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Linearly tapered bridge girder panels with steel corrugated webs near intermediate supports of continuous bridges
Thin-Walled Structures 88 (2015) 119–128
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Linearly tapered bridge girder panels with steel corrugated webs near intermediate supports of continuous bridges M.F. Hassanein n, O.F. Kharoob Department of Structural Engineering, Faculty of Engineering, Tanta University, Tanta, Egypt
art ic l e i nf o
a b s t r a c t
Article history: Received 11 July 2014 Received in revised form 20 November 2014 Accepted 20 November 2014 Available online 24 December 2014
Nowadays, girders with corrugated webs are used in bridges as an efficient alternative to conventional girders with flat stiffened webs. Particularly in bridge girders with corrugated webs (BGCWs), the corrugated webs are the main elements for bearing the shear forces. Instead of prismatic BGCWs, tapered BGCWs are currently used mainly due to their structural efficiency, providing at the same time aesthetical appearance. Available literature shows that tapered BGCWs may be classified into four typologies. Among these typologies, Case I is the most common case appearing commonly near to the intermediate supports of continuous bridges. Accordingly, in this paper, the finite element (FE) method is employed to investigate the inelastic behavior of tapered BGCWs of Case I, following to the fundamental behavior of such girders published recently by the current authors. Previously validated 3-D nonlinear FE models are developed and used in this study. The paper seeks, first, at finding the validity limit of the previously proposed design strengths for the tapered BGCWs with respect to the girder’s initial imperfection. This is made by considering different initial imperfection amplitudes. Accordingly, it is found that the proposed equation is valid with initial imperfections of hw1 =200; hw1 is the height of the long vertical edge of the web panel. However, the accuracy increases for initial imperfections similar to the web thickness (t w ) for cases of t w Z hw1 =200. Then, it investigates the effect of the aspect ratio of the web panels, different flange inclination angles and flange slenderness ratios. Finally, the paper checks the authors’ previously proposed design model using the results of the generated parametric studies. Overall, the outcomes of this study are expected to provide more insight into the behavior of tapered BGCWs and enable accurate prediction of the shear capacity of this special type of BGCWs. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Tapered corrugated steel webs Shear strength Global buckling Interactive buckling Initial imperfection Resal effect
1. Introduction Girders, in conventional design, are subjected to significant levels of bending moment and thus failure typically occurs when the applied moment due to loading at the critical section exceeds its flexural capacity. Consequently, girders are typically designed to satisfy flexural limit state and then they are checked for shear limiting criteria. However, in the case of plate girders with slender webs, the web panel buckles at a relatively low value of the applied load. Therefore, the shear may control the design of such girders. To overcome the strength reduction associated with utilizing plate girders with slender webs in bridge construction, these flat webs are often reinforced with transversal stiffeners along their spans to increase their buckling strength. More recently, girders with steel corrugated webs have been used as structural members in Bridges; for example see Fig. 1(a) which is presenting the Ginzan-Miyuki Bridge in Japan [1]. As well known, corrugated web plates have
n
Corresponding author. Mobile: þ 201228898494; fax: þ 20403315860. E-mail address: [email protected] (M.F. Hassanein).
http://dx.doi.org/10.1016/j.tws.2014.11.021 0263-8231/& 2014 Elsevier Ltd. All rights reserved.
much higher buckling strengths compared with flat web plates owing to their noteworthy out-of-plane stiffness. Accordingly, the stiffeners are eliminated and the required web thickness is reduced [2–6] by using them as the girder web. On the other hand, due to the accordion effect [7,8], the flexural strength of such girders is entirely provided by their flanges while the shear strength is carried by their webs. Such effect is characterised by the negligible axial stiffness of the such corrugated webs in the longitudinal directions of the girders. As a result, there is no interaction between shear and flexural behaviours [7,8]. Consequently, assuming a constant shear stress in the corrugated webs of such girders is commonly accepted and then it is quantified it in terms of the average shear stress. On the other hand, tapered girders are currently used in bridges mainly due to their structural efficiency, providing at the same time aesthetical appearance. An example for the application of tapered bridge girder with steel corrugated webs (BGCWs) is given in Fig. 1(b). In a tapered bridge girder such as that forming the Hondani Bridge, shown in Fig. 1(b), three typologies of girder panels can be found besides a fourth one that can be found in other bridge systems; refer to Bedynek et al. [9]. That is based on the inclination of the flange and whether the flange is under
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Fig. 1. Bridge girders with corrugated webs; (a) Ginzan-Miyuki Bridge and (b) Hondani Bridge. (a) Prismatic bridge girders. (b) Parabolic tapered bridge girders.
s Sub-panel (fold) tw
Case I: Inclined flange under compression & tension field is developed in the short diagonal
c
α
hr b
d
b
d
w is the bigger of b and c
One corrugation wave (q)
Fig. 3. Corrugation configuration and geometric notation.
2. Background and design strength
Case II: Inclined flange under tension & tension field is developed in the long diagonal
Case III: Inclined flange under compression & tension field is developed in the long diagonal
Case IV: Inclined flange under tension & tension field is developed in the short diagonal
Fig. 2. Types of tapered plate girders with corrugated webs.
tension or compression as well as the direction of the developed tension field, which may appear on the short or on the long web diagonal [9]; see Fig. 2. Accordingly, to unveil the structural behavior of the tapered BGCWs, Hassanein and Kharoob [10] investigated their fundamental behavior. Nevertheless, there is still lack of information related to their shear behavior as recommended at the end of their paper. Accordingly, this paper extends the work presented in Ref. [10] by means of conducting several parametric studies. The webs of the currently considered girders are formed from trapezoidally corrugated steel web plates which are composed of a series of longitudinal and inclined sub-panels; see Fig. 3. This is because this type of corrugated webs is the most utilized in available bridges as can be seen in Fig. 1.
Because tapered BGCWs are used extensively as the main systems in modern bridges (Fig. 1(b)) without the availability of any source describing their shear buckling behavior, the current authors studied the fundamental behavior of such girders [10]. Research presented by Hassanein and Kharoob [10] focused, first, on the critical shear buckling stress (τcr ) of the corrugated webs of tapered BGCWs. This was made by carrying out elastic bifurcation buckling analyses using ABAQUS software [11] on isolated corrugated webs with simple and fixed boundary conditions. Webs in different typologies of tapered girders with steel corrugated webs were considered; see Fig. 2. To reflect the behavior of bridges, the corrugation dimensions of the considered corrugated webs were taken typical to those used previously in Shinkai and Matsnoki bridges. However, it was found that predicting τcr values for the tapered webs based on prismatic web (having the depth of the long vertical edge of the tapered web (hw1 )) calculations is not accurate. Therefore, critical buckling stresses (τcr;Prop ) for the tapered webs were proposed based on the stresses of prismatic webs, with different equation for each typology. The paper [10] was, then, extended to investigate the nonlinear shear strengths of the tapered BGCWs. This was made based on the verifications made by the same authors in Ref. [6]. The aspect ratio of the web panels (a=hw1 ) of the girders was 2.88, while the inclination angle was fixed to 7.1251; a and hw1 are the shear span and the height of the long vertical edge. The flange slenderness was, additionally, constant throughout the investigation, while five web slenderness ratios were taken into consideration. The available design shear strength formulas for prismatic girders (by Moon et al. [3] and Sause and Braxtan [12]) were compared with the FE shear strengths of the tapered BGCWs. Based on these comparisons, design strength (τul;Prop ), based on the equation of Moon et al. [3], for different tapered BGCWs typologies was proposed, as follows: 8 1:0 : λs r0:6 > > pffiffiffi < τul;Prop 0:6Þ : 0:6 o λs r 2 1 0:614ðλ s ð1Þ ¼ CT pffiffiffi > τy 1 > : 2 o λs : λ2 s
where τy stands for the shear yielding strength of the steel material, λs represents the shear buckling parameter of the corrugated webs and C T is taken as unity for cases I and II or as the ratio hwo =hw1 for cases III and IV; hwo =hw1 is the ratio between the height of the short
M.F. Hassanein, O.F. Kharoob / Thin-Walled Structures 88 (2015) 119–128
vertical edge of the web panel to that of the long edge. It is worth pointing out that the parameter λs uses, in its calculation, the τcr;Prop value of each typology; for more details refer to Ref. [10]. According to the dimensions of Shinkai bridge with different web thickness used in Ref. [10], almost of the models had a parameter λs less than 0.6 (16 out of 20 models). The rest of models had a parameter λs just bigger than 0.6 (only 4 models). 3. Goals and objectives With the increasing utilization of tapered BGCWs worldwide [1], a better understanding of the true behavior of such girders under shear loading becomes essential. Consequently, this paper extends the paper by Hassanein and Kharoob [10] with respect to bridges of Case I. Among the four typologies shown in Fig. 2, Panels of Case I are the most common case which appears frequently near to the intermediate supports of continuous bridges; see the two examples presented in Fig. 4. Accordingly, the objective of this paper is to provide additional data to engineers and scientific community on the strength and behavior of the tapered BGCWs of Case I under shear loads. As a result, the following new points (as suggested at the end of Ref. [10]) are added to literature for the first time: 1. The paper defines the validity limit of the previously proposed design strength [10] for the tapered BGCWs regarding the initial imperfection. This was made by generating FE models with different initial imperfections to be compared with the results of the proposed design strength. 2. Models with λs greater than 0.6 (belonging to the second part of Eq. (1)) using the limiting initial imperfections, as proposed from the previous point, are generated to deepen the understanding of their behavior. 3. The paper investigates the effect of the aspect ratio of the web panel (a=hw1 ) on the shear strength of the tapered BGCWs. 4. The paper expands the pool of available results by considering tapered BGCWs with different inclination angles (γ 3 ). 5. The paper goes into the effect of flange slenderness ratio on the shear strength and behavior of the tapered BGCWs. By doing so, the vertical component of the Resal effect changes. Accordingly, its effect on the behavior of the tapered BGCWs could be explored. 6. Finally, the paper checks the proposed design model [10] by using the results of the current parametric studies as well as the considered wide range of the slenderness parameter (i.e. λs 4 0:6). 4. Validation limit of the design strength It is well known that the amplitude and the shape of the initial web geometric imperfections play a significant role in the shear strength and behavior of the girders with corrugated webs [3,8,13]. Accordingly, an initial imperfection amplitude equalling to the thickness of the corrugated web (t w ), as suggested by Driver et al. [8], was considered by Hassanein and Kharoob [10] to propose their
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design strength given herein by Eq. (1). The objective of this section is to provide the validity limit of Eq. (1) for the tapered BGCWs with regard to the initial imperfection. Accordingly, the generated girders (of different typologies) in Ref. [10] were reanalysed with two initial imperfection amplitudes of hw1 =100 (12.5 mm) and hw1 =200 (6.25 mm). The first positive Eigenmode which responds to the shear buckling mode was scaled to the above mentioned values in the nonlinear modelling. Two typical examples representing the Interactive (I) and Global (G) buckling modes for Case III models are presented herein in Fig. 5. One girder was, however, excluded because of its flexural failure (girder of Case II with web thickness of 14 mm). As can be noticed, the imperfections were linked in this analysis to the height of the long vertical edge of the web panel (hw1 ). Tables 1 and 2 show the results of those analyses and their comparison with the proposed design strength. From Table 1 it can be noticed that the FE ultimate shear load (V ul;FE ) of the tapered girders decreases with the increasing of initial geometric imperfections. Additionally, it could be observed that the girders under different initial imperfection amplitudes exhibited the same failure modes. On the other hand, the loading capacity of the webs at the descending stage is almost independent of the initial imperfections, as can be seen from Fig. 6. In this figure, the load–deflection relationships for Case III with web thickness of 8 mm are presented. Other typologies were not included in the figure because they provide similar results. This is attributed to the early occurrence of the web out-of-plane deformation due to the shear buckling before the shear yield can reduce the sensitivity of the loading capacity at the descending stage to the initial geometric imperfections. From Table 2 it can be observed that initial imperfection amplitudes of hw1 =100 provide unsafe results for tapered girders with small web thickness values (6 and 8 mm), while they are suitable for the bigger; 10, 12 and 14 mm. In contrast, hw1 =200 values provide conservative values for different typologies with different web thicknesses. However, by increasing the web thickness, the degree of conservatism of the design strength increases. Hence, it can finally be concluded that the proposed strength (Eq. (1)) is valid with initial imperfections of hw1 =200, but its accuracy increases for initial imperfections similar to the web thickness (t w ) for cases of t w Z hw1 =200.
5. Parametric study 5.1. Finite element model The finite element (FE) model presented by the authors in Ref. [10] was utilized in the current parametric study. Instead of Shinkai bridge corrugation used in Ref. [6], the Dole bridge corrugation dimensions (Fig. 7) were used in this paper to generate full-scale tapered BGCWs (Fig. 8) loaded under mid-span concentrated loads. To insure that the shear controls the failure modes of the tapered BGCWs, (1) the web thickness was chosen as 6 mm instead of 10 mm which is the actual web thickness of Dole bridge corrugation and (2) the loaded point was restrained
Fig. 4. Linearly tapered bridge girders with corrugated webs of Case I. (a) Kurobegawa Bridge. (b) Ohmi-Ohdori bridge.
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steel used was S355 according to EN 1993-1-1 [14], which has a yield (f y ) and an ultimate strength (f u ) of 355 MPa and 510 MPa, respectively. A bilinear elastic–plastic stress–strain curve with linear strain hardening was used to simulate the steel material.
Interactive (I) buckling mode: Case III-8mm
5.2. Input data Three-dimensional FE models, using ABAQUS [11] FE package, were performed on twenty seven tapered BGCWs covering the following parameters:
Close view
1. aspect ratio of the web panels (a=hw1 ); (1.49, 2.23 and 2.98), 2. angle of the inclined flanges (γ); (8, 11, 141), and 3. flange slenderness ratios (bf =2t f ); (5, 6.25 and 8.33).
Global (G) buckling mode: Case III-14mm
Buckling extends diagonally over the depth of the web Close view
Fig. 5. Typical buckling modes for: (a) interactive and (b) global buckling modes.
Table 1 FE results of the imperfection analyses. Type t w λs [mm]
Initial imperfection [mm] tw
Case I
Case II
Case III
Case IV
6 8 10 12 14 6 8 10 12 6 8 10 12 14 6 8 10 12 14
0.683 0.533 0.445 0.386 0.346 0.670 0.523 0.436 0.379 0.602 0.470 0.392 0.341 0.305 0.621 0.485 0.404 0.352 0.314
hw1 =100 (12.5 mm)
hw1 =200 (6.25 mm)
V ul;FE [kN]
Buckling mode
V ul;FE [kN]
Buckling mode
V ul;FE [kN]
Buckling mode
944 1318 1693 2144 2619 1097 1619 2010 2147 652 917 1146 1450 1686 634 894 1136 1397 1666
I G G G G I G G G I I G G G I I G G G
830 1211 1617 2128 2666 1065 1500 1992 2147 588 851 1106 1445 1688 559 829 1097 1384 1691
I G G G G I G G G I I G G G I I G G G
938 1374 1816 2306 2852 1094 1676 2023 2217 649 946 1224 1548 1798 630 925 1210 1506 1811
I G G G G I G G G I I G G G I I G G G
laterally. Initial geometric imperfections, based on the first positive shear buckling mode, were included in the nonlinear analysis of the BGCW with values of hw1 =200 as t w o hw1 =200; refer back to Section 4. The loads were applied using the modified RIKS method. S8R5 reduced integration thin shell elements were employed to discretise the models in the current nonlinear analyses. Simply supported boundary conditions were applied to end sections. The displacement history was applied to the FE models along their mid-span web points at the intersection with lower flanges. The steel material has been modelled as a von Mises material with isotropic hardening. The
The flange width (bf ) was fixed to 500 mm throughout the entire program. The webs of the girders were stiffened transversely at the supports and under the applied load. Out-standing plate stiffeners extending to the edge of the flanges of the plate girder were considered with a thickness of 25.4 mm all through the investigation. Unlike Ref. [10] where almost of the models had shear buckling parameters (λs ) less than 0.6, the values of λs of the current models were intentionally chosen to lay within second stage of the proposed pffiffiffi design model; i.e. 0:6 o λs r 2. This was made to increase the range of the available data for the tapered BGCWs in literature, as can be seen in Table 3. This table shows the dimensions considered in this study. The table also contains the ultimate shear strength (V ul;FE ) of the girders beside the maximum shear stress (τul;FE ) calculated with respect to the critical cross-section of the girders with the shorter depth (hwo ). To ensure that all the tapered BGCWs failed by shear limit state away from the interaction with the flexural limit state by means of the development of flexural plastic hinges, the ratios between the ultimate and the plastic bending moments (M ul;FE =M pl ) were computed and then added to Table 3; M pl ¼ bf t f F y ðhw1 þ t f Þ. From the table, it can be noticed that M ul;FE =M pl ratios vary between 0.21 to 0.58 indicating that the flexural capacity limit state was still away to take place. 5.3. Effect of aspect ratio of the web panels Plate girders with flat web plates require the use of a series of transverse stiffeners in their fabrication. This is because the flat web panel buckles at a relatively low value of the applied shear force. Consequently, the webs are often reinforced with transversal stiffeners along their spans to increase their buckling strengths. Accordingly, it is a well known fact that the aspect ratio of the web panel (a=hw ) plays a key role in the shear strength of such girders (see for example Refs. [15,16]). On the other hand, corrugated web plates have much higher buckling strengths compared with flat web plates because of their significant out-of-plane stiffness [2–6]. Hence, it is widely accepted that the aspect ratio of the web panels (a=hw ) has no effect on the ultimate shear stress (τul;FE ) of prismatic BGCWs. This could easily be ensured by revising the available design shear strengths [3,6,8,12] where the a=hw ratio does not exist. This was additionally checked by the current authors by adding several stiffeners (which is not the case in real bridges as can be seen in Figs. 1 and 5) within the shear spans of some prismatic girders modelled in Ref. [12] and the maximum change in the ultimate shear stress (τul;FE ) was about 2%. On the other hand, this effect did not yet investigated in the tapered BGCWs. Therefore, the effect of the a=hw1 ratio on the shear behavior of tapered BGCWs was investigated in this sub-section. However, tapered BGCWs with different a=hw1 ratios were generated herein by varying the span ratio (a) instead of dividing the shear span by using stiffeners to accord with bridge girders in reality.
M.F. Hassanein, O.F. Kharoob / Thin-Walled Structures 88 (2015) 119–128
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Table 2 Comparison between the FE results of the imperfection analyses of the proposed design strength. Type
λs
t w [mm]
Case I
6 8 10 12 14 6 8 10 12 6 8 10 12 14 6 8 10 12 14
Case II
Case III
0.683 0.533 0.445 0.386 0.346 0.670 0.523 0.436 0.379 0.602 0.470 0.392 0.341 0.305 0.621 0.485 0.404 0.352 0.314
tw
hw1 =100
hw1 =200
tw
hw1 =100
hw1 =200
197 206 212 223 234 229 253 251 224 136 143 143 151 150 132 140 142 145 149
173 189 202 222 238 222 234 249 224 123 133 138 151 151 116 130 137 144 151
195 215 227 240 255 228 262 253 231 135 148 153 161 161 131 145 151 157 162
0.96 1.00 1.03 1.09 1.14 1.12 1.23 1.23 1.09 0.66 0.70 0.70 0.74 0.73 0.64 0.68 0.69 0.71 0.73
0.84 0.92 0.99 1.08 1.16 1.08 1.14 1.21 1.09 0.60 0.65 0.67 0.73 0.74 0.57 0.63 0.67 0.70 0.74
0.95 1.05 1.11 1.17 1.24 1.11 1.28 1.23 1.13 0.66 0.72 0.75 0.79 0.79 0.64 0.71 0.74 0.77 0.79
Mid-span vertical deflection [mm] Fig. 6. Load–deflection relationship for Case III-8 mm with different initial imperfections.
s Sub-panel (fold) tw
430
α
220 430
370
430
τul;Prop τy
τul;FE =τy
Applied load [kN]
Case IV
τul;FE [MPa]
370
One corrugation wave ( q ) Fig. 7. Utilized corrugation dimensions (Dole Bridge [5]) in mm.
Fig. 8. Elevation of typical tapered BGCWs (case of γ ¼ 11 3 and a=hw1 ¼ 2:23)
0.95 1.00 1.00 1.00 1.00 0.94 1.00 1.00 1.00 0.64 0.64 0.64 0.64 0.64 0.62 0.64 0.64 0.64 0.64
In practice, the distance between vertical stiffeners is much greater than the web depth. Therefore, three ratios of a=hw1 greater than unity were considered; 1.49, 2.23 and 2.98. It can be seen from Table 3 that increasing the ratio of a=hw1 of the girder leads to a considerable decrease in the ultimate shear strength (V ul;FE ). The load against mid-span vertical deflection for selected girders is provided in Fig. 9 as sample results. These girders (G13, G14 and G15) had an inclination angle of 111. This figure shows that both the load and the initial shear stiffness considerably decrease as the aspect ratio increases. It can additionally be observed that the failure is sudden and results from buckling for such BGCWs with relatively small web thickness. After the maximum load was achieved, considerable residual strength remains after failure. On the other hand, the variation of τul;FE =τy ratio with the aspect ratio for the tapered BGCWs for different flange slenderness ratios (bf =2t f ) is given in Fig. 10. As can be noticed, the τul;FE =τy ratios of the girders with the least inclination angle (i.e. γ ¼ 81) seem not to be affected by the change in the a=hw1 ratio similar to the case of prismatic girders. For a particular value of bf =2t f ratio, this figure also demonstrates that tapered BGCWs with larger shear spans (i.e. girders with increased a=hw1 ratios) are more prone to shear deformations and, therefore, can relatively reach higher stresses with regard to their plastic shear resistances (τul;FE =τy ) compared to girders with less a=hw1 ratio specially for girders with large angle of inclination (i.e. γ ¼ 141). This is simply attributed to the very small web depth of the critical section of the girders (hwo ) compared to its length in the girders with smaller angles of inclinations; see Table 3. Fig. 11 shows the stress distribution (of the middle surface of the webs) captured at the ultimate load of girders G13, G14 and G15, as sample results. In the figure, regions with the gray contour (indicated by arrows in the figure) represent the locations exceeding the yield strength of the material (355 MPa), while the red, just surrounding the gray contour, shows the regions which have just yielded. As can be seen, the plasticity spreads near the shorter edge of the shear panels as the shear stress is maximized. Also, it should be noted that the failure mode of the BGCWs does not show the propagation of shear plastic hinges (SPHs) at their top flanges similar to those appearing in plate girders with flat web plates (see Ref. [9]). It was found [15,17] that the SPHs (Fig. 12) develop as a result of the differential shear deformation between the top and bottom flanges. Fig. 12 was taken from Ref. [18] for the
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Table 3 Full details of the current tapered BGCWs. Girder
hwo [mm]
hw1 [mm]
G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 G15 G16 G17 G18 G19 G20 G21 G22 G23 G24 G25 G26 G27
1700 1475 1250 1528 1217 905 1352 953 554 1700 1475 1250 1528 1217 905 1352 953 554 1700 1475 1250 1528 1217 905 1352 953 554
2150
γ3 8
t f [mm]
a [mm]
a=hw1
λs
M ul;FE =M pl
V ul;FE [kN]
τul;FE [MPa]
30
3200 4800 6400 3200 4800 6400 3200 4800 6400 3200 4800 6400 3200 4800 6400 3200 4800 6400 3200 4800 6400 3200 4800 6400 3200 4800 6400
1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98
1.143 1.143 1.143 1.170 1.170 1.170 1.196 1.196 1.196 1.143 1.143 1.143 1.170 1.170 1.170 1.196 1.196 1.196 1.143 1.143 1.143 1.170 1.170 1.170 1.196 1.196 1.196
0.38 0.53 0.58 0.35 0.43 0.48 0.35 0.42 0.38 0.29 0.41 0.44 0.26 0.33 0.37 0.25 0.30 0.30 0.24 0.32 0.39 0.23 0.27 0.30 0.21 0.26 0.25
1374 1255 1042 1264 1036 852 1252 1008 675 1399 1312 1052 1241 1065 883 1189 939 716 1408 1280 1172 1342 1078 909 1271 1026 759
135 142 139 138 142 157 154 176 203 137 148 140 135 146 163 147 164 216 138 145 156 146 148 167 157 179 228
11
14
8
40
11
14
8
50
11
P [kN]
14
Mid-span deflection [mm] Fig. 9. Load-mid-span deflection for girders G13, G14 and G15.
first author. As the flat web plate buckles, the top flange undergoes downward displacement greater than that of the bottom. This downward displacement, however, increases as the load increases leading to the development of diagonal tension field. Once the web has yielded, failure of the steel plate girder occurs when plastic hinges are formed in the flange due to the excessive downward displacement compared to that of the bottom. This, however, not the case in BGCWs as they possess significant out-ofplane stiffness. Accordingly, the out-of-plane deformation of the corrugated webs is relatively limited. So, no relative shear deformation exits between both flanges to form the SPHs. 5.4. Effect of angle of the inclined flange The influence of the inclined flange angle (γ) on the ultimate shear strength (V ul;FE ) and the load-mid-span deflection response of tapered BGCWs under shear loading was additionally studied. Three values of γ (8, 11 and 141) were used confirming that the taper ratio (hw1 =hwo ) of the girders does not exceed 4 which was found to cover a large proportion of existing structures [19]. Herein, the maximum value of the taper ratio was 3.88 for the case of γ ¼ 141 and a ¼ 6400 mm. The applied load-mid-span vertical deflection relationships for sample results are provided in Fig. 13. This figure shows these relationships for
girders G11, G14 and G17 which have inclined flange angles of 8, 11 and 141, respectively. It can be seen that increasing the inclined flange angle (γ) significantly reduces the ultimate shear strength of the tapered BGCWs, while the initial stiffness remains more or less the same. Additionally, it can be seen that all tapered BGCWs behave in a brittle manner irrespective of the value of γ with a considerable residual strength remaining after failure. On the other hand, the influence of the flange angle of inclination on the normalized shear strength (τul;FE =τy ) of tapered BGCWs is shown in Fig. 14 for different flange slenderness ratios. In this figure, the variation of the τul;FE =τy ratio is presented with respect to Tan ðγÞ. It is worth pointing out that tapered BGCWs with small values of γ (i.e. γ ¼ 81) have nearly the same τul;FE =τy ratios irrespective of the flange slenderness and the web aspect ratio. Also, Fig. 14 shows an increase in the τul;FE =τy ratio for tapered BGCWs with the increase of γ. However, the rate of increase becomes higher for girders with increased aspect ratio (a=hw ). This increase results from the reduction in the critical web depth (hwo ) which becomes more pronounced in girders with larger a=hw ratio. The comparison between the relationships of Fig. 14 reveals also that girders with thick flanges (bf =2t f ¼ 5) experience a greater increase in the τul;FE =τy ratio. This observation may be attributed to the fact that thicker flanges provide more fixity at the juncture between the flanges and the webs leading to the increase of the shear strength of the girders. Fig. 15 compares the stress distribution at the ultimate shear loads of the same selected girders presented previously in Fig. 13 (G11, G14 and G17). As can be seen, the portions exceeding the yield strength of the material, enveloped by rectangles in the figure, extend in a wider range from the shear span by decreasing the angle of inclination. By decreasing the value of γ, the yielded portions become concentrated near the end supports where the shear stress is maximized. 5.5. Effect of flange slenderness ratio It was shown in Section 1 that there is no interaction between shear and flexural behaviors of prismatic BGCWs. Hence, the flanges of prismatic BGCWs mainly contribute to the moment capacity, while the
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Fig. 10. Variation of τul;FE =τy with a=hw1 for tapered BGCWs for different ratios of bf =2t f
SPH
Fig. 12. Developed shear plastic hinge in plate girders with flat webs [18].
Fig. 11. Stress contour of Girders G13, G14 and G15.
corrugated web contributes to the shear capacity [7,8,20]. On the opposite, the Resal effect appearing in tapered BGCWs, from the authors’ view point, makes the interaction between shear and flexural behaviors questionable. Fig. 16 provides the Resal effect on prismatic and Case I tapered girders. As can be seen, the inclination of the flange in the tapered girder produces a vertical component that has a positive effect on the critical section’s strength for the graphed typology; additional information could be found in Ref. [9]. Hence, it could be imagined that by increasing the applied load, the resulting bending moment increases (without exceeding the flexural capacity), and hence, the vertical component arising from the Resal effect increases. This, however, affects the applied shear on the girder’s
cross-section. By varying the flange thickness (t f ), the ultimate shear load capacities of Case I tapered BGCWs were presented herein to investigate the above mentioned effect. As can be seen in Table 3, three flange thicknesses (30, 40 and 50 mm) were considered. As can be seen from Fig. 16, the Resal effect exists at section s–s where the bending moment exists, while it has no effect at the support where there is not bending moment. Accordingly, the stresses of both sections are checked in Table 4. It can be observed that effect of the flange slenderness ratio on the τul;FE =τy ratio is limited in both sections. This, however, indicates that the Resal effect on Case I panels is insignificant. On the other hand, the curves presenting the relationships between the applied load (P) against the mid-span deflection for girders G5, G14 and G23 (as sample results) are provided in Fig. 17. It can be observed that reducing the flange slenderness ratio (by using thicker flanges) increases noticeably the initial stiffness, while the
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maximum loads of the girders remain nearly unchanged. Overall, the same trend is observed for girders with different flange slenderness ratios. Generally, the girders exhibit linear elastic response followed by a sudden failure when the ultimate load is reached. Nevertheless, considerable residual strength appears after that. Regarding the stress distribution at the ultimate shear loads, the results showed that there are no obvious differences that take place by changing the flange slenderness ratios.
6. Examining the proposed design model with different studied parameters
inclination angle (γ) or reducing the flange slenderness ratio. In some cases shown in bold font in the table, the results of the design model are highly conservative. Almost of them represent BGCWs with very big flange inclination angles (i.e. γ ¼ 141). Otherwise, the design model provides suitable predictions. Nevertheless, it provides a lower bound for the shear strength of tapered BGCWs. On the other hand, the applicability of the design model for the full slenderness parameter range (greater than 0.6), was additionally explored. This was made here by varying the web thickness of the models G20, G23 and G26 from 4 mm to 14 mm with an increment of 2 mm. Accordingly, three groups (with a total of fifteen new models) were generated with a designation system of t f γ1 a=hw1 t w . Fig. 18 shows the relationship between the τul;FE =τy ratio against the
P [kN]
The accuracy of the design strength (τul;Prop ), presented in Eq. (1), for the tapered BGCWs was examined in this section respecting the different studied parameters; the aspect ratio of the web panel, flange inclination angle and the flange slenderness ratio. Accordingly, the ultimate stresses of the generated models were compared in Table 5 with that of the proposed equation (τul;FE =τul;Prop ). It can be seen that the design model provides more conservative predictions through increasing the a=hw1 ratio, raising the value of the
Mid-span deflection [mm] Fig. 13. Load-mid-span deflection for girders G13, G14 and G15.
Fig. 15. Stress contour of Girders G11, G14 and G17.
Fig. 14. Variation of τul;FE =τy with Tan ðγÞ for tapered BGCWs for different ratios of bf =2t f
M.F. Hassanein, O.F. Kharoob / Thin-Walled Structures 88 (2015) 119–128
127
Fig. 16. Resal effect on the shear strength of a girder: (a) loaded girder, (b) Prismatic girder and (c) Tapered girder of Case I.
Table 4 Resal effect on the critical section and section s–s.
G1 G10 G19 G2 G11 G20 G3 G12 G21 G4 G13 G22 G5 G14 G23 G6 G15 G24 G7 G16 G25 G8 G17 G26 G9 G18 G27
t f [mm]
30 40 50 30 40 50 30 40 50 30 40 50 30 40 50 30 40 50 30 40 50 30 40 50 30 40 50
bf =2t f
8.3 6.3 5.0 8.3 6.3 5.0 8.3 6.3 5.0 8.3 6.3 5.0 8.3 6.3 5.0 8.3 6.3 5.0 8.3 6.3 5.0 8.3 6.3 5.0 8.3 6.3 5.0
a=hw1
1.49 1.49 1.49 2.23 2.23 2.23 2.98 2.98 2.98 1.49 1.49 1.49 2.23 2.23 2.23 2.98 2.98 2.98 1.49 1.49 1.49 2.23 2.23 2.23 2.98 2.98 2.98
Critical section
Section s–s (Fig. 16)
τul;FE [MPa]
τul;FE =τy
τs s [MPa]
τs s =τy
135 137 138 142 148 145 139 140 156 138 135 146 142 146 148 157 163 167 154 147 157 176 164 179 203 216 228
0.66 0.67 0.67 0.69 0.72 0.71 0.68 0.68 0.76 0.67 0.66 0.71 0.69 0.71 0.72 0.77 0.79 0.82 0.75 0.72 0.76 0.86 0.80 0.88 0.99 1.05 1.11
107 108 109 97 102 99 81 82 91 98 96 104 80 83 84 66 68 70 97 92 99 78 73 80 52 56 59
0.52 0.53 0.53 0.47 0.50 0.48 0.39 0.40 0.44 0.48 0.47 0.51 0.39 0.40 0.41 0.32 0.33 0.34 0.47 0.45 0.48 0.38 0.36 0.39 0.26 0.27 0.29
parameter λs . The dashed line represents the design strength given by Eq. (1). It should be noticed that the change of t w results in λs ratios ranging from 0.546 to 1.756. Only one model (t w ¼p4ffiffiffimm) belongs to the third stage of the proposed design model (λs 4 2) for each series. From the figure, it can be seen that the design model is highly conservative in third stage for different angles of inclination. In spite that stage 3 of Eq. (1) requires modification to reduce its margin of safety, such modification is of practical insignificance, because these high slenderness parameters are not consistent with girders used in real bridges which do not use such small web thicknesses [5]. Accordingly, the design model may be considered suitable for the design of tapered BGCWs.
P [kN]
Girder
Mid-span deflection [mm] Fig. 17. Load-mid-span deflection for girders G5, G14 and G23.
Table 5 Comparison between ultimate and proposed shear stresses. Girder
λs
a=hw1
t f [mm]
G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 G15 G16 G17 G18 G19 G20 G21 G22 G23 G24 G25 G26 G27
1.143 1.143 1.143 1.170 1.170 1.170 1.196 1.196 1.196 1.143 1.143 1.143 1.170 1.170 1.170 1.196 1.196 1.196 1.143 1.143 1.143 1.170 1.170 1.170 1.196 1.196 1.196
1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98
30
γ3 8
11
14
40
8
11
14
50
8
11
14
τul;FE =τy
τul;Prop =τy
τul;FE =τul;Prop
0.66 0.69 0.68 0.67 0.69 0.77 0.75 0.86 0.99 0.67 0.72 0.68 0.66 0.71 0.79 0.72 0.80 1.05 0.67 0.71 0.76 0.71 0.72 0.82 0.76 0.88 1.11
0.67 0.67 0.67 0.65 0.65 0.65 0.63 0.63 0.63 0.67 0.67 0.67 0.65 0.65 0.65 0.63 0.63 0.63 0.67 0.67 0.67 0.65 0.65 0.65 0.63 0.63 0.63
0.99 1.04 1.02 1.03 1.07 1.18 1.19 1.36 1.56 1.00 1.09 1.03 1.02 1.09 1.22 1.13 1.26 1.66 1.01 1.06 1.14 1.10 1.11 1.26 1.21 1.38 1.76
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6.
7.
Fig. 18. Design model versus slenderness parameter.
7. Summary and conclusions Research on tapered bridge girders with corrugated webs (BGCWs) is very limited and is less developed than prismatic BGCWs which may attributed to the fact that tapered BGCWs are more complicated to analyse due to their continuously changing section properties. Therefore, this paper presents the nonlinear shear strength and behavior of the tapered BGCWs. It actually extends the fundamental behavior of such girders published recently by the current authors [10] with emphasis on Case I typology. Panels of Case I were found to be the most common case appearing commonly near to the intermediate supports of continuous bridges. The validity limit of the previously proposed design strengths [10] for the tapered BGCWs relating to the initial imperfection was examined first. Imperfection analyses were conducted by considering two initial imperfection amplitudes of hw1 =100 and hw1 =200; hw1 is the height of the long vertical edge of the web panel. The results showed that the proposed equation is valid with initial imperfections of hw1 =200. Nevertheless, its accuracy increases for initial imperfections similar to the web thickness (t w ) for cases of t w Z hw1 =200. Next, the paper investigated the effect of the aspect ratio of the web panel, different inclination angles and the flange slenderness on the behavior of the tapered BGCWs. Based on this parametric study, the following points may be drawn: 1. The shear failure was found to occur suddenly and results from buckling for such BGCWs with relatively small web thickness. After the maximum load was achieved, considerable residual strength remains after failure. 2. Typical shear failure mechanism was observed by varying the web panel aspect ratio, the angle of inclination or the flange slenderness. 3. The failure mode of the BGCWs does not show the propagation of shear plastic hinges (SPHs) at their top flanges similar to those appearing in plate girders with flat web plates. This is because the out-of-plane deformation of the corrugated webs is relatively limited as a result of their significant out-of-plane stiffness. So, no relative shear deformation exits between both flanges to form the SPHs. 4. The results showed that girders with the least inclination angle (i.e. γ ¼ 81) are not affected by the change in the aspect ratio of the web panel (a=hw1 ) ratio similar to the case of prismatic girders. Also, the results indicated that tapered BGCWs with such small values of γ have nearly the same τul;FE =τy ratios irrespective of the flange slenderness ratio. 5. For a particular ratio of flange slenderness (bf =2t f ), the results demonstrated that tapered BGCWs with larger shear spans (i.e.
8.
9.
girders with increased a=hw1 ratios) are more prone to shear deformations and, therefore, can relatively reach higher stresses relative to their plastic shear resistances (τul;FE =τy ) compared to girders with less a=hw1 ratio specially for girders with large angle of inclination (i.e. γ ¼ 141). Increasing the inclined flange angle (γ) was found to reduce the ultimate shear strength of the tapered BGCWs significantly, while the initial stiffness remains more or less the same. It was shown that the τul;FE =τy ratio increases for the tapered BGCWs with the increase of γ. The rate of increase becomes higher for girders with increased aspect ratio (a=hw ). This increase results from the reduction in the critical web depth (hwo ) which becomes more pronounced in girders with larger a=hw ratio. Also, girders with thick flanges (bf =2t f ¼ 5) were found to experience a greater increase in the τul;FE =τy ratio due to the more fixity developed at the juncture between the flanges and the webs leading to the increase of the shear strength of the girders. It was observed that the effect of the flange slenderness ratio on the τul;FE =τy ratio is limited in different sections of the tapered BGCWs indicating that the Resal effect on Case I panels is insignificant. The design model provides a lower bound for the shear strength of tapered BGCWs and, generally, it shows suitable predictions in almost practical cases.
References [1] Ikeda S, Sakurada M. Development of hybrid prestressed concrete bridges with corrugated steel web Plates. In: 30th conference on our world in concrete & structures, Singapore; 23–24 August, 2005. Online version: 〈http://cipremier. com/100030003〉. [2] Yi J, Gil H, Youm K, Lee H. Interactive shear buckling behavior of trapezoidally corrugated steel webs. Eng Struct 2008;30:1659–66. [3] Moon J, Yi J, Choi BH, Lee H. Shear strength and design of trapezoidally corrugated steel webs. J Constr Steel Res 2009;65:1198–205. [4] Eldib MEA. Shear buckling strength and design of curved corrugated steel webs for bridges. J Constr Steel Res 2009;65:2129–39. [5] Hassanein MF, Kharoob OF. Behavior of bridge girders with corrugated webs: (I) Real boundary conditions at the juncture of the web and flanges. Eng Struct 2013;57:554–64. [6] Hassanein MF, Kharoob OF. Behavior of bridge girders with corrugated webs: (II) Shear strength and design. Eng Struct 2013;57:544–53. [7] Hamilton RW. Behavior of welded girder with corrugated webs. PhD thesis. University of Maine; 1993. [8] Driver RG, Abbas HH, Sause R. Shear behavior of corrugated web bridge girders. J Struct Eng-ASCE 2006;132(2):195–203. [9] Bedynek A, Real E, Mirambell E. Tapered plate girders under shear: tests and numerical research. Eng Struct 2013;46:350–8. [10] Hassanein MF, Kharoob OF. Shear buckling behavior of tapered bridge girders with steel corrugated webs. Eng Struct 2014;74(2014):157–69. [11] ABAQUS standard user’s manual the Abaqus software is a product of Dassault Systèmes Simulia Corp., Providence, RI, USA Dassault Systèmes, Version 6.8, USA; 2008. [12] Sause R, Braxtan TN. Shear strength of trapezoidal corrugated steel webs. J Constr Steel Res 2011;67:223–36. [13] Elgaaly M, Hamilton RW, Seshadri A. Shear strength of beams with corrugated webs. J Struct Eng 1996;122(4):390–8. [14] EN 1993-1-1. Eurocode 3: design of steel structures—Part 1-1: General rules and rules for buildings. CEN; 2004. [15] Hassanein MF, Kharoob OF. Shear strength and behavior of transversely stiffened tubular flange plate girders. Eng Struct 2010;32(9):2617–30. [16] Hassanein MF, Kharoob OF. An extended evaluation for the shear behavior of hollow tubular flange plate girders. Thin Walled Struct 2012;56:88–102. [17] Alinia MM, Shakiba M, Habashi HR. Shear failure characteristics of steel plate girders. Thin Walled Struct 2009;47:1498–506. [18] Hassanein MF. Imperfection analysis of austenitic stainless steel plate girders failing by shear. Eng Struct 2010;32(3):704–13. [19] Marques L, Simões da Silva L, Rebelo C, Santiago A. Extension of EC3-1-1 interaction formulae for the stability verification of tapered beam-columns. J Constr Steel Res 2014;100:122–35. [20] Nie J-G, Zhu L, Tao M-X, Tang L. Shear strength of trapezoidal corrugated steel webs. J Constr Steel Res 2013;85:105–15.
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Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Linearly tapered bridge girder panels with steel corrugated webs near intermediate supports of continuous bridges M.F. Hassanein n, O.F. Kharoob Department of Structural Engineering, Faculty of Engineering, Tanta University, Tanta, Egypt
art ic l e i nf o
a b s t r a c t
Article history: Received 11 July 2014 Received in revised form 20 November 2014 Accepted 20 November 2014 Available online 24 December 2014
Nowadays, girders with corrugated webs are used in bridges as an efficient alternative to conventional girders with flat stiffened webs. Particularly in bridge girders with corrugated webs (BGCWs), the corrugated webs are the main elements for bearing the shear forces. Instead of prismatic BGCWs, tapered BGCWs are currently used mainly due to their structural efficiency, providing at the same time aesthetical appearance. Available literature shows that tapered BGCWs may be classified into four typologies. Among these typologies, Case I is the most common case appearing commonly near to the intermediate supports of continuous bridges. Accordingly, in this paper, the finite element (FE) method is employed to investigate the inelastic behavior of tapered BGCWs of Case I, following to the fundamental behavior of such girders published recently by the current authors. Previously validated 3-D nonlinear FE models are developed and used in this study. The paper seeks, first, at finding the validity limit of the previously proposed design strengths for the tapered BGCWs with respect to the girder’s initial imperfection. This is made by considering different initial imperfection amplitudes. Accordingly, it is found that the proposed equation is valid with initial imperfections of hw1 =200; hw1 is the height of the long vertical edge of the web panel. However, the accuracy increases for initial imperfections similar to the web thickness (t w ) for cases of t w Z hw1 =200. Then, it investigates the effect of the aspect ratio of the web panels, different flange inclination angles and flange slenderness ratios. Finally, the paper checks the authors’ previously proposed design model using the results of the generated parametric studies. Overall, the outcomes of this study are expected to provide more insight into the behavior of tapered BGCWs and enable accurate prediction of the shear capacity of this special type of BGCWs. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Tapered corrugated steel webs Shear strength Global buckling Interactive buckling Initial imperfection Resal effect
1. Introduction Girders, in conventional design, are subjected to significant levels of bending moment and thus failure typically occurs when the applied moment due to loading at the critical section exceeds its flexural capacity. Consequently, girders are typically designed to satisfy flexural limit state and then they are checked for shear limiting criteria. However, in the case of plate girders with slender webs, the web panel buckles at a relatively low value of the applied load. Therefore, the shear may control the design of such girders. To overcome the strength reduction associated with utilizing plate girders with slender webs in bridge construction, these flat webs are often reinforced with transversal stiffeners along their spans to increase their buckling strength. More recently, girders with steel corrugated webs have been used as structural members in Bridges; for example see Fig. 1(a) which is presenting the Ginzan-Miyuki Bridge in Japan [1]. As well known, corrugated web plates have
n
Corresponding author. Mobile: þ 201228898494; fax: þ 20403315860. E-mail address: [email protected] (M.F. Hassanein).
http://dx.doi.org/10.1016/j.tws.2014.11.021 0263-8231/& 2014 Elsevier Ltd. All rights reserved.
much higher buckling strengths compared with flat web plates owing to their noteworthy out-of-plane stiffness. Accordingly, the stiffeners are eliminated and the required web thickness is reduced [2–6] by using them as the girder web. On the other hand, due to the accordion effect [7,8], the flexural strength of such girders is entirely provided by their flanges while the shear strength is carried by their webs. Such effect is characterised by the negligible axial stiffness of the such corrugated webs in the longitudinal directions of the girders. As a result, there is no interaction between shear and flexural behaviours [7,8]. Consequently, assuming a constant shear stress in the corrugated webs of such girders is commonly accepted and then it is quantified it in terms of the average shear stress. On the other hand, tapered girders are currently used in bridges mainly due to their structural efficiency, providing at the same time aesthetical appearance. An example for the application of tapered bridge girder with steel corrugated webs (BGCWs) is given in Fig. 1(b). In a tapered bridge girder such as that forming the Hondani Bridge, shown in Fig. 1(b), three typologies of girder panels can be found besides a fourth one that can be found in other bridge systems; refer to Bedynek et al. [9]. That is based on the inclination of the flange and whether the flange is under
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Fig. 1. Bridge girders with corrugated webs; (a) Ginzan-Miyuki Bridge and (b) Hondani Bridge. (a) Prismatic bridge girders. (b) Parabolic tapered bridge girders.
s Sub-panel (fold) tw
Case I: Inclined flange under compression & tension field is developed in the short diagonal
c
α
hr b
d
b
d
w is the bigger of b and c
One corrugation wave (q)
Fig. 3. Corrugation configuration and geometric notation.
2. Background and design strength
Case II: Inclined flange under tension & tension field is developed in the long diagonal
Case III: Inclined flange under compression & tension field is developed in the long diagonal
Case IV: Inclined flange under tension & tension field is developed in the short diagonal
Fig. 2. Types of tapered plate girders with corrugated webs.
tension or compression as well as the direction of the developed tension field, which may appear on the short or on the long web diagonal [9]; see Fig. 2. Accordingly, to unveil the structural behavior of the tapered BGCWs, Hassanein and Kharoob [10] investigated their fundamental behavior. Nevertheless, there is still lack of information related to their shear behavior as recommended at the end of their paper. Accordingly, this paper extends the work presented in Ref. [10] by means of conducting several parametric studies. The webs of the currently considered girders are formed from trapezoidally corrugated steel web plates which are composed of a series of longitudinal and inclined sub-panels; see Fig. 3. This is because this type of corrugated webs is the most utilized in available bridges as can be seen in Fig. 1.
Because tapered BGCWs are used extensively as the main systems in modern bridges (Fig. 1(b)) without the availability of any source describing their shear buckling behavior, the current authors studied the fundamental behavior of such girders [10]. Research presented by Hassanein and Kharoob [10] focused, first, on the critical shear buckling stress (τcr ) of the corrugated webs of tapered BGCWs. This was made by carrying out elastic bifurcation buckling analyses using ABAQUS software [11] on isolated corrugated webs with simple and fixed boundary conditions. Webs in different typologies of tapered girders with steel corrugated webs were considered; see Fig. 2. To reflect the behavior of bridges, the corrugation dimensions of the considered corrugated webs were taken typical to those used previously in Shinkai and Matsnoki bridges. However, it was found that predicting τcr values for the tapered webs based on prismatic web (having the depth of the long vertical edge of the tapered web (hw1 )) calculations is not accurate. Therefore, critical buckling stresses (τcr;Prop ) for the tapered webs were proposed based on the stresses of prismatic webs, with different equation for each typology. The paper [10] was, then, extended to investigate the nonlinear shear strengths of the tapered BGCWs. This was made based on the verifications made by the same authors in Ref. [6]. The aspect ratio of the web panels (a=hw1 ) of the girders was 2.88, while the inclination angle was fixed to 7.1251; a and hw1 are the shear span and the height of the long vertical edge. The flange slenderness was, additionally, constant throughout the investigation, while five web slenderness ratios were taken into consideration. The available design shear strength formulas for prismatic girders (by Moon et al. [3] and Sause and Braxtan [12]) were compared with the FE shear strengths of the tapered BGCWs. Based on these comparisons, design strength (τul;Prop ), based on the equation of Moon et al. [3], for different tapered BGCWs typologies was proposed, as follows: 8 1:0 : λs r0:6 > > pffiffiffi < τul;Prop 0:6Þ : 0:6 o λs r 2 1 0:614ðλ s ð1Þ ¼ CT pffiffiffi > τy 1 > : 2 o λs : λ2 s
where τy stands for the shear yielding strength of the steel material, λs represents the shear buckling parameter of the corrugated webs and C T is taken as unity for cases I and II or as the ratio hwo =hw1 for cases III and IV; hwo =hw1 is the ratio between the height of the short
M.F. Hassanein, O.F. Kharoob / Thin-Walled Structures 88 (2015) 119–128
vertical edge of the web panel to that of the long edge. It is worth pointing out that the parameter λs uses, in its calculation, the τcr;Prop value of each typology; for more details refer to Ref. [10]. According to the dimensions of Shinkai bridge with different web thickness used in Ref. [10], almost of the models had a parameter λs less than 0.6 (16 out of 20 models). The rest of models had a parameter λs just bigger than 0.6 (only 4 models). 3. Goals and objectives With the increasing utilization of tapered BGCWs worldwide [1], a better understanding of the true behavior of such girders under shear loading becomes essential. Consequently, this paper extends the paper by Hassanein and Kharoob [10] with respect to bridges of Case I. Among the four typologies shown in Fig. 2, Panels of Case I are the most common case which appears frequently near to the intermediate supports of continuous bridges; see the two examples presented in Fig. 4. Accordingly, the objective of this paper is to provide additional data to engineers and scientific community on the strength and behavior of the tapered BGCWs of Case I under shear loads. As a result, the following new points (as suggested at the end of Ref. [10]) are added to literature for the first time: 1. The paper defines the validity limit of the previously proposed design strength [10] for the tapered BGCWs regarding the initial imperfection. This was made by generating FE models with different initial imperfections to be compared with the results of the proposed design strength. 2. Models with λs greater than 0.6 (belonging to the second part of Eq. (1)) using the limiting initial imperfections, as proposed from the previous point, are generated to deepen the understanding of their behavior. 3. The paper investigates the effect of the aspect ratio of the web panel (a=hw1 ) on the shear strength of the tapered BGCWs. 4. The paper expands the pool of available results by considering tapered BGCWs with different inclination angles (γ 3 ). 5. The paper goes into the effect of flange slenderness ratio on the shear strength and behavior of the tapered BGCWs. By doing so, the vertical component of the Resal effect changes. Accordingly, its effect on the behavior of the tapered BGCWs could be explored. 6. Finally, the paper checks the proposed design model [10] by using the results of the current parametric studies as well as the considered wide range of the slenderness parameter (i.e. λs 4 0:6). 4. Validation limit of the design strength It is well known that the amplitude and the shape of the initial web geometric imperfections play a significant role in the shear strength and behavior of the girders with corrugated webs [3,8,13]. Accordingly, an initial imperfection amplitude equalling to the thickness of the corrugated web (t w ), as suggested by Driver et al. [8], was considered by Hassanein and Kharoob [10] to propose their
121
design strength given herein by Eq. (1). The objective of this section is to provide the validity limit of Eq. (1) for the tapered BGCWs with regard to the initial imperfection. Accordingly, the generated girders (of different typologies) in Ref. [10] were reanalysed with two initial imperfection amplitudes of hw1 =100 (12.5 mm) and hw1 =200 (6.25 mm). The first positive Eigenmode which responds to the shear buckling mode was scaled to the above mentioned values in the nonlinear modelling. Two typical examples representing the Interactive (I) and Global (G) buckling modes for Case III models are presented herein in Fig. 5. One girder was, however, excluded because of its flexural failure (girder of Case II with web thickness of 14 mm). As can be noticed, the imperfections were linked in this analysis to the height of the long vertical edge of the web panel (hw1 ). Tables 1 and 2 show the results of those analyses and their comparison with the proposed design strength. From Table 1 it can be noticed that the FE ultimate shear load (V ul;FE ) of the tapered girders decreases with the increasing of initial geometric imperfections. Additionally, it could be observed that the girders under different initial imperfection amplitudes exhibited the same failure modes. On the other hand, the loading capacity of the webs at the descending stage is almost independent of the initial imperfections, as can be seen from Fig. 6. In this figure, the load–deflection relationships for Case III with web thickness of 8 mm are presented. Other typologies were not included in the figure because they provide similar results. This is attributed to the early occurrence of the web out-of-plane deformation due to the shear buckling before the shear yield can reduce the sensitivity of the loading capacity at the descending stage to the initial geometric imperfections. From Table 2 it can be observed that initial imperfection amplitudes of hw1 =100 provide unsafe results for tapered girders with small web thickness values (6 and 8 mm), while they are suitable for the bigger; 10, 12 and 14 mm. In contrast, hw1 =200 values provide conservative values for different typologies with different web thicknesses. However, by increasing the web thickness, the degree of conservatism of the design strength increases. Hence, it can finally be concluded that the proposed strength (Eq. (1)) is valid with initial imperfections of hw1 =200, but its accuracy increases for initial imperfections similar to the web thickness (t w ) for cases of t w Z hw1 =200.
5. Parametric study 5.1. Finite element model The finite element (FE) model presented by the authors in Ref. [10] was utilized in the current parametric study. Instead of Shinkai bridge corrugation used in Ref. [6], the Dole bridge corrugation dimensions (Fig. 7) were used in this paper to generate full-scale tapered BGCWs (Fig. 8) loaded under mid-span concentrated loads. To insure that the shear controls the failure modes of the tapered BGCWs, (1) the web thickness was chosen as 6 mm instead of 10 mm which is the actual web thickness of Dole bridge corrugation and (2) the loaded point was restrained
Fig. 4. Linearly tapered bridge girders with corrugated webs of Case I. (a) Kurobegawa Bridge. (b) Ohmi-Ohdori bridge.
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steel used was S355 according to EN 1993-1-1 [14], which has a yield (f y ) and an ultimate strength (f u ) of 355 MPa and 510 MPa, respectively. A bilinear elastic–plastic stress–strain curve with linear strain hardening was used to simulate the steel material.
Interactive (I) buckling mode: Case III-8mm
5.2. Input data Three-dimensional FE models, using ABAQUS [11] FE package, were performed on twenty seven tapered BGCWs covering the following parameters:
Close view
1. aspect ratio of the web panels (a=hw1 ); (1.49, 2.23 and 2.98), 2. angle of the inclined flanges (γ); (8, 11, 141), and 3. flange slenderness ratios (bf =2t f ); (5, 6.25 and 8.33).
Global (G) buckling mode: Case III-14mm
Buckling extends diagonally over the depth of the web Close view
Fig. 5. Typical buckling modes for: (a) interactive and (b) global buckling modes.
Table 1 FE results of the imperfection analyses. Type t w λs [mm]
Initial imperfection [mm] tw
Case I
Case II
Case III
Case IV
6 8 10 12 14 6 8 10 12 6 8 10 12 14 6 8 10 12 14
0.683 0.533 0.445 0.386 0.346 0.670 0.523 0.436 0.379 0.602 0.470 0.392 0.341 0.305 0.621 0.485 0.404 0.352 0.314
hw1 =100 (12.5 mm)
hw1 =200 (6.25 mm)
V ul;FE [kN]
Buckling mode
V ul;FE [kN]
Buckling mode
V ul;FE [kN]
Buckling mode
944 1318 1693 2144 2619 1097 1619 2010 2147 652 917 1146 1450 1686 634 894 1136 1397 1666
I G G G G I G G G I I G G G I I G G G
830 1211 1617 2128 2666 1065 1500 1992 2147 588 851 1106 1445 1688 559 829 1097 1384 1691
I G G G G I G G G I I G G G I I G G G
938 1374 1816 2306 2852 1094 1676 2023 2217 649 946 1224 1548 1798 630 925 1210 1506 1811
I G G G G I G G G I I G G G I I G G G
laterally. Initial geometric imperfections, based on the first positive shear buckling mode, were included in the nonlinear analysis of the BGCW with values of hw1 =200 as t w o hw1 =200; refer back to Section 4. The loads were applied using the modified RIKS method. S8R5 reduced integration thin shell elements were employed to discretise the models in the current nonlinear analyses. Simply supported boundary conditions were applied to end sections. The displacement history was applied to the FE models along their mid-span web points at the intersection with lower flanges. The steel material has been modelled as a von Mises material with isotropic hardening. The
The flange width (bf ) was fixed to 500 mm throughout the entire program. The webs of the girders were stiffened transversely at the supports and under the applied load. Out-standing plate stiffeners extending to the edge of the flanges of the plate girder were considered with a thickness of 25.4 mm all through the investigation. Unlike Ref. [10] where almost of the models had shear buckling parameters (λs ) less than 0.6, the values of λs of the current models were intentionally chosen to lay within second stage of the proposed pffiffiffi design model; i.e. 0:6 o λs r 2. This was made to increase the range of the available data for the tapered BGCWs in literature, as can be seen in Table 3. This table shows the dimensions considered in this study. The table also contains the ultimate shear strength (V ul;FE ) of the girders beside the maximum shear stress (τul;FE ) calculated with respect to the critical cross-section of the girders with the shorter depth (hwo ). To ensure that all the tapered BGCWs failed by shear limit state away from the interaction with the flexural limit state by means of the development of flexural plastic hinges, the ratios between the ultimate and the plastic bending moments (M ul;FE =M pl ) were computed and then added to Table 3; M pl ¼ bf t f F y ðhw1 þ t f Þ. From the table, it can be noticed that M ul;FE =M pl ratios vary between 0.21 to 0.58 indicating that the flexural capacity limit state was still away to take place. 5.3. Effect of aspect ratio of the web panels Plate girders with flat web plates require the use of a series of transverse stiffeners in their fabrication. This is because the flat web panel buckles at a relatively low value of the applied shear force. Consequently, the webs are often reinforced with transversal stiffeners along their spans to increase their buckling strengths. Accordingly, it is a well known fact that the aspect ratio of the web panel (a=hw ) plays a key role in the shear strength of such girders (see for example Refs. [15,16]). On the other hand, corrugated web plates have much higher buckling strengths compared with flat web plates because of their significant out-of-plane stiffness [2–6]. Hence, it is widely accepted that the aspect ratio of the web panels (a=hw ) has no effect on the ultimate shear stress (τul;FE ) of prismatic BGCWs. This could easily be ensured by revising the available design shear strengths [3,6,8,12] where the a=hw ratio does not exist. This was additionally checked by the current authors by adding several stiffeners (which is not the case in real bridges as can be seen in Figs. 1 and 5) within the shear spans of some prismatic girders modelled in Ref. [12] and the maximum change in the ultimate shear stress (τul;FE ) was about 2%. On the other hand, this effect did not yet investigated in the tapered BGCWs. Therefore, the effect of the a=hw1 ratio on the shear behavior of tapered BGCWs was investigated in this sub-section. However, tapered BGCWs with different a=hw1 ratios were generated herein by varying the span ratio (a) instead of dividing the shear span by using stiffeners to accord with bridge girders in reality.
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Table 2 Comparison between the FE results of the imperfection analyses of the proposed design strength. Type
λs
t w [mm]
Case I
6 8 10 12 14 6 8 10 12 6 8 10 12 14 6 8 10 12 14
Case II
Case III
0.683 0.533 0.445 0.386 0.346 0.670 0.523 0.436 0.379 0.602 0.470 0.392 0.341 0.305 0.621 0.485 0.404 0.352 0.314
tw
hw1 =100
hw1 =200
tw
hw1 =100
hw1 =200
197 206 212 223 234 229 253 251 224 136 143 143 151 150 132 140 142 145 149
173 189 202 222 238 222 234 249 224 123 133 138 151 151 116 130 137 144 151
195 215 227 240 255 228 262 253 231 135 148 153 161 161 131 145 151 157 162
0.96 1.00 1.03 1.09 1.14 1.12 1.23 1.23 1.09 0.66 0.70 0.70 0.74 0.73 0.64 0.68 0.69 0.71 0.73
0.84 0.92 0.99 1.08 1.16 1.08 1.14 1.21 1.09 0.60 0.65 0.67 0.73 0.74 0.57 0.63 0.67 0.70 0.74
0.95 1.05 1.11 1.17 1.24 1.11 1.28 1.23 1.13 0.66 0.72 0.75 0.79 0.79 0.64 0.71 0.74 0.77 0.79
Mid-span vertical deflection [mm] Fig. 6. Load–deflection relationship for Case III-8 mm with different initial imperfections.
s Sub-panel (fold) tw
430
α
220 430
370
430
τul;Prop τy
τul;FE =τy
Applied load [kN]
Case IV
τul;FE [MPa]
370
One corrugation wave ( q ) Fig. 7. Utilized corrugation dimensions (Dole Bridge [5]) in mm.
Fig. 8. Elevation of typical tapered BGCWs (case of γ ¼ 11 3 and a=hw1 ¼ 2:23)
0.95 1.00 1.00 1.00 1.00 0.94 1.00 1.00 1.00 0.64 0.64 0.64 0.64 0.64 0.62 0.64 0.64 0.64 0.64
In practice, the distance between vertical stiffeners is much greater than the web depth. Therefore, three ratios of a=hw1 greater than unity were considered; 1.49, 2.23 and 2.98. It can be seen from Table 3 that increasing the ratio of a=hw1 of the girder leads to a considerable decrease in the ultimate shear strength (V ul;FE ). The load against mid-span vertical deflection for selected girders is provided in Fig. 9 as sample results. These girders (G13, G14 and G15) had an inclination angle of 111. This figure shows that both the load and the initial shear stiffness considerably decrease as the aspect ratio increases. It can additionally be observed that the failure is sudden and results from buckling for such BGCWs with relatively small web thickness. After the maximum load was achieved, considerable residual strength remains after failure. On the other hand, the variation of τul;FE =τy ratio with the aspect ratio for the tapered BGCWs for different flange slenderness ratios (bf =2t f ) is given in Fig. 10. As can be noticed, the τul;FE =τy ratios of the girders with the least inclination angle (i.e. γ ¼ 81) seem not to be affected by the change in the a=hw1 ratio similar to the case of prismatic girders. For a particular value of bf =2t f ratio, this figure also demonstrates that tapered BGCWs with larger shear spans (i.e. girders with increased a=hw1 ratios) are more prone to shear deformations and, therefore, can relatively reach higher stresses with regard to their plastic shear resistances (τul;FE =τy ) compared to girders with less a=hw1 ratio specially for girders with large angle of inclination (i.e. γ ¼ 141). This is simply attributed to the very small web depth of the critical section of the girders (hwo ) compared to its length in the girders with smaller angles of inclinations; see Table 3. Fig. 11 shows the stress distribution (of the middle surface of the webs) captured at the ultimate load of girders G13, G14 and G15, as sample results. In the figure, regions with the gray contour (indicated by arrows in the figure) represent the locations exceeding the yield strength of the material (355 MPa), while the red, just surrounding the gray contour, shows the regions which have just yielded. As can be seen, the plasticity spreads near the shorter edge of the shear panels as the shear stress is maximized. Also, it should be noted that the failure mode of the BGCWs does not show the propagation of shear plastic hinges (SPHs) at their top flanges similar to those appearing in plate girders with flat web plates (see Ref. [9]). It was found [15,17] that the SPHs (Fig. 12) develop as a result of the differential shear deformation between the top and bottom flanges. Fig. 12 was taken from Ref. [18] for the
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Table 3 Full details of the current tapered BGCWs. Girder
hwo [mm]
hw1 [mm]
G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 G15 G16 G17 G18 G19 G20 G21 G22 G23 G24 G25 G26 G27
1700 1475 1250 1528 1217 905 1352 953 554 1700 1475 1250 1528 1217 905 1352 953 554 1700 1475 1250 1528 1217 905 1352 953 554
2150
γ3 8
t f [mm]
a [mm]
a=hw1
λs
M ul;FE =M pl
V ul;FE [kN]
τul;FE [MPa]
30
3200 4800 6400 3200 4800 6400 3200 4800 6400 3200 4800 6400 3200 4800 6400 3200 4800 6400 3200 4800 6400 3200 4800 6400 3200 4800 6400
1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98
1.143 1.143 1.143 1.170 1.170 1.170 1.196 1.196 1.196 1.143 1.143 1.143 1.170 1.170 1.170 1.196 1.196 1.196 1.143 1.143 1.143 1.170 1.170 1.170 1.196 1.196 1.196
0.38 0.53 0.58 0.35 0.43 0.48 0.35 0.42 0.38 0.29 0.41 0.44 0.26 0.33 0.37 0.25 0.30 0.30 0.24 0.32 0.39 0.23 0.27 0.30 0.21 0.26 0.25
1374 1255 1042 1264 1036 852 1252 1008 675 1399 1312 1052 1241 1065 883 1189 939 716 1408 1280 1172 1342 1078 909 1271 1026 759
135 142 139 138 142 157 154 176 203 137 148 140 135 146 163 147 164 216 138 145 156 146 148 167 157 179 228
11
14
8
40
11
14
8
50
11
P [kN]
14
Mid-span deflection [mm] Fig. 9. Load-mid-span deflection for girders G13, G14 and G15.
first author. As the flat web plate buckles, the top flange undergoes downward displacement greater than that of the bottom. This downward displacement, however, increases as the load increases leading to the development of diagonal tension field. Once the web has yielded, failure of the steel plate girder occurs when plastic hinges are formed in the flange due to the excessive downward displacement compared to that of the bottom. This, however, not the case in BGCWs as they possess significant out-ofplane stiffness. Accordingly, the out-of-plane deformation of the corrugated webs is relatively limited. So, no relative shear deformation exits between both flanges to form the SPHs. 5.4. Effect of angle of the inclined flange The influence of the inclined flange angle (γ) on the ultimate shear strength (V ul;FE ) and the load-mid-span deflection response of tapered BGCWs under shear loading was additionally studied. Three values of γ (8, 11 and 141) were used confirming that the taper ratio (hw1 =hwo ) of the girders does not exceed 4 which was found to cover a large proportion of existing structures [19]. Herein, the maximum value of the taper ratio was 3.88 for the case of γ ¼ 141 and a ¼ 6400 mm. The applied load-mid-span vertical deflection relationships for sample results are provided in Fig. 13. This figure shows these relationships for
girders G11, G14 and G17 which have inclined flange angles of 8, 11 and 141, respectively. It can be seen that increasing the inclined flange angle (γ) significantly reduces the ultimate shear strength of the tapered BGCWs, while the initial stiffness remains more or less the same. Additionally, it can be seen that all tapered BGCWs behave in a brittle manner irrespective of the value of γ with a considerable residual strength remaining after failure. On the other hand, the influence of the flange angle of inclination on the normalized shear strength (τul;FE =τy ) of tapered BGCWs is shown in Fig. 14 for different flange slenderness ratios. In this figure, the variation of the τul;FE =τy ratio is presented with respect to Tan ðγÞ. It is worth pointing out that tapered BGCWs with small values of γ (i.e. γ ¼ 81) have nearly the same τul;FE =τy ratios irrespective of the flange slenderness and the web aspect ratio. Also, Fig. 14 shows an increase in the τul;FE =τy ratio for tapered BGCWs with the increase of γ. However, the rate of increase becomes higher for girders with increased aspect ratio (a=hw ). This increase results from the reduction in the critical web depth (hwo ) which becomes more pronounced in girders with larger a=hw ratio. The comparison between the relationships of Fig. 14 reveals also that girders with thick flanges (bf =2t f ¼ 5) experience a greater increase in the τul;FE =τy ratio. This observation may be attributed to the fact that thicker flanges provide more fixity at the juncture between the flanges and the webs leading to the increase of the shear strength of the girders. Fig. 15 compares the stress distribution at the ultimate shear loads of the same selected girders presented previously in Fig. 13 (G11, G14 and G17). As can be seen, the portions exceeding the yield strength of the material, enveloped by rectangles in the figure, extend in a wider range from the shear span by decreasing the angle of inclination. By decreasing the value of γ, the yielded portions become concentrated near the end supports where the shear stress is maximized. 5.5. Effect of flange slenderness ratio It was shown in Section 1 that there is no interaction between shear and flexural behaviors of prismatic BGCWs. Hence, the flanges of prismatic BGCWs mainly contribute to the moment capacity, while the
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Fig. 10. Variation of τul;FE =τy with a=hw1 for tapered BGCWs for different ratios of bf =2t f
SPH
Fig. 12. Developed shear plastic hinge in plate girders with flat webs [18].
Fig. 11. Stress contour of Girders G13, G14 and G15.
corrugated web contributes to the shear capacity [7,8,20]. On the opposite, the Resal effect appearing in tapered BGCWs, from the authors’ view point, makes the interaction between shear and flexural behaviors questionable. Fig. 16 provides the Resal effect on prismatic and Case I tapered girders. As can be seen, the inclination of the flange in the tapered girder produces a vertical component that has a positive effect on the critical section’s strength for the graphed typology; additional information could be found in Ref. [9]. Hence, it could be imagined that by increasing the applied load, the resulting bending moment increases (without exceeding the flexural capacity), and hence, the vertical component arising from the Resal effect increases. This, however, affects the applied shear on the girder’s
cross-section. By varying the flange thickness (t f ), the ultimate shear load capacities of Case I tapered BGCWs were presented herein to investigate the above mentioned effect. As can be seen in Table 3, three flange thicknesses (30, 40 and 50 mm) were considered. As can be seen from Fig. 16, the Resal effect exists at section s–s where the bending moment exists, while it has no effect at the support where there is not bending moment. Accordingly, the stresses of both sections are checked in Table 4. It can be observed that effect of the flange slenderness ratio on the τul;FE =τy ratio is limited in both sections. This, however, indicates that the Resal effect on Case I panels is insignificant. On the other hand, the curves presenting the relationships between the applied load (P) against the mid-span deflection for girders G5, G14 and G23 (as sample results) are provided in Fig. 17. It can be observed that reducing the flange slenderness ratio (by using thicker flanges) increases noticeably the initial stiffness, while the
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maximum loads of the girders remain nearly unchanged. Overall, the same trend is observed for girders with different flange slenderness ratios. Generally, the girders exhibit linear elastic response followed by a sudden failure when the ultimate load is reached. Nevertheless, considerable residual strength appears after that. Regarding the stress distribution at the ultimate shear loads, the results showed that there are no obvious differences that take place by changing the flange slenderness ratios.
6. Examining the proposed design model with different studied parameters
inclination angle (γ) or reducing the flange slenderness ratio. In some cases shown in bold font in the table, the results of the design model are highly conservative. Almost of them represent BGCWs with very big flange inclination angles (i.e. γ ¼ 141). Otherwise, the design model provides suitable predictions. Nevertheless, it provides a lower bound for the shear strength of tapered BGCWs. On the other hand, the applicability of the design model for the full slenderness parameter range (greater than 0.6), was additionally explored. This was made here by varying the web thickness of the models G20, G23 and G26 from 4 mm to 14 mm with an increment of 2 mm. Accordingly, three groups (with a total of fifteen new models) were generated with a designation system of t f γ1 a=hw1 t w . Fig. 18 shows the relationship between the τul;FE =τy ratio against the
P [kN]
The accuracy of the design strength (τul;Prop ), presented in Eq. (1), for the tapered BGCWs was examined in this section respecting the different studied parameters; the aspect ratio of the web panel, flange inclination angle and the flange slenderness ratio. Accordingly, the ultimate stresses of the generated models were compared in Table 5 with that of the proposed equation (τul;FE =τul;Prop ). It can be seen that the design model provides more conservative predictions through increasing the a=hw1 ratio, raising the value of the
Mid-span deflection [mm] Fig. 13. Load-mid-span deflection for girders G13, G14 and G15.
Fig. 15. Stress contour of Girders G11, G14 and G17.
Fig. 14. Variation of τul;FE =τy with Tan ðγÞ for tapered BGCWs for different ratios of bf =2t f
M.F. Hassanein, O.F. Kharoob / Thin-Walled Structures 88 (2015) 119–128
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Fig. 16. Resal effect on the shear strength of a girder: (a) loaded girder, (b) Prismatic girder and (c) Tapered girder of Case I.
Table 4 Resal effect on the critical section and section s–s.
G1 G10 G19 G2 G11 G20 G3 G12 G21 G4 G13 G22 G5 G14 G23 G6 G15 G24 G7 G16 G25 G8 G17 G26 G9 G18 G27
t f [mm]
30 40 50 30 40 50 30 40 50 30 40 50 30 40 50 30 40 50 30 40 50 30 40 50 30 40 50
bf =2t f
8.3 6.3 5.0 8.3 6.3 5.0 8.3 6.3 5.0 8.3 6.3 5.0 8.3 6.3 5.0 8.3 6.3 5.0 8.3 6.3 5.0 8.3 6.3 5.0 8.3 6.3 5.0
a=hw1
1.49 1.49 1.49 2.23 2.23 2.23 2.98 2.98 2.98 1.49 1.49 1.49 2.23 2.23 2.23 2.98 2.98 2.98 1.49 1.49 1.49 2.23 2.23 2.23 2.98 2.98 2.98
Critical section
Section s–s (Fig. 16)
τul;FE [MPa]
τul;FE =τy
τs s [MPa]
τs s =τy
135 137 138 142 148 145 139 140 156 138 135 146 142 146 148 157 163 167 154 147 157 176 164 179 203 216 228
0.66 0.67 0.67 0.69 0.72 0.71 0.68 0.68 0.76 0.67 0.66 0.71 0.69 0.71 0.72 0.77 0.79 0.82 0.75 0.72 0.76 0.86 0.80 0.88 0.99 1.05 1.11
107 108 109 97 102 99 81 82 91 98 96 104 80 83 84 66 68 70 97 92 99 78 73 80 52 56 59
0.52 0.53 0.53 0.47 0.50 0.48 0.39 0.40 0.44 0.48 0.47 0.51 0.39 0.40 0.41 0.32 0.33 0.34 0.47 0.45 0.48 0.38 0.36 0.39 0.26 0.27 0.29
parameter λs . The dashed line represents the design strength given by Eq. (1). It should be noticed that the change of t w results in λs ratios ranging from 0.546 to 1.756. Only one model (t w ¼p4ffiffiffimm) belongs to the third stage of the proposed design model (λs 4 2) for each series. From the figure, it can be seen that the design model is highly conservative in third stage for different angles of inclination. In spite that stage 3 of Eq. (1) requires modification to reduce its margin of safety, such modification is of practical insignificance, because these high slenderness parameters are not consistent with girders used in real bridges which do not use such small web thicknesses [5]. Accordingly, the design model may be considered suitable for the design of tapered BGCWs.
P [kN]
Girder
Mid-span deflection [mm] Fig. 17. Load-mid-span deflection for girders G5, G14 and G23.
Table 5 Comparison between ultimate and proposed shear stresses. Girder
λs
a=hw1
t f [mm]
G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 G15 G16 G17 G18 G19 G20 G21 G22 G23 G24 G25 G26 G27
1.143 1.143 1.143 1.170 1.170 1.170 1.196 1.196 1.196 1.143 1.143 1.143 1.170 1.170 1.170 1.196 1.196 1.196 1.143 1.143 1.143 1.170 1.170 1.170 1.196 1.196 1.196
1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98 1.49 2.23 2.98
30
γ3 8
11
14
40
8
11
14
50
8
11
14
τul;FE =τy
τul;Prop =τy
τul;FE =τul;Prop
0.66 0.69 0.68 0.67 0.69 0.77 0.75 0.86 0.99 0.67 0.72 0.68 0.66 0.71 0.79 0.72 0.80 1.05 0.67 0.71 0.76 0.71 0.72 0.82 0.76 0.88 1.11
0.67 0.67 0.67 0.65 0.65 0.65 0.63 0.63 0.63 0.67 0.67 0.67 0.65 0.65 0.65 0.63 0.63 0.63 0.67 0.67 0.67 0.65 0.65 0.65 0.63 0.63 0.63
0.99 1.04 1.02 1.03 1.07 1.18 1.19 1.36 1.56 1.00 1.09 1.03 1.02 1.09 1.22 1.13 1.26 1.66 1.01 1.06 1.14 1.10 1.11 1.26 1.21 1.38 1.76
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6.
7.
Fig. 18. Design model versus slenderness parameter.
7. Summary and conclusions Research on tapered bridge girders with corrugated webs (BGCWs) is very limited and is less developed than prismatic BGCWs which may attributed to the fact that tapered BGCWs are more complicated to analyse due to their continuously changing section properties. Therefore, this paper presents the nonlinear shear strength and behavior of the tapered BGCWs. It actually extends the fundamental behavior of such girders published recently by the current authors [10] with emphasis on Case I typology. Panels of Case I were found to be the most common case appearing commonly near to the intermediate supports of continuous bridges. The validity limit of the previously proposed design strengths [10] for the tapered BGCWs relating to the initial imperfection was examined first. Imperfection analyses were conducted by considering two initial imperfection amplitudes of hw1 =100 and hw1 =200; hw1 is the height of the long vertical edge of the web panel. The results showed that the proposed equation is valid with initial imperfections of hw1 =200. Nevertheless, its accuracy increases for initial imperfections similar to the web thickness (t w ) for cases of t w Z hw1 =200. Next, the paper investigated the effect of the aspect ratio of the web panel, different inclination angles and the flange slenderness on the behavior of the tapered BGCWs. Based on this parametric study, the following points may be drawn: 1. The shear failure was found to occur suddenly and results from buckling for such BGCWs with relatively small web thickness. After the maximum load was achieved, considerable residual strength remains after failure. 2. Typical shear failure mechanism was observed by varying the web panel aspect ratio, the angle of inclination or the flange slenderness. 3. The failure mode of the BGCWs does not show the propagation of shear plastic hinges (SPHs) at their top flanges similar to those appearing in plate girders with flat web plates. This is because the out-of-plane deformation of the corrugated webs is relatively limited as a result of their significant out-of-plane stiffness. So, no relative shear deformation exits between both flanges to form the SPHs. 4. The results showed that girders with the least inclination angle (i.e. γ ¼ 81) are not affected by the change in the aspect ratio of the web panel (a=hw1 ) ratio similar to the case of prismatic girders. Also, the results indicated that tapered BGCWs with such small values of γ have nearly the same τul;FE =τy ratios irrespective of the flange slenderness ratio. 5. For a particular ratio of flange slenderness (bf =2t f ), the results demonstrated that tapered BGCWs with larger shear spans (i.e.
8.
9.
girders with increased a=hw1 ratios) are more prone to shear deformations and, therefore, can relatively reach higher stresses relative to their plastic shear resistances (τul;FE =τy ) compared to girders with less a=hw1 ratio specially for girders with large angle of inclination (i.e. γ ¼ 141). Increasing the inclined flange angle (γ) was found to reduce the ultimate shear strength of the tapered BGCWs significantly, while the initial stiffness remains more or less the same. It was shown that the τul;FE =τy ratio increases for the tapered BGCWs with the increase of γ. The rate of increase becomes higher for girders with increased aspect ratio (a=hw ). This increase results from the reduction in the critical web depth (hwo ) which becomes more pronounced in girders with larger a=hw ratio. Also, girders with thick flanges (bf =2t f ¼ 5) were found to experience a greater increase in the τul;FE =τy ratio due to the more fixity developed at the juncture between the flanges and the webs leading to the increase of the shear strength of the girders. It was observed that the effect of the flange slenderness ratio on the τul;FE =τy ratio is limited in different sections of the tapered BGCWs indicating that the Resal effect on Case I panels is insignificant. The design model provides a lower bound for the shear strength of tapered BGCWs and, generally, it shows suitable predictions in almost practical cases.
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