The paper deals with the resistance of longitudinally stiffened plate girders subjected to moment–shear in-teraction. Based on a verified numerical model an extensive numerical parametric study was performed. Al-together 630 girders were analysed and the results were used for reliability analysis of resistance models.Five different resistance models were considered and the corresponding partial safety factors were deter-mined. The resistance models were: moment–shear (M–V) interaction model from EN 1993-1-5, modifiedM–V interaction model, gross cross-section bending resistance model and two additional resistance modelsthat present combination of the first three resistance models. It was shown that M–V interaction can be cov-ered using EN 1993-1-5 M–V interaction resistance model with partial safety factor 1.1 or with only bendingresistance check of gross cross-section taking partial safety factor equal to 1.1.
The first resistance model for the bending–shear interaction forunstiffened girders was proposed in 1960's by Basler [1], see Fig. 1a:
VVu
� �2þ M−Mf
Mp−Mf¼ 1 for Mf <M <Meff ð1Þ
whereMf is bending capacity of flanges,Mp is plastic bending capacity ofplated girder, Meff is bending resistance of effective cross-section, Vu isshear resistance of the web, and M and V are design bending momentand shear force, respectively. The interaction formula is given for Mf ≤M ≤ Meff. The interaction check was performed at a distance of hw/2 orat mid-panel if a b hw from the high-moment end (for notations seeFig. 2). In this way the influence of moment gradient was accounted for.
Herzog [2,3] defined a tri-linear interaction diagram similar toBasler's, see Fig. 1b. The interaction of shear load and bending momentin theweb is definedwhen thebending load exceeds theflange capacity.In this case the linear interaction formula is employed. In the same waythe interaction of shear and bending in the web is treated by Fujii [4].
On the other hand, Rockey et al. [5,6] predicted the strength of thegirder under combination of bending and shear by calculating criticalbuckling stress, where the influence of both actions is taken into account.
Most of experimental tests on plated girders have been performedout of the M–V interaction range. In the literature only 9 tests in thisrange were found, but in most cases they were not documented well
enough to be used to validate the numerical model. Therefore, tobuild a verified numerical model that can be used for a detailed numer-ical analysis of theM–V interaction problemnew tests on longitudinallystiffened girders subjected to M–V interaction were conducted [7].
2. Design concepts in EN 1993-1-5
2.1. General
Plated structural elements may be designed using the followingthree approaches given in EN 1993-1-5 [8]:
• Effective width method• Reduced stress method• Finite element analysis
Detailed information on how to carry out the design using effec-tive width method and reduced stress method is given, while forthe finite element analysis only general principles are described.The main deficiency of the reduced stress method is that the resis-tance of the plate girder is defined with its weakest part. Therefore,no stress redistribution is allowed. On the other hand, the effectivewidth method takes into account stress redistribution. In this paperthe emphasis will be on verifying the effective width resistancemodel given in EN 1993-1-5.
2.2. Effective width method
The plate panel resistance according to EN 1993-1-5 is definedseparately for each effect, i.e. shear (V), and bending (M). After the
232 F. Sinur, D. Beg / Journal of Constructional Steel Research 88 (2013) 231–243
contributions of each effect have been calculated, the final resistanceis obtained, taking into account the appropriate interaction.
2.2.1. Resistance to shearTwo phenomena can be observed for slender plates: the state of
pure shear present until elastic buckling stress is achieved and thepost-buckling state with the formation of tension field which startsto form after the elastic buckling stress has been exceeded. Due tobuckling, no significant increase of principal compressive stress ispossible, whereas the principal tensile stress can still increase.
Different tension field theories of plates under shear have beendeveloped to determine their resistance; for further details see[9,10]. In EN 1993-1-5 the rotated stress field theory proposed byHöglund [11] is adopted. The advantage of the rotated stress fieldmethod is its good agreement with test results for all relevant param-eters, especially for large aspect ratio of panels (α > 3), where otherresistance models lead to very conservative results.
In this method, shear resistance Vb,Rd is determined with the con-tribution of the web Vbw,Rd and of the flanges Vbf,Rd and is limited withplastic shear resistance of the web alone:
Vb;Rd ¼ Vbw;Rd þ Vbf ;Rd≤hw⋅tw⋅η⋅f ywffiffiffi3
p⋅γM1
: ð2Þ
Fig. 2. Plated girder.
In case of the M–V interaction only the web contribution to shearresistance is taken into account and is given by:
Vbw;Rd ¼ χw⋅hw⋅tw⋅f ywffiffiffi3
p⋅γM1
ð3Þ
where χw is the reduction factor for shear buckling, hw is the webheight, tw is the web thickness, fyw is the yield strength of steel andγM1 is the partial safety factor. The recommended value of γM1 =1.1. The reduction factor takes into account the post-buckling resis-tance of the plate. Depending on the stiffness of the end post the re-duction factor reads:
• For non-rigid end post:
χw ¼ η for λw <0:83η
χw ¼ 0:83λw
for λw≥0:83η
ð4Þ
• For rigid end post:
χw ¼ η for λw ≤0:83η
χw ¼ 0:83λw
for0:83λw
≤λw≤1:08
χw ¼ 1:370:7þ λw
for λw≥1:08
ð5Þ
where η = 1.2 for steel grades up to fy = 460 MPa and η = 1.0 forhigher steel grades.
The reduction curves according to Eqs. (4) and (5), plotted inFig. 3, are based on plate slenderness λw. For longitudinally stiffenedpanel the largest slenderness λw of all sub-panels and the whole stiff-ened panel is taken into account. λw is given by:
The critical shear stress can be determined by hand calculationsaccording to formulas given in EN 1993-1-5. Alternatively, bucklingcharts and software tools may be used. The critical stress is calculatedon hinged boundary conditions.
54
Fig. 3. Reduction curves for shear buckling. Fig. 4. M–V interaction according to EN 1993-1-5.
233F. Sinur, D. Beg / Journal of Constructional Steel Research 88 (2013) 231–243
2.2.2. Resistance to bending momentThe bending resistance of the longitudinally stiffened girder is de-
termined on the basis of the effective width method. Three phenom-ena are considered when calculating the effective characteristics oflongitudinally stiffened girder: shear lag effect for wide flanges,local buckling of the web subpanels and global buckling of thewhole web panel. In parametric study the flanges were designed toavoid any reduction due to shear lag effect, which is anyway smallin the ultimate limit state.
When the effective cross-section is determined, it can be treatedas an equivalent class 3 cross-section, with the assumption of linearelastic strain and stress distribution over the reduced cross-section.The ultimate resistance is defined with the yielding in the centre ofthe flange located furthest from the centre of the cross-section:
Meff ;Rd ¼ Wy;eff ;min⋅f yγM0
; ð7Þ
where Wy,eff,min is the minimum effective section modulus.Overall buckling of a panel is related to the buckling of longitudi-
nal stiffeners and depends on the panel aspect ratio α = a/b and stiff-ness of longitudinal stiffeners γ. The plate can be treated either as“one dimensional” column-like behaviour (short plates) or as “twodimensional” plate-like behaviour (long plates). Between these twoextreme modes, where the plate-like behaviour represents theupper bound with full post-critical resistance, and the column-likebehaviour represents the lower bound with no post-buckling resis-tance, interpolation is considered.
2.2.3. M–V interactionThe bending–shear interaction for I girders is considered, if the
bending moment is higher than the bending capacity of the flangesMf,Rd or shear load is higher than 50% of the shear resistance of theplate (Fig. 4). The interaction is given by the following expression:
η1 þ 1−Mf ;Rd
Mpl;Rd
!2η3−1� �2≤1:0 ð8Þ
where
η1 ¼ MEd
Mpl;Rdandη3 ¼ VEd
Vbw;Rd:
To account for the positive effects of the moment gradient the M–Vinteraction check for longitudinally unstiffened girders can beperformed at a distance not greater than hw/2 from the edge with thelargest moment. For longitudinally stiffened girders Johansson et al.[9] suggested to perform interaction check at a distance of hwi,max/2
from the edge with the largest moment, where hwi,max is the depth ofthe largest subpanel. This recommendation was given on the basis ofengineering judgement.
In this study the distance of min (0.4a, hw/2) from the edge with thelargest moment was proposed for the interaction check. Furthermore,the interaction check was also established at a distance of hwi,max/2, asrecommended by Johansson et al. [9]. When benefits of the momentgradient are taken into account, EN1993-1-5 requires additional checkof the elastic gross cross-section bending resistance at the location ofthe maximum moment in the panel.
3. Numerical modelling and parametric study
3.1. FEA modelling
The numerical model was built up in ABAQUS [12] with the inten-tion to study the behaviour of longitudinally stiffened plated girdersunder the M–V interaction. With numerical simulations the responseof the real system can be easily obtained. However, when developingextensive work on the basis of numerical model, it is of essential im-portance to verify the results with exact theoretical solutions, if avail-able, or with experimentally obtained values.
The response of the plated girder depends on mesh density, initialimperfections of the girder (structural and geometrical) and materialmodelling. The numerical model was developed considering recom-mendations in EN 1993-1-5 Annex C, which gives guidance on theuse of FE-methods for plated structures. The verification of the nu-merical model against test results is given in [13]. The numericalmodel showed very good agreement with test results in terms of ini-tial stiffness, ultimate resistance and failure mode.
3.1.1. Initial imperfectionsIn steel structures imperfections come from two main sources:
geometric imperfections and residual stresses. Both types of imper-fections arise from the fabrication process. The amplitudes of geomet-ric imperfections are usually limited by fabrication and erectiontolerances given in the codes. For Eurocode based design the toler-ances given in EN 1090, Part 2 [14] are relevant.
In EN 1993-1-5 two options on how to model both types of initialimperfections are given. The first one is to consider explicitly geomet-ric imperfections and residual stresses in the model. The second op-tion is to consider the effect of residual stresses with increasedequivalent geometric imperfections. The influence of residual stresseson the behaviour of plate panels is rather small (see [15]).
Different patterns of equivalent geometric imperfections were ap-plied on all tested girders [13] to evaluate the influence of initial
Fig. 5. Equivalent geometric imperfections as given in EN 1993-1-5.
234 F. Sinur, D. Beg / Journal of Constructional Steel Research 88 (2013) 231–243
imperfections on girder resistance. The following imperfections werecovered:
• The first three positive buckling modes BM: BM1, BM2, BM3 (Fig. 6)• Deformed shapes DS: DS1 and DS2 (Fig. 6).• Measured imperfection of tested girders MI, see [13,16].• Combination of equivalent imperfection shapes EC1 = EG1 + 0.7EG2 + 0.7 EG3, EC2 = −EG1 − 0.7 EG2 − 0.7 EG3, according to EN1993-1-5,
where EG1 (global buckling of the panel with the maximum amplitudeat stiffener position), EG2 (local subpanel buckling) and EG3 (twisting ofthe stiffener) are imperfection shapes with amplitudes given in Fig. 5.The deformed shapes were determined in transition from global elasticto plastic response (DS1) and near the peak force (DS2). The influence ofinitial imperfection shapes was studied on girders that were experi-mentally tested [16] and are plotted in Fig. 7.
In Fig. 8 the capacity of imperfect girders is compared to the resis-tance of a geometrically perfect girder (without initial imperfections)for all the applied imperfection shapes. The highest reduction in gird-er resistance is always found for imperfection shape DS2, defined asdeformed shape. The reduction from 2.8% to maximum 4.4% isfound, see Fig. 8. The second most unfavourable imperfection shapeis EC2, where the reduction of 1.1 to 1.9% is obtained. Some of the im-perfections increase girder resistance; higher resistance is obtained insome cases of imperfection shapes EC1, BM1, BM3 and MI.
In particular case the imperfection sensitivity is not significant be-cause of the un-symmetry of the cross-section around its weak axisdue to single sided longitudinal stiffener. This results in additionalbending moments and out-of-plane deflections. In further parametricstudy the EC2 initial imperfection shape, which was found to be veryconsistent, was taken into account. The worst shape DS2 with a
BM 1 BM 2 BM 3
Fig. 6. Imperfection shapes defined as buckling modes
pronounced knee in the stiffener, is unrealistic for initial geometryand was therefore not considered in the parametric study.
3.2. FEA modelling
For the sake of parametric study the numerical model was modi-fied to reduce computational time for each analysis. The numericalmodel shown in Fig. 9 is composed of four identical panels, supportedwith one vertical support in the middle of the girder length and later-al supports at each transverse stiffener. The load, bending momentand shear force are applied at each side of the girder.
To be safe sided, an elastic–plastic material model with nominalplateau slope was used. In such model the strain hardening is nottaken into account. Nominal values for structural steel S355with elas-tic module E = 210000 MPa and yield strength fy = 355 MPa wereconsidered. To avoid numerical problems, the nominal plateau slopeof E/10000 was assumed.
3.3. Parametric study
To cover wider area of different parameters, a numerical databaseof longitudinally stiffened girders subjected to the M–V interactionwas developed and is presented herein. The results are aimed at ful-filling the existing lack of data in this particular field.
The parametric study considered the following parameters, seeTable 1:
• Flange to web cross-section ratio (Af/Aw).• Web slenderness (hw/tw).• Panel aspect ratio (α = a/hw).• Number and geometry of longitudinal stiffeners.• Stiffness of longitudinal stiffeners.
DS 1 DS 2
BM1–BM3 and as deformed shapes DS1, and DS2.
Fig. 7. Girders and notations of panels where the influence of initial imperfection is studied.
235F. Sinur, D. Beg / Journal of Constructional Steel Research 88 (2013) 231–243
• Vertical position of longitudinal stiffener (hsl).• Bending moment–shear force ratio in the panel.
The sample was divided in four groups. Within each group onlyone of the parameters was varied, see Table 1. Each panel wassubjected to 5 ratios of bending moment and shear force, as notedin Fig. 10. Four of them (i = 1, 2, 3, 4) were exposed to bending mo-ment and shear force in the area where the M–V interactionaccording to EN 1993-1-5 should be considered. The last samplewas subjected to 60% of pure shear capacity of the web Vbw. For thegirder stiffened with two longitudinal stiffeners the range of parame-ters was reduced (bold print in Table 1).
3.4. Numerical results
In Fig. 11 the load–displacement diagrams for girders stiffenedwith one longitudinal stiffener are plotted. The varied parametersalong the plotted curves are the stiffness of longitudinal stiffener
0.98
6
0.98
1
0.99
1
0.99
6
1.03
6
0.98
9
0.98
9
0.99
1
0.98
9
1.02
1
0.99
6
0.98
1 1.00
2
0.94
0.96
0.98
1.00
1.02
1.04
1.06
EC 1 EC 2 BM 1 BM 2
F im
p./F
perf
.
Imperf
SO
Fig. 8. Reduction of girder resistance for d
(γ/γ* = 0.3 — k stiffener and γ/γ* = 1.0 — strong stiffener) and theM/V ratio in the panel, where γ* denotes the minimum stiffness of thestiffener to prevent elastic global buckling of the perfect panel due toshear stress only. There is not a large difference in girder response whenapplying different cross-sections of longitudinal stiffener. As expected,closed stiffeners result in slightly higher resistance than open stiffeners.
The failure mode of girders subjected to the combination of highbending moment and high shear load is characterised by yieldingobtained in the web. This yielding is mainly caused by the shearforce and the corresponding tension field action. Normal stress distri-bution in the web and the stiffness of the longitudinal stiffener arekey parameters that influence the development of tension field ac-tion. For slender stiffeners the buckling occurs over the entire webplate, while for stocky stiffeners the buckling is present only in thesubpanels. The stiffness of stiffeners is therefore the most critical pa-rameter for which the corresponding results will be shown here. Theinfluence of other parameters can be found in [7]. The failure mecha-nism is discussed for two stiffener locations and for two typical load
0.98
1
0.98
9
0.96
3 0.98
1
0.98
8
0.98
7
0.97
2
1.00
4
0.99
3
0.98
7
0.99
4
0.97
0
0.99
3
0.99
1
1.00
1
0.98
8
0.95
6
0.99
2
BM 3 DS 1 DS 2 MIection mode
SC UO UC
ifferent imperfection shapes applied.
Fig. 9. Layout of numerical model.
236 F. Sinur, D. Beg / Journal of Constructional Steel Research 88 (2013) 231–243
cases: bending moment equal to characteristic bending capacity offlanges Mf,c (load case 1) and bending moment equal to the character-istic bending resistance of the effective cross-section Mel,eff,c (loadcase 4).
3.4.1. Longitudinal stiffener at hw/4The results for the girder stiffened with open stiffener at hw/4 are
plotted in Fig. 12. The failure mechanisms as well as the stress distri-butions are similar for both, weak and strong stiffeners.
The failure mechanism for load case 1 may be characterized withyielding line in the web panel. Another yielding line in the smallestsubpanel is found in the area next to transverse stiffener due to normalstresses from bending moments. As the bending moment increases(load case 4), the level of corresponding shear force decreases and
Table 1Range of parameters in numerical simulations.
Number of numerical simulations 140 + 40 120 + 30 100 160 + 40Total number of numerical simulations 630
therefore, the yielding of the tension field is not that pronounced. Still,tension field action can be clearly identified, but the stresses are toosmall to obtain full plastification of diagonal tension band.
3.4.2. Longitudinal stiffener at hw/2In Fig. 13 the results for girders stiffened with open stiffener at the
mid web depth are plotted. The yielding of the tension field over thewhole web panel was observed for the weak stiffener. For the stron-ger stiffener the tension field is formed in each subpanel separately.The diagonal yielding line was more pronounced in the uppersubpanel which was subjected to the interaction of shear stressesand normal compression stresses. Thus, this panel defines the resis-tance of the plated girder, since the stiffness of the longitudinal stiff-ener is high enough to prevent global buckling.
3.4.3. Stiffness of longitudinal stiffenersIn Fig. 14 the influence of the stiffener stiffness on the girder resis-
tance is shown for girders stiffened with one open and one closedstiffener at the position of hw/4 and hw/2, and for load cases 1 and 5(see Fig. 10). For the stiffener at the mid-web depth the transitionstiffness, after which almost no additional resistance is gained, isvery clear. To reach this point, the necessary stiffness γ/γ* is between0.75 and 1.00. This is due to the fact that one stiffener in the mid webdepth influences mainly the shear resistance of a girder. When thestiffener is strong enough to make a local subpanel shear resistancedecisive, further stiffness increase does not contribute to the girderresistance increase.
For the stiffener positioned in the upper part of the web (hw/4) andexposed to high compression force due to bending this transition is notso clear and continuous increase of the girder resistance is present.
In Fig. 15 the influence of stiffener stiffness is shown for a girderstiffened with two longitudinal stiffeners. The obtained results aresimilar to those obtained for the girder stiffened with one open stiff-ener at mid web depth. Also in this case the transition stiffness isfound between γ/γ* = 0.75 and γ/γ* = 1.0.
Fig. 10. Load cases in numerical simulations.
Fig. 11. Load–displacement curves for girders stiffened with one longitudinal stiffener.
237F. Sinur, D. Beg / Journal of Constructional Steel Research 88 (2013) 231–243
4. Evaluation of numerical parametric study
The characteristic resistance according to EN 1993-1-5 was cal-culated and compared to the results of numerical simulations. Themaximum resistance of the panel was evaluated at a distance of
a)
c)
Fig. 12. Von Mises stresses for girders stiff
min (0.4a, hw/2) and hwi,max/2 from the edge with the largestmoment. In this way the positive effect of the stress gradient inthe panel is taken into account.
Numerical results are plotted in the EN 1993-1-5M–V interaction di-agram. Themarks that are inside the diagram in the range ofMf,Mel,eff are
b)
d)
ened with one open stiffener at hw/4.
a) b)
c) d)
Fig. 13. Von Mises stresses for girders stiffened with one open stiffener at hw/2.
238 F. Sinur, D. Beg / Journal of Constructional Steel Research 88 (2013) 231–243
on the unsafe side and vice versa, if themarks are outside, the results aresafe. The numerical results for group II, where themain parameter is theslenderness of the web, are plotted in Fig. 16. The results are plotted fortwo web slendernesses hw/tw = 150 and hw/tw = 400. In Fig. 16a theinteraction check is performed at a distance of min (0.4a, hw/2) andin Fig. 16b at a distance of hwi,max/2. For stocky webs the currentinteraction formula is found to be unsafe for girders stiffened withstiffener at mid-web depth (see Fig. 16a, hw/tw = 150). As the inter-action check is moved towards the most stressed edge of the panel,all results are shifted towards the interaction curve and some ofthem become safe sided (see Fig. 16b, hw/tw = 150). In case ofvery slender web hw/tw = 400, the results are found safe sided al-ready when the interaction check is performed at a distance ofmin (0.4a, hw/2). Girders stiffened with one stiffener in the com-pression zone (hw/4) prove much higher resistance than obtainedwith effective width method that does not take into account thepositive effect of tension stresses in the largest subpanel whichwas decisive for the design.
Fig. 14. The influence of the stiffener stiffness on the gird
For girders stiffenedwith longitudinal stiffener in compressed part ofthe web (hw/4) the numerical results prove higher resistance than theone obtained through EN 1993-1-5 at a distance of min (0.4a, hw/2),while for girders stiffened atmidwebdepth the numerically obtained re-sistance is smaller for slenderness hw/tw ≤ 200. For the interaction checkat hwi,max/2, all numerical results, except girders with low slendernesshw/tw = 150 and with stiffener at mid web depth, prove higher resis-tance. The influence of normal tensile stresses in the largest subpanelresults in higher shear resistance, which is not taken into account withthe effective width method. Therefore, girders stiffened with stiffenerin compression zone prove much higher resistance in comparison tothe EN 1993-1-5 design rules than those stiffened with stiffener at themid web depth, where the subpanel in compression is critical as it issubjected to the interaction of shear and compressive stresses.
The results show that the current interaction formula evaluated at adistance ofmin (0.4a, hw/2) and at hwi,max/2 from the edgewith the larg-est moment may be unsafe in some cases. The shape of interaction is afunction of the web slenderness. For stocky webs the interaction
er resistance for a girder stiffened with one stiffener.
Fig. 15. The influence of the stiffener stiffness on the girder resistance for a girder stiffened with two stiffeners.
239F. Sinur, D. Beg / Journal of Constructional Steel Research 88 (2013) 231–243
diagram shape approachesMises yield criterion, while for slender webstheM–V interaction is linear. Also for all other parameters the linear in-teraction between shear force and bending moment was observed.Therefore, for the area of large bending moment and shear force alter-native interaction formula is proposed (see Fig. 17):
η1;new þ 1−Mf ;Rd
Mel;eff ;Rd
!2η3−1� �κ≤1:0 ð9Þ
a)
Fig. 16. Numerical results plotted on current form
with
η1;new ¼ MEd
Mel;eff ;Rd;η3 ¼ VEd
Vbw;Rd
κ ¼ 1:
The differences compared to previous interaction formula are:plastic bending resistance Mpl,Rd is replaced with elastic effectivebending resistance Mel,eff,Rd, and the exponent of 2 is replaced with
b)
ulation of the M–V interaction — GROUP II.
Fig. 17. M–V interaction formulation according to expression (9).
240 F. Sinur, D. Beg / Journal of Constructional Steel Research 88 (2013) 231–243
factor κ which is a function of the web slenderness and defines theshape of the interaction formula. Since most of the results show linearinteraction, parameter κ was conservatively set to 1, but it can be cal-ibrated to appropriate values between 1 and 2.
The results of statistical evaluation of both interaction models aregathered in Tables 2 and 3. The calculated parameters are the meanvalue of Vnum/Vbw,c (where Vbw,c is characteristic value of Vbw,Rd), stan-dard deviation and coefficient of variation. With new formulation themean value is increased, while the standard deviation and the coeffi-cient of variation are decreased. The results are more consistent(smaller deviation) and safe sided (see Tables 2 and 3) when usingnew M–V interaction formulation.
Table 2Statistical evaluation of the M–V interaction formulations for girders stiffened with one stif
Varied parameter Position and type of the stiffener n Mean
EN 19
Af/Aw hw/4, open 140 1.085hw/4, closed 1.085hw/2, open 1.009hw/2, closed 1.010
hw/tw hw/4, open 120 1.103hw/4, closed 1.090hw/2, open 1.014hw/2, closed 1.009
γ/γ* hw/4, open 100 1.091hw/4, closed 1.083hw/2, open 1.013hw/2, closed 0.982
α hw/4, open 160 1.112hw/4, closed 1.099hw/2, open 1.015hw/2, closed 1.012
All All 520 1.052
Table 3Statistical evaluation of the M–V interaction formulations for girders stiffened with two sti
Varied parameter Type of stiffeners n Mean value
EN 1993
Af/Aw Open 40 0.995Closed 0.984
hw/tw Open 30 0.972Closed 0.969
γ/γ* Open 40 0.999Closed 0.987
All All 110 0.986
5. Partial safety factor
For different resistance models the partial safety factors wereevaluated taking proper account of all relevant uncertaintiesaccording to Annex D of EN 1990 [17]. The following uncertainties,which are of basic importance for the determination of γM, weretaken into account:
• Uncertainty of resistance model Vδ.• Uncertainty of geometry.• Uncertainty of material properties.• Uncertainty of numerical model.
The uncertainty of material properties (log-normal distribution)and geometry (Gauss distribution) are determined on the basis ofprior knowledge. The following coefficients of variations wereconsidered:
• Vwidth = 0.005 variation coefficient for the width of the plate• Vthicknes = 0.05 variation coefficient for the thickness of the plate• Vyield = 0.07 variation coefficient for the yield strength
Additionally, the variation coefficient for the vertical position ofthe stiffener was assumed:
• Vhwi = 0.005 variation coefficient for the position of the longitudi-nal stiffener
Partial safety factor evaluated according to EN 1990, Annex D is de-termined on the basis of experimental results. In this work the experi-mental results are determined with numerical simulations. Since theresults of numerical simulations do not exactly coincidewith the exper-imental results, a coefficient of variation of numerical simulation VFEM is
241F. Sinur, D. Beg / Journal of Constructional Steel Research 88 (2013) 231–243
introduced to the calculation of partial safety factor. The calculation ofthis coefficient is given in Table 4.
Table 5
5.1. Resistance models
Five resistance models were evaluated to determine partial safetyfactors. The first resistance model rt,1 corresponds to the interactioncheck according to EN 1993-1-5. Since the interaction formulationdoes not fit the shape of interaction, a new resistance model was in-troduced in Section 4 and is denoted as resistance model rt,2. Whenthe interaction check is performed in the panel, EN 1993-1-5 requiresan additional check of bending resistance of gross cross-section at theedge with the largest moment in the panel (section 0–0, see Fig. 18).Therefore, the third resistance model rt,3 representing the bendingcheck of the gross cross-sections was introduced. Finally, the lasttwo combined models, rt,4 and rt,5, were defined as rt,4 = min(rt,1,rt,3) and rt,5 = min(rt,2, rt,3). All models were expressed in terms ofshear resistance V at the corresponding level of bending momentMEd. In resistance models characteristic values for bending resistancesMel,ell,c, Mel,eff,c and Mf,c are considered. The evaluated partial safetyfactor was then evaluated for the design shear resistance and the tar-get value is 1.1, as currently proposed in EN 1993-1-5.
The first resistance model is given with equation:
rt;1 ¼ V ¼ 1þ Mpl;c−MMpl;c−Mf ;c
!0:5 !⋅Vbw;c
2ð10Þ
Fig. 18. Position of interaction check (sections 1–1 and 2–2) and gross cross-sectioncheck (section 0–0).
where Mf,c and Mpl,c are the characteristic bending resistance of theflanges and plastic bending resistance of the cross-section, respectively.
The second resistancemodel is determinedwith Eq. (9) in Section 4:
rt;2 ¼ V ¼ 1þ Mel;eff ;c−MMel;eff ;c−Mf ;c
! !⋅Vbw;c
2: ð11Þ
The third resistance model is defined as elastic bending resistanceof a gross cross-section checked at the edge of the panel:
rt;3 ¼ V ¼ Mel;c
l; ð12Þ
where l is a distance between zero bending moment location and themost stressed edge of the panel.
The last two resistance models are defined as:
rt;4 ¼ min rt;1; rt;3� �
ð13Þ
rt;5 ¼ min rt;2; rt;3� �
: ð14Þ
All resistance models except the third one are evaluated at twodifferent distances from the edge of the panel with the largest mo-ment (x1, x2), while the third resistance model is evaluated at the
Calculated γM* values for resistance models rt,1 and rt,2 at x1 = min(a, hw/2).
242 F. Sinur, D. Beg / Journal of Constructional Steel Research 88 (2013) 231–243
edge of the panel (x0), see Fig. 18. The models were evaluated for thefollowing data sub-sets:
• Sub-set I: All analysed girders — 582 data.• Sub-set II: Only girders stiffened with longitudinal stiffener at hw/4.• Sub-set III: Only girders stiffened with longitudinal stiffener at hw/2.• Sub-set IV: Only girders stiffened with two equally spaced longitu-dinal stiffeners.
5.2. Comparison and evaluation of results
The results of the evaluated partial safety factors are gathered inTable 5 to Table 9. For resistance model rt,1 evaluated at a distancex1 = min(a, hw/2) the partial safety factors are between 1.048 and1.221 (see Table 5). The largest value was found for sub-set IV, i.e.for girders stiffened with two equidistantly spaced stiffeners. TheEN 1993-1-5 interaction model without the additional gross cross-section check is not safe enough, since the needed partial safety factoris more than the target value γM1 = 1.1. The proposal to evaluate theinteraction check at a distance of x2 = hwi,max/2 (Table 6) is found towork better, because the partial safety factor for resistance model rt,1is decreased to values from 1.045 to 1.113, which is close to the targetvalue γM1 = 1.1 currently given in EN 1993-1-5. There is a differenceof 1.1% on the unsafe side for the most unfavourable sub-set IV.
The proposed model rt,2 evaluated at a distance min(a, hw/2) re-sults in smaller safety factors than model rt,1 (Table 5). This wasexpected for two reasons: The mean values were higher and the devi-ation coefficients were smaller for resistance model rt,2 than formodel rt,1 (see Tables 2 and 3). The evaluated partial safety factorsfor all sub-sets are between 0.999 and 1.168. With the target valueof the partial safety factor γM1 = 1.1, the proposed model rt,2 doesnot fulfill reliability conditions for sub-sets I and IV. If the M–V inter-action check is performed with resistance model rt,2 at a distance ofhwi,max/2, the partial safety factors are below γM1 = 1.1. The valuesare in the range of 0.998 to 1.051 (Table 6).
The partial safety factors evaluated for resistance model rt,3, wherethe maximum bending moment should be less than elastic bendingmoment resistance of gross cross-section, are gathered in Table 7. InEN 1993-1-5 the partial safety factor is equal to 1.0, while in ourstudy the target value was 1.1 to cover the M–V interaction in thepanel. The maximum factor γM = 1.113 is found for sub-set III.
In Tables 8 and 9 the partial safety factors evaluated for resistancemodels rt,4 and rt,5 are gathered. With these two models the lowestvalue is obtained, because the theoretical resistance is defined asthe minimum values of two corresponding resistance models. Thepartial safety factor is below 1.1 for all cases except for sub-set IV
Table 9Calculated γM* values for resistance models rt,4 and rt,5 at x2 = hwi,max/2.
(for this case the partial safety factor is only slightly exceeded), if the in-teraction check is performed at a distance x1 = min (0.4a, hw/2). Thepartial safety factors for both models rt,4 and rt,5 are more consistentthan for resistance models rt,1 and rt,2.
6. Discussion and concluding remarks
The aim of this paper was to give a general view on the behaviourof longitudinally and transversally stiffened girders subjected to highbending moment and shear force, and to compare the results withthose of the existing resistance models in EN 1993-1-5 and the newproposed model. The load capacity as well as the failure mode ofthe girders depend on initial imperfections, therefore, appropriateand reasonable imperfection shapes were considered in the nonlinearanalysis to get reliable results. The numerical model was preliminarilyverified on the results of four tests [7].
The influence of initial imperfections on girder resistance is notvery significant, because the single sided longitudinal stiffeners intro-duce asymmetry of the panel and because the shear is resisted by ten-sion field action which is less sensitive to initial imperfections.
The obtained numerical resistance of girders stiffened with onestiffener in compression zone was always higher than calculatedaccording to EN 1993-1-5. The largest panel which defines shear re-sistance of girder is subjected to normal stresses. In this panel the ten-sion stresses prevail. Therefore, shear resistance is increased. Withineffective width method this influence is not considered. When sub-panels are of the same size, the critical subpanel is subjected to the in-teraction of shear stresses and normal compression stresses whichdecrease shear resistance. For girders with one stiffener at mid webdepth the numerical resistance was close to that calculated by EN1993-1-5, while for girders with two equidistantly spaced stiffenersthat prevent global buckling, the numerical resistance was smallerthan calculated by EN 1993-1-5. This shows that resistance stronglydepends on the stress distribution in the subpanels and on the stiff-ness of longitudinal stiffeners.
Within this work different existing resistance models and a newproposed resistance model were evaluated in terms of partial safetyfactors. Significant outcomes are:
• The M–V interaction check given in EN 1993-1-5 does not fulfill re-liability requirements and is too optimistic, since the required par-tial safety factor is higher than currently defined in EN 1993.
• The proposal of Johansson et al. [9] to check the M–V interaction atdistance x2 = hwi,max/2 is found to be more suitable, as the maxi-mum required partial safety factor 1.113 is slightly higher than tar-get value 1.1.
• The new proposedmodel for the M–V interaction evaluated at a dis-tance x2 = hwi,max/2 shifts most of the results on the safe side. Therequired partial safety factor is below the target value of 1.1.
• It was shown that the M–V resistance of the panel can simply bechecked by the gross cross-section resistance model evaluated atthe most stressed edge of the panel. The needed partial safety factorfor this model is 1.113, which is close to the target value of 1.1.Using this resistance model, the stability of the panel as well asthe resistance of the cross-section are checked. The advantage ofthis model is in its simplicity. The M–V interaction does not needto be checked explicitly. It is covered implicitly through the positiveeffect of the moment gradient that is always present at large shearforces.
• Resistance models rt,4 and rt,5 defined as minimum value of the twocorresponding models result in the smallest value of partial safetyfactor. Resistance model rt,4 follows the EN 1993-1-5 design rule.It was shown that this resistance model can be used if partial safetyfactor γM0 for elastic bending resistance of the cross-section is set to1.1 and not to 1.0, as it is the case in the current version of EN1993-1-5. Resistance model rt,5 is more conservative than rt,4.
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Nevertheless, partial safety factor γM0 for elastic bending resistanceshould be 1.1.
Acknowledgements
The research was carried out under the financial support providedby the Slovenian Ministry of Higher Education, Science and Technologywithin the programme of young researchers (Grant No. 6-249-1/2006).Their support is gratefully acknowledged.
References
[1] Basler K. Strength of plate girders under combined bending and shear. J Struct Div1961;87(ST7).
[2] Herzog M. Ultimate static strength of plate girders from tests. J Struct Div ASCE1974;100(ST5):849–64.
[3] HerzogM. Die Traglast unversteifer und versteifer, dünnwändiger Blechträger unterreinem Schub und Schub mit Biegung nach Versuchen; 1974 [Bauingenieur, vol.].
[4] Fujii T, Fukomoto Y, Nishino F, Okumura T. Research works on ultimate strengthof plate girders and Japanese provisions on plate girder design. Proceedings ofColloquium on design of plate and box girders for ultimate strength. London:IABSE; March 25–26 1971. p. 21–48.
[5] Rockey KC. An ultimate load method for the design of plate girders. Proceedings ofColloquium on design of plate and box girders for ultimate strength. London:IABSE; March 25–26 1971. p. 253–68.
[6] Rockey KC, Evans HR, Porter DM. Ultimate load capacity of stiffened webssubjected to shear and bending. Proc. conf. steel box girders. London; 1973.
[7] Sinur F. Behaviour of longitudinally stiffened plated girders subjected to bending–shear interaction.Faculty of Civil and Geodetic Engineering. University of Ljublja-na; 2011 [Doctoral Thesis].
[8] CEN. Eurocode 3: design of steel structures— part 1-5: plated structural elements,in EN 1993-1-5. Brussels: European Committee for Standardisation; 2006 .
[9] Johansson B, Maquoi R, Sedlacek G, Müller C, Beg D. Commentary and worked ex-amples to EN 1993-1-5 “plated structural elements”. JRC scientific and technicalreports; 2007.
[10] Beg D, Kuhlmann U, Davaine L, Braun B. In: ECCS, editor. Eurocode 3: design ofsteel structures, part 1-5—design of plated structures. , , 1st ed.Ernst & SohnWiley Company; 2010. p. 272.
[11] Höglund T. Behaviour and strength of the web of thin plate I-girders (in Swedish).Bulletin No.93 of the Division of Building Statics and Structural Engineering.Stockholm, Sweden: The Royal Institute of Technology; 1971. p. 13–30.
[13] Sinur F. Behaviour of longitudinally stiffened plated girders subjected to bending–shear interaction.Faculty of Civil and Geodetic Engineering, University of Ljubljana;July 2011 [Doctoral Thesis].
[14] CEN. EN 1090-2: 2008. — execution of steel structures and aluminium structures— part 2: technical requirements for steel structures. Brussels: European Commit-tee for Standardisation; 2008 .
[15] Sinur F, Beg D. Imperfection sensitivity analysis of longitudinally stiffened platedgirders subjected to bending–shear interaction. Proceedings of the stability andductility of steel structures conference, Rio de Janeiro, Brazil; 2010. p. 787–94.
[16] Sinur F Beg D. Moment–shear interaction of stiffened plate girders—tests and nu-merical model verification. J Constr Steel Res0143-974X June 2013;85:116–29,doi:10.1016/j.jcsr.2013.03.007.
[17] CEN. Eurocode-basis of structural design, in EN 1990. Brussels: European Commit-tee for Standardisation; 2004 .
