin o he loading pla e. The
bounda y condi ions o he base pla es a e simply speci ied
as a pinned and olle suppo .
Disc e iza ion ( ini e elemen size) was pe o med by
he p elimina y analysis o he choice o he ini e elemen
size. Figu es 4 and 5 show he g aphs o he in luence o
he size o ini e-elemen on he alue o he c i ical
buckling o ce (Fc ) and he ul ima e o ce (Fu), aking in o
accoun geome ic and physical nonlinea i y o he SP600
and PG4-2 beams. As a esul , he mos op imal size om
he poin o iew o he solu ion accu acy and he use o
compu ing powe was adop ed a ini e elemen whose
dimensions a e i e beam web hicknesses (20 mm o he
PG1-2, 10 mm o he PG4-2, and 30 mm o beams SP600
and SP1200).
Fig. 4: Analysis o he ini e elemen size o he SP600 beam
Fig. 5: Analysis o he ini e elemen size o he PG4-2 beam.
In he manu ac u e and cons uc ion o eal s uc u es,
impe ec ions a e ine i able. In gene al, geome ic and
s uc u al impe ec ions a e dis inguished. Geome ic
impe ec ions include, o example, ini ial de lec ions,
cu a u es, eccen ici ies, and ole ances o de ia ions
om nominal geome ic alues. S uc u al impe ec ions
include, o example, esidual s esses caused by he
manu ac u ing p ocess.
Since he ampli ude and dis ibu ion o esidual
s esses signi ican ly depend on he manu ac u ing me hod
and he shape o he c oss-sec ion, he assignmen o hese
impe ec ions is di icul o au oma e, while he e is no
780.5
762.7
752.5
740.9
737.1
733.8
730.8
728.6
919.0
896.6
881.1
866.5
860.5
856.1
852.5
848.6
846
725
755
785
815
845
875
905
935
0 10203040506070
F [kN]
Fini e elemen size [mm]
Fc , FEA Fu, FEA Fu, exp
17.88
17.75
16.96
16.54
16.27
16.11
161.41
159.87
146.74
141.21
137.37
135.99
154
5
35
65
95
125
155
185
0 10203040506070
F [kN]
Fini e elemen size [mm]
F
c , FEA
F
u, FEA
F
u, exp
0
100
200
300
400
500
600
700
800
900
1000
0 4 8 12 16 20
F[kN]
[mm]
(a) (b) (c) (d)
SP600
0
50
100
150
200
250
300
350
400
450
500
02468
F [kN]
[mm]
PG1-2
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enough expe imen al da a in which welding s esses we e
de e mined and hei e ec on he beha iou and esis ance
o he elemen was analyzed; he e o e, he use o
equi alen geome ic impe ec ions can be ecommended
o accoun o welding s esses. This app oach is used o
calcula e he s abili y o he common elemen s acco ding
o EN 1993-1-1 [4], a simila app oach is used o he
esis ance models unde conside a ion. The choice o his
app oach is also suppo ed by esea ch [19-21], in which
he insigni ican in luence o welding s esses is obse ed
on he esul s o calcula ions using FE models.
The mos di icul ask in speci ying impe ec ions is
choosing he shape, and ampli ude ( alue). Based on
ecommenda ions [22-26] wo op ions o aking in o
accoun impe ec ions a e conside ed – based on
eigen o ms o elas ic buckling and based on equi alen
geome ic impe ec ions (Eigenmode-a ine
impe ec ions and Manually de ined impe ec ions). The
shape o equi alen impe ec ions is assumed o be hal
sinusoidal wi h a de lec ion alue equal o min (а/200,
hw/200) [4], so ha o he beams SP600, PG1-2, PG4-2
he de lec ion was 3 mm, and o he beam SP1200 i was
6 mm.
The me hodology o impe ec ions modelling based on
eigen o ms o buckling is e y simple: i is necessa y o
pe o m a linea calcula ion o buckling o one model o
de e mine he eigen o ms o buckling, hen w i e down he
esul s o displacemen o i and use hem as he ini ial
impe ec ions o he second model o he same beam when
analysing i s geome ically and physically non-linea
wo k. The alue o impe ec ions is aken as equal o ha
o equi alen impe ec ions.
Figu e 6 shows a compa ison o he load-displacemen
cu es o he SP600 and PG1-2 beams depending on he
me hod o de ining impe ec ions, whe e i can be seen ha
he wo me hods apply o he ask a hand. The di e ence
be ween he esul s o he ul ima e load was app oxima ely
3%.
Fig. 6: Compa ison o load-displacemen cu es depending on he
me hod o de ining impe ec ions (a) Eigenmode impe ec ions,
(b) Manually de ined impe ec ions, (c) Expe imen al da a.
Common o all c oss-sec ional elemen s o nume ical
models is ha he dimensions a e la ge in wo di ec ions
and small in he hi d di ec ion. Thus, he geome y can be
idealised as an a e age su ace, which is hen subdi ided
in o shell elemen s. When calcula ing hin pla es, in
addi ion o he non-linea i y o he ma e ial, la ge
de o ma ions con ibu e o he occu ence o geome ically
non-linea e ec s. As he basic elemen o modelling he
sec ion o he beams, ou -node shell ini e elemen s wi h
a bilinea shape unc ion, deno ed as S4R, we e selec ed.
A solid wen y-node elemen , designa ed as C3D20R,
was selec ed o he compa a i e analysis, and simila FE
models we e c ea ed. Figu e 7 shows a compa ison o he
load-displacemen cu es o he S600 and PG1-2 beams
depending on he selec ed elemen ype. I can be seen
om he wo plo s ha he bes con e gence o he esul s
o he FE model wi h he expe imen al da a is achie ed
when using a solid FE. I was ound ha he di e ence
be ween he esul s o he ul ima e load be ween he FE
models was abou 2%. So i is mo e app op ia e o use he
shell S4R elemen , han an elemen o he solid C3D20R
ype, because modelling is mo e labo ious and mo e
compu ing powe is used.
Fig. 7: Compa ison o load-displacemen cu es depending on he
selec ed ini e elemen ype (a) Shell (b) Solid (c) Expe imen al
da a.
The ollowing pa ame e s a e aken o he basic FE
model apllied in he ollowing nume ical s udies:
quad ilinea ela ionship acco ding o BSK07,
mesh size equal o i e- imes web hickness (5 w),
eigenmode impe ec ions and
shell ype o ini e elemen s.
3. Resul s and Discussion
An impo an phase o he design based on FE models is
he sensi i i y analysis o he model o he alues o he
basic a iables (yield s eng h, elas ic modulus, web
hickness, impe ec ions, e c.) and model pa ame e s
(mesh size, ype o FE elemen , e c.) on he esul s (ou pu
0
100
200
300
400
500
600
700
800
900
1000
048121620
F[kN]
[mm]
(a) (b) (с)
SP600
0
50
100
150
200
250
300
350
400
450
500
02468
F [kN]
[mm]
PG1-2
0
100
200
300
400
500
600
700
800
900
1000
048121620
F[kN]
[mm]
(a) (b) (с)
SP600
0
50
100
150
200
250
300
350
400
450
500
02468
F [kN]
[mm]
PG1-2
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a iables) o he model [27]. I is necessa y o es ablish a
wha sca e o inpu da a he alidi y o he main
conclusions d awn om he modelling esul s is p ese ed.
Sensi i i y analysis should be pe o med aking in o
accoun he ac ual a iabili y o he a iables (inpu
pa ame e s). As a ule, i is mos co ec o conside
changes in he a iable ha accoun s o he mean alue
and he s anda d de ia ion. Howe e , o his pilo s udy, a
change o all a iables by plus o minus 10% is accep ed
o simpli ica ion.
When pe o ming a sensi i i y analysis o he
mechanical p ope ies o s eel, he yield s eng h,
ul ima e s eng h, and elas ic modulus alues ob ained
om expe imen al da a we e inc eased and dec eased by
10%. Fo example, he alue o he yield s eng h o he
SP600 beam a e an inc ease o 10% is: o he web yw =
422 MPa, and o he langes and s i ene s, y = 389 MPa.
Table 2 p esen s he esul s o he beam calcula ions
depending on he alue o he yield s eng h. This
pa ame e signi ican ly a ec s he alue o he ul ima e
load, bu he na u e o he de o ma ion emains unchanged.
Tab.2: Yield s eng h sensi i i y analysis.
Beams y F
exp, kN FFEA, kN Fexp /
FFEA
SP600
+10%
846
920 0.92
exp. 868 0.97
-10% 810 1.04
SP1200
+10%
1030
1086 0.94
exp. 1020 1.01
-10% 954 1.08
PG1-2
+10%
412
489 0.84
exp. 450 0.92
-10% 423 0.97
PG4-2
+10%
154
146 1.05
exp. 138 1.12
-10% 125 1.23
The alue o he ul ima e s eng h o he SP600 beam
a e an inc ease o 10% o he web is uw = 597 MPa, and
o he langes and he s i ene s - u = 571 MPa. This
pa ame e does no a ec he alue o he ul ima e load
wi h such a change.
The alue o he elas ic modulus o he SP600 beam
a e inc easing o he web is E = 194.6 GPa, and o
langes and s i ene s – E = 204.7 GPa. In Fig. 8, a
compa ison o he load-displacemen cu es o he SP600
beam is p esen ed depending on he alue o he elas ic
modulus. This pa ame e , wi h such a change, does no
signi ican ly a ec he alue o he maximum load, a
which he di e ence be ween he FE models is 3%. The
changes in he angle o inclina ion o he ini ial linea
sec ion o he cu es can be seen.
When pe o ming sensi i i y analysis o he
geome y o he FE models, he hickness o he web and
langes aken om expe imen al da a inc eased and
dec eased by 10%. The web hickness o he SP600 beam
a e an inc ease o 10% is 6.6 mm. In Fig. 9, you can see
a compa ison o he load-displacemen cu es o he
SP600 beam depending on he alue o he web hickness.
Table 3 shows he esul s o beam calcula ions a e
changing he hickness o he web. This pa ame e
signi ican ly a ec s he maximum load alue, bu he
na u e o he de o ma ion emains unchanged.
Fig. 8: Compa ison o he load-displacemen cu es depending on he
alue o he elas ic modulus (a) Basic FE model, (b) The alue
o he elas ic modulus a e an inc ease o 10%, (c) Expe imen al
da a.
Fig. 9: Compa ison o he load-displacemen cu es depending on he
alue o he web hickness (a) Basic FE model ( w = 6 mm), (b)
FE model ( w = 6.6 mm), (c) Expe imen al da a ( w = 6 mm).
Tab.3: Web hickness sensi i i y analysis.
Beam w Fexp, kN FFEA, kN Fexp / FFEA
SP600
exp.
846
868 0.97
+10% 1004 0.84
-10% 730 1.16
SP1200
exp.
1030
1020 1.01
+10% 1184 0.87
-10% 866 1.19
PG1-2
exp.
412
450 0.92
+10% 520 0.79
-10% 392 1.05
PG4-2
exp.
154
138 1.12
+10% 155 0.99
-10% 123 1.25
0
100
200
300
400
500
600
700
800
900
1000
0 4 8 12 16 20
F
[kN]
[mm
]
(a) (b) (с)
SP600
0
200
400
600
800
1000
1200
0 4 8 12 16 20
F
[kN]
[mm
]
(a) (b) (с)
SP600
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The hickness o he langes o he SP600 beam a e
an inc ease o 10% is 22 mm. Fig. 10 shows a compa ison
o he load-displacemen cu es o he SP600 beam
depending on he alue o he hickness o he langes. This
pa ame e , wi h such a change, does no signi ican ly a ec
he alue o he maximum load, a which he di e ence
be ween he wo models is abou 2% and he na u e o he
de o ma ion emains unchanged.
Fig. 10: Compa ison o he load-displacemen cu es depending on he
alue o he hickness o he langes (a) Basic FE model
( = 20 mm), (b) FE model ( = 22 mm), (c) Expe imen al da a
( = 20 mm).
Sensi i i y analysis o he ini ial impe ec ions was
pe o med o he i s and second eigen o ms and he
combina ion o eigen o ms. I can be no ed ha o hese
beams, he choice be ween he i s , second, and combined
eigen o ms o loss o s abili y does no signi ican ly a ec
he alue o he maximum load and he na u e o
de o ma ion. Table 4 p o ides he esul s o sensi i i y
analysis ocused on he e ec o he shape and magni ude
o equi alen impe ec ions on ul ima e model esis ance.
Tab.4: Sensi i i y o ul ima e model esis ance o ini ial impe ec ions.
Beam Eigen o ms Valu e,
mm Fexp, kN FFEA,
kN
Fexp/
FFEA
SP600
1- s 3
846
868 0.97
6 852 0.99
2- s
3 886 0.95
Comb. 3 882 0.96
SP1200
1- s 6
1030
1020 1.01
12 1014 1.02
2- s
6 1010 1.02
Comb. 6 1040 0.99
PG1-2
1- s 3
412
450 0.91
6 445 0.93
2- s
3 418 0.98
Comb. 3 418 0.98
PG4-2
1- s 2.5
154
138 1.11
5 137 1.12
2- s
2.5 123 1.25
Comb. 2.5 121 1.27
As a esul , FE models we e cons uc ed and da a we e
ob ained o compa ison wi h he esul s o he
expe imen s. G aphs o e ical mo emen o beams unde
he combined ac ion o local and shea o ces a e shown in
Fig.11. The esul s o he expe imen al and he FE models
a e summa ized in Tab. 5. The models showed close
nume ical con e gence wi h he expe imen al esul s.
Tab.5: Resul s o expe imen al da a and FE models
Beam
F
ex
p
.
,
kN
F
FEA., kN
F
ex
p
/
F
FEA
SP600 846 868 0.97
SP1200 1030 1020 1.01
PG1-2 412 450 0.92
PG4-2 154 137 1.12
Fig. 11: G aphs o he e ical mo emen o he beams unde he
combined ac ion o local and shea o ces (a) Basic FE model,
(b) Expe imen al da a.
The pu pose o his a icle is o apply a gene ic FE
model, and no o de elop he bes - alida ed model. In his
ligh , a di e ence o 10% may seem easonable. The main
easons o he di e ences be ween he FE models and
expe imen al da a may be he shape and magni ude o
impe ec ions. Mo eo e , also inaccu acies in
expe imen al measu emen s o yield s eng h, elas ic
modulus and web hickness may play a ole. Addi ional
assump ions and app oxima ions adop ed he e and
possibly a ec ing model p edic ions a e lis ed below:
Only some mechanical p ope ies o s eel we e
speci ied In [1],, he e o e, he ensile s eng h is
assumed o be u = 1.35 y acco ding o [17], he
modulus o elas ici y and he Poisson's a io is
assumed o be E = 210 GPa and ν = 0.3.
This s udy is limi ed ega ding a numbe o samples
0
100
200
300
400
500
600
700
800
900
1000
0 4 8 12 16 20
F
[kN]
[mm
]
(a) (b) (с)
SP600
0
50
100
150
200
250
300
350
400
450
500
0 2 4 6 8 10 12 14
F [kN]
[mm]
PG1-2
0
20
40
60
80
100
120
140
160
02468
F[kN]
[mm]
PG4-2
0
100
200
300
400
500
600
700
800
900
1000
0 4 8 12 16 20
F[kN]
[mm]
(a) (b)
SP600
0
200
400
600
800
1000
1200
0 8 16 24
F[kN]
[mm]
SP1200
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o e i y he esul s. The ob ained indings hus
need o be e i ied by u he compa isons.
Analysis co e s only h ee ma e ial models.
No sensi i i y analysis wi h espec o he
eccen ici y o loading applica ion is made. I is
assumed ha his eccen ici y has a signi ican
e ec o a pa ch load; addi ional in es iga ions
based on measu emen s a e needed.
4. Conclusions
This a icle in es iga es he impo ance o selec ion o
pa ame e s o FE models, which should be conside ed
when assessing he esis ance o s eel elemen s wi h a
lexible hin web. The main pa ame e s a ec ing he esul
o modelling a hin-walled elemen include he selec ion o
he s ess-s ain cu e, speci ica ion o ma e ial p ope ies,
ype and size o he ini e elemen s, and shape and
magni ude o ini ial impe ec ions. A e analysing he
models o ma e ials, i is p oposed o use a quad ilinea
ela ionship wi h yielding and s ain ha dening. I appea s
ha a lowe alue o he ul ima e load is eached when he
load-displacemen cu e wi hou s ain ha dening is
applied. Howe e , he di e ence be ween he esul s o he
FE models may be e y small (abou 1% in he cases unde
in es iga ion). The analysis o he mesh size shows ha he
mos op imal size is abou i e web hicknesses o he
elemen . The use o solid elemen s does no lead o a
signi ican inc ease in he accu acy o he model; he e o e,
i is ecommended o use shell elemen s. The impe ec ions
a e ecommended o be based on he i s eigen o ms o
buckling.
The s udy indica es ha he ini e elemen me hod is
pe ec ly sui able o sol ing p oblems ela ed o he
s abili y o beam web unde combined load, makes i
possible o ake in o accoun a wide ange o ac o s and
alida e agains ull-scale expe imen al da a.
The sensi i i y analysis e eals ha domina ing is he
a iabili y o he hickness o he web (wi h a change o ±
10%, he load-bea ing capaci y changes by ± 16%) and he
s eng h o he yield (wi h a change o ± 10%, he load-
bea ing capaci y changes by ± 7%). These a iables should
hus be con olled du ing manu ac u ing and aken in o
accoun when calib a ing pa ial ac o s.
I should be no ed ha , in addi ion o esea ch in he
ield o he p inciples o cons uc ing FE models and hei
e i ica ion wi h expe imen al da a, i is necessa y o
de elop c i e ia o limi s a es and sa e y o ma o FE
applica ions [28].
5. Acknowledgemen s
This s udy has been suppo ed by he Technology Agency o he
Czech Republic unde G an CK03000125 and by he Minis y
o Educa ion, You h and Spo s o he Czech Republic unde
G an CZ.02.2.69/0.0/0.0/18_053/0016980.
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Abou Au ho s
Vi ali NADOLSKI was bo n in Minsk, Bela us. He
ecei ed his Ph.D. deg ee om B es S a e Technical
Uni e si y in 2015. His esea ch in e es s include
eliabili y analyses, design based on FEA, esis ance
models o s eel s uc u es.
Jana MARKOVÁ was bo n in P ague, Czech Republic.
She was appoin ed Associa e P o esso a Facul y o Ci il
Enginee ing, CTU in P ague in 2007. He esea ch
in e es s include basis o s uc u al design and assessmen
o exis ing s uc u es, s uc u al eliabili y, load modelling
wi h special ocus on clima ic ac ion e ec s, in es iga ions
o clima e change e ec s on s uc u al eliabili y, and
applica ions o p obabilis ic me hods in s uc u al design.
Vladisla PODYMAKO was bo n in Mogile , Bela us.
He ecei ed his M.Sc. om Bela usian Na ional Technical
Uni e si y in 2022. His esea ch in e es s include he
es ima ion o ul ima e esis ance o s eel s uc u es using
o Fini e Elemen Me hod o analysis.
Mi osla SÝKORA was bo n in České Budějo ice,
Czech Republic. He ecei ed his M.Sc. in 2001 and Ph.D.
in 2005 om he Facul y o Ci il Enginee ing, CTU in
P ague. In 2015 he ecei ed he academic deg ee o
Associa e P o esso a CTU in P ague. His esea ch
in e es s include a basis o s uc u al design, s uc u al
eliabili y, p obabilis ic op imisa ion, load modelling, isk
assessmen o echnical sys ems, and applica ions o
p obabilis ic me hods in s uc u al design.