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Probabilistic ultimate buckling strength of stiffened plates, considering thick and high-performance steel
Journal of Constructional Steel Research 138 (2017) 184–195
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Journal of Constructional Steel Research
Probabilistic ultimate buckling strength of stiffened plates, considering thick and high-performance steel Mahmudur Rahman a,⁎, Yoshiaki Okui a, Takahiro Shoji a, Masato Komuro b a b
Department of Civil and Environmental Engineering, Saitama University, 255 Shimo-Okubo, Sakura-ku, Saitama City, Saitama 338-8570, Japan Department of Civil Engineering, Muroran Institute of Technology, 27-1 Mizumoto-cho, Muroran, Hokkaido 050-8585, Japan
a r t i c l e
i n f o
Article history: Received 9 February 2017 Received in revised form 30 May 2017 Accepted 8 July 2017 Available online xxxx Keywords: Stiffened steel plate Initial imperfection FE analysis Response surface Monte Carlo simulation
a b s t r a c t The probabilistic distribution of ultimate buckling strength for stiffened steel plates subjected to a distributed axial stress was obtained using Monte Carlo simulations in association with the response surface method. The plates of both normal and high-performance steel (SBHS) were taken into account, and their thickness was varied from 10 to 90 mm. The ultimate buckling strength was determined by nonlinear elasto-plastic finite element (FE) analysis, considering geometric and material nonlinearity. The initial out-of-plane deflection and residual stress were considered as two independent random variables upon which the ultimate buckling strength depends. The response surface, showing the variation of ultimate strength due to the initial deflection and residual stress, was estimated using the nonlinear FE results. Based on the obtained statistical distribution, partial safety factors for the ultimate buckling strength were proposed. © 2017 Elsevier Ltd. All rights reserved.
Symbol List
Al E Il Ne RR a b bs fN hr kr n pf t
cross-sectional area of longitudinal stiffener modulus of elasticity moment of inertia of a longitudinal stiffener with respect to its base number of elements per half subpanel width reduced slenderness parameter length of the stiffened plate in between two transverse stiffeners overall width of the stiffened plate width of a subpanel in between two longitudinal stiffeners nominal strength of the stiffened plate height of the longitudinal stiffener buckling coefficient number of subpanels divided by the number of longitudinal stiffeners probability of non-exceedance thickness of the panel plate
⁎ Corresponding author. E-mail addresses: [email protected] (M. Rahman), [email protected] (Y. Okui), [email protected] (T. Shoji), [email protected] (M. Komuro).
http://dx.doi.org/10.1016/j.jcsr.2017.07.004 0143-974X/© 2017 Elsevier Ltd. All rights reserved.
tr t0 Δ α α0 βT γ γl γl , req δl δ01 ε εl εy μ ν σ σcr σe σrc σT σy
thickness of the longitudinal stiffener critical thickness of the panel plate to avoid local buckling maximum initial out-of-plane deflection for local mode deflection aspect ratio = a/b critical aspect ratio target reliability index partial safety factor relative stiffness of the longitudinal stiffener required relative stiffness of the longitudinal stiffener according to the Japanese Specification for Highway Bridges (JSHB) cross-sectional area ratio of one longitudinal stiffener to the panel plate = Al/bt maximum initial out-of-plane deflection magnitude for a whole-plate deflection shape engineering strain local strain in the longitudinal direction of the stiffened plate yield strain mean value Poisson's ratio engineering stress or standard deviation (as mentioned in the text) ultimate buckling strength elastic buckling strength compressive residual stress tensile strength yield strength
M. Rahman et al. / Journal of Constructional Steel Research 138 (2017) 184–195
1. Introduction Stiffened plates are often used to construct different parts of steel bridges, such as the bottom flange of box girders, and the box sections used as truss members or columns. Under compression, such thin plate components exhibit local buckling and may fail with sudden collapse. The collapse of large steel box girder bridges during the 1970s led to extensive research on buckling behavior and the ultimate load carrying capacity of steel plates, including stiffened plates [1]. Rigorous experimental and numerical studies were carried out during the 1970–80s to investigate the ultimate buckling strength, considering the effect of initial imperfections. For example, Komatsu et al. [2] measured the initial deflection and residual stress for 28 stiffened plate specimens including high-strength steel. The residual stress distribution inside stiffened plates was also reported. Komatsu et al. [3] obtained statistical data on the initial deflection and the ultimate buckling strength of steel bridge members. Komatsu and Nara [4] also investigated fundamental modes of initial deflection and their individual effect on the ultimate strength. Employing a semi-analytical finite element (FE) analysis, Nara and Komatsu [5] proposed ultimate buckling strength curves corresponding to 1%, 5% and 10% probability of non-exceedance. Three different numbers of longitudinal stiffeners were considered to obtain the buckling strength curves. Stochastic variation of the initial out-of-plane deflection was taken into account, but the residual stress was assumed to be constant. Furthermore, Nara et al. [6] investigated the effect of the relative stiffness of longitudinal stiffeners on the ultimate strength, employing elasto-plastic finite displacement theory. The numerical results were compared with the strength curves specified by German design codes DASt Ri-012 [7] and DIN4114 [8]. It was found that stiffened plates, satisfying relative stiffness requirement of JSHB [9–10] showed lower ultimate strengths compared to DASt Ri-012 and DIN4114 code. Stiffened plates with twice the required relative stiffness yield the same ultimate strength as DASt Ri-012 at a reduced slenderness parameter RR = 0.7. The reduced slenderness parameter RR is defined as b RR ¼ t
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ y 12 1−ν 2 E π 2 kr
ð1Þ
where b is the overall width of the plate, t is the thickness of the plate, σy, E and ν represent the yield strength, modulus of elasticity and Poisson's ratio for the steel, respectively, and the buckling coefficient kr = 4n2, where n is the number of subpanels divided by the number of longitudinal stiffeners.
185
Kanai and Otsuka [11] carried out experiments on 43 stiffened plates under uniaxial loading and proposed an ultimate strength curve, which has been adopted in the current provision of JSHB [10]. Fukumoto et al. [12] also conducted experiments on stiffened plates with low reduced slenderness parameter (RR = 0.46–0.78). Fig. 1 shows the standard ultimate strength curve for a stiffened plate, as per the current JSHB provision, compared with the aforementioned experimental results. Here, σcr is the ultimate buckling strength. Previous research in the 1970–80s, generally dealt with relatively thin plates (about 10 mm thick). The experimental data of Kanai and Otsuka [11], which is the basis of the JSHB strength curve, were also obtained from experiments conducted with thin plates. Since 1996, the JSHB limit for the maximum thickness of steel plates that can be used in steel bridge construction has increased from 50 mm to 100 mm [9–10]. However, the effect on the ultimate load bearing capacity due to use of such thicker plates has not yet been investigated. In 2008, Steel for Bridge High-performance Structure (SBHS) was incorporated into the Japanese Industrial Standards (JIS). SBHS offers advantages over ordinary steel, such as a higher yield strength, better weldability and ease of fabrication [13]. Nevertheless, its inelastic behavior differs from that for ordinary steel since it has a high yieldto-tensile strength ratio and almost no yield plateau [14]. The current strength curve for JSHB does not account for the effects of SBHS steel. There is one more reason to re-examine the current JSHB strength curve. The present practice in developing design codes, i.e., AASHTO LRFD [15] and Eurocode [16], is to develop reliability-based design criteria with partial safety factors (PSF) so as to account for uncertainties originating from individual sources. The current JSHB code does not adopt the partial factor format. To develop a reliability-based strength curve for ultimate buckling strength, it is necessary to obtain probabilistic information, such as a probability density function, a mean value and a standard deviation for ultimate buckling strength. Even though the past study of Nara and Komatsu [5] proposed ultimate strength values for 1%, 5% and 10% fractile, the effects of thick plates and SBHS steel are still unknown. This paper investigates the probabilistic distribution of ultimate buckling strength for longitudinally stiffened plates with 2 equidistant flat plate longitudinal stiffeners, satisfying the relative stiffness requirement of JSHB (γl/γl,req = 1) and with an aspect ratio α = 1. This kind of plates is generally used in the bottom flange of a steel box girder bridge. The ratio of relative stiffness to the required relative stiffness of a longitudinal stiffener (γl/γl ,req) is calculated according to the JSHB [10] as presented in the Appendix-A. The plates of both normal and highperformance steel (SBHS) were taken into account, and their thickness was varied from 10 to 90 mm. Based on the probabilistic distribution obtained from Monte Carlo simulation (MCS), PSFs are proposed for the ultimate buckling strength.
1.2 1
a
a
0.8
0.4
0 0
hr tr
Longitudinal Stiffeners
bs
b
cr
σ /σ
y
Transverse Stiffeners
0.6
0.2
t
Current JSHB Provision Exp. Kanai et al. Exp. Fukumoto et al. 0.2
0.4
0.6
X
0.8
1
1.2
a 2
a 2
a
1.4
RR Fig. 1. Standard ultimate strength curve of stiffened plate in JSHB.
Fig. 2. Model geometry.
Section X-X
X
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800
SM490Y SM570 SBHS500 SBHS700
σ [MPa]
600
400
200
0 0
0.05
0.1
ε
0.15
0.2
Fig. 4. Idealized stress-stain relations for different steel grades. Fig. 3. Boundary conditions.
2. Finite element modelling and elastic buckling analysis
2.3. Elastic buckling analysis
2.1. Model geometry and boundary conditions
An eigenvalue buckling analysis was performed prior to the nonlinear analysis in order to determine the elastic buckling strength and buckling mode. A compressive load was applied through a forced displacement. The first two buckling modes and the normalized elastic
Schematic diagram of the stiffened plate model is shown in Fig. 2. Due to symmetric geometric and loading conditions, and in order to reduce the computational time, the shaded rectangular area shown in Fig. 2 was modeled in ABAQUS. The plate members were modeled using 4-node, quadrilateral, stress-displacement shell element S4R, available in the ABAQUS element library [17], and which is suitable for large scale displacement analysis. Fig. 3 illustrates the boundary conditions for the stiffened plate model. Here Ux, Uy, Uz denote translational degrees of freedom, and URx, URy, URz denote rotational degrees of freedom with respect to the X, Y and Z axes. Boundary conditions corresponding to a simple support were applied along the both side longitudinal edges. Instead of modeling a transverse stiffener, translational movement in the Z direction was restrained, assuming a transverse stiffener with sufficient rigidity.
U, U3 1.00 0.71 0.43 0.14 −0.14 −0.43 −0.71 −1.00
Z
2.2. Material model
Y
As the current study discusses the member safety factor, it is necessary to exclude the effect of uncertainties originating from the material properties, such as the yield strength, modulus of elasticity and Poisson's ratio. For this reason, material properties are considered as a deterministic standardized value in this study. The yield strength for steel with different plate thickness and material grades was obtained from the JSHB code as shown in Table 1. The modulus of elasticity and Poison's ratio were considered to be identical for all steel grades, and were taken to be 200 GPa and 0.3, respectively. The inelastic characteristics of the four different steel grades were determined from idealized uniaxial stress (σ)-stain (ε) relationships, as shown in Fig. 4, based on test data [18].
(a) 1st buckling mode ( σ e σ y = 1.25). U, U3 1.00 0.71 0.43 0.14 −0.14 −0.43 −0.71 −1.00
Table 1 Yield strength [MPa] for different steel grades.
Z Y
Steel grade
SM490Y SM570 SBHS500 SBHS700
X
Plate thickness (mm) t ≤ 16
16 b t ≤ 40
40 b t ≤ 75
t N 75
365 460 500 700
355 450 500 700
335 430 500 700
325 420 500 700
X
(b) 2nd buckling mode ( σ e σ y = 1.58). Fig. 5. Buckling modes and σe/σy values for stiffened plate with RR = 0.8, t = 30 mm and steel grade SBHS500.
M. Rahman et al. / Journal of Constructional Steel Research 138 (2017) 184–195
(+)
0.9
σy
Tension (+)
187
σ rc
(-)
Compression
0.8
σcr/σy
(a)
σrc
0.6σy
RR 0.6
0.7
RR 1.2
Z dir.
(+)
(-)
0.6 (+)
σy Compression
0.5 2
Tension
4
6
8
10
12
14
Ne (b) Fig. 7. Mesh dependency analysis result. Fig. 6. Distribution of residual stresses at (a) subpanels and (b) longitudinal stiffeners.
buckling strength (σe/σy) for a stiffened plate model is presented in Fig. 5 as an example.
the input out-of-plane deflection magnitude was adjusted by trial and error until the desired deflection magnitude was obtained.
3.2. Mesh dependency analysis 3. Nonlinear elasto-plastic FE analysis Material as well as geometric nonlinearity was considered in the nonlinear elasto-plastic FE analyses. Mises plasticity and the isotropic strain hardening theory were applied to model the material nonlinearity. The nonlinear analyses were carried out in two steps. In the first step, initial imperfections were simulated to reproduce the initial condition. In the second step, a compressive load was applied to simulate subsequent loading. 3.1. Simulation of initial imperfections Initial imperfections, i.e., out-of-plane deflections and residual stresses, were simulated simultaneously in each stiffened plate model. The initial out-of-plane deflections were simulated through linear superposition of buckling modes obtained from elastic buckling analysis. Residual stresses were included directly in each element following the distribution pattern, as shown in Fig. 6. Variations of the residual stress distribution along the plate thickness direction were not considered because of the negligible effect on the load carrying capacity [19]. The magnitude of the imperfections was simulated based on the mean value (μ) and the standard deviation (σ), as presented in Table 2, obtained from past studies [3,5 and 20]. Here, δ01 represents the magnitude of an out-of-plane deflection with half-sine wave shape, representing the first elastic buckling mode (whole-plate mode deflection), |Δ| is the magnitude of local mode out-of-plane deflection with similar shape of Fig. 5 (b), a is the plate length, bs is the subpanel width and σrc represents the compressive residual stress. At the first step of nonlinear analysis, due to the incorporation of residual stress, the input out-of-plane deflection magnitude yields higher deflection magnitude than the input value. To overcome this problem,
A mesh dependency analysis was performed to ensure the convergence of the nonlinear elasto-plastic FE results. Two different stiffened plate models with RR = 0.6 and 1.2 were considered for the analysis. Both models have a thickness of 30 mm, material grade SM570 and mean values of the residual stress and initial deflection. The number of elements (Ne) per half subpanel width (which is the half wavelength of the buckle with a local buckling mode) was considered equal to the number of elements along the height of a longitudinal stiffener. Fig. 7 shows the result of mesh dependency analysis. It was found that a convergent result can be obtained if Ne is greater or equal to ten. Based on this result, the value of Ne was taken equal to ten for the subsequent analysis. This result is also consistent with the rule of thumb for shell elements, that there should be at least six elements in the expected half wavelength of a buckle [21]. 1
3
4
2
b
a/2 a
Table 2 Statistical parameters for initial imperfections obtained from the past studies [3,5 and 20].
X Imperfections
Mean (μ)
Standard deviation (σ)
1000δ01/a 150|Δ|/bs σrc/σy
0.096 0.138 0.230
0.426 0.107 0.145
Y
bs
Fig. 8. Geometry of test model (after Komatsu et al. [22]).
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Table 3 Cross sectional dimensions and material properties of test model. Cross sectional dimensions a (mm) 2000
Material properties
b (mm) 810
t (mm) 10.60
bs (mm) 270
hr (mm) 91
tr (mm) 10.60
σy (MPa) 262.82
σT (MPa) 440.32
The nonlinear FE analysis technique was verified using the experimental result of a selected test model as shown in Fig. 8, reported by Komatsu et al. [22]. The test model with two equidistant flat plate longitudinal stiffeners has the geometric and material properties as presented in Table 3. Initial out-of-plane deflections and residual stresses were measured in the panel plate and longitudinal stiffeners. Reported highest magnitude of δ01 was b/306.12 while the average compressive residual stress magnitude (σrc) for panel plate and stiffener was 0.41 σy and 0.35 σy respectively. In the FE model, the magnitude of δ01 and σrc were considered the same as the reported magnitude, following the shape and distribution pattern mentioned in Section 3.1. Boundary conditions assumed for the FE model corresponds to the experimental condition of a four side simply supported plate. Strain gauges were attached at four locations of the test model as shown in Fig. 8 to measure the local strain (εl) along the longitudinal direction. Fig. 9 shows comparisons of the normalized stress σ/σy versus normalized local strain εl/εy curves from the FE analysis and experimental result. The ultimate buckling strength obtained from FE analysis was found to be 1.16% lower than the experimental result. Moreover, good agreement in the normalized stress-strain curves were observed, which validates FE analysis result. 3.4. Parametric study To obtain probabilistic information, it is necessary to determine the ultimate buckling strength for several stiffened plate models, with varying parameters such as the material grades, reduced slenderness parameter RR, and plate thickness. The present study employed a parametric analysis for 96 stiffened plate models with 4 different material grades (SM490Y, SM570, SBHS500 and SBHS700) with RR values of 0.4 to 1.4, and plate thicknesses of 10 to 90 mm. Each model encompasses 12 sets of combinations of initial deflection and residual stress, as shown in Fig. 10. The total number of FE analyses is 1152.
Verification of FE Model 0.8
cr
0.6
42.70
Results of the nonlinear elasto-plastic analysis are represented by normalized stress (σ/σy) – strain (ε/εy) curves in Fig. 11, where σ and σy represent the compressive stress and the yield stress, and ε and εy denote the applied axial strain and the yield strain, respectively. In this figure, the residual stress and initial out-of-plane deflection are assigned to respective mean value, as an example. The ultimate buckling strength is determined from the peak stress value in the stress-strain curve. Fig. 11 shows the effect of RR on the buckling behavior of the stiffened plates. Normalized stress-strain curves are shown for SM570 and SBHS500, with RR ranging from 0.4 to 1.4. It can be seen that for RR ≤ 0.6, the plates exhibit inelastic buckling, whereas for RR ≥ 0.8, unstable snap-through behavior is observed. 4.2. Effect of plate thickness Fig. 12 shows the effect of the plate thickness. Here, the normalized ultimate buckling strength for stiffened plates with material grades SM570 and SBHS500 are plotted for different plate thickness and different RR values. It was found that the plate thickness does not significantly affect the ultimate strengths, provided the residual stress distribution along the plate thickness direction is considered constant. 4.3. Effect of material grade Fig. 13 shows the effect of the material grade. Here, the normalized ultimate buckling strength for stiffened plates with t = 20 mm are plotted for four different material grades and different RR values. It is observed that for RR ≤ 0.6, higher steel grades exhibit a lower load carrying capacity (σcr/σy), while for RR ≥ 0.8 the pattern is irregular. For RR =
μ−σ
y
1000δ01/a
0.4 1 FE 1 Exp. 2 FE 2 Exp. 3,4 FE 3,4 Exp.
0.3 0.2 0.1 0
εl/εy
2
μ
μ+σ
μ+2σ
μ+3σ μ+3σ
1.2
FE =0.7403 Exp.=0.7490
0.5
σ/σy
0.29
4.1. Effect of reduced slenderness parameter RR
σ /σ
−2
% of elongation
4. Discussion of nonlinear analysis results
0.7
−4
ν
Details of the dimensions of the stiffened plate models are presented in Appendix B.
3.3. Verification of the FE model
0
E (GPa) 203.98
4
μ+2σ
0.8
μ+σ
0.4
μ
0
μ−σ
−0.4 6
Fig. 9. Normalized stress (σ/σy) versus normalized local strain (εl/εy) curves from experiment and FE analysis.
−0.8 0
μ−2σ
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
σrc/σy
Fig. 10. Combination of initial imperfections.
M. Rahman et al. / Journal of Constructional Steel Research 138 (2017) 184–195
189
RS μ ID μ
RS μ ID μ 1
1.2
0.8
1 0.8
σcr/σy
RR 0.4 RR 0.6 RR 0.8
0.5
1
RR 1.4 1.5
ε/εy
0
2
355
SBHS700
0 0
0.2
SBHS500
RR 1.2
SM570 SBHS500
RR 0.8
0.4
RR 1.0
0.2
RR 0.6
SM570
0.4
RR 0.4
0.6
SM490Y
σ/σy
0.6
RR 1.0 RR 1.2 RR 1.4
450 500 700 Nominal Yield Strength (MPa)
Fig. 11. Effect of RR on normalized stress-strain curves for mean values of the residual stress and initial deflection, and a thickness of 30 mm.
Fig. 13. Effect of material grade on normalized ultimate buckling strength for mean values of the residual stress and initial deflection, and a thickness of 20 mm.
0.8, there is no significant effect of material grade. Moreover, except for RR = 1.4, SBHS700 shows a lower load carrying capacity than SBHS500.
tolerance for respective modes. Due to including both positive and negative out-of-plane deflection, where positive means a deflection towards the stiffener and negative means a deflection in the opposite side of the stiffener, mean (μ) value of 1000δ01/a is very small i.e. 0.096. To avoid such low magnitude of whole-plate mode compared to the local mode, μ + σ was used for both whole-plate mode and local mode while investigating the effect ultimate strength variation due to addition of local mode. On the other hand, it is rational to use the mean value (μ) of residual stress. Fig. 15(a) and (b) illustrates the shape of initial out-of-plane deflection for Case-1 and Case-2 respectively, simulated in the first step of the nonlinear analysis. Table 5 presents the comparison of ultimate buckling strengths for Case-1 and Case-2. It was observed that the effect of addition of local mode deflection is not significant. The variation of ultimate strengths due to incorporation of the local deflection mode is b1%. Moreover, Fig. 16 shows the normalized compressive stress-strain curves for Case-1 and Case-2 at different RR values, which confirms that there is no significant deviation in the trend of the curves. Hence, only wholeplate mode (the lowest eigenmode) is considered for subsequent analyses.
4.4. Effect of local mode initial out-of-plane deflection Up to previous section, only the whole-plate mode initial out-ofplane deflection was considered. In this section, the effect of local mode is presented. Fig. 5(a) and (b) shows the first and second eigenmodes which are identical to the two fundamental modes of initial out-of-plane deflections: whole-plate mode and local mode as shown in Fig. 14(a) and (b), respectively. To identify the effect of local mode, a nonlinear FE analysis was carried out for two cases i.e. Case-1 and Case-2, covering the full range of reduced slenderness parameter (RR = 0.4 to 1.4) considered in this study. Table 4 shows details of the imperfection combinations expressed in terms of respective mean value (μ) and the standard deviation (σ). Both cases account for the residual stress and whole-plate mode while Case-2 has the additional local mode. Unit values of the normalized expression for whole-plate mode (1000δ01/a) and local mode (150 | Δ |/bs) represent the fabrication
5. Response surface
1.2
The variation in the ultimate buckling strength depends on the initial out-of-plane deflection and the residual stress. To identify the effect of
1
Longitudinal Stiffeners
0.6
δ 01
σcr/σy
0.8 RR 0.4 RR 0.6 0.4
RR 0.8
(a)
RR 1.0
0 0
RR 1.2
SBHS500 SM570 10
20
30
40
50
60
Longitudinal Stiffeners
RR 1.4 70
80
90 100
Thickness (mm) Fig. 12. Effect of plate thickness on normalized ultimate buckling strength for mean values of the residual stress and initial deflection, and for material grades SM570 and SBHS500.
(b)
Δ
0.2
Fig. 14. Initial deflection (a) Whole-plate deflection, (b) Local deflection.
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Table 4 Combination of imperfections in terms of respective mean (μ) and standard deviation (σ) for Case-1 and Case-2.
Case-1 Case-2
Residual stress
Initial out-of-plane deflection
(σrc/σy)
Whole-plate mode (1000δ01/a)
Local mode (150|Δ|/bs)
μ μ
μ +σ μ +σ
0 μ+σ
these two variables, the response surface method [23,24] was employed using the nonlinear FE analysis results. In this method, the ultimate buckling strength is expressed as a response function of the initial deflection and the residual stress. The main objective of using the response surface is to reduce the computational cost of the numerical analysis. In the present study, the following second-order polynomial response function is employed: σ cr X ¼ pij xi1 x2j σy
ði ¼ 0; 1; 2; j ¼ 0; 2; i þ j ≤4Þ
ð2Þ
where, x1 = normalized residual stress (σrc/σy), x2 = normalized initial deflection (1000δ01/a), and pij are the coefficients of the polynomial, determined by a nonlinear multiple regression analysis. The symmetricity of the out-of-plane deflection, obtained in the first buckling mode, yields the same ultimate strength for a similar magnitude of positive and negative out-of-plane deflection, where positive means a deflection towards the stiffener and negative means a
Table 5 Comparison of ultimate buckling strengths for Case-1 and Case-2. σcr/σy
RR
0.4 0.6 0.8 1.0 1.2 1.4
Case-1
Case-2
0.920 0.826 0.760 0.624 0.492 0.390
0.918 0.827 0.753 0.620 0.491 0.392
% difference with respect to Case-1 0.20% −0.11% 0.92% 0.66% 0.24% −0.38%
deflection in the opposite side of the stiffener. Accordingly, an even function with respect to x2 is employed in Eq. (2). Fig. 17(a) illustrates an example of the response surface for RR = 1.0, accounting for all of the material grades and plate thicknesses considered in this study. The mesh in the figure shows the response surface and the solid dots represent the nonlinear FE analysis results. To further clarify the variation in ultimate strength, Fig. 17(b) and (c) was extracted from the results shown in Fig. 17(a). Fig. 17(b) shows the strength variation with respect to the residual stress for a mean (μ) value of the initial deflection. Similarly, Fig. 17(c) presents the effect of variation of the initial deflection for a mean (μ) value of the residual stress. It was found that the ultimate strength, represented by the response surface function (RSF), decreases with increasing magnitude of the initial deflection and residual stress. 6. Monte Carlo simulation An MCS was performed to obtain probabilistic information on the ultimate buckling strength, where the initial out-of-plane deflection and the residual stress were considered as two independent random variables. 6.1. Generation of random variables The two random variables were generated in the MCS based on their probability density function (PDF) shown in Figs. 18 and 19. Fig. 18 shows the reported histogram for positive and negative initial out-of-plane deflections [5], fitted using two exponential distributions f+(x) and f−(x), under the following constraints: þ
Z Y
−
f ð0Þ ¼ f ð0Þ
X
ð3:1Þ
1
(a)
0.8
σ/σy
0.6 RR 0.4 RR 0.6
0.4
RR 0.8 RR 1.0
0.2
Case−1 Case−2
Z Y
X
(b) Fig. 15. Simulated initial deflection shape for (a) Case-1 and (b) Case-2.
0 0
0.5
1
ε/εy
RR 1.2 RR 1.4 1.5
2
Fig. 16. Normalized compressive stress-strain curves for Case-1 and Case-2 at different RR values.
(a)
Relative Frequency/width of histogram bins
M. Rahman et al. / Journal of Constructional Steel Research 138 (2017) 184–195
Response Surface; RR 1.0 SM490Y SM570 SBHS500 SBHS700
σcr/σy
1 0.8 0.6 0.4 0.2 −1
0 0.2 0
0.4 0.6
1000δ01/a
(b)
1
Residual Stress Variation; RR 1.0 RSF SM490Y SM570 SBHS500 SBHS700
1.1 1 σcr/σy
σrc/σy
0.9
0.6
(c)
0.4 σrc/σy
λ (+ve)=8.4993 λ (−ve)=7.1244
3 2.5 2 1.5 1 0.5 0
−0.5
0
0.5 1 1000δ01/a
1.5
2
0.8
Initial Deflection Variation; RR 1.0 RSF SM490Y SM570 SBHS500 SBHS700
1.1 1 σcr/σy
0.6
Experimental Data Exponential Distribution
3.5
exponential distributions, respectively. The +/− signs used as superscripts in Eqs. (3.1) and (3.2) denote the respective functions or parameters for positive and negative deflections. The obtained PDF was employed to model the probabilistic distribution of initial deflections in the MCS. In the MCS, the maximum value of 1000δ01/a was restricted to ± 1 in order to maintain the allowable deflection specified in the JSHB. A lognormal distribution with the same mean value and standard deviation reported by Fukumoto and Itoh [20] was employed in order to generate the random values of compressive residual stress shown in Fig. 19.
0.7
0.2
4
Fig. 18. PDF for initial deflection used in MCS (after Nara & Komatsu, 1988).
0.8
0.5 0
191
0.9
6.2. Calculation of ultimate strength After randomly generating pairs of initial deflection and residual stress values, the response surface function was used to determine the ultimate buckling strength for each pair of values. This process was repeated for a number of iterations, until the μ and σ values for the ultimate strength obtained from the MCS converged. In this study, 100,000 iterations were performed for this convergence to occur.
0.8 0.7 0.6 −0.5
0 1000δ01/a
0.5
0.2
1
Fig. 17. (a) Response surface, (b) Effect of variation of residual stress for a mean value of the initial deflection and (c) Effect of variation of initial deflection for a mean value of the residual stress, for different steel grades at RR = 1.0.
XZ
þ
f ðxþ Þ dx þ
Z
−
f ðx− Þ dx ¼ 1
ð3:2Þ
where þ
þ
þ
−λþ xþ
f ðx Þ ¼ ηλ exp −
Relative Frequency
0.5 −1
Lognormal Distribution μ = 0.23 σ = 0.145
0.15
0.1
0.05
ð3:2:1Þ − −
f ðx− Þ ¼ ð1−ηÞλ− exp−λ
x
ð3:2:2Þ
Here, η is the ratio of positive experimental data points to the total number of experimental data points, and x and λ represent the initial deflection magnitude and rate parameter for the
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
σrc/σy
1
Fig. 19. PDF for compressive residual stress used in MCS (after Fukumoto et al., 1984).
192
M. Rahman et al. / Journal of Constructional Steel Research 138 (2017) 184–195
1.2
Table 6 Mean (μ) and standard deviation (σ) for normalized ultimate buckling strength (σcr/σy). Steel grades
Statistical parameter
RR 0.4
0.6
0.8
1.0
1.2
1.4
SM490Y
μ σ μ σ μ σ μ σ μ σ
0.9507 0.0102 0.9403 0.0055 0.9206 0.0075 0.8894 0.0140 0.9276 0.0085
0.8869 0.0366 0.8702 0.0246 0.8714 0.0303 0.8269 0.0358 0.8658 0.0304
0.7599 0.0590 0.7895 0.0533 0.7844 0.0541 0.8015 0.0499 0.7849 0.0515
0.6269 0.0652 0.6751 0.0560 0.6792 0.0561 0.6888 0.0563 0.6686 0.0504
0.4870 0.0390 0.5244 0.0349 0.5452 0.0412 0.5125 0.0301 0.5155 0.0313
0.3843 0.0262 0.4030 0.0236 0.4041 0.0214 0.4077 0.0209 0.3992 0.0201
SBHS500 SBHS700 All steel grades
0.8
σcr/σy
SM570
1
JSHB Europl
0.6
Eurocl Eurointr
0.4
AASHTO Canadian Nara pf=5%
0.2
MCS μ−pf=5−95% Exp. Kanai
Table 6 shows the μ and σ values for the ultimate strength, obtained from the MCS. Fig. 20(a) and (b) shows the relative frequency distribution for the ultimate buckling strength for RR = 0.6 and 1.0, respectively. The solid curve represents the normal distribution to fit the ultimate buckling strength using the mean value and standard deviation
JSHB
0.8
σcr/σy
0.4
0.6
0.8
1
1.2
1.4
RR Fig. 21. Comparison of MCS results with JSHB, AASHTO, Eurocode, Canadian Code, study of Nara et al. and experimental results of Kanai et al.
6.3. Discussion on MCS results
μ=0.86577,σ=0.030314
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0. 0
0.2
obtained from the MCS. The red horizontal line represents the deterministic value for the ultimate strength specified by JSHB.
1.0 0.9
0 0
0.05
0.1 0.15 0.2 Relative Frequency
0.25
0.3
(a)
A summary of the MCS results is presented in Fig. 21, where the top and bottom error bars represent 95% and 5% probabilities of nonexceedance for the normalized ultimate buckling strength, respectively, and the midpoint shows the mean value. The green solid line represents the ultimate buckling strength curve corresponding to a 5% probability of non-exceedance, as reported by Nara et al. The MCS results were compared with JSHB [10], AASHTO LRFD [15], Eurocode [16], Canadian Highway Bridge Design Code [25], study of Nara et al. [5] as well as the experimental results of Kanai et al. [11]. The predictions from all of the sources in the Fig. 21 corresponds to a stiffened plate which is similar to our FE model. According to the effective width method, Eurocode provides a reduction factor ρ for plate-like buckling and χc for column-like buckling, and interpolates between ρ and χc to obtain the final reduction factor ρc. The predictions from Eurocode, shown in Fig. 21, corresponds to the plate-like buckling, column-like buckling as well as
1.0 0.9
0.1
μ−βTσ
0.8 μ=0.66877,σ=0.050615
0.08
Relative Frequency
σcr /σy
0.7 0.6
JSHB
0.5 0.4 0.3 0.2 0.1 0. 0
μ
βTσ
0.06
0.04
Probability of Non−exceedance
0.02
0.05
0.1 0.15 Relative Frequency
0.2
0.25
(b) Fig. 20. Relative frequency distribution for ultimate buckling strength at (a) RR = 0.6 and (b) RR = 1.0.
0 0.5
0.6
0.7
σcr/σy
0.8
Fig. 22. Explanation of Eq. (4).
0.9
1
M. Rahman et al. / Journal of Constructional Steel Research 138 (2017) 184–195
Partial Safety Factor (γ)
1.25
8. Conclusions
pf 5% pf 3%
1.2
pf 1%
1.15 1.1 1.05 1 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
RR Fig. 23. Graphical representation of PSFs.
the interpolated final strength. It was observed that, our FE model shows a strong deviation to the column-like buckling. The bottom error bars (5% probabilities of non-exceedance) of the MCS results agree with those of Nara et al. and those of Eurocode, except for RR values in the range of 0.8–1.2. In this range, the buckling strength is strongly affected by imperfections, warranted from the relatively higher standard deviations for the MCS result for RR = 0.8 and 1.0. Compared to the MCS results, JSHB, AASHTO and Canadian Code overestimate the ultimate strength for RR ≤ 0.6, and underestimate it for RR ≥ 0.6. It is interesting to note that predictions from AASHTO and Canadian Code are less than the JSHB for RR ≥ 0.4. It is widely reported that for stiffened plates with more than one longitudinal stiffener, AASHTO LRFD provides very conservative result [26,27]. Experimental results of Kanai et al. show fair agreement with MCS results in the range of 0.5 ≤ RR ≤0.9. Around RR = 0.8, scattered experimental results were found which explains the higher standard deviation of MCS result. For RR ≤ 0.5 andRR ≥ 1.2, experimental results are higher than the MCS results. 7. Proposal for PSFs PSFs for ultimate load carrying capacity of stiffened plates were determined using the reliability indexing method. The following equation shows the relationship between the PSF and the reliability index:
μ−βT σ ¼
1 f γ N
193
ð4Þ
where, μ and σ are the mean and the standard deviation of the normalized ultimate buckling strength, respectively, βT is the target reliability index, γ is the partial safety factor and fN is the corresponding nominal strength. Fig. 22 illustrates Eq. (4) graphically for RR = 0.8. If the nominal strength is considered to be the mean value of the buckling strength, the corresponding partial safety factor can be determined from Eq. (4). Assuming that the probability density function for the ultimate buckling strength is normally distributed, the target reliability index βT is 1.64, 1.88 and 2.33 for 5%, 3% and 1% probability of non-exceedance, respectively. The calculated PSFs for ultimate buckling strengths with 5%, 3% and 1% probability of non-exceedance (pf) are shown in Fig. 23, for the case when the nominal strength is equal to the mean value. The maximum PSF was obtained for RR = 1.0.
This paper discusses the ultimate buckling strengths of stiffened steel plates by employing numerical and probabilistic approaches. Stiffened steel plates that are commonly used in the bottom flange of a box girder bridge with an aspect ratio α = 1.0 and 2 flat plate longitudinal stiffener, satisfying the relative stiffness requirement of JSHB was considered in this study. A total of 1152 FE analysis was carried out for 96 stiffened plate models with varying parameters. Monte Carlo simulation was applied to determine the probabilistic ultimate buckling strength for compressive stiffened steel plates. The response surface was used in the MCS to account for the variability of the initial deflection and residual stress simultaneously. The effect of thick plates and SHBS steels on the ultimate strength was investigated. It was found that the plate thickness does not significantly affect the ultimate strengths but the effect of SBHS steels is significant. Analysis of Table 6 reveals that, compared to the conventional steels, SBHS steels have higher mean ultimate strengths for RR ≥ 0.7. For example, at RR = 1.2, SBHS500 shows 10.67% higher mean strength than SM490Y. However, for RR b 0.7, mean strengths of SBHS steels are lower than that of conventional steels. The MCS results were compared with different design codes, study of Nara et al. [5] as well as experimental results of Kanai et al. [11]. MCS results showed very good agreement with those of Nara et al. and Eurocode, except for RR values in the range of 0.8–1.2. The higher prediction of MCS results at this range possibly due to consideration of SBHS steels. Moreover, the buckling strength is also strongly affected by imperfections at this region. Experimental results [11] also show fair agreement with MCS results at 0.5 ≤ RR ≤0.9. The probability density function for the ultimate buckling strength was obtained for different RR values in the range of 0.4 to 1.4. Finally, the PSF for several non-exceedance probabilities was proposed for different RR values. The results of the present study can be used as an important baseline for deriving a reliability-based ultimate strength curve. In this study, we restricted our attention to the ultimate strengths. For RR ≥ 1.0, large out-of-plane deflection, due to elastic buckling, may occur before ultimate buckling. Restriction of this out-of-plane deflection is important from a serviceability point of view, and serviceability limit strength is a topic for future research. Moreover, probabilistic research on plate-like and column-like buckling is of future interest. Acknowledgment This work was supported by The Japan Iron and Steel Federation. For some of the numerical analysis, the authors would like to express their gratitude to Mr. Daiki Miazaki, a former student of Saitama University. Appendix A. Required relative stiffness of longitudinal stiffeners in JSHB According to the design provision of JSHB, the required relative stiffness for a flat longitudinal stiffener, γl ,req, is given by the following equations: If α≤ α0 2 2 α þ1 t0 ; γl;req ¼ 4α 2 n ð1 þ nδl Þ− n t γl;req ¼ 4α 2 n ð1 þ nδl Þ−
α2 þ 1 ; n
for t ≥t 0
for tbt 0
ðA:1:1Þ
ðA:1:2Þ
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M. Rahman et al. / Journal of Constructional Steel Research 138 (2017) 184–195
If α N α0 γ l;req
Appendix B (continued) (continued)
2( 3 )2 2 14 t 0 ¼ ð1 þ nδl Þ−1 −15; 2n2 t n
γ l;req ¼
i 2 1 h 2 2n ð1 þ nδl Þ−1 −1 ; n
RR
for t ≥t 0
ðA:2:1Þ 1.2
ðA:2:2Þ
for tbt 0
1.4
In the preceding equations, the aspect ratio α = a/b, and the critical pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aspect ratio α 0 ¼ 4 1 þ nγ l, where n is the number of subpanels divided by the longitudinal stiffeners, and γl is the relative stiffness of the longitudinal stiffener given by
3 γ l ¼ Il = bt =11
ðA:3Þ
where Il is the moment of inertia for a longitudinal stiffener with respect to the base of the longitudinal stiffener, and t is the thickness of the panel plate. In Eqs. (A.1) and (A.2), δl denotes the cross-sectional area ratio for the longitudinal stiffener to panel plate (Al/bt), where Al is the cross-sectional area of one longitudinal stiffener and t0 is the critical thickness of the panel plate to avoid local buckling, given by b t0 ¼ nπ
SBHS500 0.4
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 1−ν 2 σ y E
ðA:4Þ
where the symbols are the same as those in Eq. (1).
0.6
0.8
1.0
1.2
Appendix B. Detailed dimensions of stiffened plate models 1.4 RR SM490Y 0.4
0.6
0.8
1.0
1.2
1.4
SM570 0.4
0.6
0.8
1.0
t (mm)
b (mm)
tr (mm)
hr (mm)
γl/γl,req
20 30 50 70 90 10 20 30 50 70 10 20 30 50 10 20 30 10 20 30 10 20 30
1079 1635 2788 3903 5094 801 1618 2437 4181 5854 1068 2158 3249 5575 1335 2697 4062 1602 3237 4874 1869 3776 5686
12 18 30 41 54 9 15 22 37 51 9 16 23 39 9 17 24 9 18 26 9 18 26
120 180 310 430 560 70 150 220 370 520 75 160 240 400 80 170 250 85 170 260 90 180 270
1.06 1.03 1.05 1.00 1.01 1.05 1.09 1.00 1.01 1.01 1.01 1.09 1.05 1.04 1.00 1.13 1.01 1.02 1.01 1.03 1.05 1.04 1.01
20 30 50 70 90 10 20 30 50 70 10 20 30 50 10
959 1443 2460 3445 4481 714 1439 2165 3691 5167 952 1918 2886 4921 1189
12 17 29 40 52 9 16 23 38 52 9 17 25 40 9
105 160 270 380 490 70 140 210 360 500 75 150 230 380 80
1.06 1.05 1.03 1.03 1.01 1.15 1.04 1.00 1.06 1.01 1.11 1.05 1.10 1.02 1.11
SBHS700 0.4
0.6
0.8
1.0
1.2
1.4
t (mm)
b (mm)
tr (mm)
hr (mm)
γl/γl, req
20 30 10 20 30 10 20 30
2398 3608 1427 2877 4329 1665 3357 5051
17 26 10 18 27 10 19 28
160 240 80 170 250 85 170 260
1.05 1.06 1.17 1.12 1.06 1.08 1.03 1.07
20 30 50 70 90 10 20 30 50 70 10 20 30 50 10 20 30 50 10 20 30 10 20 30
913 1369 2282 3194 4107 684 1369 2053 3422 4791 913 1825 2738 4563 1141 2282 3422 5704 1369 2738 4107 1597 3194 4791
11 17 28 38 49 9 15 23 37 51 9 16 25 40 9 17 26 42 10 17 27 9 18 28
100 150 250 350 460 70 140 210 350 490 75 150 220 370 80 160 240 390 80 180 250 85 170 260
1.01 1.04 1.03 1.01 1.07 1.18 1.03 1.04 1.02 1.00 1.14 1.04 1.01 1.00 1.14 1.09 1.11 1.01 1.07 1.31 1.10 1.02 1.02 1.12
20 30 50 70 90 20 30 50 70 90 10 20 30 50 70 10 20 30 50 10 20 30 50 10 20 30
771 1157 1928 2706 3471 1157 1736 2893 4050 5207 771 1543 2314 3857 5399 964 1928 2893 4821 1157 2314 3471 5785 1350 2700 4050
11 16 26 36 46 15 23 37 52 66 9 17 26 43 60 10 19 27 45 10 20 28 47 10 20 29
85 125 210 290 380 120 180 300 420 540 70 140 210 350 490 75 150 220 370 75 150 230 380 80 160 240
1.15 1.06 1.06 1.00 1.06 1.02 1.04 1.01 1.01 1.00 1.08 1.03 1.05 1.04 1.04 1.19 1.14 1.03 1.05 1.03 1.03 1.03 1.01 1.09 1.09 1.06
References [1] N.E. Shanmugam, M. Arockiasamy, Local buckling of stiffened plates in offshore structures, J. Constr. Steel Res. 38 (1) (1996) 41–59. [2] S. Komatsu, M. Ushio, T. Kitada, An experimental study on residual stress and initial deformations of stiffened plates, Proc. JSCE (265) (1976) 25–35 (In Japanese). [3] S. Komatsu, T. Kitada, E. Watanabe, M. Akiraumi, K. Nara, Statistical study of the initial deformation and the ultimate strengths of steel bridge members, JSSC 16 (179) (1980) 10–43 (In Japanese). [4] S. Komatsu, S. Nara, Statistical study on steel plate members, J. Struct. Eng. ASCE 109 (4) (1983) 977–992. [5] S. Nara, S. Komatsu, Statistical study on ultimate strength curves of longitudinally stiffened plates under uniaxial compression, J. Struct. Mech. Earthq. Eng. 392 (I-9) (1988) 289–296 (In Japanese).
M. Rahman et al. / Journal of Constructional Steel Research 138 (2017) 184–195 [6] S. Nara, S. Komatsu, T. Kitada, A study on ultimate compressive strength properties of longitudinally stiffened continuous plates under uniaxial compression, J. Struct. Mech. Earthq. Eng. JSCE 392 (I-9) (1988) 273–280 (In Japanese). [7] Deutscher Ausschuss fur Stahlbau-Richtlinien: DASt Ri-0121978 (In German). [8] Din 4114 Blatt 1: Stahlbau, Stabilitatsfalle (Knickung, Kippung), Berechnungsgrundlagen, Varschrichten, Juli1952 (In German). [9] JSHB, Specification for Highway Bridges: Part-II Steel Bridges, Japan Road Association, 1980. [10] JSHB, Specification for Highway Bridges: Part-II Steel Bridges, Japan Road Association, 2012. [11] M. Kanai, K. Otsuka, Design method of stiffened plate, J. JSCE 19 (10) (1977) 498–503 (In Japanese). [12] Y. Fukumoto, K. Yamaguchi, T. Usami, Inelastic buckling strength of stiffened plates in compression, IABSE Proc. P-8/77 (1977) 1–15. [13] T. Yoneyama, T. Fujii, Fabrication and erection of Tokyo Gate Bridge, Proceedings of IABSE-JSCE Joint Conference on Advances in Bridge Engineering-III, Dhaka 2015, pp. 268–277. [14] D.K. Shin, E.Y. Cho, K. Kim, Ultimate flexural strengths of plate girders subjected to web local buckling, Int. J. Steel Struct. 13 (2) (2013) 291–303. [15] AASHTO LRFD Bridge Design Specifications. 4th ed, American Association of State Highway and Transportation Officials, 2007. [16] Eurocode 3- Design of Steel Structures- Part1-5: Plated Structural Elements, EN 1993-1-5, The European Union, 2006. [17] Abaqus 6.14 Documentation, Dassault Systèmes Simulia Corp, 2014. [18] Research committee for rational structure and design for steel bridges, Rationalization of Strength Design for Steel Bridges, Japan Society of Steel Construction, JSSC Technical Report No. 98, 2013 (In Japanese).
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[19] T. Miyashita, N. Inaba, S. Hirayama, C. Liu, M. Nagai, Measurement method for welding residual stress in steel I-shaped girder with thick flange and its influence on load-carrying capacity for bending, J. JSCE 3 (2015) 191–208. [20] Y. Fukumoto, Y. Itoh, Basic compressive strength of steel plates from test data, Proc. JSCE Struct. Eng./Earthq. Eng. 344 (I-1) (1984) 129–139. [21] B. Johansson, R. Maquoi, G. Sedlacek, C. Müller, D. Beg, Commentary and Worked Examples to EN 1993-1-5 “Plated Structural Elements”, JRC – ECCS, Luxembourg, 2007. [22] S. Komatsu, M. Ushio, T. Kitada, S. Nara, An experimental study on the ultimate strength of orthogonally stiffened plates under uniaxial compression, Proc. JSCE (288) (1979) 13–27 (In Japanese). [23] B. Gaspar, A. Naess, B.J. Leira, C.G. Soares, System reliability analysis of a stiffened panel under combined uniaxial compression and lateral pressure loads, Struct. Saf. 39 (2012) 30–43. [24] D.V. Duc, Y. Okui, K. Hagiwara, M. Nagai, Probabilistic distributions of plate buckling strength for normal and bridge high performance steels, Int. J. Steel Struct. 13 (3) (2013) 557–567. [25] Canadian Highway Bridge Design Code, S6-14, Canadian Standard Association (CSA) Group, 2016. [26] B.H. Choi, M.O. Hwang, T.Y. Yoon, Experimental study of inelastic buckling strength and stiffness requirements for longitudinally stiffened panels, Eng. Struct. 31 (2009) 1141–1153. [27] D.D. Thang, M.S. Koo, A. Hameed, Y.D. Jeong, Stiffness requirement for flat-bar longitudinal stiffener of box-girder compression flanges, Int. J. Steel Struct. 9 (2009) 115–122.
Contents lists available at ScienceDirect
Journal of Constructional Steel Research
Probabilistic ultimate buckling strength of stiffened plates, considering thick and high-performance steel Mahmudur Rahman a,⁎, Yoshiaki Okui a, Takahiro Shoji a, Masato Komuro b a b
Department of Civil and Environmental Engineering, Saitama University, 255 Shimo-Okubo, Sakura-ku, Saitama City, Saitama 338-8570, Japan Department of Civil Engineering, Muroran Institute of Technology, 27-1 Mizumoto-cho, Muroran, Hokkaido 050-8585, Japan
a r t i c l e
i n f o
Article history: Received 9 February 2017 Received in revised form 30 May 2017 Accepted 8 July 2017 Available online xxxx Keywords: Stiffened steel plate Initial imperfection FE analysis Response surface Monte Carlo simulation
a b s t r a c t The probabilistic distribution of ultimate buckling strength for stiffened steel plates subjected to a distributed axial stress was obtained using Monte Carlo simulations in association with the response surface method. The plates of both normal and high-performance steel (SBHS) were taken into account, and their thickness was varied from 10 to 90 mm. The ultimate buckling strength was determined by nonlinear elasto-plastic finite element (FE) analysis, considering geometric and material nonlinearity. The initial out-of-plane deflection and residual stress were considered as two independent random variables upon which the ultimate buckling strength depends. The response surface, showing the variation of ultimate strength due to the initial deflection and residual stress, was estimated using the nonlinear FE results. Based on the obtained statistical distribution, partial safety factors for the ultimate buckling strength were proposed. © 2017 Elsevier Ltd. All rights reserved.
Symbol List
Al E Il Ne RR a b bs fN hr kr n pf t
cross-sectional area of longitudinal stiffener modulus of elasticity moment of inertia of a longitudinal stiffener with respect to its base number of elements per half subpanel width reduced slenderness parameter length of the stiffened plate in between two transverse stiffeners overall width of the stiffened plate width of a subpanel in between two longitudinal stiffeners nominal strength of the stiffened plate height of the longitudinal stiffener buckling coefficient number of subpanels divided by the number of longitudinal stiffeners probability of non-exceedance thickness of the panel plate
⁎ Corresponding author. E-mail addresses: [email protected] (M. Rahman), [email protected] (Y. Okui), [email protected] (T. Shoji), [email protected] (M. Komuro).
http://dx.doi.org/10.1016/j.jcsr.2017.07.004 0143-974X/© 2017 Elsevier Ltd. All rights reserved.
tr t0 Δ α α0 βT γ γl γl , req δl δ01 ε εl εy μ ν σ σcr σe σrc σT σy
thickness of the longitudinal stiffener critical thickness of the panel plate to avoid local buckling maximum initial out-of-plane deflection for local mode deflection aspect ratio = a/b critical aspect ratio target reliability index partial safety factor relative stiffness of the longitudinal stiffener required relative stiffness of the longitudinal stiffener according to the Japanese Specification for Highway Bridges (JSHB) cross-sectional area ratio of one longitudinal stiffener to the panel plate = Al/bt maximum initial out-of-plane deflection magnitude for a whole-plate deflection shape engineering strain local strain in the longitudinal direction of the stiffened plate yield strain mean value Poisson's ratio engineering stress or standard deviation (as mentioned in the text) ultimate buckling strength elastic buckling strength compressive residual stress tensile strength yield strength
M. Rahman et al. / Journal of Constructional Steel Research 138 (2017) 184–195
1. Introduction Stiffened plates are often used to construct different parts of steel bridges, such as the bottom flange of box girders, and the box sections used as truss members or columns. Under compression, such thin plate components exhibit local buckling and may fail with sudden collapse. The collapse of large steel box girder bridges during the 1970s led to extensive research on buckling behavior and the ultimate load carrying capacity of steel plates, including stiffened plates [1]. Rigorous experimental and numerical studies were carried out during the 1970–80s to investigate the ultimate buckling strength, considering the effect of initial imperfections. For example, Komatsu et al. [2] measured the initial deflection and residual stress for 28 stiffened plate specimens including high-strength steel. The residual stress distribution inside stiffened plates was also reported. Komatsu et al. [3] obtained statistical data on the initial deflection and the ultimate buckling strength of steel bridge members. Komatsu and Nara [4] also investigated fundamental modes of initial deflection and their individual effect on the ultimate strength. Employing a semi-analytical finite element (FE) analysis, Nara and Komatsu [5] proposed ultimate buckling strength curves corresponding to 1%, 5% and 10% probability of non-exceedance. Three different numbers of longitudinal stiffeners were considered to obtain the buckling strength curves. Stochastic variation of the initial out-of-plane deflection was taken into account, but the residual stress was assumed to be constant. Furthermore, Nara et al. [6] investigated the effect of the relative stiffness of longitudinal stiffeners on the ultimate strength, employing elasto-plastic finite displacement theory. The numerical results were compared with the strength curves specified by German design codes DASt Ri-012 [7] and DIN4114 [8]. It was found that stiffened plates, satisfying relative stiffness requirement of JSHB [9–10] showed lower ultimate strengths compared to DASt Ri-012 and DIN4114 code. Stiffened plates with twice the required relative stiffness yield the same ultimate strength as DASt Ri-012 at a reduced slenderness parameter RR = 0.7. The reduced slenderness parameter RR is defined as b RR ¼ t
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ y 12 1−ν 2 E π 2 kr
ð1Þ
where b is the overall width of the plate, t is the thickness of the plate, σy, E and ν represent the yield strength, modulus of elasticity and Poisson's ratio for the steel, respectively, and the buckling coefficient kr = 4n2, where n is the number of subpanels divided by the number of longitudinal stiffeners.
185
Kanai and Otsuka [11] carried out experiments on 43 stiffened plates under uniaxial loading and proposed an ultimate strength curve, which has been adopted in the current provision of JSHB [10]. Fukumoto et al. [12] also conducted experiments on stiffened plates with low reduced slenderness parameter (RR = 0.46–0.78). Fig. 1 shows the standard ultimate strength curve for a stiffened plate, as per the current JSHB provision, compared with the aforementioned experimental results. Here, σcr is the ultimate buckling strength. Previous research in the 1970–80s, generally dealt with relatively thin plates (about 10 mm thick). The experimental data of Kanai and Otsuka [11], which is the basis of the JSHB strength curve, were also obtained from experiments conducted with thin plates. Since 1996, the JSHB limit for the maximum thickness of steel plates that can be used in steel bridge construction has increased from 50 mm to 100 mm [9–10]. However, the effect on the ultimate load bearing capacity due to use of such thicker plates has not yet been investigated. In 2008, Steel for Bridge High-performance Structure (SBHS) was incorporated into the Japanese Industrial Standards (JIS). SBHS offers advantages over ordinary steel, such as a higher yield strength, better weldability and ease of fabrication [13]. Nevertheless, its inelastic behavior differs from that for ordinary steel since it has a high yieldto-tensile strength ratio and almost no yield plateau [14]. The current strength curve for JSHB does not account for the effects of SBHS steel. There is one more reason to re-examine the current JSHB strength curve. The present practice in developing design codes, i.e., AASHTO LRFD [15] and Eurocode [16], is to develop reliability-based design criteria with partial safety factors (PSF) so as to account for uncertainties originating from individual sources. The current JSHB code does not adopt the partial factor format. To develop a reliability-based strength curve for ultimate buckling strength, it is necessary to obtain probabilistic information, such as a probability density function, a mean value and a standard deviation for ultimate buckling strength. Even though the past study of Nara and Komatsu [5] proposed ultimate strength values for 1%, 5% and 10% fractile, the effects of thick plates and SBHS steel are still unknown. This paper investigates the probabilistic distribution of ultimate buckling strength for longitudinally stiffened plates with 2 equidistant flat plate longitudinal stiffeners, satisfying the relative stiffness requirement of JSHB (γl/γl,req = 1) and with an aspect ratio α = 1. This kind of plates is generally used in the bottom flange of a steel box girder bridge. The ratio of relative stiffness to the required relative stiffness of a longitudinal stiffener (γl/γl ,req) is calculated according to the JSHB [10] as presented in the Appendix-A. The plates of both normal and highperformance steel (SBHS) were taken into account, and their thickness was varied from 10 to 90 mm. Based on the probabilistic distribution obtained from Monte Carlo simulation (MCS), PSFs are proposed for the ultimate buckling strength.
1.2 1
a
a
0.8
0.4
0 0
hr tr
Longitudinal Stiffeners
bs
b
cr
σ /σ
y
Transverse Stiffeners
0.6
0.2
t
Current JSHB Provision Exp. Kanai et al. Exp. Fukumoto et al. 0.2
0.4
0.6
X
0.8
1
1.2
a 2
a 2
a
1.4
RR Fig. 1. Standard ultimate strength curve of stiffened plate in JSHB.
Fig. 2. Model geometry.
Section X-X
X
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800
SM490Y SM570 SBHS500 SBHS700
σ [MPa]
600
400
200
0 0
0.05
0.1
ε
0.15
0.2
Fig. 4. Idealized stress-stain relations for different steel grades. Fig. 3. Boundary conditions.
2. Finite element modelling and elastic buckling analysis
2.3. Elastic buckling analysis
2.1. Model geometry and boundary conditions
An eigenvalue buckling analysis was performed prior to the nonlinear analysis in order to determine the elastic buckling strength and buckling mode. A compressive load was applied through a forced displacement. The first two buckling modes and the normalized elastic
Schematic diagram of the stiffened plate model is shown in Fig. 2. Due to symmetric geometric and loading conditions, and in order to reduce the computational time, the shaded rectangular area shown in Fig. 2 was modeled in ABAQUS. The plate members were modeled using 4-node, quadrilateral, stress-displacement shell element S4R, available in the ABAQUS element library [17], and which is suitable for large scale displacement analysis. Fig. 3 illustrates the boundary conditions for the stiffened plate model. Here Ux, Uy, Uz denote translational degrees of freedom, and URx, URy, URz denote rotational degrees of freedom with respect to the X, Y and Z axes. Boundary conditions corresponding to a simple support were applied along the both side longitudinal edges. Instead of modeling a transverse stiffener, translational movement in the Z direction was restrained, assuming a transverse stiffener with sufficient rigidity.
U, U3 1.00 0.71 0.43 0.14 −0.14 −0.43 −0.71 −1.00
Z
2.2. Material model
Y
As the current study discusses the member safety factor, it is necessary to exclude the effect of uncertainties originating from the material properties, such as the yield strength, modulus of elasticity and Poisson's ratio. For this reason, material properties are considered as a deterministic standardized value in this study. The yield strength for steel with different plate thickness and material grades was obtained from the JSHB code as shown in Table 1. The modulus of elasticity and Poison's ratio were considered to be identical for all steel grades, and were taken to be 200 GPa and 0.3, respectively. The inelastic characteristics of the four different steel grades were determined from idealized uniaxial stress (σ)-stain (ε) relationships, as shown in Fig. 4, based on test data [18].
(a) 1st buckling mode ( σ e σ y = 1.25). U, U3 1.00 0.71 0.43 0.14 −0.14 −0.43 −0.71 −1.00
Table 1 Yield strength [MPa] for different steel grades.
Z Y
Steel grade
SM490Y SM570 SBHS500 SBHS700
X
Plate thickness (mm) t ≤ 16
16 b t ≤ 40
40 b t ≤ 75
t N 75
365 460 500 700
355 450 500 700
335 430 500 700
325 420 500 700
X
(b) 2nd buckling mode ( σ e σ y = 1.58). Fig. 5. Buckling modes and σe/σy values for stiffened plate with RR = 0.8, t = 30 mm and steel grade SBHS500.
M. Rahman et al. / Journal of Constructional Steel Research 138 (2017) 184–195
(+)
0.9
σy
Tension (+)
187
σ rc
(-)
Compression
0.8
σcr/σy
(a)
σrc
0.6σy
RR 0.6
0.7
RR 1.2
Z dir.
(+)
(-)
0.6 (+)
σy Compression
0.5 2
Tension
4
6
8
10
12
14
Ne (b) Fig. 7. Mesh dependency analysis result. Fig. 6. Distribution of residual stresses at (a) subpanels and (b) longitudinal stiffeners.
buckling strength (σe/σy) for a stiffened plate model is presented in Fig. 5 as an example.
the input out-of-plane deflection magnitude was adjusted by trial and error until the desired deflection magnitude was obtained.
3.2. Mesh dependency analysis 3. Nonlinear elasto-plastic FE analysis Material as well as geometric nonlinearity was considered in the nonlinear elasto-plastic FE analyses. Mises plasticity and the isotropic strain hardening theory were applied to model the material nonlinearity. The nonlinear analyses were carried out in two steps. In the first step, initial imperfections were simulated to reproduce the initial condition. In the second step, a compressive load was applied to simulate subsequent loading. 3.1. Simulation of initial imperfections Initial imperfections, i.e., out-of-plane deflections and residual stresses, were simulated simultaneously in each stiffened plate model. The initial out-of-plane deflections were simulated through linear superposition of buckling modes obtained from elastic buckling analysis. Residual stresses were included directly in each element following the distribution pattern, as shown in Fig. 6. Variations of the residual stress distribution along the plate thickness direction were not considered because of the negligible effect on the load carrying capacity [19]. The magnitude of the imperfections was simulated based on the mean value (μ) and the standard deviation (σ), as presented in Table 2, obtained from past studies [3,5 and 20]. Here, δ01 represents the magnitude of an out-of-plane deflection with half-sine wave shape, representing the first elastic buckling mode (whole-plate mode deflection), |Δ| is the magnitude of local mode out-of-plane deflection with similar shape of Fig. 5 (b), a is the plate length, bs is the subpanel width and σrc represents the compressive residual stress. At the first step of nonlinear analysis, due to the incorporation of residual stress, the input out-of-plane deflection magnitude yields higher deflection magnitude than the input value. To overcome this problem,
A mesh dependency analysis was performed to ensure the convergence of the nonlinear elasto-plastic FE results. Two different stiffened plate models with RR = 0.6 and 1.2 were considered for the analysis. Both models have a thickness of 30 mm, material grade SM570 and mean values of the residual stress and initial deflection. The number of elements (Ne) per half subpanel width (which is the half wavelength of the buckle with a local buckling mode) was considered equal to the number of elements along the height of a longitudinal stiffener. Fig. 7 shows the result of mesh dependency analysis. It was found that a convergent result can be obtained if Ne is greater or equal to ten. Based on this result, the value of Ne was taken equal to ten for the subsequent analysis. This result is also consistent with the rule of thumb for shell elements, that there should be at least six elements in the expected half wavelength of a buckle [21]. 1
3
4
2
b
a/2 a
Table 2 Statistical parameters for initial imperfections obtained from the past studies [3,5 and 20].
X Imperfections
Mean (μ)
Standard deviation (σ)
1000δ01/a 150|Δ|/bs σrc/σy
0.096 0.138 0.230
0.426 0.107 0.145
Y
bs
Fig. 8. Geometry of test model (after Komatsu et al. [22]).
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Table 3 Cross sectional dimensions and material properties of test model. Cross sectional dimensions a (mm) 2000
Material properties
b (mm) 810
t (mm) 10.60
bs (mm) 270
hr (mm) 91
tr (mm) 10.60
σy (MPa) 262.82
σT (MPa) 440.32
The nonlinear FE analysis technique was verified using the experimental result of a selected test model as shown in Fig. 8, reported by Komatsu et al. [22]. The test model with two equidistant flat plate longitudinal stiffeners has the geometric and material properties as presented in Table 3. Initial out-of-plane deflections and residual stresses were measured in the panel plate and longitudinal stiffeners. Reported highest magnitude of δ01 was b/306.12 while the average compressive residual stress magnitude (σrc) for panel plate and stiffener was 0.41 σy and 0.35 σy respectively. In the FE model, the magnitude of δ01 and σrc were considered the same as the reported magnitude, following the shape and distribution pattern mentioned in Section 3.1. Boundary conditions assumed for the FE model corresponds to the experimental condition of a four side simply supported plate. Strain gauges were attached at four locations of the test model as shown in Fig. 8 to measure the local strain (εl) along the longitudinal direction. Fig. 9 shows comparisons of the normalized stress σ/σy versus normalized local strain εl/εy curves from the FE analysis and experimental result. The ultimate buckling strength obtained from FE analysis was found to be 1.16% lower than the experimental result. Moreover, good agreement in the normalized stress-strain curves were observed, which validates FE analysis result. 3.4. Parametric study To obtain probabilistic information, it is necessary to determine the ultimate buckling strength for several stiffened plate models, with varying parameters such as the material grades, reduced slenderness parameter RR, and plate thickness. The present study employed a parametric analysis for 96 stiffened plate models with 4 different material grades (SM490Y, SM570, SBHS500 and SBHS700) with RR values of 0.4 to 1.4, and plate thicknesses of 10 to 90 mm. Each model encompasses 12 sets of combinations of initial deflection and residual stress, as shown in Fig. 10. The total number of FE analyses is 1152.
Verification of FE Model 0.8
cr
0.6
42.70
Results of the nonlinear elasto-plastic analysis are represented by normalized stress (σ/σy) – strain (ε/εy) curves in Fig. 11, where σ and σy represent the compressive stress and the yield stress, and ε and εy denote the applied axial strain and the yield strain, respectively. In this figure, the residual stress and initial out-of-plane deflection are assigned to respective mean value, as an example. The ultimate buckling strength is determined from the peak stress value in the stress-strain curve. Fig. 11 shows the effect of RR on the buckling behavior of the stiffened plates. Normalized stress-strain curves are shown for SM570 and SBHS500, with RR ranging from 0.4 to 1.4. It can be seen that for RR ≤ 0.6, the plates exhibit inelastic buckling, whereas for RR ≥ 0.8, unstable snap-through behavior is observed. 4.2. Effect of plate thickness Fig. 12 shows the effect of the plate thickness. Here, the normalized ultimate buckling strength for stiffened plates with material grades SM570 and SBHS500 are plotted for different plate thickness and different RR values. It was found that the plate thickness does not significantly affect the ultimate strengths, provided the residual stress distribution along the plate thickness direction is considered constant. 4.3. Effect of material grade Fig. 13 shows the effect of the material grade. Here, the normalized ultimate buckling strength for stiffened plates with t = 20 mm are plotted for four different material grades and different RR values. It is observed that for RR ≤ 0.6, higher steel grades exhibit a lower load carrying capacity (σcr/σy), while for RR ≥ 0.8 the pattern is irregular. For RR =
μ−σ
y
1000δ01/a
0.4 1 FE 1 Exp. 2 FE 2 Exp. 3,4 FE 3,4 Exp.
0.3 0.2 0.1 0
εl/εy
2
μ
μ+σ
μ+2σ
μ+3σ μ+3σ
1.2
FE =0.7403 Exp.=0.7490
0.5
σ/σy
0.29
4.1. Effect of reduced slenderness parameter RR
σ /σ
−2
% of elongation
4. Discussion of nonlinear analysis results
0.7
−4
ν
Details of the dimensions of the stiffened plate models are presented in Appendix B.
3.3. Verification of the FE model
0
E (GPa) 203.98
4
μ+2σ
0.8
μ+σ
0.4
μ
0
μ−σ
−0.4 6
Fig. 9. Normalized stress (σ/σy) versus normalized local strain (εl/εy) curves from experiment and FE analysis.
−0.8 0
μ−2σ
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
σrc/σy
Fig. 10. Combination of initial imperfections.
M. Rahman et al. / Journal of Constructional Steel Research 138 (2017) 184–195
189
RS μ ID μ
RS μ ID μ 1
1.2
0.8
1 0.8
σcr/σy
RR 0.4 RR 0.6 RR 0.8
0.5
1
RR 1.4 1.5
ε/εy
0
2
355
SBHS700
0 0
0.2
SBHS500
RR 1.2
SM570 SBHS500
RR 0.8
0.4
RR 1.0
0.2
RR 0.6
SM570
0.4
RR 0.4
0.6
SM490Y
σ/σy
0.6
RR 1.0 RR 1.2 RR 1.4
450 500 700 Nominal Yield Strength (MPa)
Fig. 11. Effect of RR on normalized stress-strain curves for mean values of the residual stress and initial deflection, and a thickness of 30 mm.
Fig. 13. Effect of material grade on normalized ultimate buckling strength for mean values of the residual stress and initial deflection, and a thickness of 20 mm.
0.8, there is no significant effect of material grade. Moreover, except for RR = 1.4, SBHS700 shows a lower load carrying capacity than SBHS500.
tolerance for respective modes. Due to including both positive and negative out-of-plane deflection, where positive means a deflection towards the stiffener and negative means a deflection in the opposite side of the stiffener, mean (μ) value of 1000δ01/a is very small i.e. 0.096. To avoid such low magnitude of whole-plate mode compared to the local mode, μ + σ was used for both whole-plate mode and local mode while investigating the effect ultimate strength variation due to addition of local mode. On the other hand, it is rational to use the mean value (μ) of residual stress. Fig. 15(a) and (b) illustrates the shape of initial out-of-plane deflection for Case-1 and Case-2 respectively, simulated in the first step of the nonlinear analysis. Table 5 presents the comparison of ultimate buckling strengths for Case-1 and Case-2. It was observed that the effect of addition of local mode deflection is not significant. The variation of ultimate strengths due to incorporation of the local deflection mode is b1%. Moreover, Fig. 16 shows the normalized compressive stress-strain curves for Case-1 and Case-2 at different RR values, which confirms that there is no significant deviation in the trend of the curves. Hence, only wholeplate mode (the lowest eigenmode) is considered for subsequent analyses.
4.4. Effect of local mode initial out-of-plane deflection Up to previous section, only the whole-plate mode initial out-ofplane deflection was considered. In this section, the effect of local mode is presented. Fig. 5(a) and (b) shows the first and second eigenmodes which are identical to the two fundamental modes of initial out-of-plane deflections: whole-plate mode and local mode as shown in Fig. 14(a) and (b), respectively. To identify the effect of local mode, a nonlinear FE analysis was carried out for two cases i.e. Case-1 and Case-2, covering the full range of reduced slenderness parameter (RR = 0.4 to 1.4) considered in this study. Table 4 shows details of the imperfection combinations expressed in terms of respective mean value (μ) and the standard deviation (σ). Both cases account for the residual stress and whole-plate mode while Case-2 has the additional local mode. Unit values of the normalized expression for whole-plate mode (1000δ01/a) and local mode (150 | Δ |/bs) represent the fabrication
5. Response surface
1.2
The variation in the ultimate buckling strength depends on the initial out-of-plane deflection and the residual stress. To identify the effect of
1
Longitudinal Stiffeners
0.6
δ 01
σcr/σy
0.8 RR 0.4 RR 0.6 0.4
RR 0.8
(a)
RR 1.0
0 0
RR 1.2
SBHS500 SM570 10
20
30
40
50
60
Longitudinal Stiffeners
RR 1.4 70
80
90 100
Thickness (mm) Fig. 12. Effect of plate thickness on normalized ultimate buckling strength for mean values of the residual stress and initial deflection, and for material grades SM570 and SBHS500.
(b)
Δ
0.2
Fig. 14. Initial deflection (a) Whole-plate deflection, (b) Local deflection.
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Table 4 Combination of imperfections in terms of respective mean (μ) and standard deviation (σ) for Case-1 and Case-2.
Case-1 Case-2
Residual stress
Initial out-of-plane deflection
(σrc/σy)
Whole-plate mode (1000δ01/a)
Local mode (150|Δ|/bs)
μ μ
μ +σ μ +σ
0 μ+σ
these two variables, the response surface method [23,24] was employed using the nonlinear FE analysis results. In this method, the ultimate buckling strength is expressed as a response function of the initial deflection and the residual stress. The main objective of using the response surface is to reduce the computational cost of the numerical analysis. In the present study, the following second-order polynomial response function is employed: σ cr X ¼ pij xi1 x2j σy
ði ¼ 0; 1; 2; j ¼ 0; 2; i þ j ≤4Þ
ð2Þ
where, x1 = normalized residual stress (σrc/σy), x2 = normalized initial deflection (1000δ01/a), and pij are the coefficients of the polynomial, determined by a nonlinear multiple regression analysis. The symmetricity of the out-of-plane deflection, obtained in the first buckling mode, yields the same ultimate strength for a similar magnitude of positive and negative out-of-plane deflection, where positive means a deflection towards the stiffener and negative means a
Table 5 Comparison of ultimate buckling strengths for Case-1 and Case-2. σcr/σy
RR
0.4 0.6 0.8 1.0 1.2 1.4
Case-1
Case-2
0.920 0.826 0.760 0.624 0.492 0.390
0.918 0.827 0.753 0.620 0.491 0.392
% difference with respect to Case-1 0.20% −0.11% 0.92% 0.66% 0.24% −0.38%
deflection in the opposite side of the stiffener. Accordingly, an even function with respect to x2 is employed in Eq. (2). Fig. 17(a) illustrates an example of the response surface for RR = 1.0, accounting for all of the material grades and plate thicknesses considered in this study. The mesh in the figure shows the response surface and the solid dots represent the nonlinear FE analysis results. To further clarify the variation in ultimate strength, Fig. 17(b) and (c) was extracted from the results shown in Fig. 17(a). Fig. 17(b) shows the strength variation with respect to the residual stress for a mean (μ) value of the initial deflection. Similarly, Fig. 17(c) presents the effect of variation of the initial deflection for a mean (μ) value of the residual stress. It was found that the ultimate strength, represented by the response surface function (RSF), decreases with increasing magnitude of the initial deflection and residual stress. 6. Monte Carlo simulation An MCS was performed to obtain probabilistic information on the ultimate buckling strength, where the initial out-of-plane deflection and the residual stress were considered as two independent random variables. 6.1. Generation of random variables The two random variables were generated in the MCS based on their probability density function (PDF) shown in Figs. 18 and 19. Fig. 18 shows the reported histogram for positive and negative initial out-of-plane deflections [5], fitted using two exponential distributions f+(x) and f−(x), under the following constraints: þ
Z Y
−
f ð0Þ ¼ f ð0Þ
X
ð3:1Þ
1
(a)
0.8
σ/σy
0.6 RR 0.4 RR 0.6
0.4
RR 0.8 RR 1.0
0.2
Case−1 Case−2
Z Y
X
(b) Fig. 15. Simulated initial deflection shape for (a) Case-1 and (b) Case-2.
0 0
0.5
1
ε/εy
RR 1.2 RR 1.4 1.5
2
Fig. 16. Normalized compressive stress-strain curves for Case-1 and Case-2 at different RR values.
(a)
Relative Frequency/width of histogram bins
M. Rahman et al. / Journal of Constructional Steel Research 138 (2017) 184–195
Response Surface; RR 1.0 SM490Y SM570 SBHS500 SBHS700
σcr/σy
1 0.8 0.6 0.4 0.2 −1
0 0.2 0
0.4 0.6
1000δ01/a
(b)
1
Residual Stress Variation; RR 1.0 RSF SM490Y SM570 SBHS500 SBHS700
1.1 1 σcr/σy
σrc/σy
0.9
0.6
(c)
0.4 σrc/σy
λ (+ve)=8.4993 λ (−ve)=7.1244
3 2.5 2 1.5 1 0.5 0
−0.5
0
0.5 1 1000δ01/a
1.5
2
0.8
Initial Deflection Variation; RR 1.0 RSF SM490Y SM570 SBHS500 SBHS700
1.1 1 σcr/σy
0.6
Experimental Data Exponential Distribution
3.5
exponential distributions, respectively. The +/− signs used as superscripts in Eqs. (3.1) and (3.2) denote the respective functions or parameters for positive and negative deflections. The obtained PDF was employed to model the probabilistic distribution of initial deflections in the MCS. In the MCS, the maximum value of 1000δ01/a was restricted to ± 1 in order to maintain the allowable deflection specified in the JSHB. A lognormal distribution with the same mean value and standard deviation reported by Fukumoto and Itoh [20] was employed in order to generate the random values of compressive residual stress shown in Fig. 19.
0.7
0.2
4
Fig. 18. PDF for initial deflection used in MCS (after Nara & Komatsu, 1988).
0.8
0.5 0
191
0.9
6.2. Calculation of ultimate strength After randomly generating pairs of initial deflection and residual stress values, the response surface function was used to determine the ultimate buckling strength for each pair of values. This process was repeated for a number of iterations, until the μ and σ values for the ultimate strength obtained from the MCS converged. In this study, 100,000 iterations were performed for this convergence to occur.
0.8 0.7 0.6 −0.5
0 1000δ01/a
0.5
0.2
1
Fig. 17. (a) Response surface, (b) Effect of variation of residual stress for a mean value of the initial deflection and (c) Effect of variation of initial deflection for a mean value of the residual stress, for different steel grades at RR = 1.0.
XZ
þ
f ðxþ Þ dx þ
Z
−
f ðx− Þ dx ¼ 1
ð3:2Þ
where þ
þ
þ
−λþ xþ
f ðx Þ ¼ ηλ exp −
Relative Frequency
0.5 −1
Lognormal Distribution μ = 0.23 σ = 0.145
0.15
0.1
0.05
ð3:2:1Þ − −
f ðx− Þ ¼ ð1−ηÞλ− exp−λ
x
ð3:2:2Þ
Here, η is the ratio of positive experimental data points to the total number of experimental data points, and x and λ represent the initial deflection magnitude and rate parameter for the
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
σrc/σy
1
Fig. 19. PDF for compressive residual stress used in MCS (after Fukumoto et al., 1984).
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1.2
Table 6 Mean (μ) and standard deviation (σ) for normalized ultimate buckling strength (σcr/σy). Steel grades
Statistical parameter
RR 0.4
0.6
0.8
1.0
1.2
1.4
SM490Y
μ σ μ σ μ σ μ σ μ σ
0.9507 0.0102 0.9403 0.0055 0.9206 0.0075 0.8894 0.0140 0.9276 0.0085
0.8869 0.0366 0.8702 0.0246 0.8714 0.0303 0.8269 0.0358 0.8658 0.0304
0.7599 0.0590 0.7895 0.0533 0.7844 0.0541 0.8015 0.0499 0.7849 0.0515
0.6269 0.0652 0.6751 0.0560 0.6792 0.0561 0.6888 0.0563 0.6686 0.0504
0.4870 0.0390 0.5244 0.0349 0.5452 0.0412 0.5125 0.0301 0.5155 0.0313
0.3843 0.0262 0.4030 0.0236 0.4041 0.0214 0.4077 0.0209 0.3992 0.0201
SBHS500 SBHS700 All steel grades
0.8
σcr/σy
SM570
1
JSHB Europl
0.6
Eurocl Eurointr
0.4
AASHTO Canadian Nara pf=5%
0.2
MCS μ−pf=5−95% Exp. Kanai
Table 6 shows the μ and σ values for the ultimate strength, obtained from the MCS. Fig. 20(a) and (b) shows the relative frequency distribution for the ultimate buckling strength for RR = 0.6 and 1.0, respectively. The solid curve represents the normal distribution to fit the ultimate buckling strength using the mean value and standard deviation
JSHB
0.8
σcr/σy
0.4
0.6
0.8
1
1.2
1.4
RR Fig. 21. Comparison of MCS results with JSHB, AASHTO, Eurocode, Canadian Code, study of Nara et al. and experimental results of Kanai et al.
6.3. Discussion on MCS results
μ=0.86577,σ=0.030314
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0. 0
0.2
obtained from the MCS. The red horizontal line represents the deterministic value for the ultimate strength specified by JSHB.
1.0 0.9
0 0
0.05
0.1 0.15 0.2 Relative Frequency
0.25
0.3
(a)
A summary of the MCS results is presented in Fig. 21, where the top and bottom error bars represent 95% and 5% probabilities of nonexceedance for the normalized ultimate buckling strength, respectively, and the midpoint shows the mean value. The green solid line represents the ultimate buckling strength curve corresponding to a 5% probability of non-exceedance, as reported by Nara et al. The MCS results were compared with JSHB [10], AASHTO LRFD [15], Eurocode [16], Canadian Highway Bridge Design Code [25], study of Nara et al. [5] as well as the experimental results of Kanai et al. [11]. The predictions from all of the sources in the Fig. 21 corresponds to a stiffened plate which is similar to our FE model. According to the effective width method, Eurocode provides a reduction factor ρ for plate-like buckling and χc for column-like buckling, and interpolates between ρ and χc to obtain the final reduction factor ρc. The predictions from Eurocode, shown in Fig. 21, corresponds to the plate-like buckling, column-like buckling as well as
1.0 0.9
0.1
μ−βTσ
0.8 μ=0.66877,σ=0.050615
0.08
Relative Frequency
σcr /σy
0.7 0.6
JSHB
0.5 0.4 0.3 0.2 0.1 0. 0
μ
βTσ
0.06
0.04
Probability of Non−exceedance
0.02
0.05
0.1 0.15 Relative Frequency
0.2
0.25
(b) Fig. 20. Relative frequency distribution for ultimate buckling strength at (a) RR = 0.6 and (b) RR = 1.0.
0 0.5
0.6
0.7
σcr/σy
0.8
Fig. 22. Explanation of Eq. (4).
0.9
1
M. Rahman et al. / Journal of Constructional Steel Research 138 (2017) 184–195
Partial Safety Factor (γ)
1.25
8. Conclusions
pf 5% pf 3%
1.2
pf 1%
1.15 1.1 1.05 1 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
RR Fig. 23. Graphical representation of PSFs.
the interpolated final strength. It was observed that, our FE model shows a strong deviation to the column-like buckling. The bottom error bars (5% probabilities of non-exceedance) of the MCS results agree with those of Nara et al. and those of Eurocode, except for RR values in the range of 0.8–1.2. In this range, the buckling strength is strongly affected by imperfections, warranted from the relatively higher standard deviations for the MCS result for RR = 0.8 and 1.0. Compared to the MCS results, JSHB, AASHTO and Canadian Code overestimate the ultimate strength for RR ≤ 0.6, and underestimate it for RR ≥ 0.6. It is interesting to note that predictions from AASHTO and Canadian Code are less than the JSHB for RR ≥ 0.4. It is widely reported that for stiffened plates with more than one longitudinal stiffener, AASHTO LRFD provides very conservative result [26,27]. Experimental results of Kanai et al. show fair agreement with MCS results in the range of 0.5 ≤ RR ≤0.9. Around RR = 0.8, scattered experimental results were found which explains the higher standard deviation of MCS result. For RR ≤ 0.5 andRR ≥ 1.2, experimental results are higher than the MCS results. 7. Proposal for PSFs PSFs for ultimate load carrying capacity of stiffened plates were determined using the reliability indexing method. The following equation shows the relationship between the PSF and the reliability index:
μ−βT σ ¼
1 f γ N
193
ð4Þ
where, μ and σ are the mean and the standard deviation of the normalized ultimate buckling strength, respectively, βT is the target reliability index, γ is the partial safety factor and fN is the corresponding nominal strength. Fig. 22 illustrates Eq. (4) graphically for RR = 0.8. If the nominal strength is considered to be the mean value of the buckling strength, the corresponding partial safety factor can be determined from Eq. (4). Assuming that the probability density function for the ultimate buckling strength is normally distributed, the target reliability index βT is 1.64, 1.88 and 2.33 for 5%, 3% and 1% probability of non-exceedance, respectively. The calculated PSFs for ultimate buckling strengths with 5%, 3% and 1% probability of non-exceedance (pf) are shown in Fig. 23, for the case when the nominal strength is equal to the mean value. The maximum PSF was obtained for RR = 1.0.
This paper discusses the ultimate buckling strengths of stiffened steel plates by employing numerical and probabilistic approaches. Stiffened steel plates that are commonly used in the bottom flange of a box girder bridge with an aspect ratio α = 1.0 and 2 flat plate longitudinal stiffener, satisfying the relative stiffness requirement of JSHB was considered in this study. A total of 1152 FE analysis was carried out for 96 stiffened plate models with varying parameters. Monte Carlo simulation was applied to determine the probabilistic ultimate buckling strength for compressive stiffened steel plates. The response surface was used in the MCS to account for the variability of the initial deflection and residual stress simultaneously. The effect of thick plates and SHBS steels on the ultimate strength was investigated. It was found that the plate thickness does not significantly affect the ultimate strengths but the effect of SBHS steels is significant. Analysis of Table 6 reveals that, compared to the conventional steels, SBHS steels have higher mean ultimate strengths for RR ≥ 0.7. For example, at RR = 1.2, SBHS500 shows 10.67% higher mean strength than SM490Y. However, for RR b 0.7, mean strengths of SBHS steels are lower than that of conventional steels. The MCS results were compared with different design codes, study of Nara et al. [5] as well as experimental results of Kanai et al. [11]. MCS results showed very good agreement with those of Nara et al. and Eurocode, except for RR values in the range of 0.8–1.2. The higher prediction of MCS results at this range possibly due to consideration of SBHS steels. Moreover, the buckling strength is also strongly affected by imperfections at this region. Experimental results [11] also show fair agreement with MCS results at 0.5 ≤ RR ≤0.9. The probability density function for the ultimate buckling strength was obtained for different RR values in the range of 0.4 to 1.4. Finally, the PSF for several non-exceedance probabilities was proposed for different RR values. The results of the present study can be used as an important baseline for deriving a reliability-based ultimate strength curve. In this study, we restricted our attention to the ultimate strengths. For RR ≥ 1.0, large out-of-plane deflection, due to elastic buckling, may occur before ultimate buckling. Restriction of this out-of-plane deflection is important from a serviceability point of view, and serviceability limit strength is a topic for future research. Moreover, probabilistic research on plate-like and column-like buckling is of future interest. Acknowledgment This work was supported by The Japan Iron and Steel Federation. For some of the numerical analysis, the authors would like to express their gratitude to Mr. Daiki Miazaki, a former student of Saitama University. Appendix A. Required relative stiffness of longitudinal stiffeners in JSHB According to the design provision of JSHB, the required relative stiffness for a flat longitudinal stiffener, γl ,req, is given by the following equations: If α≤ α0 2 2 α þ1 t0 ; γl;req ¼ 4α 2 n ð1 þ nδl Þ− n t γl;req ¼ 4α 2 n ð1 þ nδl Þ−
α2 þ 1 ; n
for t ≥t 0
for tbt 0
ðA:1:1Þ
ðA:1:2Þ
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M. Rahman et al. / Journal of Constructional Steel Research 138 (2017) 184–195
If α N α0 γ l;req
Appendix B (continued) (continued)
2( 3 )2 2 14 t 0 ¼ ð1 þ nδl Þ−1 −15; 2n2 t n
γ l;req ¼
i 2 1 h 2 2n ð1 þ nδl Þ−1 −1 ; n
RR
for t ≥t 0
ðA:2:1Þ 1.2
ðA:2:2Þ
for tbt 0
1.4
In the preceding equations, the aspect ratio α = a/b, and the critical pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aspect ratio α 0 ¼ 4 1 þ nγ l, where n is the number of subpanels divided by the longitudinal stiffeners, and γl is the relative stiffness of the longitudinal stiffener given by
3 γ l ¼ Il = bt =11
ðA:3Þ
where Il is the moment of inertia for a longitudinal stiffener with respect to the base of the longitudinal stiffener, and t is the thickness of the panel plate. In Eqs. (A.1) and (A.2), δl denotes the cross-sectional area ratio for the longitudinal stiffener to panel plate (Al/bt), where Al is the cross-sectional area of one longitudinal stiffener and t0 is the critical thickness of the panel plate to avoid local buckling, given by b t0 ¼ nπ
SBHS500 0.4
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 1−ν 2 σ y E
ðA:4Þ
where the symbols are the same as those in Eq. (1).
0.6
0.8
1.0
1.2
Appendix B. Detailed dimensions of stiffened plate models 1.4 RR SM490Y 0.4
0.6
0.8
1.0
1.2
1.4
SM570 0.4
0.6
0.8
1.0
t (mm)
b (mm)
tr (mm)
hr (mm)
γl/γl,req
20 30 50 70 90 10 20 30 50 70 10 20 30 50 10 20 30 10 20 30 10 20 30
1079 1635 2788 3903 5094 801 1618 2437 4181 5854 1068 2158 3249 5575 1335 2697 4062 1602 3237 4874 1869 3776 5686
12 18 30 41 54 9 15 22 37 51 9 16 23 39 9 17 24 9 18 26 9 18 26
120 180 310 430 560 70 150 220 370 520 75 160 240 400 80 170 250 85 170 260 90 180 270
1.06 1.03 1.05 1.00 1.01 1.05 1.09 1.00 1.01 1.01 1.01 1.09 1.05 1.04 1.00 1.13 1.01 1.02 1.01 1.03 1.05 1.04 1.01
20 30 50 70 90 10 20 30 50 70 10 20 30 50 10
959 1443 2460 3445 4481 714 1439 2165 3691 5167 952 1918 2886 4921 1189
12 17 29 40 52 9 16 23 38 52 9 17 25 40 9
105 160 270 380 490 70 140 210 360 500 75 150 230 380 80
1.06 1.05 1.03 1.03 1.01 1.15 1.04 1.00 1.06 1.01 1.11 1.05 1.10 1.02 1.11
SBHS700 0.4
0.6
0.8
1.0
1.2
1.4
t (mm)
b (mm)
tr (mm)
hr (mm)
γl/γl, req
20 30 10 20 30 10 20 30
2398 3608 1427 2877 4329 1665 3357 5051
17 26 10 18 27 10 19 28
160 240 80 170 250 85 170 260
1.05 1.06 1.17 1.12 1.06 1.08 1.03 1.07
20 30 50 70 90 10 20 30 50 70 10 20 30 50 10 20 30 50 10 20 30 10 20 30
913 1369 2282 3194 4107 684 1369 2053 3422 4791 913 1825 2738 4563 1141 2282 3422 5704 1369 2738 4107 1597 3194 4791
11 17 28 38 49 9 15 23 37 51 9 16 25 40 9 17 26 42 10 17 27 9 18 28
100 150 250 350 460 70 140 210 350 490 75 150 220 370 80 160 240 390 80 180 250 85 170 260
1.01 1.04 1.03 1.01 1.07 1.18 1.03 1.04 1.02 1.00 1.14 1.04 1.01 1.00 1.14 1.09 1.11 1.01 1.07 1.31 1.10 1.02 1.02 1.12
20 30 50 70 90 20 30 50 70 90 10 20 30 50 70 10 20 30 50 10 20 30 50 10 20 30
771 1157 1928 2706 3471 1157 1736 2893 4050 5207 771 1543 2314 3857 5399 964 1928 2893 4821 1157 2314 3471 5785 1350 2700 4050
11 16 26 36 46 15 23 37 52 66 9 17 26 43 60 10 19 27 45 10 20 28 47 10 20 29
85 125 210 290 380 120 180 300 420 540 70 140 210 350 490 75 150 220 370 75 150 230 380 80 160 240
1.15 1.06 1.06 1.00 1.06 1.02 1.04 1.01 1.01 1.00 1.08 1.03 1.05 1.04 1.04 1.19 1.14 1.03 1.05 1.03 1.03 1.03 1.01 1.09 1.09 1.06
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