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Influence of designer-assumed initial conditions on the numerical modelling of steel plate girders subjected to patch loading
ARTICLE IN PRESS Thin-Walled Structures 47 (2009) 391–402
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Influence of designer-assumed initial conditions on the numerical modelling of steel plate girders subjected to patch loading R. Chaco´n , E. Mirambell, E. Real Construction Engineering Department, Universitat Polite`cnica de Catalunya, C/Jordi Girona 1-3, 08034 Barcelona, Spain
a r t i c l e in fo
abstract
Article history: Received 15 October 2007 Received in revised form 12 June 2008 Accepted 4 September 2008 Available online 19 October 2008
European design rules EN1993-1-5 allows the usage of FE-Analyses as reliable tools in the verification of limit states of plated structures. Throughout the modelling of these structures, certain initial conditions must be assumed. Consequently, engineering judgment is needed to some degree. This paper presents an assessment of the influence of these conditions on the ultimate load capacity of steel plate girders when subjected to patch loading. The basis of the research is a thorough comparison between theory and design rules by means of experimental and numerical results. The results obtained prove usefulness for practical and academic purposes alike. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Patch loading Initial geometric imperfections Residual stresses Ultimate limit states
1. Introduction Thin-walled plate girders are used when it is necessary for a structural element to support high loads, above which a normal rolled section would either not be structurally viable or would become uneconomical. A plate girder may be designed as homogenous (fyf ¼ fyw) or as hybrid (fyf4fyw). This latter alternative is popular as the girder yields a greater flexural capacity at a lower cost compared to a homogeneous girder [1]. In both hybrid and homogeneous configurations, the welding process between web and flanges adds considerable imperfections to the initially straight plates. Additionally, if transversal and/or longitudinal stiffeners are systematically welded to the girders, the magnitudes and shapes of these imperfections may randomly change. These imperfections may either be geometrical or structural. Geometrical imperfections are often characterised by initially ‘‘imperfect’’ plates following a certain shape, whereas structural imperfections are characterised by a residual stress pattern. Historically, it has been shown that thin-walled structures are ‘‘imperfection sensitive’’, i.e., critical buckling/ultimate loads depend upon the actual initial imperfection. An out-of-flatness of the plate with small initial amplitude exerts an influence on the basic equilibrium path. If the initial shape is related to instability modes of the panels, the equilibrium path follows this shape and merges into the post-critical path [2]. The random nature of the
Corresponding author.
E-mail address: [email protected] (R. Chaco´n). 0263-8231/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2008.09.001
shapes and magnitudes of the initial imperfections, though, results in difficulties. Fortunately, it has been shown that if these imperfections are kept within allowable limits (usually related to fabrication tolerances), these difficulties may be controlled. Over the years, the field of measurement techniques has become a high-tech industry. Different measurement techniques have emerged [3,4]. Indeed, a vast amount of data can nowadays be measured, stored and processed. Therefore, truly realistic geometrical maps of the plates can be drawn from the obtained results. This enhancement is accompanied with the fact that precision tolerances are nowadays tighter and more stringent. As a result, the whole industry gives a quite reliable performance. Therefore, the randomness of the phenomena can be at least, kept on the aforementioned allowable limits. Presently, European design rules EN1993-1-5 [5] (Annex C) include the possibility of using FE-Analyses as reliable tools in the verification of limit states. As a result, plated steel structures are usually verified by numerical simulations performed on available reliable codes. The requested verifications are fairly generic for any type of loading and/or geometry. In order to present a systematic procedure for any type of modelling of plated structures, certain initial conditions must be assumed. Moreover, in recent years, pre- and post-processing techniques have become increasingly user-friendly, which facilitates the former expensive and time-consuming modelling. In this particular case, the focal point is on the simulation of the patch loading phenomena on plate girders. This simulation requires a definition of a geometrically and structurally imperfect three-dimensional (3D) model as well as a plastic definition of the constitutive equation of the steel. For the sake of capturing the
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E
Nomenclature
Young modulus slenderness gM1 partial factor % FT percentage of maximum allowable fabrication tolerance w maximum amplitude of the imperfection dog initial sweep of the girder dow initial out of flatness of the web y initial twist of the flanges FRd design resistance to transverse forces Fu predicted resistance to transverse forces according to numerical results Fu,(w ¼ 80%) predicted resistance according to numerical results (standardised) DFu maximum difference between extreme values of Fu
lF
hw a Ss tf tw ts fyf fyw fys ly
wF Fy Fcr
clear web depth between flanges width of web panel (distance between transverse stiffeners) length of stiff bearing thickness of the flanges thickness of the web thickness of the transverse stiffeners flange yield strength web yield strength stiffener yield strength effective loaded length reduction factor plastic resistance critical buckling load
local buckling phenomena, the model must be based upon shell elements geometries. There exists a considerable amount of literature on the influence of initial geometrical imperfections on the resistance of girders subjected to patch loading [6–10]. Some authors [11], however, consider that the correlation between theory, imperfection measurements and ultimate limit states verification in European standards is judged to be still incomplete. Furthermore, as far as known by the authors, scarce studies include the effect of the structural imperfections (secondorder imperfection sensitivity) of welded girders on their ultimate load capacities when subjected to patch loading [12–15]. Preliminary, the drawn conclusions point out that these structural imperfections (idealized as residual stresses) do not play a decisive role. The present work is aimed at assessing the influence of the designer-assumed initial conditions on the general response of steel plate girders when subjected to patch loading (Fig. 1) focusing on the current design rules implemented in EN1993-1-5 [5]. On one hand, an experimental program on four steel plate girders subjected to patch loading is presented. Before the tests, the initial geometry of the web and flange panels in all four girders was obtained by means of a 3D co-ordinate measuring device [16]. On the other hand, the numerical model implemented in the multi-purpose code ABAQUS 6.5 [17] which includes geometrical and material nonlinearities, was used as a simulation tool in order to develop a parametric study which allowed presenting a vaster assessment of this influence. The ultimate load capacity of the steel plate girders subjected to patch loading was numerically obtained by means of three different methods. The first method is by the introduction of realistic precise shapes of initial imperfections previously measured for each girder together with a typical idealized residual
Web yielding
Stocky panel
Web buckling
Slender panel
Fig. 1. Effect of a concentrated load on a girder (a) web yielding and (b) web buckling.
stress pattern. The second method is by the introduction of initial imperfections based upon different eigenmodes (together with structural imperfections, i.e., an idealized residual stress pattern). The third, by using the equivalent initial imperfections proposed in EN1993-1-5 [5], which includes both geometric and structural imperfections. The results obtained show the effect of the amplitude and the shape of the geometrical imperfection as well as the effect of the structural imperfections on the resistance of plate girders when subjected to patch loading. The drawn conclusions may give guidance to practicing engineers and researchers on the designerassumed conditions and the potential influence on ultimate limit states verifications for the particular case of study.
2. Design provisions 2.1. EN1993-1-5. resistance to transverse forces Patch loading phenomena has been widely analysed since the early sixties. Several failure mechanisms and critical buckling loads have been proposed throughout the last decades for the case of stiffened and unstiffened panels subjected to concentrated loads [18–21]. In early ages, the normal approach in EN1993-1-1 [22] used upper and lower limits for the resistance, upper limit based on yielding and lower limit based on instability. The results from tests had, however, historically shown that there is not any clear distinction between these two bounds. Instead, a blurred transition between yielding and behaviour influenced by buckling is observed, similarly to other instability-based problems. Presently, EN1993-1-5 [5] defines this gradual transition by means of the w-reduction factor for both stiffened and unstiffened webs. Mechanical models based on first-order limit analysis are the most outstanding approaches for defining the plastic resistance for patch loading [18,20]. In Fig. 2, one can observe the four-hinge model from which ultimate load resistance is typically obtained. Other approaches include semi-circular yield lines in the web which have been experimentally observed and thus, theoretically added to the hinge-based resistances. On the other hand, the limit related to instability has generally been obtained by numerical simulations using the classical elastic critical theory in a locally loaded simply supported plate. The general equation currently included in EN1993-1-5 [5] is based upon a plastic resistance derived from these models (Fig. 2) and partially reduced by means of wF (Eq. (1)). In this equation, ‘‘ly’’ is the yield-prone effectively loaded length. This length
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Ss+2tf
Sy/2
a
bf
Mo
Mi
tf
Mo
bf Mi
fywtw
Y δow
Mo
tf
Mi
X
hw
Sy/2
393
10εtw tw Ζ
Z
1y Fig. 2. Model of resistance [19].
θ physically represents the distance between outer hinges and is calculated from geometrical and mechanical properties of the girders by using (2) and (3). wF takes into account instability by means of (4) and (5) and can be obtained with Eq. (6). Buckling coefficient kf varies whether the web panels are longitudinally stiffened or unstiffened. F Rd ¼
f yw wF ly t w
gM1
ly ¼ Ss þ 2t f
m2 ¼ 0:02
lF ¼
2 hw if lX0:5 otherwise m2 ¼ 0 tf
sffiffiffiffiffiffi Fy F cr
F cr ¼ 0:9kf E
wF ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! f yf bf 1þ þ m2 f yw t w
0:5
lF
(1)
(2)
(3)
(4) t 3w hw
p1:0
(5)
(6)
2.2. EN1993-1-5-Annex C. FE-analyses Presently, European design rules include the possibility of using FE-Analyses as reliable tools in the calculation of plated structures. The modelling may be either based upon a refined analysis by including geometrical eigenmode-based imperfections as well as residual stresses or based upon equivalent geometric imperfections which should include both effects. In fact, there is a decision to be taken in this latter definition and hence, engineering judge is needed to some degree. Firstly, the refined analysis is discussed. This analysis can be based upon critical eigenmodes of the structure conveniently scaled a maximum amplitude w. EN1993-1-5-Annex C [5] rules recommend a value for w at least of 80% of the maximum allowed fabrication tolerances (80% FT), provided that the chosen deviations lead to the lowest resistance for each case. This percentage may vary from one National Annex to another. In the particular case of patch loading, the initial out-of-flatness of the web in the directly loaded panel should be the most significant deviation to be considered. EN1993-1-5-Annex C [5], however, suggests this percentage generically for all cases in design of plated structures. Presently, fabrication tolerances limit this out-of-flatness to the lowest value between two magnitudes:
tw, thickness of the web hw/100, being hw the clear web depth between flanges
X Y
Y
δog L
Z
Fig. 3. Introduction of equivalent geometric imperfections. (a) Initial out-offlatness; (b) initial sweep of the girder; and (c) initial twist of the compression flange.
The second possibility is defined by the equivalent geometric imperfections, which should be based upon the following considerations: (a) An initial out-of-flatness of the web panels dow (Fig. 3-a). In accordance with EN1993-1-5-Annex C [5], the maximum initial out-of flatness of the web panel is the minimum value between (a/200;hw/200). (b) An initial sweep of the girder within bearings dog (Fig. 3-b). This represents a lateral deviation in the longitudinal direction. The minimum value to be introduced following EN1993-1-1 [22] is L/100 for girders using buckling curve ‘‘d’’. (c) An initial twisting y of the top flange and/or longitudinal stiffening (Fig. 3-c). The magnitude of this twisting must be at least 1/50 radians. These equivalent imperfections may be substituted by appropriate forces acting on the member. In this paper, the equivalent imperfections are treated as geometrical and are not substituted by any force.
3. Experimental program 3.1. Description An experimental program over four steel girders subjected to concentrated loads was carried out at the Structural Technology Laboratory of the School of Civil Engineering of Barcelona, UPC [23]. The length of the span in all four girders was 2500 mm. The web height was 500 mm and a 150 200 mm2 patch was used for introducing the load. Three different steel plates were used for the fabrication of these girders, the first plate of 4 mm thickness for the web (fyw ¼ 325 N/mm2), the second of 20 mm thickness for the flanges (bf ¼ 200 mm and fyf ¼ 454 N/mm2) and the third one of 20 mm for the stiffeners (two-sided transverse stiffener, bs ¼ 200 mm, fys ¼ 310 N/mm2). The mechanical properties of all plates were obtained by means of tensile coupon tests. 3.2. Test configuration The tests were carried out under displacement control using a MTS hydraulic jack. Loads, displacements and strains at key points
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were measured during the development of the tests. Uniaxial gauges and rosettes and displacement transducers were placed for measuring strains and vertical and horizontal deflections. Other outputs were obtained from the jack as well as from the load-cell. A MGC-Plus data acquisition system was used in order to record readings following a one-hertz frequency. In Fig. 4, a general view of the test configuration is displayed. 3.3. Measuring the initial shapes of the girders Before the tests, initial shapes of the web and flange panels in all four girders were obtained by means of a 3D co-ordinate measuring device. The 3D data format is commonly referred to as ‘‘point cloud’’, which leads to a series of co-ordinates {xi,yi,zi} of each point. The data allow the reproduction of the original shape for both the web and flanges of all steel plate girders. The measurement principle is to obtain the position of infrared LEDs by means of cameras through triangulations using a portable device as shown in Fig. 5. 3D data from all web and flange panels were obtained following a hundred millimetres square grid previously drawn in all four girders. Two singularities were observed after plotting the results. Firstly, random shapes of the initial imperfections of the webs were noticed (Fig. 6). Secondly, the initial imperfection of the flanges was judged to be negligible. A considerable twisting relative to a vertical reference was, however, noticed in specimen 1VPL1500.
3.4. Test results Geometrical data and results obtained are given in Table 1. Notice that distance between transversal stiffeners in the directly loaded panel (distance ‘‘a’’) is varied from one specimen to another. As long as the distance ‘‘a’’ is decreased, ultimate load capacity of the girders is increased, except for specimen 1VPL1500, in which results were certainly influenced by the excessive twisting of the top flange occurring throughout the test. This specimen was discarded for subsequent studies, since it did not reproduce the desired phenomena.
4. Numerical modelling The pre-processors ABAQUS-CAE [17] and GID [24], which are capable of reproducing intricate geometries, were used to assemble suitable meshes for developing the geometrical models. Several nonlinear analyses were performed on the steel plate girders using quad-dominant S4 shell elements for web, flanges and stiffeners. A convergence analysis led to quite dense meshes in all cases (25 mm square meshes for the coarsest). Both geometric and material nonlinear effects were considered. The adopted constitutive equation was the one observed in tensile coupon tests. The nonlinear solution strategy used is the Modified Riks [25] algorithm implemented in ABAQUS. 4.1. Structural imperfections
Rigid frame
Hydraulic Jack Directly loaded panel
Hinge Patch load a
Protection
Bearing
Fig. 4. Test configuration.
Fig. 5. Measurement of initial imperfections.
Double I-shaped stocky beam Concrete slab
In accordance to EN1993-1-5-Annex C [5], structural imperfections should be defined in terms of residual stresses by a typical stress pattern from the fabrication process. Residual stress pattern is a parameter that defies generalisations [26]; each particular case might present different features. In this study, a typical idealized residual stress pattern for plate girders is assumed. Longitudinal residual stresses are incorporated into the girder models using the stress pattern idealization depicted in Fig. 7-a. The pattern follows the European Convention for Constructional Steelwork Recommendations (ECCS [27]), which reflect the two primary causes of longitudinal residual stress in welded I-girders: flame-cutting of the plates and longitudinal welding in the webto-flange juncture. Essentially, the residual stresses are equal to the yield stress of the plate within a small width at the heataffected zones; this is primarily a tensile stress. Then, a smaller self-equilibrating compressive stress is generated within the other regions of the plates. The modelling of such idealized pattern might be rather tough in systematic parametric studies. Fortunately, ECCS provides simplified equations for estimating the widths that are effectively stressed. These equations take into account flame-cutting, welding at junctures, geometrical properties of the added weld metal as well as the efficiency in the welding process. If these equations are faithfully followed, one result is that the predicted residual stresses are highly dependent upon the cross-section size. Within this study, the stress pattern is assumed to present a near-constant tensile stress within a rather wide zone of the borders of the flanges (0.18 fyf), a higher value of tensile stress in the web-to-flange juncture (0.33 fyf in the flange and 0.63 fyw in the web) and an additional compressive stress aimed to equilibrate the whole section (0.20 fyf in the flange and 0.15 fyw in the web). These values (Fig. 7-b) are considered structurally sound in other studies [28] and were systematically included as typical patterns in all girders of the present work. Considering that the initial stress state may not be in exact equilibrium, an initial step is included to allow ABAQUS to check for equilibrium and iterate, if necessary, to achieve it.
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1VPL2500
1VPL1500 500 Web-Depth (mm)
500 Web-Depth (mm)
395
400 300 200 100
400 300 200 100 0
0 0
1
2 3 4 Measured out-of-the plane imperfection (mm)
5
0
1
2 3 4 Measured out-of-the plane imperfection (mm)
5
1VPL750 1VPL450
Web-Depth (mm)
500
520
400 Web-Depth (mm)
300 200 100 0 -2 0 2 Measured out-of-the plane imperfection (mm)
-4
420
4
320 220 120 -3
20 -2 -1 -80 0 Measured out-of-the plane imperfection (mm)
1
Fig. 6. Measurements of the imperfections at mid-span section of each girder.
Table 1 Test data Specimen
a (mm)
a/hw
Fu,
1VPL2500 1VPL1500 1VPL750 1VPL450
2500 1500 750 450
5.00 3.00 1.50 0.90
217.23 190.00 251.80 426.00
test
(kN)
FRd (kN)
Vertical deflection (mm)
162.59 164.48 173.06 173.10
2.61 5.61 13.08 10.20
Fig. 8. Introduction of the precise shape of the girders. (a) Point cloud; (b) plate (nurbs surface); and (c) assembled model.
+0.18 fyf
+0.33 fyf +0.18 fyf
The residual stress profile is simpler than the one depicted above and lead to accurate results. 4.2. Initial geometrical imperfections
-0.20 fyf
-0.20 fyf +0.63 fyw
Three different initial ‘‘imperfect’’ geometries were modelled in ABAQUS.
-0.15 fyw
-0.20 fyf
+0.18 fyf
-0.20 fyf
+0.18 fyf +0.33 fyf
Fig. 7. Residual stresses. (a) Typical idealized pattern; and (b) assumed pattern.
Other studies which included residual stresses in the patch loading verifications [13–16] used the simplified pattern suggested by the Swedish design code for steel structures [29].
4.2.1. Precise shape initial imperfection The precise shape of the girders was introduced from 3D experimental data previously obtained. The data include global co-ordinates of points previously located on grids in the webs and flanges of the girders. The points are referenced to a global coordinate system. From these results, a graphical reproduction of the girders is sketched with a computer-aid-design (CAD) tool (see Fig. 8-a). The next step (Fig. 8-b) is the development of surfaces containing these points. For this purpose, two assumptions were necessary. First, surface bound points (specifically NURBS surfaces) were developed aiming to numerically smooth the vertices of all lines of the webs. NURBS surfaces were developed with GID and then exported as an IGES part to ABAQUS-CAE. Second, perfect vertical and horizontal straight surfaces were assumed for the flanges and transversal stiffeners of the girders.
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The tf/tw ratio of the tested girders (tf/tw ¼ 5) was judged to be sufficiently high for considering null relative deformation amid points of the flanges. Once the plates were exported into ABAQUS-CAE, this preprocessor was used for assembling all plates as a whole (Fig. 8-c). An advancing-front algorithm using quad-dominant elements was used for meshing the assemblies. The shell-element based mesh fitted to the precise shape imperfection of the girders was then used for each nonlinear analysis including both material and geometric effects. It is noteworthy that in this particular case a very dense mesh was necessary to model these prototypes (5 mm S4 elements with triangular S3 elements when needed). 4.2.2. Eigenmode-based geometries Imperfections consisting of multiple superimposed buckling modes are usually introduced in plate girder simulations. Initially straight plates are assembled and then perturbed with an eigenvalue analysis. This procedure is quite advantageous, since there is no need of modelling intricate geometries. The method requires the model definitions for the eigenvalue prediction analysis and the nonlinear procedure to be identical. The mesh may be perturbed by a given eigenmode or different superimposed eigenmodes scaled so that the largest perturbation is a known value w. Eigenvalue analyses give guidance on both the elastic critical buckling load of the system and the potential critical shape of each mode. The first step consists in obtaining four eigenmodes for each girder by means of an eigenvalue extraction. Table 2 shows useful information about critical loads and shapes of the girders in study after performing an eigenvalue extraction in girders 1VPL2500, 1VPL750 and 1VPL450. For each case, top and frontal views of the perturbed specimens are displayed. Furthermore, numerical values of the elastic critical loads are included in the table.
Focusing on specimen 1VPL2500, it is observed that eigenmode M1 leads to an initially C-shaped web vertically below the patch load, whereas eigenmode M2 leads to an initially S-shaped web. These eigenmodes are deemed as being symmetric (S). It is also observed that for eigenmodes M3 and M4, the cross-section vertically below the patch load presents null deformation. These eigenmodes are referred to as antisymmetric (AS) and would hardly lead to the lowest resistance as required in EN1993-1-5Annex C [5]. Eigenvalue extraction performed on 1VPL750 leads to slightly different results. Two S and one AS eigenmodes are observed (M1, M2 and M3, respectively), whereas the fourth shape is an eigenmode related to shear buckling in the adjacent panels (SB). It is noteworthy that eigenmode 1 leads to an initially C-shaped web in the loaded cross-section, whereas eigenmode 2 leads to an initially S-shaped web. On 1VPL450, only one S C-shaped wave in the loaded panel was found in the first eigenmode. The following eigenmodes (M2, M3, and M4) are related to shear buckling in adjacent panels. Predictably, these geometries do not lead to the lowest resistance as required by EN1993-1-5-Annex C [5]. 4.2.3. Equivalent geometric imperfections EN1993-1-5-Annex C [5] allows modelling plated structures from equivalent geometric imperfections. This procedure may be advantageous when the structures to be modelled are idealized with one-dimensional finite elements. Global bow imperfections can be easily represented by mathematical functions (e.g. sinusoidal laws). When local buckling is prone to occur, however, 3D shell-based idealizations are needed. EN1993-1-5Annex C [5] suggests a combination of equivalent imperfections involving local and global imperfections of the panels. For this particular study, three types of geometric imperfections are included in the finite element analysis following the EN1993-1-5
Table 2 Numerical data from eigenvalue analyses IVPL2500
IVPL750
IVPL450
M1-161,84 kN (S)
M1-180,72 kN (S)
M1-325,04 kN (S)
M2-239,16 kN (S)
M2-271,40 kN (S)
M2-436,28 kN (SB)
M3-340,86 kN (AS)
M3-432,48 kN (AS)
M3-436,31 kN (SB)
M4-353,46 kN (AS)
M4-436,13 kN (SB)
M4-449,29 kN (SB)
Critical loads (kN) and critical shapes Mk.
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5. Results
500 450 400 Ultimate load capacity (kN)
procedure described in Section 2.2. The girders were assembled in ABAQUS-CAE as a whole and then an advancing-front algorithm using quad-dominant elements was used for meshing the assemblies. Even though these imperfections shapes are thought to be a reasonable approximation of the actual imperfections that occur in typical welded girders, it is necessary to assume certain geometrical incompatibilities. First, an initial constant twist of the flanges leads to an odd geometrical inconsistency. The transverse stiffeners happen to be trapezoidally -shaped (Fig. 3-c). Furthermore, the webs happen to be C-shaped in all cases. The actual shapes of the girders were found as being either C-shaped or S-shaped. This designer-assumed condition may influence the response of the girders. Third, the orientation of transversal stiffeners located along the swept length of the girders ought to follow (or not) a certain assumed pattern in the modelling. This pattern (or its absence) is arbitrary and designer-assumed.
397
350 300 250 200 150 Experimental results ABAQUS-Precise shape
100
ABAQUS-Equivalent Imperfection ABAQUS-1st Eigenmode
The influence of the initial conditions on the general response of steel plate girders when subjected to patch loading was tackled within three separated sections. Firstly, an appraisal of the influence of the shape of the initial geometry is presented (structural imperfections were included when needed). In this case, the calculations were performed on girders similar to those depicted in Table 1. Secondly, simulations were separately performed on models with and without including structural imperfections. Thus, the influence of the latter condition was also assessed. Thirdly, the influence of the maximum amplitude of the geometric shape was assessed by means of a large parametric study varying significant parameters, among others, the web slenderness. In this particular case, a refined analysis including both geometrical and structural imperfections was deployed on each model of the study. 5.1. Influence of the shape of the initial imperfection The first appraisal is extracted from results concerning ultimate load capacity of the experimentally and numerically tested girders. For this purpose, the results are graphically presented in Fig. 9 as a function of the distance between transverse stiffeners ‘‘a’’. Generally, the numerical results are in good agreement with the experimentally obtained values. Nevertheless, the geometries which are based upon some eigenmodes lead to undesired overestimations, i.e., these modes do not reproduce the required lowest resistance. Matter-of-fact, it is observed in Table 2 that these modes do not show any deformation in the directly loaded panel (or seem related to other instability modes different from patch loading). Table 3 numerically displays the ratio between experimental and FE-obtained results for each case. Precise shape, equivalent imperfection and the S eigenmodes lead to results on ultimate load capacity thought to be satisfactory. Perfect match between both results is observed for the 1st eigenvalue. It is finally confirmed that values corresponding to AS and shearbuckling like modes (generally, the 3rd eigenmode) do not yield the lowest resistance. The general response of the girders is also studied by using load-deflection plots. The first analysed prototype was 1VPL2500 (a ¼ 2500 mm). Fig. 10 shows differently obtained responses when varying the initial shape. Continuous lines correspond to experimental results whereas dashed lines to numerical ones. The response seems to be appropriately captured in all cases save for 3rd eigenmode (which confirms prior results). Linear branches
50
ABAQUS-2nd Eigenmode ABAQUS-3rd Eigenmode
0 0
500
1000 1500 Distance "a" (mm)
2000
2500
Fig. 9. Ultimate load capacity of the girder for different initial conditions.
as well as peak zones are similar. The precise shape, the 1st and the 2nd eigenmodes accurately reproduce the post-peak branch of the plots. The equivalent imperfection shows a slightly different response than other cases. Secondly, specimen 1VPL750 was studied (a ¼ 750 mm). Fig. 11 shows similar plots. Noticeably, linear branches are appropriately captured in all cases. Post-buckling behaviour is however, less satisfactory. Only refined analyses performed by introducing either the 1st or the 2nd eigenvalues as initial imperfection lead to the same response observed experimentally. Precise shape gives quite accurate results, but numerical difficulties were encountered throughout the analysis of such specimen. Other reproductions such as equivalent geometric imperfections give satisfactory results regarding ultimate load capacity, but do not appropriately capture the observed post-buckling branch. 3rd eigenmode yields to overestimated results and should not be taken as a suitable initial geometry. Clearly, it does not yield to the lowest resistance. It is worth emphasising that even if 1st and 2nd eigenmodes lead to very similar results regarding ultimate load capacity, results concerning elastic critical loads substantially differ. It would seem that both initially C-Shaped and S-shaped webs yield to similar results after buckling. This fact might be due to a widely studied phenomenon in instability of thin-walled structures referred to as secondary bifurcation [30]. Finally, the prototype 1VPL450 was analysed (a ¼ 450 mm). Fig. 12 shows the obtained results. Expectedly, linear branches are well-captured in all cases. The experimental curve presents a certain loss of linearity for an approximate value of 270 kN which is also captured by the 1st eigenvalue-based model. The peak load as well as the post-peak branch are fairly adjusted over this numerical reproduction. Equivalent geometric imperfections do not reproduce the response but rather, underestimate the girder response. Unfortunately, the precise shape geometry also yielded to numerical difficulties due to modelling. As previously explained, the second eigenmode is related to other instability phenomena such as shear buckling of the adjacent panels. As a result, this eigenmode does not yield the desired lowest resistance.
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Table 3 Experimental vs. numerical results Fu,
ABAQUS/Fu, experimental
Girder
Precise shape
Equivalent imperfection
1st eigenmode
2nd eigenmode
3rd eigenmode
1VPL2500 1VPL750 1VPL450
0.98 1.03 0.94
0.96 1.01 0.89
1.00 1.00 1.00
0.97 0.95 1.10
1.44 1.40 1.10
250
250
200
200 Load (kN)
Load (kN)
Influence of the initial shape imperfections.
150 100 50
100 50
0
0 0
5
10 15 20 Vertical displacement (mm)
25
250
400
200
300
Load (kN)
Load (kN)
150
150 100 50
0
5
10 15 20 Vertical displacement (mm)
25
0
5
10 15 20 Vertical displacement (mm)
25
200 100 0
0 0
5
10 15 20 Vertical displacement (mm)
25
300
300
250
250 Load (kN)
Load (kN)
Fig. 10. Load-deflection curve (1VPL2500). Influence of the initial shape imperfection. (a) Precise shape; (b) equivalent imperfections; (c) 1st and 2nd eigenmodes; and (d) 3rd eigenmode.
200 150 100 50
150 100 50
0
0 0
10 20 Vertical displacement (mm)
30
0
10 20 Vertical displacement (mm)
30
0
10 20 Vertical displacement (mm)
30
400
300 Load (kN)
250 Load (kN)
200
200 150 100 50
300 200 100 0
0 0
10 15 20 25 Vertical displacement (mm)
30
Fig. 11. Load-deflection curve (1VPL750). Influence of the initial shape imperfection. (a) Precise shape; (b) equivalent imperfections; (c) 1st and 2nd eigenmodes; and (d) 3rd eigenmode.
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500
500
400
400 Load (kN)
Load (kN)
´n et al. / Thin-Walled Structures 47 (2009) 391–402 R. Chaco
300 200
300 200 100
100 0
0 0
10 20 Vertical displacement (mm)
30
500
500
400
400 Load (kN)
Load (kN)
399
300 200 100
0
10 20 Vertical displacement (mm)
30
0
10 20 Vertical displacement (mm)
30
300 200 100
0
0 0
20 10 30 Vertical displacement (mm)
40
Fig. 12. Load-deflection curve (1VPL450). Influence of the initial shape imperfection. (a) Precise shape; (b) equivalent imperfections; (c) 1st eigenmode; and (d) 2nd eigenmode.
5.2. Influence of the structural imperfections Furthermore, an appraisal of the influence of initial structural imperfections is presented. Fig. 13 shows the structural response of three cases of study. The left side of the plot (Fig. 13-a) is sketched from numerical data in which no residual stresses are considered. The right side (Fig. 13-b) displays the same responses of the same models but in this case, a typical pattern of residual stresses is included within the plates as structural imperfection. These analyses were performed on girders in which the 1st eigenmode was included as initial geometry. The geometry was scaled following the EN1993-1-5-Annex C [5], i.e., 80% FT was adopted as the largest perturbation w. Table 4 shows the numerical-to-experimental ratio on ultimate load capacity. Noticeably, a 6,66% overestimation of ultimate load is observed in case 1VPL2500. The two other cases present results judged to be satisfactory whether residual stresses are included or disregarded. It is, however, noticeable that the load-deflection plot is slightly rather different in 1VPL2500. The results show that the residual stresses do not play a decisive role on the ultimate load capacity. This remark was also pinpointed by other authors when using different initial residual stress patterns [13,14]. Further research on this topic would eventually clarify and/or confirm these statements.
5.3. Influence of the magnitude of the maximum amplitude The second designer-assumed feature in the analyses is the maximum amplitude of the web out-of-flatness (w). Numerical simulations presented in this study so far are focused on a reproduction of a half-scale experimental program. However, one could argue that the maximum out-of-flatness of the webs in plate girders due to welding may be dependent on the tf/tw ratio and/or the slenderness of the web panel (hw/tw). An appraisal of
this influence without considering these additional conditions was judged to be unfair. Consequently, the web thickness was also included as an additional parameter. For this purpose, a wide deterministic parametric study was developed. Numerical simulations were performed on girders in which the maximum amplitude of the buckling shape varies approximately from 0% to 250% of the maximum allowable outof-flatness according to fabrication tolerances. The mechanical properties of the girders involved in this parametric study were identical to the properties of the tested girders. Geometrically, the length and clear web depth were held constant and equal to the experimental prototypes. The web slenderness was, however, varied ranging from 71 to 166.66. The variation was performed via web thickness (ranging from 3 to 6 mm). The study Case 1 includes the stockiest web panel, whereas the study Case 4 includes the most slender one. Case 3 coincides with the experimental geometry (highlighted in Table 5). In this study, all the calculations were performed on an initial shape based upon the first mode previously obtained in the perturbation analysis. Initial structural imperfections were included by the simplified pattern of residual stresses depicted in Section 4.1. A total amount of 120 simulations were performed. Noticeably, the web thickness tw together with w were the principal parameters within the set of variation. Ultimate load capacity of the girders was obtained for each case. Four plots are sketched in Fig. 14. In the y-axis (vertical), the ratio Fu/Fu,80%, is displayed (ultimate load Fu of the girders standardised to the Fu,80%, in which 80% of the fabrication tolerances is suggested as maximum amplitude). In the x-axis (horizontal), the maximum amplitude of the shape imperfection is sketched as a percentage of the allowed fabrication tolerances. The Eq. (7) indicates the mathematical relationship governing these plots. Fu ¼ f ðw %FTÞ F u;w¼80%
(7)
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400
Table 5 Parametric study. Influence of the maximum amplitude
450 1VPL450
Case
tw (mm)
tf/tw
Web slenderness (hw/tw)
Aspect ratio (a/hw)
w-f (%FT)
1 2 3 4
6 5 4 3
3.33 4 5 6.67
71 100 125 166.66
5-1.25-0.9 5-1.25-0.9 5-1.25-0.9 5-1.25-0.9
[0,10,30y250] [0,10,30y250] [0,10,30y250] [0,10,30y250]
400 350
Load (kN)
300 1VPL750 250 200 150 1VPL2500 100 50 0 0
5
10 15 20 Vertical displacement (mm)
25
30
450 1VPL450 400 350 300 Load (kN)
1VPL750 250 200 150 1VPL2500 100 50 0 0
5
10 15 20 Vertical displacement (mm)
25
30
Fig. 13. Load-deflection curves. Influence of the initial structural imperfections. (a) Residual stresses not taken into consideration and (b) residual stresses included.
Table 4 Influence of initial structural imperfections Residual stresses Girder
Not taken into consideration [(Fu,num/Fu,exp)1,0] 100
Taken into consideration [(Fu,num/Fu,exp)1,0] 100
1VPL2500 1VPL750 1VPL450
6.66 0.22 0.29
0.00 0.40 0.20
Three cases of study. Experimental vs. numerical results.
Each plot corresponds to a given value of web slenderness. Within each plot, three different aspect ratios are sketched. Each continuous line represents a variation of w, as a percentage of FT, i.e., the designer-assumed amplitudes. A vertical line indicating the suggested minimum value in EN1993-1-5-Annex C [5] is also displayed in four plots. Noticeably, two different trends are observed within all curves. Stocky girders (hw/tw ¼ 71 and 100) present a decreasing tendency in the plot, whereas slender girders (hw/tw ¼ 125 and 166,66) present a quite stable tendency. For all cases, very small magnitudes of the initial imperfection (0% FT) lead to undesired erroneous overestimations in ultimate load capacity. This fact might be due to pathological numerical difficulties and potential bifurcation of equilibrium. This fact should warn about the potential overestimations, the designer might commit by neglecting this parameter. Focusing on results obtained in stocky girders, it is observed that for most cases, an increment in the amplitude values leads to a decrement in the resistance of the girders. In general, this trend is more noticeable for high aspect ratios of the panel (a/hw) than for closely spaced transversal stiffeners, in which this sensitivity is insignificant. It is then found that the magnitude should be based upon conservative percentage of the maximum geometric fabrication tolerances. For the recommended values of 80% FT, the results of ultimate load capacity obtained are considered structurally sound. Oddly enough, slender girders do not present such dependency. Provided that the initial magnitude is greater than a certain value, the tendency can be considered as being horizontal. It is likely that failure is instability-dependent in such cases and the potential bifurcation of equilibrium occurs even for small values of w. For these cases, it is also found that the magnitude can be based upon the recommended percentage of the maximum geometric fabrication tolerances. 80% FT leads to sound results of ultimate load capacity of slender plate girders. Table 6 shows, for each case, a numerical appraisal of such influence. One can define the maximum difference existing between extreme values of Fu (DFu ¼ Fu,10%Fu,250%) as the maximum realistic loss of load-carrying capacity as long as the largest perturbation w is increased (results related to Fu,0% are discarded due to the encountered numerical difficulties and values of w4250% FT are considered unlikely). DFu is pointed out in Fig. 14. Results show that girders assembled with stocky web panels present a greater sensitivity to this magnitude than girders assembled with slender web panels, in which the dependency seems to be negligible. This fact is particularly noticeably in girders with high a/hw ratios. Furthermore, it shows that for slender panels presenting small aspect ratios (a/hw), an increment of w may lead to a slight increment of Fu. This latter increment is rather small and should not be considered as significant.
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hw /tw=100 1.40
1.30
1.30 80%
1.10 1.00
ΔFu
Fu / Fu 80%
Fu / Fu 80%
hw /tw=71 1.40 1.20
401
80%
1.20
a/hw=5.0
1.10 1.00
ΔFu
0.90
0.90 0.80
a/hw=1.25 a/hw=0.9
0.80 50
0
100 150 200 w = f(%FT)
250
300
0
50
100 150 200 w = f(%FT)
250
300
y x hw /tw =166,66
80%
ΔFu
0
50
100 150 200 w = f(%FT)
250
Fu / Fu (80%)
Fu /Fu (80%)
hw /tw=125 1.40 1.30 1.20 1.10 1.00 0.90 0.80
300
1.40 1.30 1.20 1.10 1.00 0.90 0.80
80% a/hw=5.0 a/hw=1.25 a/hw=0.9
ΔFu
0
50
100 150 w = f(%FT)
200
250
Fig. 14. Influence of the maximum amplitude of the initial shape imperfection.
Table 6 Parametric study Case
Web slenderness (hw/tw)
Aspect ratio (a/hw)
DFu,% (Fu,10%–Fu,250%)
Maximum difference
1
71
5.00 1.25 0.90
11.87 4.59 0.95
2
100
5.00 1.25 0.90
11.85 6.77 2.72
3
125
5.00 1.25 0.90
4.13 2.39 2.36
4
166.66
5.00 1.25 0.90
2.34 1.39 0.00
Influence of the maximum amplitude.
6. Conclusions The current European guidelines EN1993-1-Annex C [5] allow designers using FE-Analyses as reliable tools on the calculation of plated structures. The guides overtly request the usage of initial designer-assumed conditions which are necessary for the development of appropriate simulations. The guidelines suggest two potential initial conditions for any analysis of plated structures; a refined analysis which includes both geometrical and structural imperfections or an equivalent geometric shape that includes both initial conditions. The numerical modelling of both possibilities differs considerably from one to another. These conditions were assessed for the particular case of study by using an experimental basis of three steel plate girders in which initial shape imperfection was known in advance.
The drawn conclusions may give guidance to practicing engineers and researchers alike on the designer-assumed conditions and the potential influence on ultimate limit states verifications for the particular case of steel plate girders subjected to patch loading. First, it has been shown that the initial shape of the girder does not play a primary role on the ultimate load capacity of the girders as far as realistic symmetric (S) shapes are introduced directly below the patch load. It is worth pointing out that the experimentally obtained response of the girders is perfectly reproduced when 1st eigenmode-based geometries are used as initial geometrical conditions. The introduction of equivalent geometric imperfections leads to satisfactory and conservative results concerning ultimate load capacity. However, the modelling is judged rather intricate and several decisions involving engineering judgement are needed. This procedure may be conservatively used if more refined analysis is not feasible. Due to simplicity and accuracy, the usage of eigenmode-based geometries is highly recommended. Second, the structural imperfections are introduced in the girders by means of a typical residual stress pattern found in literature. It has been noticed that if these patterns are included in the analysis, the results are more refined. The difference between considering or discarding these imperfections does not play a decisive role on the response of the girders. Other studies have also pointed out similar remarks. It is, however, recommended to address further research concerning this particular topic by means of vaster parametric studies including a set of variation of different patterns. Third, the influence of the maximum amplitude magnitude w suggested in EN1993-1-5-Annex C [5] has been assessed. The maximum allowable fabrication tolerances are used as a comparative magnitude. It has been shown that whether these amplitudes are kept within reasonable limits related to fabrication tolerances, the influence of the magnitude is negligible. It has been also observed that in stocky girders, an increment in the amplitude values leads to a decrement in the resistance of the girders, whereas in slender girder this tendency is not noticeable. In all cases, for the specific recommended values of
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80% of the allowed fabrication tolerances, results of ultimate load capacity obtained are considered structurally sound.
Acknowledgements The authors wish to gratefully acknowledge the financial support provided by Spanish Ministry of Science and Education, as a part of the Research Project BIA 2004-04673 and the grant provided to the first author. Likewise, the authors are grateful to Arcelor and Tadarsa for supplying and manufacturing the girders and to Abrox for measuring the initial imperfections of the girders. References [1] Veljkovic M, Johansson B. Design of hybrid steel girders. J Constr Steel Res 2004;60(3–5):535–47. [2] Dubas P, Gehri E. Behaviour and design of steel plated structures. ECCS-Tech Comm 8 1986. [3] Bernard ES, Coleman R, Bridge RQ. Measurement and assessment of geometric imperfections in thin-walled structures. Thin-Walled Struct 1999;33:103–26. [4] Singer J, Abramovich H. The development of shell imperfection measurement techniques. Thin-Walled Struct 1995;23:379–98. [5] EN 1993-1-5 Eurocode 3: design of steel structures, Part 1.5 Plated structural elements. 2005. [6] Bergfelt A. Patch loading on a slender web-influence of horizontal and vertical web stiffeners on the load carrying capacity. Chalmers University of Technology, S79:1. 1979. Go¨teborg, Sweden (in Swedish). [7] Dubas P, Tschamper H. Stabilite´ des aˆmes soumises a` une charge concentre´e et a` une flexion globale. Constr Met 1990;2 [in French]. [8] Seitz M, Kulhman U. Longitudinally Stiffened Girder Webs Subjected To Patch Loading. Steelbridge 2004. Millau, June p. 23–25. [9] Martı´nez J, Graciano C, Casanova E. Imperfection sensitivity of plate girder webs under patch loading. In: Proceedings of the Third International Conference on Structural Engineering, Mechanics and Computation (SEMC). September 2007. Cape Town, South Africa, 1–8. [10] Chacon R, Mirambell E, Real E. Influence of initial imperfections on the resistance of hybrid steel plate girders subjected to concentrated loads. In: Proceedings of the 6th International Conference Steel and Aluminium Structures ICSAS’07. July 2007. Oxford, U.K. p. 736–743.
[11] Davaine L. Formulations de la re´sistance au lancement d’une aˆme me´tallique de pont raidie longitudinalement. Ph.D. Thesis. 2005. INSA, Rennes, France. (in French). [12] Granath P, Lagerqvist O. Behaviour of girder webs subjected to patch loading. J Constr Steel Res 1999;50(1):49–69. [13] Granath P. Behavior of slender plate girders subjected to patch loading. J Constr Steel Res 1997;42(1):1–19. [14] Gozzi J. Patch loading resistance of plated girders. Doctoral thesis 2007:30. Lulea˚ University of Technology, ISRN: LTU-DT-07/30–SE. [15] Clarin M. Plate buckling resistance—patch loading of longitudinally stiffened webs and local buckling. Doctoral thesis 2007:31. Lulea˚ University of Technology, ISRN: LTU-DT-07/31–SE. [16] Chacon R, Mirambell E, Real E. Algunas consideraciones sobre la resistencia de las vigas armadas hı´bridas sometidas a cargas concentradas. Hormigo´n y Acero 2007;3rd trimester:43–57 [in Spanish]. [17] ABAQUS. 6.5 version manuals. USA: Abaqus Inc.; 2005. [18] Roberts TM, Rockey KC. A mechanism solution for predicting the collapse loads of slender plate girders when subjected to in-plane patch loading. Proc Inst Civ Eng 1979;67(2):155–75. [19] Elgaaly M. Web design under compressive edge loads. Eng J 1983;20(4): 153–71. [20] Lagerqvist O, Johansson B. Resistance of I-girders to concentrated loads. J Constr Steel Res 1996;39(2):87–119. [21] Graciano C, Johansson B. Resistance of longitudinally stiffened I-girders subjected to concentrated loads. J Constr Steel Res 2003;59(5): 561–86. [22] EN1993-1-1. Eurocode 3. Design of steel structures, Part 1.1 General rules. 2005. [23] Chacon R, Mirambell E, Real E. Resistance of transversally stiffened hybrid steel plate girders subjected to concentrated loads. In: Proceedings SDSS. International Colloquium on Stability and Ductility of Steel Structures, Lisbon; 2006. [24] GiD (v. 7) Pre and post processor. CIMNE. 2006 Barcelona, Spain. [25] Riks E. An incremental approach to the solution of snapping and buckling problems. Int J Solids Struct 1979;15(7):529–51. [26] Barth K, White D. Finite element evaluation of pier moment-rotation characteristics in continuous-span steel I-girders. Eng Struct 1998;20(8): 761–78. [27] ECCS, Ultimate Limit State Calculations of Sway Frames with Rigid Joints (1st ed), ECCS-Committee 8-Stability, Technical Working Group 8.2-System, European Convention for Constructional Steelwork, 1984, 33, p. 20. [28] Barth K, Righman J, Freeman L. Assessment of AASHTO LFRD specifications for hybrid HPS 690W steel I-girders. J Bridge Eng 2007;12(3):380–8. [29] BSK 99. Boverkets handbook om stalkonstruktioner. Boverket; 1999 [in Swedish]. [30] Nakamura T, Uetani K. The secondary buckling and post-secondary-buckling behaviours of rectangular plates. Int J Mech Sci 1979;21(5):265–86.
Contents lists available at ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Influence of designer-assumed initial conditions on the numerical modelling of steel plate girders subjected to patch loading R. Chaco´n , E. Mirambell, E. Real Construction Engineering Department, Universitat Polite`cnica de Catalunya, C/Jordi Girona 1-3, 08034 Barcelona, Spain
a r t i c l e in fo
abstract
Article history: Received 15 October 2007 Received in revised form 12 June 2008 Accepted 4 September 2008 Available online 19 October 2008
European design rules EN1993-1-5 allows the usage of FE-Analyses as reliable tools in the verification of limit states of plated structures. Throughout the modelling of these structures, certain initial conditions must be assumed. Consequently, engineering judgment is needed to some degree. This paper presents an assessment of the influence of these conditions on the ultimate load capacity of steel plate girders when subjected to patch loading. The basis of the research is a thorough comparison between theory and design rules by means of experimental and numerical results. The results obtained prove usefulness for practical and academic purposes alike. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Patch loading Initial geometric imperfections Residual stresses Ultimate limit states
1. Introduction Thin-walled plate girders are used when it is necessary for a structural element to support high loads, above which a normal rolled section would either not be structurally viable or would become uneconomical. A plate girder may be designed as homogenous (fyf ¼ fyw) or as hybrid (fyf4fyw). This latter alternative is popular as the girder yields a greater flexural capacity at a lower cost compared to a homogeneous girder [1]. In both hybrid and homogeneous configurations, the welding process between web and flanges adds considerable imperfections to the initially straight plates. Additionally, if transversal and/or longitudinal stiffeners are systematically welded to the girders, the magnitudes and shapes of these imperfections may randomly change. These imperfections may either be geometrical or structural. Geometrical imperfections are often characterised by initially ‘‘imperfect’’ plates following a certain shape, whereas structural imperfections are characterised by a residual stress pattern. Historically, it has been shown that thin-walled structures are ‘‘imperfection sensitive’’, i.e., critical buckling/ultimate loads depend upon the actual initial imperfection. An out-of-flatness of the plate with small initial amplitude exerts an influence on the basic equilibrium path. If the initial shape is related to instability modes of the panels, the equilibrium path follows this shape and merges into the post-critical path [2]. The random nature of the
Corresponding author.
E-mail address: [email protected] (R. Chaco´n). 0263-8231/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2008.09.001
shapes and magnitudes of the initial imperfections, though, results in difficulties. Fortunately, it has been shown that if these imperfections are kept within allowable limits (usually related to fabrication tolerances), these difficulties may be controlled. Over the years, the field of measurement techniques has become a high-tech industry. Different measurement techniques have emerged [3,4]. Indeed, a vast amount of data can nowadays be measured, stored and processed. Therefore, truly realistic geometrical maps of the plates can be drawn from the obtained results. This enhancement is accompanied with the fact that precision tolerances are nowadays tighter and more stringent. As a result, the whole industry gives a quite reliable performance. Therefore, the randomness of the phenomena can be at least, kept on the aforementioned allowable limits. Presently, European design rules EN1993-1-5 [5] (Annex C) include the possibility of using FE-Analyses as reliable tools in the verification of limit states. As a result, plated steel structures are usually verified by numerical simulations performed on available reliable codes. The requested verifications are fairly generic for any type of loading and/or geometry. In order to present a systematic procedure for any type of modelling of plated structures, certain initial conditions must be assumed. Moreover, in recent years, pre- and post-processing techniques have become increasingly user-friendly, which facilitates the former expensive and time-consuming modelling. In this particular case, the focal point is on the simulation of the patch loading phenomena on plate girders. This simulation requires a definition of a geometrically and structurally imperfect three-dimensional (3D) model as well as a plastic definition of the constitutive equation of the steel. For the sake of capturing the
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E
Nomenclature
Young modulus slenderness gM1 partial factor % FT percentage of maximum allowable fabrication tolerance w maximum amplitude of the imperfection dog initial sweep of the girder dow initial out of flatness of the web y initial twist of the flanges FRd design resistance to transverse forces Fu predicted resistance to transverse forces according to numerical results Fu,(w ¼ 80%) predicted resistance according to numerical results (standardised) DFu maximum difference between extreme values of Fu
lF
hw a Ss tf tw ts fyf fyw fys ly
wF Fy Fcr
clear web depth between flanges width of web panel (distance between transverse stiffeners) length of stiff bearing thickness of the flanges thickness of the web thickness of the transverse stiffeners flange yield strength web yield strength stiffener yield strength effective loaded length reduction factor plastic resistance critical buckling load
local buckling phenomena, the model must be based upon shell elements geometries. There exists a considerable amount of literature on the influence of initial geometrical imperfections on the resistance of girders subjected to patch loading [6–10]. Some authors [11], however, consider that the correlation between theory, imperfection measurements and ultimate limit states verification in European standards is judged to be still incomplete. Furthermore, as far as known by the authors, scarce studies include the effect of the structural imperfections (secondorder imperfection sensitivity) of welded girders on their ultimate load capacities when subjected to patch loading [12–15]. Preliminary, the drawn conclusions point out that these structural imperfections (idealized as residual stresses) do not play a decisive role. The present work is aimed at assessing the influence of the designer-assumed initial conditions on the general response of steel plate girders when subjected to patch loading (Fig. 1) focusing on the current design rules implemented in EN1993-1-5 [5]. On one hand, an experimental program on four steel plate girders subjected to patch loading is presented. Before the tests, the initial geometry of the web and flange panels in all four girders was obtained by means of a 3D co-ordinate measuring device [16]. On the other hand, the numerical model implemented in the multi-purpose code ABAQUS 6.5 [17] which includes geometrical and material nonlinearities, was used as a simulation tool in order to develop a parametric study which allowed presenting a vaster assessment of this influence. The ultimate load capacity of the steel plate girders subjected to patch loading was numerically obtained by means of three different methods. The first method is by the introduction of realistic precise shapes of initial imperfections previously measured for each girder together with a typical idealized residual
Web yielding
Stocky panel
Web buckling
Slender panel
Fig. 1. Effect of a concentrated load on a girder (a) web yielding and (b) web buckling.
stress pattern. The second method is by the introduction of initial imperfections based upon different eigenmodes (together with structural imperfections, i.e., an idealized residual stress pattern). The third, by using the equivalent initial imperfections proposed in EN1993-1-5 [5], which includes both geometric and structural imperfections. The results obtained show the effect of the amplitude and the shape of the geometrical imperfection as well as the effect of the structural imperfections on the resistance of plate girders when subjected to patch loading. The drawn conclusions may give guidance to practicing engineers and researchers on the designerassumed conditions and the potential influence on ultimate limit states verifications for the particular case of study.
2. Design provisions 2.1. EN1993-1-5. resistance to transverse forces Patch loading phenomena has been widely analysed since the early sixties. Several failure mechanisms and critical buckling loads have been proposed throughout the last decades for the case of stiffened and unstiffened panels subjected to concentrated loads [18–21]. In early ages, the normal approach in EN1993-1-1 [22] used upper and lower limits for the resistance, upper limit based on yielding and lower limit based on instability. The results from tests had, however, historically shown that there is not any clear distinction between these two bounds. Instead, a blurred transition between yielding and behaviour influenced by buckling is observed, similarly to other instability-based problems. Presently, EN1993-1-5 [5] defines this gradual transition by means of the w-reduction factor for both stiffened and unstiffened webs. Mechanical models based on first-order limit analysis are the most outstanding approaches for defining the plastic resistance for patch loading [18,20]. In Fig. 2, one can observe the four-hinge model from which ultimate load resistance is typically obtained. Other approaches include semi-circular yield lines in the web which have been experimentally observed and thus, theoretically added to the hinge-based resistances. On the other hand, the limit related to instability has generally been obtained by numerical simulations using the classical elastic critical theory in a locally loaded simply supported plate. The general equation currently included in EN1993-1-5 [5] is based upon a plastic resistance derived from these models (Fig. 2) and partially reduced by means of wF (Eq. (1)). In this equation, ‘‘ly’’ is the yield-prone effectively loaded length. This length
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Ss+2tf
Sy/2
a
bf
Mo
Mi
tf
Mo
bf Mi
fywtw
Y δow
Mo
tf
Mi
X
hw
Sy/2
393
10εtw tw Ζ
Z
1y Fig. 2. Model of resistance [19].
θ physically represents the distance between outer hinges and is calculated from geometrical and mechanical properties of the girders by using (2) and (3). wF takes into account instability by means of (4) and (5) and can be obtained with Eq. (6). Buckling coefficient kf varies whether the web panels are longitudinally stiffened or unstiffened. F Rd ¼
f yw wF ly t w
gM1
ly ¼ Ss þ 2t f
m2 ¼ 0:02
lF ¼
2 hw if lX0:5 otherwise m2 ¼ 0 tf
sffiffiffiffiffiffi Fy F cr
F cr ¼ 0:9kf E
wF ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! f yf bf 1þ þ m2 f yw t w
0:5
lF
(1)
(2)
(3)
(4) t 3w hw
p1:0
(5)
(6)
2.2. EN1993-1-5-Annex C. FE-analyses Presently, European design rules include the possibility of using FE-Analyses as reliable tools in the calculation of plated structures. The modelling may be either based upon a refined analysis by including geometrical eigenmode-based imperfections as well as residual stresses or based upon equivalent geometric imperfections which should include both effects. In fact, there is a decision to be taken in this latter definition and hence, engineering judge is needed to some degree. Firstly, the refined analysis is discussed. This analysis can be based upon critical eigenmodes of the structure conveniently scaled a maximum amplitude w. EN1993-1-5-Annex C [5] rules recommend a value for w at least of 80% of the maximum allowed fabrication tolerances (80% FT), provided that the chosen deviations lead to the lowest resistance for each case. This percentage may vary from one National Annex to another. In the particular case of patch loading, the initial out-of-flatness of the web in the directly loaded panel should be the most significant deviation to be considered. EN1993-1-5-Annex C [5], however, suggests this percentage generically for all cases in design of plated structures. Presently, fabrication tolerances limit this out-of-flatness to the lowest value between two magnitudes:
tw, thickness of the web hw/100, being hw the clear web depth between flanges
X Y
Y
δog L
Z
Fig. 3. Introduction of equivalent geometric imperfections. (a) Initial out-offlatness; (b) initial sweep of the girder; and (c) initial twist of the compression flange.
The second possibility is defined by the equivalent geometric imperfections, which should be based upon the following considerations: (a) An initial out-of-flatness of the web panels dow (Fig. 3-a). In accordance with EN1993-1-5-Annex C [5], the maximum initial out-of flatness of the web panel is the minimum value between (a/200;hw/200). (b) An initial sweep of the girder within bearings dog (Fig. 3-b). This represents a lateral deviation in the longitudinal direction. The minimum value to be introduced following EN1993-1-1 [22] is L/100 for girders using buckling curve ‘‘d’’. (c) An initial twisting y of the top flange and/or longitudinal stiffening (Fig. 3-c). The magnitude of this twisting must be at least 1/50 radians. These equivalent imperfections may be substituted by appropriate forces acting on the member. In this paper, the equivalent imperfections are treated as geometrical and are not substituted by any force.
3. Experimental program 3.1. Description An experimental program over four steel girders subjected to concentrated loads was carried out at the Structural Technology Laboratory of the School of Civil Engineering of Barcelona, UPC [23]. The length of the span in all four girders was 2500 mm. The web height was 500 mm and a 150 200 mm2 patch was used for introducing the load. Three different steel plates were used for the fabrication of these girders, the first plate of 4 mm thickness for the web (fyw ¼ 325 N/mm2), the second of 20 mm thickness for the flanges (bf ¼ 200 mm and fyf ¼ 454 N/mm2) and the third one of 20 mm for the stiffeners (two-sided transverse stiffener, bs ¼ 200 mm, fys ¼ 310 N/mm2). The mechanical properties of all plates were obtained by means of tensile coupon tests. 3.2. Test configuration The tests were carried out under displacement control using a MTS hydraulic jack. Loads, displacements and strains at key points
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were measured during the development of the tests. Uniaxial gauges and rosettes and displacement transducers were placed for measuring strains and vertical and horizontal deflections. Other outputs were obtained from the jack as well as from the load-cell. A MGC-Plus data acquisition system was used in order to record readings following a one-hertz frequency. In Fig. 4, a general view of the test configuration is displayed. 3.3. Measuring the initial shapes of the girders Before the tests, initial shapes of the web and flange panels in all four girders were obtained by means of a 3D co-ordinate measuring device. The 3D data format is commonly referred to as ‘‘point cloud’’, which leads to a series of co-ordinates {xi,yi,zi} of each point. The data allow the reproduction of the original shape for both the web and flanges of all steel plate girders. The measurement principle is to obtain the position of infrared LEDs by means of cameras through triangulations using a portable device as shown in Fig. 5. 3D data from all web and flange panels were obtained following a hundred millimetres square grid previously drawn in all four girders. Two singularities were observed after plotting the results. Firstly, random shapes of the initial imperfections of the webs were noticed (Fig. 6). Secondly, the initial imperfection of the flanges was judged to be negligible. A considerable twisting relative to a vertical reference was, however, noticed in specimen 1VPL1500.
3.4. Test results Geometrical data and results obtained are given in Table 1. Notice that distance between transversal stiffeners in the directly loaded panel (distance ‘‘a’’) is varied from one specimen to another. As long as the distance ‘‘a’’ is decreased, ultimate load capacity of the girders is increased, except for specimen 1VPL1500, in which results were certainly influenced by the excessive twisting of the top flange occurring throughout the test. This specimen was discarded for subsequent studies, since it did not reproduce the desired phenomena.
4. Numerical modelling The pre-processors ABAQUS-CAE [17] and GID [24], which are capable of reproducing intricate geometries, were used to assemble suitable meshes for developing the geometrical models. Several nonlinear analyses were performed on the steel plate girders using quad-dominant S4 shell elements for web, flanges and stiffeners. A convergence analysis led to quite dense meshes in all cases (25 mm square meshes for the coarsest). Both geometric and material nonlinear effects were considered. The adopted constitutive equation was the one observed in tensile coupon tests. The nonlinear solution strategy used is the Modified Riks [25] algorithm implemented in ABAQUS. 4.1. Structural imperfections
Rigid frame
Hydraulic Jack Directly loaded panel
Hinge Patch load a
Protection
Bearing
Fig. 4. Test configuration.
Fig. 5. Measurement of initial imperfections.
Double I-shaped stocky beam Concrete slab
In accordance to EN1993-1-5-Annex C [5], structural imperfections should be defined in terms of residual stresses by a typical stress pattern from the fabrication process. Residual stress pattern is a parameter that defies generalisations [26]; each particular case might present different features. In this study, a typical idealized residual stress pattern for plate girders is assumed. Longitudinal residual stresses are incorporated into the girder models using the stress pattern idealization depicted in Fig. 7-a. The pattern follows the European Convention for Constructional Steelwork Recommendations (ECCS [27]), which reflect the two primary causes of longitudinal residual stress in welded I-girders: flame-cutting of the plates and longitudinal welding in the webto-flange juncture. Essentially, the residual stresses are equal to the yield stress of the plate within a small width at the heataffected zones; this is primarily a tensile stress. Then, a smaller self-equilibrating compressive stress is generated within the other regions of the plates. The modelling of such idealized pattern might be rather tough in systematic parametric studies. Fortunately, ECCS provides simplified equations for estimating the widths that are effectively stressed. These equations take into account flame-cutting, welding at junctures, geometrical properties of the added weld metal as well as the efficiency in the welding process. If these equations are faithfully followed, one result is that the predicted residual stresses are highly dependent upon the cross-section size. Within this study, the stress pattern is assumed to present a near-constant tensile stress within a rather wide zone of the borders of the flanges (0.18 fyf), a higher value of tensile stress in the web-to-flange juncture (0.33 fyf in the flange and 0.63 fyw in the web) and an additional compressive stress aimed to equilibrate the whole section (0.20 fyf in the flange and 0.15 fyw in the web). These values (Fig. 7-b) are considered structurally sound in other studies [28] and were systematically included as typical patterns in all girders of the present work. Considering that the initial stress state may not be in exact equilibrium, an initial step is included to allow ABAQUS to check for equilibrium and iterate, if necessary, to achieve it.
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1VPL2500
1VPL1500 500 Web-Depth (mm)
500 Web-Depth (mm)
395
400 300 200 100
400 300 200 100 0
0 0
1
2 3 4 Measured out-of-the plane imperfection (mm)
5
0
1
2 3 4 Measured out-of-the plane imperfection (mm)
5
1VPL750 1VPL450
Web-Depth (mm)
500
520
400 Web-Depth (mm)
300 200 100 0 -2 0 2 Measured out-of-the plane imperfection (mm)
-4
420
4
320 220 120 -3
20 -2 -1 -80 0 Measured out-of-the plane imperfection (mm)
1
Fig. 6. Measurements of the imperfections at mid-span section of each girder.
Table 1 Test data Specimen
a (mm)
a/hw
Fu,
1VPL2500 1VPL1500 1VPL750 1VPL450
2500 1500 750 450
5.00 3.00 1.50 0.90
217.23 190.00 251.80 426.00
test
(kN)
FRd (kN)
Vertical deflection (mm)
162.59 164.48 173.06 173.10
2.61 5.61 13.08 10.20
Fig. 8. Introduction of the precise shape of the girders. (a) Point cloud; (b) plate (nurbs surface); and (c) assembled model.
+0.18 fyf
+0.33 fyf +0.18 fyf
The residual stress profile is simpler than the one depicted above and lead to accurate results. 4.2. Initial geometrical imperfections
-0.20 fyf
-0.20 fyf +0.63 fyw
Three different initial ‘‘imperfect’’ geometries were modelled in ABAQUS.
-0.15 fyw
-0.20 fyf
+0.18 fyf
-0.20 fyf
+0.18 fyf +0.33 fyf
Fig. 7. Residual stresses. (a) Typical idealized pattern; and (b) assumed pattern.
Other studies which included residual stresses in the patch loading verifications [13–16] used the simplified pattern suggested by the Swedish design code for steel structures [29].
4.2.1. Precise shape initial imperfection The precise shape of the girders was introduced from 3D experimental data previously obtained. The data include global co-ordinates of points previously located on grids in the webs and flanges of the girders. The points are referenced to a global coordinate system. From these results, a graphical reproduction of the girders is sketched with a computer-aid-design (CAD) tool (see Fig. 8-a). The next step (Fig. 8-b) is the development of surfaces containing these points. For this purpose, two assumptions were necessary. First, surface bound points (specifically NURBS surfaces) were developed aiming to numerically smooth the vertices of all lines of the webs. NURBS surfaces were developed with GID and then exported as an IGES part to ABAQUS-CAE. Second, perfect vertical and horizontal straight surfaces were assumed for the flanges and transversal stiffeners of the girders.
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The tf/tw ratio of the tested girders (tf/tw ¼ 5) was judged to be sufficiently high for considering null relative deformation amid points of the flanges. Once the plates were exported into ABAQUS-CAE, this preprocessor was used for assembling all plates as a whole (Fig. 8-c). An advancing-front algorithm using quad-dominant elements was used for meshing the assemblies. The shell-element based mesh fitted to the precise shape imperfection of the girders was then used for each nonlinear analysis including both material and geometric effects. It is noteworthy that in this particular case a very dense mesh was necessary to model these prototypes (5 mm S4 elements with triangular S3 elements when needed). 4.2.2. Eigenmode-based geometries Imperfections consisting of multiple superimposed buckling modes are usually introduced in plate girder simulations. Initially straight plates are assembled and then perturbed with an eigenvalue analysis. This procedure is quite advantageous, since there is no need of modelling intricate geometries. The method requires the model definitions for the eigenvalue prediction analysis and the nonlinear procedure to be identical. The mesh may be perturbed by a given eigenmode or different superimposed eigenmodes scaled so that the largest perturbation is a known value w. Eigenvalue analyses give guidance on both the elastic critical buckling load of the system and the potential critical shape of each mode. The first step consists in obtaining four eigenmodes for each girder by means of an eigenvalue extraction. Table 2 shows useful information about critical loads and shapes of the girders in study after performing an eigenvalue extraction in girders 1VPL2500, 1VPL750 and 1VPL450. For each case, top and frontal views of the perturbed specimens are displayed. Furthermore, numerical values of the elastic critical loads are included in the table.
Focusing on specimen 1VPL2500, it is observed that eigenmode M1 leads to an initially C-shaped web vertically below the patch load, whereas eigenmode M2 leads to an initially S-shaped web. These eigenmodes are deemed as being symmetric (S). It is also observed that for eigenmodes M3 and M4, the cross-section vertically below the patch load presents null deformation. These eigenmodes are referred to as antisymmetric (AS) and would hardly lead to the lowest resistance as required in EN1993-1-5Annex C [5]. Eigenvalue extraction performed on 1VPL750 leads to slightly different results. Two S and one AS eigenmodes are observed (M1, M2 and M3, respectively), whereas the fourth shape is an eigenmode related to shear buckling in the adjacent panels (SB). It is noteworthy that eigenmode 1 leads to an initially C-shaped web in the loaded cross-section, whereas eigenmode 2 leads to an initially S-shaped web. On 1VPL450, only one S C-shaped wave in the loaded panel was found in the first eigenmode. The following eigenmodes (M2, M3, and M4) are related to shear buckling in adjacent panels. Predictably, these geometries do not lead to the lowest resistance as required by EN1993-1-5-Annex C [5]. 4.2.3. Equivalent geometric imperfections EN1993-1-5-Annex C [5] allows modelling plated structures from equivalent geometric imperfections. This procedure may be advantageous when the structures to be modelled are idealized with one-dimensional finite elements. Global bow imperfections can be easily represented by mathematical functions (e.g. sinusoidal laws). When local buckling is prone to occur, however, 3D shell-based idealizations are needed. EN1993-1-5Annex C [5] suggests a combination of equivalent imperfections involving local and global imperfections of the panels. For this particular study, three types of geometric imperfections are included in the finite element analysis following the EN1993-1-5
Table 2 Numerical data from eigenvalue analyses IVPL2500
IVPL750
IVPL450
M1-161,84 kN (S)
M1-180,72 kN (S)
M1-325,04 kN (S)
M2-239,16 kN (S)
M2-271,40 kN (S)
M2-436,28 kN (SB)
M3-340,86 kN (AS)
M3-432,48 kN (AS)
M3-436,31 kN (SB)
M4-353,46 kN (AS)
M4-436,13 kN (SB)
M4-449,29 kN (SB)
Critical loads (kN) and critical shapes Mk.
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5. Results
500 450 400 Ultimate load capacity (kN)
procedure described in Section 2.2. The girders were assembled in ABAQUS-CAE as a whole and then an advancing-front algorithm using quad-dominant elements was used for meshing the assemblies. Even though these imperfections shapes are thought to be a reasonable approximation of the actual imperfections that occur in typical welded girders, it is necessary to assume certain geometrical incompatibilities. First, an initial constant twist of the flanges leads to an odd geometrical inconsistency. The transverse stiffeners happen to be trapezoidally -shaped (Fig. 3-c). Furthermore, the webs happen to be C-shaped in all cases. The actual shapes of the girders were found as being either C-shaped or S-shaped. This designer-assumed condition may influence the response of the girders. Third, the orientation of transversal stiffeners located along the swept length of the girders ought to follow (or not) a certain assumed pattern in the modelling. This pattern (or its absence) is arbitrary and designer-assumed.
397
350 300 250 200 150 Experimental results ABAQUS-Precise shape
100
ABAQUS-Equivalent Imperfection ABAQUS-1st Eigenmode
The influence of the initial conditions on the general response of steel plate girders when subjected to patch loading was tackled within three separated sections. Firstly, an appraisal of the influence of the shape of the initial geometry is presented (structural imperfections were included when needed). In this case, the calculations were performed on girders similar to those depicted in Table 1. Secondly, simulations were separately performed on models with and without including structural imperfections. Thus, the influence of the latter condition was also assessed. Thirdly, the influence of the maximum amplitude of the geometric shape was assessed by means of a large parametric study varying significant parameters, among others, the web slenderness. In this particular case, a refined analysis including both geometrical and structural imperfections was deployed on each model of the study. 5.1. Influence of the shape of the initial imperfection The first appraisal is extracted from results concerning ultimate load capacity of the experimentally and numerically tested girders. For this purpose, the results are graphically presented in Fig. 9 as a function of the distance between transverse stiffeners ‘‘a’’. Generally, the numerical results are in good agreement with the experimentally obtained values. Nevertheless, the geometries which are based upon some eigenmodes lead to undesired overestimations, i.e., these modes do not reproduce the required lowest resistance. Matter-of-fact, it is observed in Table 2 that these modes do not show any deformation in the directly loaded panel (or seem related to other instability modes different from patch loading). Table 3 numerically displays the ratio between experimental and FE-obtained results for each case. Precise shape, equivalent imperfection and the S eigenmodes lead to results on ultimate load capacity thought to be satisfactory. Perfect match between both results is observed for the 1st eigenvalue. It is finally confirmed that values corresponding to AS and shearbuckling like modes (generally, the 3rd eigenmode) do not yield the lowest resistance. The general response of the girders is also studied by using load-deflection plots. The first analysed prototype was 1VPL2500 (a ¼ 2500 mm). Fig. 10 shows differently obtained responses when varying the initial shape. Continuous lines correspond to experimental results whereas dashed lines to numerical ones. The response seems to be appropriately captured in all cases save for 3rd eigenmode (which confirms prior results). Linear branches
50
ABAQUS-2nd Eigenmode ABAQUS-3rd Eigenmode
0 0
500
1000 1500 Distance "a" (mm)
2000
2500
Fig. 9. Ultimate load capacity of the girder for different initial conditions.
as well as peak zones are similar. The precise shape, the 1st and the 2nd eigenmodes accurately reproduce the post-peak branch of the plots. The equivalent imperfection shows a slightly different response than other cases. Secondly, specimen 1VPL750 was studied (a ¼ 750 mm). Fig. 11 shows similar plots. Noticeably, linear branches are appropriately captured in all cases. Post-buckling behaviour is however, less satisfactory. Only refined analyses performed by introducing either the 1st or the 2nd eigenvalues as initial imperfection lead to the same response observed experimentally. Precise shape gives quite accurate results, but numerical difficulties were encountered throughout the analysis of such specimen. Other reproductions such as equivalent geometric imperfections give satisfactory results regarding ultimate load capacity, but do not appropriately capture the observed post-buckling branch. 3rd eigenmode yields to overestimated results and should not be taken as a suitable initial geometry. Clearly, it does not yield to the lowest resistance. It is worth emphasising that even if 1st and 2nd eigenmodes lead to very similar results regarding ultimate load capacity, results concerning elastic critical loads substantially differ. It would seem that both initially C-Shaped and S-shaped webs yield to similar results after buckling. This fact might be due to a widely studied phenomenon in instability of thin-walled structures referred to as secondary bifurcation [30]. Finally, the prototype 1VPL450 was analysed (a ¼ 450 mm). Fig. 12 shows the obtained results. Expectedly, linear branches are well-captured in all cases. The experimental curve presents a certain loss of linearity for an approximate value of 270 kN which is also captured by the 1st eigenvalue-based model. The peak load as well as the post-peak branch are fairly adjusted over this numerical reproduction. Equivalent geometric imperfections do not reproduce the response but rather, underestimate the girder response. Unfortunately, the precise shape geometry also yielded to numerical difficulties due to modelling. As previously explained, the second eigenmode is related to other instability phenomena such as shear buckling of the adjacent panels. As a result, this eigenmode does not yield the desired lowest resistance.
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Table 3 Experimental vs. numerical results Fu,
ABAQUS/Fu, experimental
Girder
Precise shape
Equivalent imperfection
1st eigenmode
2nd eigenmode
3rd eigenmode
1VPL2500 1VPL750 1VPL450
0.98 1.03 0.94
0.96 1.01 0.89
1.00 1.00 1.00
0.97 0.95 1.10
1.44 1.40 1.10
250
250
200
200 Load (kN)
Load (kN)
Influence of the initial shape imperfections.
150 100 50
100 50
0
0 0
5
10 15 20 Vertical displacement (mm)
25
250
400
200
300
Load (kN)
Load (kN)
150
150 100 50
0
5
10 15 20 Vertical displacement (mm)
25
0
5
10 15 20 Vertical displacement (mm)
25
200 100 0
0 0
5
10 15 20 Vertical displacement (mm)
25
300
300
250
250 Load (kN)
Load (kN)
Fig. 10. Load-deflection curve (1VPL2500). Influence of the initial shape imperfection. (a) Precise shape; (b) equivalent imperfections; (c) 1st and 2nd eigenmodes; and (d) 3rd eigenmode.
200 150 100 50
150 100 50
0
0 0
10 20 Vertical displacement (mm)
30
0
10 20 Vertical displacement (mm)
30
0
10 20 Vertical displacement (mm)
30
400
300 Load (kN)
250 Load (kN)
200
200 150 100 50
300 200 100 0
0 0
10 15 20 25 Vertical displacement (mm)
30
Fig. 11. Load-deflection curve (1VPL750). Influence of the initial shape imperfection. (a) Precise shape; (b) equivalent imperfections; (c) 1st and 2nd eigenmodes; and (d) 3rd eigenmode.
ARTICLE IN PRESS
500
500
400
400 Load (kN)
Load (kN)
´n et al. / Thin-Walled Structures 47 (2009) 391–402 R. Chaco
300 200
300 200 100
100 0
0 0
10 20 Vertical displacement (mm)
30
500
500
400
400 Load (kN)
Load (kN)
399
300 200 100
0
10 20 Vertical displacement (mm)
30
0
10 20 Vertical displacement (mm)
30
300 200 100
0
0 0
20 10 30 Vertical displacement (mm)
40
Fig. 12. Load-deflection curve (1VPL450). Influence of the initial shape imperfection. (a) Precise shape; (b) equivalent imperfections; (c) 1st eigenmode; and (d) 2nd eigenmode.
5.2. Influence of the structural imperfections Furthermore, an appraisal of the influence of initial structural imperfections is presented. Fig. 13 shows the structural response of three cases of study. The left side of the plot (Fig. 13-a) is sketched from numerical data in which no residual stresses are considered. The right side (Fig. 13-b) displays the same responses of the same models but in this case, a typical pattern of residual stresses is included within the plates as structural imperfection. These analyses were performed on girders in which the 1st eigenmode was included as initial geometry. The geometry was scaled following the EN1993-1-5-Annex C [5], i.e., 80% FT was adopted as the largest perturbation w. Table 4 shows the numerical-to-experimental ratio on ultimate load capacity. Noticeably, a 6,66% overestimation of ultimate load is observed in case 1VPL2500. The two other cases present results judged to be satisfactory whether residual stresses are included or disregarded. It is, however, noticeable that the load-deflection plot is slightly rather different in 1VPL2500. The results show that the residual stresses do not play a decisive role on the ultimate load capacity. This remark was also pinpointed by other authors when using different initial residual stress patterns [13,14]. Further research on this topic would eventually clarify and/or confirm these statements.
5.3. Influence of the magnitude of the maximum amplitude The second designer-assumed feature in the analyses is the maximum amplitude of the web out-of-flatness (w). Numerical simulations presented in this study so far are focused on a reproduction of a half-scale experimental program. However, one could argue that the maximum out-of-flatness of the webs in plate girders due to welding may be dependent on the tf/tw ratio and/or the slenderness of the web panel (hw/tw). An appraisal of
this influence without considering these additional conditions was judged to be unfair. Consequently, the web thickness was also included as an additional parameter. For this purpose, a wide deterministic parametric study was developed. Numerical simulations were performed on girders in which the maximum amplitude of the buckling shape varies approximately from 0% to 250% of the maximum allowable outof-flatness according to fabrication tolerances. The mechanical properties of the girders involved in this parametric study were identical to the properties of the tested girders. Geometrically, the length and clear web depth were held constant and equal to the experimental prototypes. The web slenderness was, however, varied ranging from 71 to 166.66. The variation was performed via web thickness (ranging from 3 to 6 mm). The study Case 1 includes the stockiest web panel, whereas the study Case 4 includes the most slender one. Case 3 coincides with the experimental geometry (highlighted in Table 5). In this study, all the calculations were performed on an initial shape based upon the first mode previously obtained in the perturbation analysis. Initial structural imperfections were included by the simplified pattern of residual stresses depicted in Section 4.1. A total amount of 120 simulations were performed. Noticeably, the web thickness tw together with w were the principal parameters within the set of variation. Ultimate load capacity of the girders was obtained for each case. Four plots are sketched in Fig. 14. In the y-axis (vertical), the ratio Fu/Fu,80%, is displayed (ultimate load Fu of the girders standardised to the Fu,80%, in which 80% of the fabrication tolerances is suggested as maximum amplitude). In the x-axis (horizontal), the maximum amplitude of the shape imperfection is sketched as a percentage of the allowed fabrication tolerances. The Eq. (7) indicates the mathematical relationship governing these plots. Fu ¼ f ðw %FTÞ F u;w¼80%
(7)
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400
Table 5 Parametric study. Influence of the maximum amplitude
450 1VPL450
Case
tw (mm)
tf/tw
Web slenderness (hw/tw)
Aspect ratio (a/hw)
w-f (%FT)
1 2 3 4
6 5 4 3
3.33 4 5 6.67
71 100 125 166.66
5-1.25-0.9 5-1.25-0.9 5-1.25-0.9 5-1.25-0.9
[0,10,30y250] [0,10,30y250] [0,10,30y250] [0,10,30y250]
400 350
Load (kN)
300 1VPL750 250 200 150 1VPL2500 100 50 0 0
5
10 15 20 Vertical displacement (mm)
25
30
450 1VPL450 400 350 300 Load (kN)
1VPL750 250 200 150 1VPL2500 100 50 0 0
5
10 15 20 Vertical displacement (mm)
25
30
Fig. 13. Load-deflection curves. Influence of the initial structural imperfections. (a) Residual stresses not taken into consideration and (b) residual stresses included.
Table 4 Influence of initial structural imperfections Residual stresses Girder
Not taken into consideration [(Fu,num/Fu,exp)1,0] 100
Taken into consideration [(Fu,num/Fu,exp)1,0] 100
1VPL2500 1VPL750 1VPL450
6.66 0.22 0.29
0.00 0.40 0.20
Three cases of study. Experimental vs. numerical results.
Each plot corresponds to a given value of web slenderness. Within each plot, three different aspect ratios are sketched. Each continuous line represents a variation of w, as a percentage of FT, i.e., the designer-assumed amplitudes. A vertical line indicating the suggested minimum value in EN1993-1-5-Annex C [5] is also displayed in four plots. Noticeably, two different trends are observed within all curves. Stocky girders (hw/tw ¼ 71 and 100) present a decreasing tendency in the plot, whereas slender girders (hw/tw ¼ 125 and 166,66) present a quite stable tendency. For all cases, very small magnitudes of the initial imperfection (0% FT) lead to undesired erroneous overestimations in ultimate load capacity. This fact might be due to pathological numerical difficulties and potential bifurcation of equilibrium. This fact should warn about the potential overestimations, the designer might commit by neglecting this parameter. Focusing on results obtained in stocky girders, it is observed that for most cases, an increment in the amplitude values leads to a decrement in the resistance of the girders. In general, this trend is more noticeable for high aspect ratios of the panel (a/hw) than for closely spaced transversal stiffeners, in which this sensitivity is insignificant. It is then found that the magnitude should be based upon conservative percentage of the maximum geometric fabrication tolerances. For the recommended values of 80% FT, the results of ultimate load capacity obtained are considered structurally sound. Oddly enough, slender girders do not present such dependency. Provided that the initial magnitude is greater than a certain value, the tendency can be considered as being horizontal. It is likely that failure is instability-dependent in such cases and the potential bifurcation of equilibrium occurs even for small values of w. For these cases, it is also found that the magnitude can be based upon the recommended percentage of the maximum geometric fabrication tolerances. 80% FT leads to sound results of ultimate load capacity of slender plate girders. Table 6 shows, for each case, a numerical appraisal of such influence. One can define the maximum difference existing between extreme values of Fu (DFu ¼ Fu,10%Fu,250%) as the maximum realistic loss of load-carrying capacity as long as the largest perturbation w is increased (results related to Fu,0% are discarded due to the encountered numerical difficulties and values of w4250% FT are considered unlikely). DFu is pointed out in Fig. 14. Results show that girders assembled with stocky web panels present a greater sensitivity to this magnitude than girders assembled with slender web panels, in which the dependency seems to be negligible. This fact is particularly noticeably in girders with high a/hw ratios. Furthermore, it shows that for slender panels presenting small aspect ratios (a/hw), an increment of w may lead to a slight increment of Fu. This latter increment is rather small and should not be considered as significant.
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hw /tw=100 1.40
1.30
1.30 80%
1.10 1.00
ΔFu
Fu / Fu 80%
Fu / Fu 80%
hw /tw=71 1.40 1.20
401
80%
1.20
a/hw=5.0
1.10 1.00
ΔFu
0.90
0.90 0.80
a/hw=1.25 a/hw=0.9
0.80 50
0
100 150 200 w = f(%FT)
250
300
0
50
100 150 200 w = f(%FT)
250
300
y x hw /tw =166,66
80%
ΔFu
0
50
100 150 200 w = f(%FT)
250
Fu / Fu (80%)
Fu /Fu (80%)
hw /tw=125 1.40 1.30 1.20 1.10 1.00 0.90 0.80
300
1.40 1.30 1.20 1.10 1.00 0.90 0.80
80% a/hw=5.0 a/hw=1.25 a/hw=0.9
ΔFu
0
50
100 150 w = f(%FT)
200
250
Fig. 14. Influence of the maximum amplitude of the initial shape imperfection.
Table 6 Parametric study Case
Web slenderness (hw/tw)
Aspect ratio (a/hw)
DFu,% (Fu,10%–Fu,250%)
Maximum difference
1
71
5.00 1.25 0.90
11.87 4.59 0.95
2
100
5.00 1.25 0.90
11.85 6.77 2.72
3
125
5.00 1.25 0.90
4.13 2.39 2.36
4
166.66
5.00 1.25 0.90
2.34 1.39 0.00
Influence of the maximum amplitude.
6. Conclusions The current European guidelines EN1993-1-Annex C [5] allow designers using FE-Analyses as reliable tools on the calculation of plated structures. The guides overtly request the usage of initial designer-assumed conditions which are necessary for the development of appropriate simulations. The guidelines suggest two potential initial conditions for any analysis of plated structures; a refined analysis which includes both geometrical and structural imperfections or an equivalent geometric shape that includes both initial conditions. The numerical modelling of both possibilities differs considerably from one to another. These conditions were assessed for the particular case of study by using an experimental basis of three steel plate girders in which initial shape imperfection was known in advance.
The drawn conclusions may give guidance to practicing engineers and researchers alike on the designer-assumed conditions and the potential influence on ultimate limit states verifications for the particular case of steel plate girders subjected to patch loading. First, it has been shown that the initial shape of the girder does not play a primary role on the ultimate load capacity of the girders as far as realistic symmetric (S) shapes are introduced directly below the patch load. It is worth pointing out that the experimentally obtained response of the girders is perfectly reproduced when 1st eigenmode-based geometries are used as initial geometrical conditions. The introduction of equivalent geometric imperfections leads to satisfactory and conservative results concerning ultimate load capacity. However, the modelling is judged rather intricate and several decisions involving engineering judgement are needed. This procedure may be conservatively used if more refined analysis is not feasible. Due to simplicity and accuracy, the usage of eigenmode-based geometries is highly recommended. Second, the structural imperfections are introduced in the girders by means of a typical residual stress pattern found in literature. It has been noticed that if these patterns are included in the analysis, the results are more refined. The difference between considering or discarding these imperfections does not play a decisive role on the response of the girders. Other studies have also pointed out similar remarks. It is, however, recommended to address further research concerning this particular topic by means of vaster parametric studies including a set of variation of different patterns. Third, the influence of the maximum amplitude magnitude w suggested in EN1993-1-5-Annex C [5] has been assessed. The maximum allowable fabrication tolerances are used as a comparative magnitude. It has been shown that whether these amplitudes are kept within reasonable limits related to fabrication tolerances, the influence of the magnitude is negligible. It has been also observed that in stocky girders, an increment in the amplitude values leads to a decrement in the resistance of the girders, whereas in slender girder this tendency is not noticeable. In all cases, for the specific recommended values of
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80% of the allowed fabrication tolerances, results of ultimate load capacity obtained are considered structurally sound.
Acknowledgements The authors wish to gratefully acknowledge the financial support provided by Spanish Ministry of Science and Education, as a part of the Research Project BIA 2004-04673 and the grant provided to the first author. Likewise, the authors are grateful to Arcelor and Tadarsa for supplying and manufacturing the girders and to Abrox for measuring the initial imperfections of the girders. References [1] Veljkovic M, Johansson B. Design of hybrid steel girders. J Constr Steel Res 2004;60(3–5):535–47. [2] Dubas P, Gehri E. Behaviour and design of steel plated structures. ECCS-Tech Comm 8 1986. [3] Bernard ES, Coleman R, Bridge RQ. Measurement and assessment of geometric imperfections in thin-walled structures. Thin-Walled Struct 1999;33:103–26. [4] Singer J, Abramovich H. The development of shell imperfection measurement techniques. Thin-Walled Struct 1995;23:379–98. [5] EN 1993-1-5 Eurocode 3: design of steel structures, Part 1.5 Plated structural elements. 2005. [6] Bergfelt A. Patch loading on a slender web-influence of horizontal and vertical web stiffeners on the load carrying capacity. Chalmers University of Technology, S79:1. 1979. Go¨teborg, Sweden (in Swedish). [7] Dubas P, Tschamper H. Stabilite´ des aˆmes soumises a` une charge concentre´e et a` une flexion globale. Constr Met 1990;2 [in French]. [8] Seitz M, Kulhman U. Longitudinally Stiffened Girder Webs Subjected To Patch Loading. Steelbridge 2004. Millau, June p. 23–25. [9] Martı´nez J, Graciano C, Casanova E. Imperfection sensitivity of plate girder webs under patch loading. In: Proceedings of the Third International Conference on Structural Engineering, Mechanics and Computation (SEMC). September 2007. Cape Town, South Africa, 1–8. [10] Chacon R, Mirambell E, Real E. Influence of initial imperfections on the resistance of hybrid steel plate girders subjected to concentrated loads. In: Proceedings of the 6th International Conference Steel and Aluminium Structures ICSAS’07. July 2007. Oxford, U.K. p. 736–743.
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