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Intermediate transverse stiffener requirements of high-strength steel plate girders considering postbuckling capacity
Engineering Structures 196 (2019) 109289
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Intermediate transverse stiffener requirements of high-strength steel plate girders considering postbuckling capacity
T
Yong Xiaoa,b, Xuan Yi Xuea, , Fei Fei Suna,c, Guo Qiang Lia,c ⁎
a
College of Civil Engineering, Tongji University, Shanghai 200092, China Chongqing Steel Structure Limited Company, Chongqing 400000, China c State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China b
ARTICLE INFO
ABSTRACT
Keywords: High-strength steel Intermediate transverse stiffener Plate girder Postbuckling capacity AASHTO LRFD Out-of-plane deflection
The postbuckling capacity of HSS plate girders is larger than that of mild steel plate girders because of the differences in material properties. Hence, to fully develop the postbuckling capacity, the intermediate transverse stiffener requirement of HSS plate girders could be different from that of mild steel plate girders. The plate girders concerned in this study are without a longitudinal stiffener. To investigate the aforementioned issue, the minimum moment of inertia requirements of intermediate transverse stiffeners in HSS and mild steel plate girders are studied through the finite element analysis (FEA). After the comparison between the FEA results and the results from the design equations in AASHTO LRFD, it states that the design equations in AASHTO LRFD could not be directly used in the design of intermediate transverse stiffeners in HSS plate girders. Hence, based on the FEA results, a new equation is proposed to predict the minimum moment of inertia requirement of intermediate transverse stiffeners in HSS plate girders. Furthermore, because of the difference in the resistant capacity between HSS and mild steel plate girders, the maximum additional out-of-plane deflection of the intermediate transverse stiffener in HSS plate girders corresponding to the ultimate shear resistance is significantly larger than that of the intermediate transverse stiffener in mild steel plate girders. Hence, there is a noteworthy adverse effect caused by the additional out-of-plane deflection of the intermediate transverse stiffener on the shear resistance of HSS plate girders with the maximum limit value set.
1. Introduction Depending on the development of the postbuckling capacity in plate girders, there are differences in the requirement of the intermediate transverse stiffener. When the postbuckling capacity of plate girders is not considered, the purpose of the intermediate transverse stiffener is to improve the local buckling resistance of the web panel. Therefore, intermediate transverse stiffeners are designed to have a proper moment of inertia which are able to maintain the nodal line when the elastic buckling occurs in the web panel. However, when the postbuckling capacity is taken into account, the function of the intermediate transverse stiffeners could be more complicated. To clarify this issue, Basler [1] proposed a theory of intermediate transvers stiffeners which states that a compressed axial force caused by the tension field action develops on the intermediate transverse stiffener. After that, Stanway et al. investigated the behaviour of the intermediate transverse stiffener attached on a web plate and proposed a design model [2,3]. Xie et al. conducted a numerical study of the intermediate transverse stiffener
⁎
and investigated the stress distribution at the postbuckling stage [4]. Kim et al proposed a comprehensive theoretical study of the intermediate transverse stiffener requirements during the postbuckling stage [5,6]. Furthermore, Lee et al. introduced some experimental, numerical and theoretical studies of the intermediate transverse stiffener [7,8]. A new design rule of the intermediate transverse stiffener was subsequently proposed by these authors [9]. Based on the above studies, a general agreement was that there is no the aforementioned compressed axial force developing in the intermediate transverse stiffener during the postbuckling stage. However, it is uncertain as these studies were purely based on the initial investigations on mild steel plate girders. Given the rapid advancements in thermal and mechanical processing procedures for structural steel production, high-strength steel (HSS) is increasingly used in components and structures [10,11]. It is worth noting that in general, the steel whose yield strength is larger than 460 MPa could be considered as HSS. In order to promote the use of HSS, recently, Xiao et al. conducted a series of numerical studies of HSS plate girders. It has proven that because of the difference in the
Corresponding author. E-mail address: [email protected] (X.Y. Xue).
https://doi.org/10.1016/j.engstruct.2019.109289 Received 19 March 2019; Received in revised form 30 May 2019; Accepted 7 June 2019 Available online 12 June 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.
Engineering Structures 196 (2019) 109289
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Nomenclature
Vu,
E Fyw Fcrs
a bs bf bt b d0 hw k L ts tf tw ρt µ
Fys It,FEA It1,AASHTO It2,AASHTO J Vu Vr
elastic modulus yield strength of the web panel local buckling stress for the stiffener Fcrs = 0.31E/(bt/ ts)2 ≤ Fys yield strength of the intermediate transverse stiffener moment of inertia of intermediate transverse stiffeners predicted by FEA moment of inertia of intermediate transverse stiffeners predicted by Eq. (1) in AASHTO LRFD moment of inertia of intermediate transverse stiffeners predicted by Eq. (2) in AASHTO LRFD stiffener bending rigidity parameter ultimate shear resistance reduced shear resistance
material property between mild steel and HSS, the postbuckling capacity of HSS plate girders is much larger than that of mild steel plate girders [12]. Hence, the intermediate transverse stiffener requirements of HSS plate girders can be different from that of mild steel plate girders. However, the resistant properties and deformation of the transverse stiffener are not concerned as long as it is rigid enough to develop the full postbuckling capacity of plate girders [12]. To consider the postbuckling capacity of HSS plate girders in the practical design, it is necessary to investigate the intermediate transverse stiffener requirement to develop the full postbuckling capacity of HSS plate girders. This study aims to clarify the aforementioned issue through the finite element analysis (FEA). In Section 2, a numerical modelling method was introduced. To investigate the intermediate transverse stiffener requirement for developing the postbuckling capacity of HSS plate girders, the sizes and dimensions of specimens in FEA were determined. In Section 3, the failure modes of HSS plate girders with different intermediate transverse stiffeners were identified. The difference in the intermediate transverse stiffener requirements between mild steel and HSS plate girders was compared. In Section4, based on the FEA results, a new equation was proposed to predict the minimum moment of inertia requirement of the intermediate transverse stiffener in HSS plate girders. Then, the design equations in AASHTO LRFD and the new equation proposed in this study were validated. In Section 5, the influence caused by the differences in the resistant properties of HSS and mild steel plate girders on the requirement of intermediate
max
ultimate shear resistance of the plate girder with the thickest intermediate transverse stiffener distance between the transverse stiffeners width of intermediate transverse stiffeners width of the flange for symmetric transverse stiffener, bt = 2bs the smaller of a and hw smaller of the adjacent web panel widths depth of the web panel shear buckling coefficient length of the plate girder thickness of the intermediate transverse stiffener thickness of the flange thickness of the web panel the larger of Fyw/Fcrs and 1 poisson’s ratio
transverse stiffeners was discussed. Furthermore, the out-of-plane deflection of intermediate transverse stiffeners in HSS plate girders was studied. 2. Numerical study 2.1. Numerical modelling method In order to investigate the intermediate transverse stiffener properties, a full plate girder nonlinear finite element model is proposed, as shown in Fig. 1. A concentrated load is applied at the top flange in the mid-span area. Two pairs of parallel intermediate transverse stiffeners are selected as the tested stiffeners, as shown in Fig. 1. There are two reference points which are coupled with the end-face of the plate girder. Then, to simulate the simply supported boundary condition, the degrees-of-freedom of reference points are limited, as listed in Table 1. The directions 1, 2, 3 indicate the x, y, z directions, separately. To avoid the occurrence of lateral-torsional buckling, the out-of-plane freedom of the compressed flange is restrained. In practical structures, the out-ofplane freedom of compressed flange is usually constrained by other components, such as floors. Hence, the aforementioned boundary condition assumption is consistent with the actual engineering condition. The initial imperfection of the plate girder is included in the FEA of this study. After the finite element buckling analysis of the ideal
Concentrated load
Tested stiffener
RF2
Tested stiffener RF1
2 3 1 Fig. 1. Full plate girder finite element model.
2
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Table 1 Boundary conditions of finite element models (free: 1, fixed: 0). Reference points
U1
U2
U3
UR1
UR2
UR3
RF1 RF2
0 1
0 0
0 0
0 0
1 1
1 1
RF1 denotes the reference point 1.
component, the first buckling mode is identified, as shown in Fig. 2. Then, the first buckling mode is selected to include the initial imperfection. Given that the displacement hw/100 is the largest permitted initial displacement according to the AASHTO Bridge Welding Code [13], hw/100 is introduced in the FEA as the largest out-of-plane displacement. 2.2. Idealized material model
Fig. 3. Idealized material models of Q235, Q345, Q550, and Q690 steels.
To investigate the difference in the requirement of the intermediate transverse stiffener in mild steel and HSS plate girders, Q235/Q345/ Q550/Q690 steels are considered in the FEA. The steels Q235/Q345 are mild steels and the steels Q550/Q690 are HSS. For the stress-strain behaviour of HSS, there is no clear yield point nor distinct yield plateau. Based on FEA results of HSS and mild steel plate girders introduced in [12], it was suggested that because of the aforementioned stress-strain behaviour, the strain hardening effect has barely any influence on the postbuckling capacity for mild steel plate girders. However, the strain hardening effect does have a clear strengthening effect on the postbuckling capacity of HSS plate girders. To take into account of this effect, a multilinear material model considering the strain hardening effect was introduced to carry out the FEA, as shown in Fig. 3. The multilinear model is extensively used for strength analyses of the HSS plate girders [14,15]. In this study, the FEA is conducted by using ABAQUS FE software. The S4R element which is a four node reduced integration quadrilateral shell element is selected to investigate the intermediate transverse stiffener requirement of mild steel and HSS plate girders. The S4R element in ABAQUS contains three rotational and three translational degrees-of-freedom per node, which means the membrane and flexural behaviours can be simulated by the S4R element. Hence, to investigate the properties of the intermediate transverse stiffeners in HSS plate girders, the elastic, plastic, and large strain behaviours can be included in the numerical analyses.
study. To avoid the confusion, the number used in Table 2 is consistent with that in [8]. Multilinear material model considering the strain hardening effect is chosen to conduct the FEA. The yield strength of the steel is 298.1 MPa. After FEA, it is clear to see that the ultimate shear resistances of the FEA Vu,FEA agree well with that of the experiment Vu,test, as shown in Table 2. Furthermore, the deformed shape of PG3 FE model is similar with that of PG3 specimen in the experiment, as shown in Fig. 4. To quantify the maximum out-of-plane deflection of the web panel along the intermediate transverse stiffener, the maximum additional out-of-plane deflection vs. V/Vu curves of the PG3 specimen and the PG3 FE model are compared, as shown in Fig. 5. It indicates that the maximum additional out-of-plane deflection vs. V/Vu curve of the PG3 FE model agrees well with that of the PG3 specimen. Hence, after the aforementioned careful validation, the numerical modelling method proposed in this study could be used to investigate the requirements of intermediate transverse stiffeners for HSS plate girders considering the postbuckling capacity. 2.4. Determination of specimen dimensions In order to prevent the buckling failure of the transverse stiffeners at the mid-span and end post, the transverse stiffener width bs = 97.5 mm and the transverse stiffener thickness ts = 20 mm are determined. The transverse stiffener width bs = 97.5 mm = (bf -tw)/2. Based on the results of the numerical study conducted in [12], the transverse stiffener with width bs = 97.5 mm and the thickness ts = 20 mm is rigid enough to prevent the buckling failure mode. For the tested intermediate
2.3. Validation of numerical modelling method The experiment conducted by Lee et al. is selected [8] to validate the accuracy of the numerical modelling method introduced in this
Fig. 2. First buckling mode.
3
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Table 2 Dimensions and section properties of specimens. Model
hw/mm
a/mm
tf/mm
tw/mm
bs/mm
ts/mm
Vu,test/kN
Vu,FEA/kN
Vu,FEA/Vu,test
PG2 PG3 PG5
600 600 600
600 600 450
15 15 15
3.2 4 3.2
40 30 45
3.2 4 3.2
204.5 323.5 241.6
200.2 312.5 228.7
0.98 0.97 0.95
transverse stiffeners which are in the absence of an axial load, the transverse stiffener width bs = 40 mm and the transverse stiffener thickness is varied to investigate the minimum moment of inertia requirement. The dimensions of pate girders are shown in Table 3 and Fig. 6, where the P-1.5-750 denotes that the aspect ratio a/hw = 1.5 and the hw = 750. It is easy to see that different ratios hw/tw and a/hw are considered. Based on the conclusion in [12], the plate girders in this study are assumed to have a clear postbuckling capacity. Furthermore, to prevent the influence of the bending moment on the shear resistance, the flanges of all specimens are strengthened. Then, the bending moment resistance of the flanges Mf is always larger than the bending moment produced by the external loading, which means the influence of the bending moment on the shear resistance could be neglected. 2.5. Convergence study A convergence test is conducted to verify a suitable mesh element size. The plate girder P-1-750 in Table 3 is modelled. Given that the shape of the shell element S4R selected is a square, the length of the square is selected to quantify the size of the shell element. Considering that the element numbers of the web panel and the flange are much more than that of transverse stiffeners, the mesh element size of the web panel and the flange has a larger influence on the computational cost. Hence, for the mesh element of the web panel, different mesh element sizes are considered, including 5 mm, 15 mm, 25 mm, 45 mm, 65 mm, 85 mm and 105 mm. The mesh element size of tested transverse stiffeners is 10 mm. The mesh element size of the mid-span and end post transverse stiffener are 20 mm. After FEA, the results for the finite element model with different mesh element sizes are shown in Fig. 7, where Vu = Vu,i × i Vu,5 ×5 and i = 5, 15, 25, 45, 65, 85 and 105. The iteration error for the finite element model with the mesh element size of 45 mm is less than 5%, as shown in Fig. 7. Furthermore, the tie constraint in ABAQUS is selected to simulate the weld between web, flanges and transverse stiffeners. It reaches the convergence much faster when the mesh element size of the web and flange is larger than that of transverse stiffeners. Hence, the mesh element size 45 mm is supposed to be suitable.
Maximum additional out-of-plane deflection (mm)
Fig. 4. (a) Deformed shape of PG3 specimen [8]; (a) Deformed shape of PG3 FE model.
3. Discussions of finite element analysis results
0.8
3.1. Failure modes 0.6
The transverse stiffeners with different ts are introduced in the Q550 P-2-750 plate girder to carry out FEA. Then, there are two different failure modes observed in this study, i.e., intermediate transverse stiffener buckling mode and flange plastic hinge mode, as shown in Fig. 8. When the moment of inertia of the intermediate transverse stiffener is small, the intermediate transverse stiffener fails to maintain the nodal line during the postbuckling stage which leads to the intermediate transverse stiffener buckling mode, as shown in Fig. 8(a). Then, the postbuckling capacity of the plate girder will not be fully developed. However, when the moment of inertia of the intermediate transverse stiffener is large, the nodal line of the intermediate transverse stiffener can be maintained during the postbuckling stage. The out-of-plane deflection of the web panel at the intermediate transverse stiffener area can be restrained effectively. Then, with increases in the shear loading,
0.4
0.2
0.0
Experiment results of PG3 in [8] FEA results 0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
V/Vu
Fig. 5. Maximum additional out-of-plane deflection vs. V/Vu curves of PG3 specimen and FE model. 4
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Table 3 Specimen dimensions and section properties of specimens. Number
L/mm
hw/mm
tw/mm
bf/mm
tf/mm
a/mm
hw/tw
a/hw
bs/mm
P-0.5-750 P-1-750 P-1.5-750 P-2-750 P-0.5-1000 P-1-1000 P-1.5-1000 P-2-1000 P-0.5-1250 P-1-1250 P-1.5-1250 P-2-1250 P-0.5-1500 P-1-1500 P-1.5-1500 P-2-1500
1950 3450 4950 6450 2450 4450 6450 8450 2950 5450 7950 10,450 3450 6450 9450 12,450
750 750 750 750 1000 1000 1000 1000 1250 1250 1250 1250 1500 1500 1500 1500
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200
30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30
375 750 1125 1500 500 1000 1500 2000 625 1250 1875 2500 750 1500 2250 3000
150 150 150 150 200 200 200 200 250 250 250 250 300 300 300 300
0.5 1 1.5 2 0.5 1 1.5 2 0.5 1 1.5 2 0.5 1 1.5 2
40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40
Fig. 6. Dimensions of plate girders.
girders [12]. However, for the complete development of the ultimate shear resistance, the current design equations for intermediate transverse stiffeners have all been based on the study of mild steel plate girders. Therefore, they may not be suitable for the design of intermediate transverse stiffeners for HSS plate girders as discussed earlier. To clarify the aforementioned issue, the P-1-750 plate girder is selected to perform the FEA. The Q235/Q345/Q550/Q690 P-1-750 plate girders are considered to investigate the difference between the requirements of intermediate transverse stiffeners in mild steel and HSS plate girders. In FEA, the intermediate transverse stiffeners with different ts are applied in the plate girder to investigate the minimum requirement of moment of inertia. Based on results of FEA, the shear resistance vs. midspan displacement curves of mild steel and HSS plate girders with different intermediate transverse stiffeners are obtained, as shown in Figs. 9 and 10. It indicates that the intermediate transverse stiffener thickness has barely any effect on the initial stiffness. Furthermore, when the thickness of intermediate transverse stiffeners is small, the ultimate shear resistance cannot be fully developed during the postbuckling stage, as shown in Figs. 9 and 10. For Q690 P-1-750 plate girders, the shear resistance curve of Q690 P-1-750 plate girder with ts = 7 mm is almost the same with that of Q690 P-1-750 plate girder with ts = 20 mm which means that the postbuckling capacity is fully developed. Moreover, the failure modes of Q690 P-1-750 plate girders with ts = 6.5 mm and 7 mm are proposed, as shown in Fig. 11. It is easy to see that the failure mode of the Q690 P-1-750 plate girder with ts = 6.5 mm is the transverse stiffener buckling mode and that of Q690 P-1-750 plate girder with ts = 7 mm is the flange plastic hinge mode. Therefore, in this study, the intermediate transverse stiffener with ts = 7 mm is supposed to meet the requirement of the Q690 P-1-750 plate girder. The similar method is adopted to determine the minimum moment of inertia requirement of the intermediate transverse stiffener in mild steel and HSS plate girders.
Fig. 7. Result of the convergence test for different mesh element sizes.
plastic hinges develop in the flanges, as shown in Fig. 8(b). Hence, for the flange plastic hinge mode, the ultimate shear capacity of the plate girder can be completely developed. Therefore, when the failure mode of the plate girder is the flange plastic hinge mode, it is supposed that the moment of inertia of the intermediate transverse stiffener can meet the minimum requirement. 3.2. Differences between HSS and mild steel plate girders Given that there is no clear yield plateau for HSS, the postbuckling capacity of HSS plate girders is higher than that of mild steel plate
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Fig. 8. Failure modes of the Q550 P-2-750 plate girders with different ts: (a) intermediate transverse stiffener buckling mode: ts = 2.5 mm; (b) flange plastic hinge mode: ts = 3 mm.
Based on the aforementioned method, the minimum requirements of intermediate transverse stiffeners thickness for Q235/Q345/Q550/ Q690 P-1-750 plate girders are 4 mm/4.5 mm/6.5 mm/7 mm, respectively. Hence, to fully develop the postbuckling capacity of plate girders, the minimum moment of inertia requirement of the intermediate transverse stiffeners increases with the increase in the steel strength grade.
4. Discussions of AASHTO LRFD design method and new prediction formula 4.1. Specifications in AASHTO LRFD design method Considering differences in the depth ratio hw/tw, the shear strength curve of plate girder can be divided into three zones: (1) yield zone; (2) inelastic buckling zone and (3) elastic buckling zone [16]. In this study, the web panels in HSS plate girders are classified into the elastic buckling zone, as shown in Fig. 16. In AASHTO LRFD [16], there are two design equations for the intermediate transverse stiffener in plate girders without a longitudinal stiffener, as shown in Eqs. (1) and (2). It states that transverse stiffeners adjacent to web panels in which neither panel supports shear forces being larger than the shear buckling resistance, the moment of inertia of the transverse stiffener shall satisfy the smaller of Eqs. (1) and (2). When the postbuckling capacity is considered, the moment of inertia of transverse stiffeners shall satisfy Eq. (2) [16]. After a careful investigation, it is suggested that for Eq. (1), the intermediate transverse stiffener is designed to provide nodal lines until the web reaches the theoretical buckling stress. For Eq. (2), the intermediate transverse stiffener is designed to ensure that the web panel is able to develop the shear yield strength [9]. For HSS plate girders in the elastic buckling zone, there is a noteworthy postbuckling shear capacity. Hence, the minimum requirement of the moment of inertia for the intermediate transverse stiffener in HSS plate girder should be somewhere between Eqs. (1) and (2). After the calculation, results from Eqs. (1) and (2) are shown in Tables 4 and 5. Based on the comparison between the results of Eqs. (1) and (2) and the results of FEA, it is clear to see that the requirement of the intermediate transverse stiffener in HSS plate girders considering the postbuckling capacity is larger than the results obtained from Eq. (1) and much lower
3.3. Parametric studies of HSS plate girders In Section 3.2, the method of determining the minimum moment of inertia requirement of the intermediate transverse stiffener in mild steel and HSS plate girders is introduced. After the FEA on HSS plate girders with different intermediate transverse stiffeners, the ratios Vu/Vu, max are presented, as shown in Figs. 12–15, where Vu denotes the ultimate shear resistance, Vu, max denotes the ultimate shear resistance of the plate girder with the thickest intermediate transverse stiffener. Furthermore, It denotes the moment of inertia of the intermediate transverse stiffener and It1, AASHTO denotes the minimum moment of inertia requirement of the intermediate transverse stiffener calculated by Eq. (1) in AASHTO LRFD [16]. Hence, it is clear to see that with increases in moment of inertia of the intermediate transverse stiffener, the postbuckling capacity of HSS plate girders develops, gradually. Based on the results of FEA, the minimum moment of inertia requirements of the intermediate transverse stiffener in HSS plate girders listed in Table 3 are obtained, as shown in Tables 4 and 5. It is evident to see that when tw is the same, with increases in hw/tw, the minimum moment of inertia requirement increases, correspondingly. When hw is the same, with increases in a/hw, the minimum moment of inertia requirement decreases, correspondingly.
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Fig. 10. Shear resistance vs. mid-span displacement curves of HSS P-1-750 plate girders with different tested stiffeners: (a) Q550 HSS plate girders; (b) Q690 HSS plate girders.
Fig. 9. Shear resistance vs. mid-span displacement curves of mild steel P-1-750 plate girders with different tested stiffeners: (a) Q235 mild steel plate girders; (b) Q345 mild steel plate girders.
It 2, AASHTO
than the results from Eq. (2). Furthermore, considering the results of FEA, the steel strength grades, the aspect ratio a/hw and the depth ratio hw/tw do have an effect on the minimum moment of inertia requirement of the intermediate transverse stiffener for HSS plate girders. However, 2.5 for Eq. (1), J = (d / h )2 2 0.5 where d0 denotes the smaller of the 0 w adjacent web panel widths. Hence, when a/hw > 1, the results from Eq. (1) are not influenced by the ratio a/hw which is inconsistent with the FEA results. For Eq. (2), the influence of the aspect ratio a/hw and the depth ratio hw/tw is not fully taken into account. Therefore, given the aforementioned discussion, the design equations in AASHTO LRFD [16] could not be directly applicable in the design of the intermediate transverse stiffener for HSS plate girders considering the postbuckling capacity.
It1, AASHTO
bt w3 J
h w4
1.3 t
40
Fyw
1.5
E
(2)
4.2. New prediction formula Based on the comparison conducted in Section 4.1, there is a certain relationship between the results of Eq. (1) and the results of the FEA. Therefore, in order to predict the minimum moment of inertia requirement of the transverse stiffener for HSS plate girders, Eq. (1) is modified by including the influence of the aspect ratio a/hw, depth ratio hw/tw, and steel strength grade. Then, a new equation is proposed in this study to predict the minimum moment of inertia requirement of the intermediate transverse stiffener for HSS plate girders, as shown in Eq. (3). Compared with Eq. (1), b is replaced by a, where b denotes the smaller of a and hw. In addition, a denotes the distance between the transverse stiffeners. Moreover, there is a new equation of J introduced
(1)
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Tested stiffener
ts=6.5 mm
Tested stiffener
ts=7 mm Fig. 11. Failure modes of Q690 P-1-750 plate girders with ts = 6.5 mm and 7 mm.
Fig. 12. FEA results of Q550 and Q690 plate girders with a/hw = 0.5.
Fig. 13. FEA results of Q550 and Q690 plate girders with a/hw = 1.
in Eq. (3). After the numerical regression, the new equation of J is proposed, as shown in Eq. (4). It is worth noting that in order to develop a significant postbuckling capacity, the aspect ratio a/hw and depth ratio hw/tw of HSS plate girder should not be too large [12]. Given the conclusion proposed in [12] and the numerical analyses conducted in this study, to develop a considerable postbuckling capacity, the intermediate transverse stiffener spacing is suggested to be less than 2hw which means the aspect ratio a/hw ≤ 2. Hence, Eq. (3) proposed in this study could be applied to the HSS plate girder whose hw/tw ≤ 300 and a/hw ≤ 2 which are frequently used in the practical design and
construction. In order to validate the accuracy of Eq. (3), a comparison between the results of FEA and those of Eq. (3) is carried out, as shown in Figs. 17–19. The errors of the Eq. (3) results are all less than 10%. For results of It, Eq. (3)/It, FEA, the average value is 0.991 and the coefficient of variation is Cv, Eq. (3) = 0.039. Furthermore, to show the predictive power of the proposed formula, the independent simulations are performed. The Q550/Q690 plate girders with a/hw = 0.8 and hw/ tw = 275 are simulated. The comparison between the results of FEA and Eq. (3) is conducted, as shown in Table 6, Figs. 20 and 21. It turns out that the errors are acceptable. The failure modes could be predicted
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It
(3)
Jat w3
J = [162.98 - 218.65(a / hw ) + 104.59(a /hw )2 - 17.43(a/ hw )3 + 0.031(hw /tw )](fyw / E )0.4
(4) 5. Performance of intermediate transverse stiffeners 5.1. Discussion of resistant capacities The resistant capacity of HSS plate girders is different from that of mild steel plate girders. Therefore, to fully develop the postbuckling capacity, the requirement of the intermediate transverse stiffener in HSS plate girders might be different from that of the mild steel plate girders. Based on the analyses conducted in Section 3.2, the Q235/ Q345/Q550/Q690 P-1-750 plate girders with the corresponding minimum intermediate transverse stiffeners are included in the FEA to clarify the aforementioned issue. Then, the shear resistance vs. midspan displacement curves are obtained, as shown in Fig. 22. The resistant curve form of HSS plate girders is different from that of mild steel plate girders. Because that there is no clear yield plateau for HSS, the strain hardening is beneficial to the postbuckling capacity. Then, for the resistant curve of HSS plate girders, after the elasticity region, there is a significant increase in the shear resistant capacity. However, for mild steels, there is a clear yield plateau. Hence, the strain hardening has barely any effect on the postbuckling capacity. After the elasticity region, no significant increase in the shear resistant capacity is observed. Therefore, the maximum additional out-of-plane deflection of the intermediate transverse stiffener in HSS plate girders corresponding to the ultimate shear resistance is significantly larger than that of the intermediate transverse stiffener in mild steel plate girders, as shown in Fig. 23. However, in EC3 [17], it states that the maximum additional deflection should not exceed b/300, where b denotes the height between the centroids of the flanges or span of the transverse stiffener. In this study, b is considered as the height between the centroids of the flanges. Therefore, when the postbuckling capacity can be fully developed, the additional out-of-plane deflection of the intermediate transverse stiffener in HSS plate girders might not meet the requirement in EC3 [17]. However, for mild steel plate girders, such problem does not need to be concerned, as shown in Fig. 23. Hence, even though there are some studies about the minimum moment of inertia requirement and the behaviour of the intermediate transverse stiffener in mild steel plate girders, little attention has been paid to investigating the adverse effect caused by the additional out-of-plane deflection of the
Fig. 14. FEA results of Q550 and Q690 plate girders with a/hw = 1.5.
Fig. 15. FEA results of Q550 and Q690 plate girders with a/hw = 2.
accurately. Hence, the proposed formula could be used to predict the minimum moment of inertia requirement of the intermediate transverse stiffener for Q550 and Q690 HSS plate girders considering the postbuckling capacity effectively. Table 4 Comparison between the results of FEA and AASHTO LRFD for Q550 plate girders. Number
ts/mm
It,FEA/mm4
It1,AASHTO/mm4
It2,AASHTO/mm4
It,FEA/It1,AASHTO
It,FEA/It2,AASHTO
P-0.5-750 P-1-750 P-1.5-750 P-2-750 P-0.5-1000 P-1-1000 P-1.5-1000 P-2-1000 P-0.5-1250 P-1-1250 P-1.5-1250 P-2-1250 P-0.5-1500 P-1-1500 P-1.5-1500 P-2-1500
7 6.5 4.5 3 10 9 6 4.5 13 11.5 8 7 16 14 11 10
358,240 332,651 230,297 153,531 511,771 460,594 307,063 230,297 665,302 588,536 409,417 358,240 818,833 716,479 562,948 511,771
375,000 46,875 46,875 46,875 500,000 62,500 62,500 62,500 625,000 78,125 78,125 78,125 750,000 93,750 93,750 93,750
1,381,419 1,674,965 4,357,404 12,504,465 3,605,280 3,605,280 6,518,416 13,771,549 8,801,954 8,801,954 8,801,954 10,659,098 18,251,732 18,251,732 18,251,732 18,251,732
0.955 7.097 4.913 3.275 1.024 7.370 4.913 3.685 1.064 7.533 5.241 4.585 1.092 7.642 6.005 5.459
0.259 0.199 0.053 0.012 0.142 0.128 0.047 0.017 0.076 0.067 0.047 0.034 0.045 0.039 0.031 0.028
9
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Table 5 Comparison between the results of FEA and AASHTO LRFD for Q690 plate girders. Number
ts/mm
It,FEA/mm4
It1,AASHTO/mm4
It2,AASHTO/mm4
It,FEA/It1,AASHTO
It,FEA/It2,AASHTO
P-0.5-750 P-1-750 P-1.5-750 P-2-750 P-0.5-1000 P-1-1000 P-1.5-1000 P-2-1000 P-0.5-1250 P-1-1250 P-1.5-1250 P-2-1250 P-0.5-1500 P-1-1500 P-1.5-1500 P-2-1500
7.5 7 4.5 3.5 10.5 9.5 6.5 5 13.5 12 8.5 7.5 16.5 15 11.5 10.5
383,828 358,240 230,297 179,120 537,359 486,182 332,651 255,885 690,891 614,125 435,005 383,828 844,422 767,656 588,536 537,359
375,000 46,875 46,875 46,875 500,000 62,500 62,500 62,500 625,000 78,125 78,125 78,125 750,000 93,750 93,750 93,750
2,178,628 2,606,678 8,222,233 15,803,940 5,066,040 5,066,040 9,989,011 19,759,467 12,368,263 12,368,263 12,368,263 16,810,398 25,646,830 25,646,830 25,646,830 25,646,830
1.024 7.642 4.913 3.821 1.075 7.779 5.322 4.094 1.105 7.861 5.568 4.913 1.126 8.188 6.278 5.732
0.176 0.137 0.028 0.011 0.106 0.096 0.033 0.013 0.056 0.050 0.035 0.023 0.033 0.030 0.023 0.021
Vu Vp
Elastic Buckling curve
I Yield
0
II
Inelastic Buckling
Elastic Buckling
hw tw
Fig. 16. Shear strength curve of plate girders in AASHTO LRFD [16].
Fig. 18. Comparison of It for a/hw = 0.5/1/1.5/2 and Fyw = 690 MPa: Eq. (3) vs. FEA results.
Fig. 17. Comparison of It for a/hw = 0.5/1/1.5/2 and Fyw = 550 MPa: Eq. (3) vs. FEA results.
Fig. 19. Variation of It,Eq. and hw/tw.
(3)/It,FEA
vs. a/hw for plate girders with different HSS
the issue discussed above, the adverse effect and the behaviour of the intermediate transverse stiffener in HSS plate girders are investigated in the following section.
intermediate transverse stiffener on the shear resistance of plate girders. Given the discussions done before, for HSS plate girders, the aforementioned adverse effect should be taken into account. To clarify 10
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more obvious than that of Q550 plate girders. It is worth noting that in some situations, the shear resistance of the HSS plate girder is reduced by almost half. Hence, it is meaningful to pay attention to the aforementioned shear resistance reduction of HSS plate girders in the practical applications.
Table 6 Comparison between the results of FEA and Eq. (3). Number
Steel strength
tw/mm
ts/mm
It,FEA/mm4
It,Eq.(3)/mm4
It,Eq.(3)/It,FEA
P-0.8-1375 P-0.8-1375
Q550 Q690
5 5
13.5 14
690,891 716,479
699,067 765,444
1.011 1.068
5.3. Out-of-plane deflection Considering the FEA results of the Q690 P-1-750 plate girder, the maximum additional out-of-plane deflection of the intermediate transverse stiffener whose ts = 7 mm vs. mid-span displacement of the plate girder curve is obtained, as shown in Fig. 25(a). Before the buckling resistance, the maximum additional out-of-plane deflection develops slowly. Then, after the buckling resistance, the maximum out-of-plane deflection develops faster. For the different states in Fig. 25(a), the outof-plane deflection distribution curves of the intermediate transverse stiffener are plotted along the web depth, as shown in Fig. 25(b). The out-of-plane deflection distribution curves are unimodal. The peak point is reached in the middle of the web depth. For the Q690 P-1-750 plate girder, when ts = 7 mm, the postbuckling capacity of the plate girder can be fully developed. However, the maximum additional out-of-plane deflection corresponding to the ultimate resistance is obviously larger than b/300 = 2.6 mm, as shown in Fig. 26. When ts = 10 mm and 20 mm, the maximum out-of-plane deflection can be effectively reduced, as shown in Fig. 26. Moreover, when ts = 20 mm, the maximum additional out-of-plane deflection is always less than b/300 = 2.6 mm. Hence, for the actual design of HSS plate girders, the increase in the moment of inertia of the intermediate
5.2. Reduction of shear resistance For HSS plate girders, before the ultimate shear resistance is fully developed, the intermediate transverse stiffener will develop an unacceptable additional out-of-plane deflection. Hence, it is necessary to investigate the reduction of the shear resistance of the HSS plate girder, according to the additional out-of-plane deflection of the intermediate transverse stiffener. Based on the FEA results of Q550 and Q690 plate girders in Section 3.3, the ultimate shear resistance and the reduced shear resistance corresponding to the limit state of the intermediate transverse stiffener are obtained, as shown in Table 7. The reduced shear resistance denotes the shear resistance corresponding to the condition where the additional out-of-plane deflection of the intermediate transverse stiffener reaches b/300, as shown in Fig. 24. It indicates that with increases in the aspect ratio a/hw and depth ratio hw/ tw, the shear resistance reduction caused by the additional out-of-plane deflection of the intermediate transverse stiffener increases, gradually. Furthermore, the shear resistance reduction of Q690 plate girders is
Fig. 20. Failure modes of the Q550 P-0.8-1375 plate girders with different ts: (a) intermediate transverse stiffener buckling mode: ts = 13 mm; (b) flange plastic hinge mode: ts = 13.5 mm.
11
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Fig. 21. Failure modes of the Q690 P-0.8-1375 plate girders with different ts: (a) intermediate transverse stiffener buckling mode: ts = 13.5 mm; (b) flange plastic hinge mode: ts = 14 mm.
Fig. 22. Shear resistance vs. mid-span displacement curves of Q235/Q345/ Q550/Q690 plate girders.
Fig. 23. Maximum additional out-of-plane deflection vs. mid-span displacement curves of intermediate transverse stiffeners in Q235/Q345/Q550/Q690 plate girders.
transverse stiffener can effectively reduce the additional out-of-plane deflection. To investigate the effect of the initial imperfection on the additional out-of-plane deflection of the intermediate transverse stiffener, different initial imperfections are introduced in the FEA of Q690 P-1-750 plate girder whose ts = 10 mm. In the early stage, there is barely any difference in the FEA results of plate girders with different initial
imperfections, as shown in Fig. 27. With increases in the mid-span displacement, the difference in the additional out-of-plane deflection increases. To a certain extent, reducing the initial imperfection of the plate girder can help to reduce the additional out-of-plane deflection of the intermediate transverse stiffener. 12
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Table 7 Reduced shear resistance and ultimate shear resistance of Q550 and Q690 plate girders. Number
Steel strength grades
Thickness of the transverse stiffener ts (mm)
Ultimate shear resistance Vu (kN)
Reduced shear resistance Vr (kN)
Vr/Vu
P-0.5-750 P-1-750 P-1.5-750 P-2-750 P-1-1000 P-1-1250 P-1-1500 P-0.5-750 P-1-750 P-1.5-750 P-2-750 P-1-1000 P-1-1250 P-1-1500
Q550 Q550 Q550 Q550 Q550 Q550 Q550 Q690 Q690 Q690 Q690 Q690 Q690 Q690
7 6.5 4.5 3 9 11.5 14 7.5 7 4.5 3.5 9.5 12.5 15
1450.4 1141.2 888.1 760.9 1280.9 1399.8 1570.1 1746.9 1365.4 1054.1 902.6 1535.6 1679.3 1883.2
1403.3 1052.1 671.9 504.3 955.9 825.9 745.7 1624.5 1223.9 688.6 505.1 998.6 915.4 1087.9
0.97 0.92 0.76 0.66 0.75 0.59 0.47 0.93 0.90 0.65 0.56 0.65 0.55 0.58
Fig. 24. Reduced shear resistance and ultimate shear resistance of the HSS plate girder.
6. Conclusions
Fig. 25. (a) Maximum additional out-of-plane deflection vs. mid-span displacement curve of the transverse stiffener in Q690 P-1-750-7 plate girder; (b) Additional out-of-plane deflection of the transverse stiffener in Q690 P-1-750-7 and P-1-750-10 plate girders.
This paper conducted a thorough investigation on the properties and minimum requirements for the intermediate transverse stiffener of HSS plate girders with the consideration of the postbuckling capacity. The main contributions of this study are summarized as the followings:
corresponding to the ultimate shear resistance is significantly larger than that of the mild steel plate girders. Hence, for HSS plate girders, this adverse effect on the shear resistance of plate girders needs to be considered. 5. It has proven that with increases in the aspect ratio a/hw and depth ratio hw/tw, the shear resistance reduction caused by the additional out-of-plane deflection of the intermediate transverse stiffener increases, gradually. Furthermore, the shear resistance reduction of Q690 plate girders is more obvious than that of Q550 plate girders. It is meaningful to pay attention to the aforementioned shear resistance reduction for HSS plate girders in the practical applications. 6. For the actual design of HSS plate girders, the increase in the moment of inertia of the intermediate transverse stiffener can effectively reduce the additional out-of-plane deflection. To a certain extent, reducing the initial imperfection of the plate girder can help to reduce the additional out-of-plane deflection of the intermediate transverse stiffener.
1. To fully develop the postbuckling capacity of plate girders, the minimum moment of inertia requirement of intermediate transverse stiffeners in HSS plate girders is larger than that of the requirement in the mild steel plate girders. 2. With the comparison between the FEA results and the results predicted by the design equations in AASHTO LRFD, it indicates that the design equations in AASHTO LRFD could not be directly used in the design of the intermediate transverse stiffener for HSS plate girders when the postbuckling capacity is considered. 3. A new equation is proposed in this study to predict the minimum moment of inertia requirement of the intermediate transverse stiffener for HSS plate girders. A comparison between the FEA results and the results from the new equation is conducted to validate the accuracy of the new equation. 4. With the difference in the resistant capacity between HSS and mild steel plate girders, the maximum additional out-of-plane deflection of the intermediate transverse stiffener in HSS plate girders 13
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Acknowledgements This study was financially supported by the Shanghai Recruitment Program of Global Experts. This research was conducted at the State Key Laboratory for Disaster Reduction in Civil Engineering of Tongji University. References [1] Basler K. Strength of plate girders in shear. J Struct Div ASCE 1961;87:151–81. [2] Stanway GS, Chapman JC, Dowling PJ. A design model for intermediate stiffeners. Proc Inst Civ Eng Struct Build 1996;116:54–68. [3] Stanway GS, Chapman JC, Dowling PJ. Behaviour of a web plate in shear with an intermediate stiffener. Proc Inst Civ Eng Struct Build 1993;99:327–44. [4] Xie M, Chapman JC. Design of web stiffeners: Axial forces. J Constr Steel Res 2003;59:1035–56. https://doi.org/10.1016/S0143-974X(02)00115-3. [5] Kim YD, White DW. Transverse stiffener requirements to develop shear-buckling and postbuckling resistance of steel I-girders. J Struct Eng 2014;140:04013098. https://doi.org/10.1061/(ASCE)ST.1943-541X.0000867. [6] Kim YD, Asce SM, Jung S, White DW, Asce M. Transverse stiffener requirements in straight and horizontally curved steel I-girders. J Bridg Eng 2007;12:174–83. https://doi.org/10.1061/(ASCE)1084-0702(2007) 12:2(174). [7] Lee SC, Asce M, Yoo CH, Asce M, Yoon DY. Behavior of intermediate transverse stiffeners attached on web panels. J Struct Eng 2002;128:337–45. [8] Lee SC, Asce M, Yoo CH, Asce F, Yoon DY. New design rule for intermediate transverse stiffeners attached on web panels. J Struct Eng 2004;129:1607–14. [9] Lee SC, Lee DS, Yoo CH. Design of intermediate transverse stiffeners for shear web panels. Eng Struct 2014;75:27–38. https://doi.org/10.1016/j.engstruct.2014.05. 037. [10] Sun FF, Xue XY, Xiao Y, Le YM, Li GQ. Effect of welding and complex loads on the high-strength steel T-stub connection. J Constr Steel Res 2018;150:76–86. https:// doi.org/10.1016/j.jcsr.2018.08.002. [11] Bjorhovde R. Development and use of high performance steel. J Constr Steel Res 2004;60:393–400. https://doi.org/10.1016/S0143-974X(03)00118-4. [12] Xiao Y, Xue XY, Sun FF, Li GQ. Postbuckling shear capacity of high-strength steel plate girders. J Constr Steel Res 2018;150:475–90. https://doi.org/10.1016/j.jcsr. 2018.08.032. [13] AASHTO/AWS. Bridge welding code, ANSI/AASHTO/AWS D1.5M/D1.5:2002, A Joint Publication of American Association of State Highway and Transportation Officials, Inc., Washington, DC, and American Welding Society, Miami; 2002. [14] Shin DK, Cho EY, Kim K. Ultimate flexural strengths of plate girders subjected to web local buckling. Int J Steel Struct 2013;13:291–303. https://doi.org/10.1007/ s13296-013-2008-3. [15] Choi YS, Kim D, Lee SC. Ultimate shear behavior of web panels of HSB800 plate girders. Constr Build Mater 2015;101:828–37. https://doi.org/10.1016/j. conbuildmat.2015.10.118. [16] American Association of State Highway and Transportation Officials (AASHTO). AASHTO LRFD bridge design specifications, 5th ed.,. Washington, DC; 2010. [17] European Committee for Standardisation (ECS), Eurocode 3: Design of Steel Structures, Part 1-5: Plated Structural Elements,. Brussels, Belgium; 2006.
Fig. 26. Maximum additional out-of-plane deflection vs. mid-span displacement curves of the transverse stiffener in Q690 P-1-750 plate girders with different transverse stiffeners.
Fig. 27. Maximum additional out-of-plane deflection vs. mid-span displacement curves of the transverse stiffener in Q690 P-1-750-10 plate girders with different initial imperfections.
14
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Intermediate transverse stiffener requirements of high-strength steel plate girders considering postbuckling capacity
T
Yong Xiaoa,b, Xuan Yi Xuea, , Fei Fei Suna,c, Guo Qiang Lia,c ⁎
a
College of Civil Engineering, Tongji University, Shanghai 200092, China Chongqing Steel Structure Limited Company, Chongqing 400000, China c State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China b
ARTICLE INFO
ABSTRACT
Keywords: High-strength steel Intermediate transverse stiffener Plate girder Postbuckling capacity AASHTO LRFD Out-of-plane deflection
The postbuckling capacity of HSS plate girders is larger than that of mild steel plate girders because of the differences in material properties. Hence, to fully develop the postbuckling capacity, the intermediate transverse stiffener requirement of HSS plate girders could be different from that of mild steel plate girders. The plate girders concerned in this study are without a longitudinal stiffener. To investigate the aforementioned issue, the minimum moment of inertia requirements of intermediate transverse stiffeners in HSS and mild steel plate girders are studied through the finite element analysis (FEA). After the comparison between the FEA results and the results from the design equations in AASHTO LRFD, it states that the design equations in AASHTO LRFD could not be directly used in the design of intermediate transverse stiffeners in HSS plate girders. Hence, based on the FEA results, a new equation is proposed to predict the minimum moment of inertia requirement of intermediate transverse stiffeners in HSS plate girders. Furthermore, because of the difference in the resistant capacity between HSS and mild steel plate girders, the maximum additional out-of-plane deflection of the intermediate transverse stiffener in HSS plate girders corresponding to the ultimate shear resistance is significantly larger than that of the intermediate transverse stiffener in mild steel plate girders. Hence, there is a noteworthy adverse effect caused by the additional out-of-plane deflection of the intermediate transverse stiffener on the shear resistance of HSS plate girders with the maximum limit value set.
1. Introduction Depending on the development of the postbuckling capacity in plate girders, there are differences in the requirement of the intermediate transverse stiffener. When the postbuckling capacity of plate girders is not considered, the purpose of the intermediate transverse stiffener is to improve the local buckling resistance of the web panel. Therefore, intermediate transverse stiffeners are designed to have a proper moment of inertia which are able to maintain the nodal line when the elastic buckling occurs in the web panel. However, when the postbuckling capacity is taken into account, the function of the intermediate transverse stiffeners could be more complicated. To clarify this issue, Basler [1] proposed a theory of intermediate transvers stiffeners which states that a compressed axial force caused by the tension field action develops on the intermediate transverse stiffener. After that, Stanway et al. investigated the behaviour of the intermediate transverse stiffener attached on a web plate and proposed a design model [2,3]. Xie et al. conducted a numerical study of the intermediate transverse stiffener
⁎
and investigated the stress distribution at the postbuckling stage [4]. Kim et al proposed a comprehensive theoretical study of the intermediate transverse stiffener requirements during the postbuckling stage [5,6]. Furthermore, Lee et al. introduced some experimental, numerical and theoretical studies of the intermediate transverse stiffener [7,8]. A new design rule of the intermediate transverse stiffener was subsequently proposed by these authors [9]. Based on the above studies, a general agreement was that there is no the aforementioned compressed axial force developing in the intermediate transverse stiffener during the postbuckling stage. However, it is uncertain as these studies were purely based on the initial investigations on mild steel plate girders. Given the rapid advancements in thermal and mechanical processing procedures for structural steel production, high-strength steel (HSS) is increasingly used in components and structures [10,11]. It is worth noting that in general, the steel whose yield strength is larger than 460 MPa could be considered as HSS. In order to promote the use of HSS, recently, Xiao et al. conducted a series of numerical studies of HSS plate girders. It has proven that because of the difference in the
Corresponding author. E-mail address: [email protected] (X.Y. Xue).
https://doi.org/10.1016/j.engstruct.2019.109289 Received 19 March 2019; Received in revised form 30 May 2019; Accepted 7 June 2019 Available online 12 June 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.
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Nomenclature
Vu,
E Fyw Fcrs
a bs bf bt b d0 hw k L ts tf tw ρt µ
Fys It,FEA It1,AASHTO It2,AASHTO J Vu Vr
elastic modulus yield strength of the web panel local buckling stress for the stiffener Fcrs = 0.31E/(bt/ ts)2 ≤ Fys yield strength of the intermediate transverse stiffener moment of inertia of intermediate transverse stiffeners predicted by FEA moment of inertia of intermediate transverse stiffeners predicted by Eq. (1) in AASHTO LRFD moment of inertia of intermediate transverse stiffeners predicted by Eq. (2) in AASHTO LRFD stiffener bending rigidity parameter ultimate shear resistance reduced shear resistance
material property between mild steel and HSS, the postbuckling capacity of HSS plate girders is much larger than that of mild steel plate girders [12]. Hence, the intermediate transverse stiffener requirements of HSS plate girders can be different from that of mild steel plate girders. However, the resistant properties and deformation of the transverse stiffener are not concerned as long as it is rigid enough to develop the full postbuckling capacity of plate girders [12]. To consider the postbuckling capacity of HSS plate girders in the practical design, it is necessary to investigate the intermediate transverse stiffener requirement to develop the full postbuckling capacity of HSS plate girders. This study aims to clarify the aforementioned issue through the finite element analysis (FEA). In Section 2, a numerical modelling method was introduced. To investigate the intermediate transverse stiffener requirement for developing the postbuckling capacity of HSS plate girders, the sizes and dimensions of specimens in FEA were determined. In Section 3, the failure modes of HSS plate girders with different intermediate transverse stiffeners were identified. The difference in the intermediate transverse stiffener requirements between mild steel and HSS plate girders was compared. In Section4, based on the FEA results, a new equation was proposed to predict the minimum moment of inertia requirement of the intermediate transverse stiffener in HSS plate girders. Then, the design equations in AASHTO LRFD and the new equation proposed in this study were validated. In Section 5, the influence caused by the differences in the resistant properties of HSS and mild steel plate girders on the requirement of intermediate
max
ultimate shear resistance of the plate girder with the thickest intermediate transverse stiffener distance between the transverse stiffeners width of intermediate transverse stiffeners width of the flange for symmetric transverse stiffener, bt = 2bs the smaller of a and hw smaller of the adjacent web panel widths depth of the web panel shear buckling coefficient length of the plate girder thickness of the intermediate transverse stiffener thickness of the flange thickness of the web panel the larger of Fyw/Fcrs and 1 poisson’s ratio
transverse stiffeners was discussed. Furthermore, the out-of-plane deflection of intermediate transverse stiffeners in HSS plate girders was studied. 2. Numerical study 2.1. Numerical modelling method In order to investigate the intermediate transverse stiffener properties, a full plate girder nonlinear finite element model is proposed, as shown in Fig. 1. A concentrated load is applied at the top flange in the mid-span area. Two pairs of parallel intermediate transverse stiffeners are selected as the tested stiffeners, as shown in Fig. 1. There are two reference points which are coupled with the end-face of the plate girder. Then, to simulate the simply supported boundary condition, the degrees-of-freedom of reference points are limited, as listed in Table 1. The directions 1, 2, 3 indicate the x, y, z directions, separately. To avoid the occurrence of lateral-torsional buckling, the out-of-plane freedom of the compressed flange is restrained. In practical structures, the out-ofplane freedom of compressed flange is usually constrained by other components, such as floors. Hence, the aforementioned boundary condition assumption is consistent with the actual engineering condition. The initial imperfection of the plate girder is included in the FEA of this study. After the finite element buckling analysis of the ideal
Concentrated load
Tested stiffener
RF2
Tested stiffener RF1
2 3 1 Fig. 1. Full plate girder finite element model.
2
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Table 1 Boundary conditions of finite element models (free: 1, fixed: 0). Reference points
U1
U2
U3
UR1
UR2
UR3
RF1 RF2
0 1
0 0
0 0
0 0
1 1
1 1
RF1 denotes the reference point 1.
component, the first buckling mode is identified, as shown in Fig. 2. Then, the first buckling mode is selected to include the initial imperfection. Given that the displacement hw/100 is the largest permitted initial displacement according to the AASHTO Bridge Welding Code [13], hw/100 is introduced in the FEA as the largest out-of-plane displacement. 2.2. Idealized material model
Fig. 3. Idealized material models of Q235, Q345, Q550, and Q690 steels.
To investigate the difference in the requirement of the intermediate transverse stiffener in mild steel and HSS plate girders, Q235/Q345/ Q550/Q690 steels are considered in the FEA. The steels Q235/Q345 are mild steels and the steels Q550/Q690 are HSS. For the stress-strain behaviour of HSS, there is no clear yield point nor distinct yield plateau. Based on FEA results of HSS and mild steel plate girders introduced in [12], it was suggested that because of the aforementioned stress-strain behaviour, the strain hardening effect has barely any influence on the postbuckling capacity for mild steel plate girders. However, the strain hardening effect does have a clear strengthening effect on the postbuckling capacity of HSS plate girders. To take into account of this effect, a multilinear material model considering the strain hardening effect was introduced to carry out the FEA, as shown in Fig. 3. The multilinear model is extensively used for strength analyses of the HSS plate girders [14,15]. In this study, the FEA is conducted by using ABAQUS FE software. The S4R element which is a four node reduced integration quadrilateral shell element is selected to investigate the intermediate transverse stiffener requirement of mild steel and HSS plate girders. The S4R element in ABAQUS contains three rotational and three translational degrees-of-freedom per node, which means the membrane and flexural behaviours can be simulated by the S4R element. Hence, to investigate the properties of the intermediate transverse stiffeners in HSS plate girders, the elastic, plastic, and large strain behaviours can be included in the numerical analyses.
study. To avoid the confusion, the number used in Table 2 is consistent with that in [8]. Multilinear material model considering the strain hardening effect is chosen to conduct the FEA. The yield strength of the steel is 298.1 MPa. After FEA, it is clear to see that the ultimate shear resistances of the FEA Vu,FEA agree well with that of the experiment Vu,test, as shown in Table 2. Furthermore, the deformed shape of PG3 FE model is similar with that of PG3 specimen in the experiment, as shown in Fig. 4. To quantify the maximum out-of-plane deflection of the web panel along the intermediate transverse stiffener, the maximum additional out-of-plane deflection vs. V/Vu curves of the PG3 specimen and the PG3 FE model are compared, as shown in Fig. 5. It indicates that the maximum additional out-of-plane deflection vs. V/Vu curve of the PG3 FE model agrees well with that of the PG3 specimen. Hence, after the aforementioned careful validation, the numerical modelling method proposed in this study could be used to investigate the requirements of intermediate transverse stiffeners for HSS plate girders considering the postbuckling capacity. 2.4. Determination of specimen dimensions In order to prevent the buckling failure of the transverse stiffeners at the mid-span and end post, the transverse stiffener width bs = 97.5 mm and the transverse stiffener thickness ts = 20 mm are determined. The transverse stiffener width bs = 97.5 mm = (bf -tw)/2. Based on the results of the numerical study conducted in [12], the transverse stiffener with width bs = 97.5 mm and the thickness ts = 20 mm is rigid enough to prevent the buckling failure mode. For the tested intermediate
2.3. Validation of numerical modelling method The experiment conducted by Lee et al. is selected [8] to validate the accuracy of the numerical modelling method introduced in this
Fig. 2. First buckling mode.
3
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Table 2 Dimensions and section properties of specimens. Model
hw/mm
a/mm
tf/mm
tw/mm
bs/mm
ts/mm
Vu,test/kN
Vu,FEA/kN
Vu,FEA/Vu,test
PG2 PG3 PG5
600 600 600
600 600 450
15 15 15
3.2 4 3.2
40 30 45
3.2 4 3.2
204.5 323.5 241.6
200.2 312.5 228.7
0.98 0.97 0.95
transverse stiffeners which are in the absence of an axial load, the transverse stiffener width bs = 40 mm and the transverse stiffener thickness is varied to investigate the minimum moment of inertia requirement. The dimensions of pate girders are shown in Table 3 and Fig. 6, where the P-1.5-750 denotes that the aspect ratio a/hw = 1.5 and the hw = 750. It is easy to see that different ratios hw/tw and a/hw are considered. Based on the conclusion in [12], the plate girders in this study are assumed to have a clear postbuckling capacity. Furthermore, to prevent the influence of the bending moment on the shear resistance, the flanges of all specimens are strengthened. Then, the bending moment resistance of the flanges Mf is always larger than the bending moment produced by the external loading, which means the influence of the bending moment on the shear resistance could be neglected. 2.5. Convergence study A convergence test is conducted to verify a suitable mesh element size. The plate girder P-1-750 in Table 3 is modelled. Given that the shape of the shell element S4R selected is a square, the length of the square is selected to quantify the size of the shell element. Considering that the element numbers of the web panel and the flange are much more than that of transverse stiffeners, the mesh element size of the web panel and the flange has a larger influence on the computational cost. Hence, for the mesh element of the web panel, different mesh element sizes are considered, including 5 mm, 15 mm, 25 mm, 45 mm, 65 mm, 85 mm and 105 mm. The mesh element size of tested transverse stiffeners is 10 mm. The mesh element size of the mid-span and end post transverse stiffener are 20 mm. After FEA, the results for the finite element model with different mesh element sizes are shown in Fig. 7, where Vu = Vu,i × i Vu,5 ×5 and i = 5, 15, 25, 45, 65, 85 and 105. The iteration error for the finite element model with the mesh element size of 45 mm is less than 5%, as shown in Fig. 7. Furthermore, the tie constraint in ABAQUS is selected to simulate the weld between web, flanges and transverse stiffeners. It reaches the convergence much faster when the mesh element size of the web and flange is larger than that of transverse stiffeners. Hence, the mesh element size 45 mm is supposed to be suitable.
Maximum additional out-of-plane deflection (mm)
Fig. 4. (a) Deformed shape of PG3 specimen [8]; (a) Deformed shape of PG3 FE model.
3. Discussions of finite element analysis results
0.8
3.1. Failure modes 0.6
The transverse stiffeners with different ts are introduced in the Q550 P-2-750 plate girder to carry out FEA. Then, there are two different failure modes observed in this study, i.e., intermediate transverse stiffener buckling mode and flange plastic hinge mode, as shown in Fig. 8. When the moment of inertia of the intermediate transverse stiffener is small, the intermediate transverse stiffener fails to maintain the nodal line during the postbuckling stage which leads to the intermediate transverse stiffener buckling mode, as shown in Fig. 8(a). Then, the postbuckling capacity of the plate girder will not be fully developed. However, when the moment of inertia of the intermediate transverse stiffener is large, the nodal line of the intermediate transverse stiffener can be maintained during the postbuckling stage. The out-of-plane deflection of the web panel at the intermediate transverse stiffener area can be restrained effectively. Then, with increases in the shear loading,
0.4
0.2
0.0
Experiment results of PG3 in [8] FEA results 0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
V/Vu
Fig. 5. Maximum additional out-of-plane deflection vs. V/Vu curves of PG3 specimen and FE model. 4
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Table 3 Specimen dimensions and section properties of specimens. Number
L/mm
hw/mm
tw/mm
bf/mm
tf/mm
a/mm
hw/tw
a/hw
bs/mm
P-0.5-750 P-1-750 P-1.5-750 P-2-750 P-0.5-1000 P-1-1000 P-1.5-1000 P-2-1000 P-0.5-1250 P-1-1250 P-1.5-1250 P-2-1250 P-0.5-1500 P-1-1500 P-1.5-1500 P-2-1500
1950 3450 4950 6450 2450 4450 6450 8450 2950 5450 7950 10,450 3450 6450 9450 12,450
750 750 750 750 1000 1000 1000 1000 1250 1250 1250 1250 1500 1500 1500 1500
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200
30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30
375 750 1125 1500 500 1000 1500 2000 625 1250 1875 2500 750 1500 2250 3000
150 150 150 150 200 200 200 200 250 250 250 250 300 300 300 300
0.5 1 1.5 2 0.5 1 1.5 2 0.5 1 1.5 2 0.5 1 1.5 2
40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40
Fig. 6. Dimensions of plate girders.
girders [12]. However, for the complete development of the ultimate shear resistance, the current design equations for intermediate transverse stiffeners have all been based on the study of mild steel plate girders. Therefore, they may not be suitable for the design of intermediate transverse stiffeners for HSS plate girders as discussed earlier. To clarify the aforementioned issue, the P-1-750 plate girder is selected to perform the FEA. The Q235/Q345/Q550/Q690 P-1-750 plate girders are considered to investigate the difference between the requirements of intermediate transverse stiffeners in mild steel and HSS plate girders. In FEA, the intermediate transverse stiffeners with different ts are applied in the plate girder to investigate the minimum requirement of moment of inertia. Based on results of FEA, the shear resistance vs. midspan displacement curves of mild steel and HSS plate girders with different intermediate transverse stiffeners are obtained, as shown in Figs. 9 and 10. It indicates that the intermediate transverse stiffener thickness has barely any effect on the initial stiffness. Furthermore, when the thickness of intermediate transverse stiffeners is small, the ultimate shear resistance cannot be fully developed during the postbuckling stage, as shown in Figs. 9 and 10. For Q690 P-1-750 plate girders, the shear resistance curve of Q690 P-1-750 plate girder with ts = 7 mm is almost the same with that of Q690 P-1-750 plate girder with ts = 20 mm which means that the postbuckling capacity is fully developed. Moreover, the failure modes of Q690 P-1-750 plate girders with ts = 6.5 mm and 7 mm are proposed, as shown in Fig. 11. It is easy to see that the failure mode of the Q690 P-1-750 plate girder with ts = 6.5 mm is the transverse stiffener buckling mode and that of Q690 P-1-750 plate girder with ts = 7 mm is the flange plastic hinge mode. Therefore, in this study, the intermediate transverse stiffener with ts = 7 mm is supposed to meet the requirement of the Q690 P-1-750 plate girder. The similar method is adopted to determine the minimum moment of inertia requirement of the intermediate transverse stiffener in mild steel and HSS plate girders.
Fig. 7. Result of the convergence test for different mesh element sizes.
plastic hinges develop in the flanges, as shown in Fig. 8(b). Hence, for the flange plastic hinge mode, the ultimate shear capacity of the plate girder can be completely developed. Therefore, when the failure mode of the plate girder is the flange plastic hinge mode, it is supposed that the moment of inertia of the intermediate transverse stiffener can meet the minimum requirement. 3.2. Differences between HSS and mild steel plate girders Given that there is no clear yield plateau for HSS, the postbuckling capacity of HSS plate girders is higher than that of mild steel plate
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Fig. 8. Failure modes of the Q550 P-2-750 plate girders with different ts: (a) intermediate transverse stiffener buckling mode: ts = 2.5 mm; (b) flange plastic hinge mode: ts = 3 mm.
Based on the aforementioned method, the minimum requirements of intermediate transverse stiffeners thickness for Q235/Q345/Q550/ Q690 P-1-750 plate girders are 4 mm/4.5 mm/6.5 mm/7 mm, respectively. Hence, to fully develop the postbuckling capacity of plate girders, the minimum moment of inertia requirement of the intermediate transverse stiffeners increases with the increase in the steel strength grade.
4. Discussions of AASHTO LRFD design method and new prediction formula 4.1. Specifications in AASHTO LRFD design method Considering differences in the depth ratio hw/tw, the shear strength curve of plate girder can be divided into three zones: (1) yield zone; (2) inelastic buckling zone and (3) elastic buckling zone [16]. In this study, the web panels in HSS plate girders are classified into the elastic buckling zone, as shown in Fig. 16. In AASHTO LRFD [16], there are two design equations for the intermediate transverse stiffener in plate girders without a longitudinal stiffener, as shown in Eqs. (1) and (2). It states that transverse stiffeners adjacent to web panels in which neither panel supports shear forces being larger than the shear buckling resistance, the moment of inertia of the transverse stiffener shall satisfy the smaller of Eqs. (1) and (2). When the postbuckling capacity is considered, the moment of inertia of transverse stiffeners shall satisfy Eq. (2) [16]. After a careful investigation, it is suggested that for Eq. (1), the intermediate transverse stiffener is designed to provide nodal lines until the web reaches the theoretical buckling stress. For Eq. (2), the intermediate transverse stiffener is designed to ensure that the web panel is able to develop the shear yield strength [9]. For HSS plate girders in the elastic buckling zone, there is a noteworthy postbuckling shear capacity. Hence, the minimum requirement of the moment of inertia for the intermediate transverse stiffener in HSS plate girder should be somewhere between Eqs. (1) and (2). After the calculation, results from Eqs. (1) and (2) are shown in Tables 4 and 5. Based on the comparison between the results of Eqs. (1) and (2) and the results of FEA, it is clear to see that the requirement of the intermediate transverse stiffener in HSS plate girders considering the postbuckling capacity is larger than the results obtained from Eq. (1) and much lower
3.3. Parametric studies of HSS plate girders In Section 3.2, the method of determining the minimum moment of inertia requirement of the intermediate transverse stiffener in mild steel and HSS plate girders is introduced. After the FEA on HSS plate girders with different intermediate transverse stiffeners, the ratios Vu/Vu, max are presented, as shown in Figs. 12–15, where Vu denotes the ultimate shear resistance, Vu, max denotes the ultimate shear resistance of the plate girder with the thickest intermediate transverse stiffener. Furthermore, It denotes the moment of inertia of the intermediate transverse stiffener and It1, AASHTO denotes the minimum moment of inertia requirement of the intermediate transverse stiffener calculated by Eq. (1) in AASHTO LRFD [16]. Hence, it is clear to see that with increases in moment of inertia of the intermediate transverse stiffener, the postbuckling capacity of HSS plate girders develops, gradually. Based on the results of FEA, the minimum moment of inertia requirements of the intermediate transverse stiffener in HSS plate girders listed in Table 3 are obtained, as shown in Tables 4 and 5. It is evident to see that when tw is the same, with increases in hw/tw, the minimum moment of inertia requirement increases, correspondingly. When hw is the same, with increases in a/hw, the minimum moment of inertia requirement decreases, correspondingly.
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Fig. 10. Shear resistance vs. mid-span displacement curves of HSS P-1-750 plate girders with different tested stiffeners: (a) Q550 HSS plate girders; (b) Q690 HSS plate girders.
Fig. 9. Shear resistance vs. mid-span displacement curves of mild steel P-1-750 plate girders with different tested stiffeners: (a) Q235 mild steel plate girders; (b) Q345 mild steel plate girders.
It 2, AASHTO
than the results from Eq. (2). Furthermore, considering the results of FEA, the steel strength grades, the aspect ratio a/hw and the depth ratio hw/tw do have an effect on the minimum moment of inertia requirement of the intermediate transverse stiffener for HSS plate girders. However, 2.5 for Eq. (1), J = (d / h )2 2 0.5 where d0 denotes the smaller of the 0 w adjacent web panel widths. Hence, when a/hw > 1, the results from Eq. (1) are not influenced by the ratio a/hw which is inconsistent with the FEA results. For Eq. (2), the influence of the aspect ratio a/hw and the depth ratio hw/tw is not fully taken into account. Therefore, given the aforementioned discussion, the design equations in AASHTO LRFD [16] could not be directly applicable in the design of the intermediate transverse stiffener for HSS plate girders considering the postbuckling capacity.
It1, AASHTO
bt w3 J
h w4
1.3 t
40
Fyw
1.5
E
(2)
4.2. New prediction formula Based on the comparison conducted in Section 4.1, there is a certain relationship between the results of Eq. (1) and the results of the FEA. Therefore, in order to predict the minimum moment of inertia requirement of the transverse stiffener for HSS plate girders, Eq. (1) is modified by including the influence of the aspect ratio a/hw, depth ratio hw/tw, and steel strength grade. Then, a new equation is proposed in this study to predict the minimum moment of inertia requirement of the intermediate transverse stiffener for HSS plate girders, as shown in Eq. (3). Compared with Eq. (1), b is replaced by a, where b denotes the smaller of a and hw. In addition, a denotes the distance between the transverse stiffeners. Moreover, there is a new equation of J introduced
(1)
7
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Tested stiffener
ts=6.5 mm
Tested stiffener
ts=7 mm Fig. 11. Failure modes of Q690 P-1-750 plate girders with ts = 6.5 mm and 7 mm.
Fig. 12. FEA results of Q550 and Q690 plate girders with a/hw = 0.5.
Fig. 13. FEA results of Q550 and Q690 plate girders with a/hw = 1.
in Eq. (3). After the numerical regression, the new equation of J is proposed, as shown in Eq. (4). It is worth noting that in order to develop a significant postbuckling capacity, the aspect ratio a/hw and depth ratio hw/tw of HSS plate girder should not be too large [12]. Given the conclusion proposed in [12] and the numerical analyses conducted in this study, to develop a considerable postbuckling capacity, the intermediate transverse stiffener spacing is suggested to be less than 2hw which means the aspect ratio a/hw ≤ 2. Hence, Eq. (3) proposed in this study could be applied to the HSS plate girder whose hw/tw ≤ 300 and a/hw ≤ 2 which are frequently used in the practical design and
construction. In order to validate the accuracy of Eq. (3), a comparison between the results of FEA and those of Eq. (3) is carried out, as shown in Figs. 17–19. The errors of the Eq. (3) results are all less than 10%. For results of It, Eq. (3)/It, FEA, the average value is 0.991 and the coefficient of variation is Cv, Eq. (3) = 0.039. Furthermore, to show the predictive power of the proposed formula, the independent simulations are performed. The Q550/Q690 plate girders with a/hw = 0.8 and hw/ tw = 275 are simulated. The comparison between the results of FEA and Eq. (3) is conducted, as shown in Table 6, Figs. 20 and 21. It turns out that the errors are acceptable. The failure modes could be predicted
8
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It
(3)
Jat w3
J = [162.98 - 218.65(a / hw ) + 104.59(a /hw )2 - 17.43(a/ hw )3 + 0.031(hw /tw )](fyw / E )0.4
(4) 5. Performance of intermediate transverse stiffeners 5.1. Discussion of resistant capacities The resistant capacity of HSS plate girders is different from that of mild steel plate girders. Therefore, to fully develop the postbuckling capacity, the requirement of the intermediate transverse stiffener in HSS plate girders might be different from that of the mild steel plate girders. Based on the analyses conducted in Section 3.2, the Q235/ Q345/Q550/Q690 P-1-750 plate girders with the corresponding minimum intermediate transverse stiffeners are included in the FEA to clarify the aforementioned issue. Then, the shear resistance vs. midspan displacement curves are obtained, as shown in Fig. 22. The resistant curve form of HSS plate girders is different from that of mild steel plate girders. Because that there is no clear yield plateau for HSS, the strain hardening is beneficial to the postbuckling capacity. Then, for the resistant curve of HSS plate girders, after the elasticity region, there is a significant increase in the shear resistant capacity. However, for mild steels, there is a clear yield plateau. Hence, the strain hardening has barely any effect on the postbuckling capacity. After the elasticity region, no significant increase in the shear resistant capacity is observed. Therefore, the maximum additional out-of-plane deflection of the intermediate transverse stiffener in HSS plate girders corresponding to the ultimate shear resistance is significantly larger than that of the intermediate transverse stiffener in mild steel plate girders, as shown in Fig. 23. However, in EC3 [17], it states that the maximum additional deflection should not exceed b/300, where b denotes the height between the centroids of the flanges or span of the transverse stiffener. In this study, b is considered as the height between the centroids of the flanges. Therefore, when the postbuckling capacity can be fully developed, the additional out-of-plane deflection of the intermediate transverse stiffener in HSS plate girders might not meet the requirement in EC3 [17]. However, for mild steel plate girders, such problem does not need to be concerned, as shown in Fig. 23. Hence, even though there are some studies about the minimum moment of inertia requirement and the behaviour of the intermediate transverse stiffener in mild steel plate girders, little attention has been paid to investigating the adverse effect caused by the additional out-of-plane deflection of the
Fig. 14. FEA results of Q550 and Q690 plate girders with a/hw = 1.5.
Fig. 15. FEA results of Q550 and Q690 plate girders with a/hw = 2.
accurately. Hence, the proposed formula could be used to predict the minimum moment of inertia requirement of the intermediate transverse stiffener for Q550 and Q690 HSS plate girders considering the postbuckling capacity effectively. Table 4 Comparison between the results of FEA and AASHTO LRFD for Q550 plate girders. Number
ts/mm
It,FEA/mm4
It1,AASHTO/mm4
It2,AASHTO/mm4
It,FEA/It1,AASHTO
It,FEA/It2,AASHTO
P-0.5-750 P-1-750 P-1.5-750 P-2-750 P-0.5-1000 P-1-1000 P-1.5-1000 P-2-1000 P-0.5-1250 P-1-1250 P-1.5-1250 P-2-1250 P-0.5-1500 P-1-1500 P-1.5-1500 P-2-1500
7 6.5 4.5 3 10 9 6 4.5 13 11.5 8 7 16 14 11 10
358,240 332,651 230,297 153,531 511,771 460,594 307,063 230,297 665,302 588,536 409,417 358,240 818,833 716,479 562,948 511,771
375,000 46,875 46,875 46,875 500,000 62,500 62,500 62,500 625,000 78,125 78,125 78,125 750,000 93,750 93,750 93,750
1,381,419 1,674,965 4,357,404 12,504,465 3,605,280 3,605,280 6,518,416 13,771,549 8,801,954 8,801,954 8,801,954 10,659,098 18,251,732 18,251,732 18,251,732 18,251,732
0.955 7.097 4.913 3.275 1.024 7.370 4.913 3.685 1.064 7.533 5.241 4.585 1.092 7.642 6.005 5.459
0.259 0.199 0.053 0.012 0.142 0.128 0.047 0.017 0.076 0.067 0.047 0.034 0.045 0.039 0.031 0.028
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Table 5 Comparison between the results of FEA and AASHTO LRFD for Q690 plate girders. Number
ts/mm
It,FEA/mm4
It1,AASHTO/mm4
It2,AASHTO/mm4
It,FEA/It1,AASHTO
It,FEA/It2,AASHTO
P-0.5-750 P-1-750 P-1.5-750 P-2-750 P-0.5-1000 P-1-1000 P-1.5-1000 P-2-1000 P-0.5-1250 P-1-1250 P-1.5-1250 P-2-1250 P-0.5-1500 P-1-1500 P-1.5-1500 P-2-1500
7.5 7 4.5 3.5 10.5 9.5 6.5 5 13.5 12 8.5 7.5 16.5 15 11.5 10.5
383,828 358,240 230,297 179,120 537,359 486,182 332,651 255,885 690,891 614,125 435,005 383,828 844,422 767,656 588,536 537,359
375,000 46,875 46,875 46,875 500,000 62,500 62,500 62,500 625,000 78,125 78,125 78,125 750,000 93,750 93,750 93,750
2,178,628 2,606,678 8,222,233 15,803,940 5,066,040 5,066,040 9,989,011 19,759,467 12,368,263 12,368,263 12,368,263 16,810,398 25,646,830 25,646,830 25,646,830 25,646,830
1.024 7.642 4.913 3.821 1.075 7.779 5.322 4.094 1.105 7.861 5.568 4.913 1.126 8.188 6.278 5.732
0.176 0.137 0.028 0.011 0.106 0.096 0.033 0.013 0.056 0.050 0.035 0.023 0.033 0.030 0.023 0.021
Vu Vp
Elastic Buckling curve
I Yield
0
II
Inelastic Buckling
Elastic Buckling
hw tw
Fig. 16. Shear strength curve of plate girders in AASHTO LRFD [16].
Fig. 18. Comparison of It for a/hw = 0.5/1/1.5/2 and Fyw = 690 MPa: Eq. (3) vs. FEA results.
Fig. 17. Comparison of It for a/hw = 0.5/1/1.5/2 and Fyw = 550 MPa: Eq. (3) vs. FEA results.
Fig. 19. Variation of It,Eq. and hw/tw.
(3)/It,FEA
vs. a/hw for plate girders with different HSS
the issue discussed above, the adverse effect and the behaviour of the intermediate transverse stiffener in HSS plate girders are investigated in the following section.
intermediate transverse stiffener on the shear resistance of plate girders. Given the discussions done before, for HSS plate girders, the aforementioned adverse effect should be taken into account. To clarify 10
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more obvious than that of Q550 plate girders. It is worth noting that in some situations, the shear resistance of the HSS plate girder is reduced by almost half. Hence, it is meaningful to pay attention to the aforementioned shear resistance reduction of HSS plate girders in the practical applications.
Table 6 Comparison between the results of FEA and Eq. (3). Number
Steel strength
tw/mm
ts/mm
It,FEA/mm4
It,Eq.(3)/mm4
It,Eq.(3)/It,FEA
P-0.8-1375 P-0.8-1375
Q550 Q690
5 5
13.5 14
690,891 716,479
699,067 765,444
1.011 1.068
5.3. Out-of-plane deflection Considering the FEA results of the Q690 P-1-750 plate girder, the maximum additional out-of-plane deflection of the intermediate transverse stiffener whose ts = 7 mm vs. mid-span displacement of the plate girder curve is obtained, as shown in Fig. 25(a). Before the buckling resistance, the maximum additional out-of-plane deflection develops slowly. Then, after the buckling resistance, the maximum out-of-plane deflection develops faster. For the different states in Fig. 25(a), the outof-plane deflection distribution curves of the intermediate transverse stiffener are plotted along the web depth, as shown in Fig. 25(b). The out-of-plane deflection distribution curves are unimodal. The peak point is reached in the middle of the web depth. For the Q690 P-1-750 plate girder, when ts = 7 mm, the postbuckling capacity of the plate girder can be fully developed. However, the maximum additional out-of-plane deflection corresponding to the ultimate resistance is obviously larger than b/300 = 2.6 mm, as shown in Fig. 26. When ts = 10 mm and 20 mm, the maximum out-of-plane deflection can be effectively reduced, as shown in Fig. 26. Moreover, when ts = 20 mm, the maximum additional out-of-plane deflection is always less than b/300 = 2.6 mm. Hence, for the actual design of HSS plate girders, the increase in the moment of inertia of the intermediate
5.2. Reduction of shear resistance For HSS plate girders, before the ultimate shear resistance is fully developed, the intermediate transverse stiffener will develop an unacceptable additional out-of-plane deflection. Hence, it is necessary to investigate the reduction of the shear resistance of the HSS plate girder, according to the additional out-of-plane deflection of the intermediate transverse stiffener. Based on the FEA results of Q550 and Q690 plate girders in Section 3.3, the ultimate shear resistance and the reduced shear resistance corresponding to the limit state of the intermediate transverse stiffener are obtained, as shown in Table 7. The reduced shear resistance denotes the shear resistance corresponding to the condition where the additional out-of-plane deflection of the intermediate transverse stiffener reaches b/300, as shown in Fig. 24. It indicates that with increases in the aspect ratio a/hw and depth ratio hw/ tw, the shear resistance reduction caused by the additional out-of-plane deflection of the intermediate transverse stiffener increases, gradually. Furthermore, the shear resistance reduction of Q690 plate girders is
Fig. 20. Failure modes of the Q550 P-0.8-1375 plate girders with different ts: (a) intermediate transverse stiffener buckling mode: ts = 13 mm; (b) flange plastic hinge mode: ts = 13.5 mm.
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Fig. 21. Failure modes of the Q690 P-0.8-1375 plate girders with different ts: (a) intermediate transverse stiffener buckling mode: ts = 13.5 mm; (b) flange plastic hinge mode: ts = 14 mm.
Fig. 22. Shear resistance vs. mid-span displacement curves of Q235/Q345/ Q550/Q690 plate girders.
Fig. 23. Maximum additional out-of-plane deflection vs. mid-span displacement curves of intermediate transverse stiffeners in Q235/Q345/Q550/Q690 plate girders.
transverse stiffener can effectively reduce the additional out-of-plane deflection. To investigate the effect of the initial imperfection on the additional out-of-plane deflection of the intermediate transverse stiffener, different initial imperfections are introduced in the FEA of Q690 P-1-750 plate girder whose ts = 10 mm. In the early stage, there is barely any difference in the FEA results of plate girders with different initial
imperfections, as shown in Fig. 27. With increases in the mid-span displacement, the difference in the additional out-of-plane deflection increases. To a certain extent, reducing the initial imperfection of the plate girder can help to reduce the additional out-of-plane deflection of the intermediate transverse stiffener. 12
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Table 7 Reduced shear resistance and ultimate shear resistance of Q550 and Q690 plate girders. Number
Steel strength grades
Thickness of the transverse stiffener ts (mm)
Ultimate shear resistance Vu (kN)
Reduced shear resistance Vr (kN)
Vr/Vu
P-0.5-750 P-1-750 P-1.5-750 P-2-750 P-1-1000 P-1-1250 P-1-1500 P-0.5-750 P-1-750 P-1.5-750 P-2-750 P-1-1000 P-1-1250 P-1-1500
Q550 Q550 Q550 Q550 Q550 Q550 Q550 Q690 Q690 Q690 Q690 Q690 Q690 Q690
7 6.5 4.5 3 9 11.5 14 7.5 7 4.5 3.5 9.5 12.5 15
1450.4 1141.2 888.1 760.9 1280.9 1399.8 1570.1 1746.9 1365.4 1054.1 902.6 1535.6 1679.3 1883.2
1403.3 1052.1 671.9 504.3 955.9 825.9 745.7 1624.5 1223.9 688.6 505.1 998.6 915.4 1087.9
0.97 0.92 0.76 0.66 0.75 0.59 0.47 0.93 0.90 0.65 0.56 0.65 0.55 0.58
Fig. 24. Reduced shear resistance and ultimate shear resistance of the HSS plate girder.
6. Conclusions
Fig. 25. (a) Maximum additional out-of-plane deflection vs. mid-span displacement curve of the transverse stiffener in Q690 P-1-750-7 plate girder; (b) Additional out-of-plane deflection of the transverse stiffener in Q690 P-1-750-7 and P-1-750-10 plate girders.
This paper conducted a thorough investigation on the properties and minimum requirements for the intermediate transverse stiffener of HSS plate girders with the consideration of the postbuckling capacity. The main contributions of this study are summarized as the followings:
corresponding to the ultimate shear resistance is significantly larger than that of the mild steel plate girders. Hence, for HSS plate girders, this adverse effect on the shear resistance of plate girders needs to be considered. 5. It has proven that with increases in the aspect ratio a/hw and depth ratio hw/tw, the shear resistance reduction caused by the additional out-of-plane deflection of the intermediate transverse stiffener increases, gradually. Furthermore, the shear resistance reduction of Q690 plate girders is more obvious than that of Q550 plate girders. It is meaningful to pay attention to the aforementioned shear resistance reduction for HSS plate girders in the practical applications. 6. For the actual design of HSS plate girders, the increase in the moment of inertia of the intermediate transverse stiffener can effectively reduce the additional out-of-plane deflection. To a certain extent, reducing the initial imperfection of the plate girder can help to reduce the additional out-of-plane deflection of the intermediate transverse stiffener.
1. To fully develop the postbuckling capacity of plate girders, the minimum moment of inertia requirement of intermediate transverse stiffeners in HSS plate girders is larger than that of the requirement in the mild steel plate girders. 2. With the comparison between the FEA results and the results predicted by the design equations in AASHTO LRFD, it indicates that the design equations in AASHTO LRFD could not be directly used in the design of the intermediate transverse stiffener for HSS plate girders when the postbuckling capacity is considered. 3. A new equation is proposed in this study to predict the minimum moment of inertia requirement of the intermediate transverse stiffener for HSS plate girders. A comparison between the FEA results and the results from the new equation is conducted to validate the accuracy of the new equation. 4. With the difference in the resistant capacity between HSS and mild steel plate girders, the maximum additional out-of-plane deflection of the intermediate transverse stiffener in HSS plate girders 13
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Fig. 26. Maximum additional out-of-plane deflection vs. mid-span displacement curves of the transverse stiffener in Q690 P-1-750 plate girders with different transverse stiffeners.
Fig. 27. Maximum additional out-of-plane deflection vs. mid-span displacement curves of the transverse stiffener in Q690 P-1-750-10 plate girders with different initial imperfections.
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