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Elastic Web Buckling Stress and Ultimate Strength of H-Section Beams Dominated by Web Buckling.

1. Introduction

The main girder in a steel frame bears large bending moment and shear force due to the horizontal load and the secondary beam load as shown in Figure 1; the global buckling and local buckling may occur at the end of the main girder. Current stability design methods of the H-section girder are mainly based on the calculation of elastic buckling stress. The elastic global buckling stress mainly depends on the interval of lateral supports and the width of flange, while the elastic local buckling stress mainly depends on the width-thickness ratio and depth-thickness ratio. However, the load condition and boundary condition may have a large influence on the buckling stress as well.

The theoretic studies on elastic local buckling have a long history: the traditional approach is to study the elastic buckling of a rectangular flat plate under assumed stress conditions and with various boundary conditions by using the energy method [1-5]. Yuan and Jin [6] proposed an extended Knatorovich method to solve the buckling problem of flat plates with various boundary conditions under compression and pure shear force and derived high-accuracy buckling coefficients. Kang and Leissa [7] formulated an exact solution procedure for the buckling analysis of flat plates with various boundary conditions under combined compression and bending force. Jana and Bhaskar [8] carried out buckling analyses of flat plates under nonuniform uniaxial compression by using Galerkin's method. To analyze the buckling stress of a web plate under complex tress conditions, Ritz's energy method using Fourier series functions is commonly adopted. Ikarashi and Suzuki [9] conducted Ritz's energy method to analyze the buckling stress of web plates with simply supported and clamped edges under combined bending and shear force, discovering that the results rapidly converged to the true solution as the number of Fourier series terms increased. Liu and Pavlovic [10, 11] conducted Ritz's energy method to analyze the buckling stress of simply supported flat plates under patch compression and arbitrary loads, founding that the idealization of simple supports yielded sound agreements to many plates problems, when a plate was attached to other plates.

As above, efforts have been carried out on the elastic buckling of the rectangular flat plates. However, precise solutions for the buckling stress can only be obtained under simply assumed stress conditions, and it is still difficult to calculate the precise buckling stress of plates under complex stress condition, especially for the web of an H-section beam under combined bending and shear force as shown in Figure 1. To calculate the buckling stress of plates under complex stress condition, the approximate method is often used. Based on the parametric study, Suzuki and Ikarashi [12] proposed a series of approximate equations to calculate the buckling stress of web under combined compression, bending, and shear force. The equations were found to be too complex to be applied for practical uses, due to the number of parameters involved. It is highly necessary to develop a design formula with high accuracies and simple calculations.

It is assumed that the local stability and the load bearing capacity of an H-section beam are mainly dependent on the width-thickness ratio of the flange and depth-thickness ratio of the web. Kadono et al. [13] proposed the equivalent width-thickness ratio which can be regarded as a major parameter to approximate the load bearing capacity of an H-section beam. Kimura [14] proposed design equations for the ultimate strength and plastic deformation capacity by using the equivalent width-thickness ratio, which were useful when the buckling of the beam was dominated by flange buckling. However, this method has not been proved applicable for the web buckling dominant H-section beams.

To evaluate the ultimate strength of web buckling dominant H-section beams, the high-accuracy equation for the elastic buckling stress of web is required. This study aims to solve the eigenvalue problem for an H-section beam web under combined bending and shear force by using Ritz's energy method and find out involved parameters. A parametric study is conducted to reveal the effect of parameters on the elastic buckling stress and to propose approximate equations for practical use. Based on the test results, the direct strength method is attempted to derive design equation for the ultimate strength of web buckling dominant H-section beams by using the proposed equations of elastic buckling stress.

2. Buckling Analysis

2.1. Finite Element Analysis. The loading condition of the part of an H-section beam (main girder) between the column and the secondary beam as shown in Figure 1 can be approximately regarded as Figure 2. The internal moment on the beam is assumed to be linearly varied along the length direction axis (x-axis), which can be expressed as

M(x)= (1 - [[beta]/L] - x)[M.sub.b], (1)

where Mb is the left end bending moment, [beta] is moment gradient (0 [less than or equal to] [beta] [less than or equal to] 2), and the right end bending moment is (1 - [beta])Mb. The relationship between Mb and shear load [Q.sub.s] is expressed as

[beta][M.sub.b] = [Q.sub.s] L. (2)

In this study, numerical simulations are conducted using the finite element program ABAQUS to analyze the H-section beam. The boundary condition and load condition of an H-section beam between the column and the secondary beam (Figure 1) are assumed as shown in Figure 3. The left beam end is connected with the column, and the right end with a stiffener is connected with the secondary beam. For simplification, a cantilever beam model using shell elements is used in FEA (Figure 3). The left end is completely fixed, the edges of web and flange on the right end are set as rigid edges to form a rigid plane, and the displacement along z-direction of the rigid plane is also constrained. On such boundary condition, the shear load [Q.sub.s] and moment (1 - [beta])Mb are acting on the center point of the rigid plane in which [Q.sub.s] and [M.sub.b] satisfy equation (2). The FEA includes linear elastic buckling analysis and large deformation analysis. The elastic buckling analysis is performed first, and then the distribution of the initial geometric imperfection is assumed as the first elastic buckling mode obtained from the elastic buckling analysis.

The load condition of a beam end (Figure 1) is close to a cantilever beam with [beta] = 1. For this reason, numerous experimental studies on the H-section cantilever beams have been reported [14-30], which have important reference significance for the study of local buckling. The shear span L may be close to the interval of the secondary beam (Figure 1), and the normal value of L/D should be approximately 4-6. According to the Code for Design of Steel Structure [31], for normal strength steel (Q235), L/B should be no more than 16 to ensure the elastic global stability. In this paper, 158 sets of reported experimental data [14-30] of H-section cantilever beams (with [beta] = 1) are collected as shown in Table 1. According to the data, it can be confirmed that, in most cases, 4L/D [less than or equal to] 6 and L/B [less than or equal to] 16. For comparison, two test results [30] and FEA results with normal dimensions are shown in Figure 4 (L/D = 1400/350 = 4, L/B = 1400/ 175 = 8, [t.sub.w] = 4.5, and [t.sub.f] = 16) and Figure 5 (L/D = 1400/ 350 = 4, L/B = 1400/175 = 8, [t.sub.w] = 6, and [t.sub.f] = 9). The load versus deflection response of the beam is generally affected by the initial geometric imperfection (IMP) and residual stress. As shown in Figures 4(a)-4(c) and Figures 5(a)-5(c), for both web buckling dominant and flange buckling dominant beams, the FEA results agree well with the test results, by employing proper initial geometric imperfections (with IMP [approximately equal to] D/400). However, the actual distribution of the initial geometric imperfections and the residual stresses involved in the beams is unclear in the reported paper [14-30], and it is difficult to exactly analyze the load versus deflection response of the beams. In addition, the load versus deflection response is slightly affected by the loading program (including monotonic loading and various kinds of cyclic loading) as indicated by Kimura [14]. Thus, it is difficult to include these effects in the evaluation method to estimate the ultimate strength or the plastic deformation capacity of a beam. To estimate the ultimate strength of a beam, a simple calculation method is preferable for practical use.

Although the initial geometric imperfection, residual stress, loading program, and so on may cause a large deviation in the test, the relatively conservative evaluation method should be produced. In the following research, the direct strength method is conducted to investigate the correlation between the buckling slenderness ratio and the ultimate strength based on the test results [14-30] subjected to the web buckling dominant beam. The primary study is to propose high-accuracy formulas to calculate the elastic buckling stress.

2.2. Theoretic Analysis. To analyze the elastic buckling stress of a web, a simplified model conducted by the theoretical energy method is used here, in which the H-section beam web is regarded as a single web with the boundary condition shown in Figure 6. The end edges AB and CD are set as rigid edges and constrained by a pin and a roller, respectively. The out-of-plane displacement of the other two longer edges AC and BD is constrained to keep straight, and the rotation around the x-axis is constrained as well. As shown in Figure 7, the bending stress [sigma] (x, y) in the web can be expressed as

[sigma] (x, y) = (1 - [[beta]/L[ x)(1 - [2/d] y) [[sigma].sub.b] (3)

where [sigma] b is the maximum value of bending normal stress. [M.sub.b] can be expressed as

[M.sub.b] = [[A.sub.w]d/6 + [A.sub.f](d + [t.sub.f])] [[sigma].sub.b] (4)

where [A.sub.w] = d[t.sub.w] is the section area of web and [A.sub.f] = 2b[t.sub.f] is the section area of flange. The shear stress is assumed to be uniformly distributed, the shear force [Q.sub.s] is expressed as

[Q.sub.s] = [A.sub.w] [[tau].sub.s], (5)

where [tau]s is the web shear stress. According to the above expression, the ratio of [tau]s to [sigma]b can be approximately expressed as

[mathematical expression not reproducible] (6)

where [lambda] w = L/d is the aspect ratio of web. The total potential energy of the web under combined bending and shear force is

[PI] = U - [V.sub.b] - [V.sub.s], (7)

where U is the strain energy and [V.sub.b] and [V.sub.s] are the external work due to bending and shear force, respectively. U, [V.sub.b], and [V.sub.s] can be expressed as follows:

[mathematical expression not reproducible] (8)

[mathematical expression not reproducible] (9)

[mathematical expression not reproducible] (10)

where W is the out-of-plane displacement function and [D.sub.w] is the flexural rigidity:

[D.sub.w] = E[t.sup.3.sub.w]/12(1 - [v.sup.2]), (11)

where E is Young's modulus and v is the Poisson ratio (u = 0.3). [[sigma].sub.crw] and [[tau].sub.crw] in equations (9) and (10) are the critical value of [sigma]b and [tau]s, which are defined as

[[sigma].sub.crw] = [k.sub.bw] [[[pi].sub.2]E/12(1 - [v.sup.2])] [1/[(d/[t.sub.w]).sup.2]], (12)

[[tau].sub.crw] = [k.sub.sw] [[[pi].sub.2]E/12(1 - [v.sup.2])] [1/[(d/[t.sub.w]).sup.2]], (13)

where [k.sub.bw] and [k.sub.sw] denote the buckling coefficients due to [[sigma].sub.crw], [[tau].sub.crw], and [k.sub.bw]/[k.sub.sw] = [alpha] (equation (6)). The out-of-plane displacement W is expressed by a double Fourier series function as follows:

W = [M.summation over (m=1)] [N.summation over (n=1)] [e.sub.m,n] * [f.sub.m] (x)[g.sub.n] (x), (14)

where [e.sub.mn] is the series coefficient and [f.sub.m] (x) and [g.sub.n] (y) are the Fourier series functions. Assuming the edges of web are clamped, the functions [f.sub.m] (x) and [g.sub.n] (y) can be expressed as

[f.sub.m](x) = sin [[pi]x/L] * sin [m[pi]x/L], [g.sub.n](y) = sin [[pi]y/d] * sin [n[pi]x/d], (15)

Here, W, [f.sub.m] (x), and [g.sub.n] (y) in above equations are replaced with [omega], [[mu].sub.m] (x), and [v.sub.n] (y), respectively, as follows:

[omega] = [M.summation over (m=1)] [N.summation over (n=1)] [e.sub.m,n] * [[mu].sub.m] (x)[v.sub.n] (y), [[mu].sub.m](x) - sin[pi]x * sin m[pi]x, [[v].sub.m](y) - sin[pi]y * sin n[pi]y. (16)

Equations (8)-(10) can be written as the following equivalent equations:

[mathematical expression not reproducible] (17)

[mathematical expression not reproducible] (18)

[mathematical expression not reproducible] (19)

According to the stationary value theory of total potential energy, the conditional expression for the critical state of local stability is

[mathematical expression not reproducible] (20)

To solve the buckling coefficients [k.sub.bw] and [k.sub.sw] in equation (20), a generalized eigenvalue analysis is required. The involved parameters to be input are the aspect ratio [[lambda].sub.w], moment gradient [beta], and stress ratio [alpha] = [k.sub.sw]/[k.sub.bw] (equation (6)).

The theoretic analysis using Ritz's energy method is presented above. The energy method using Fourier series functions showed sound convergences, small amount of computations, and high accuracies in the previous studies [9]. Moreover, the energy method has been validated using the finite element analysis (FEA), which showed that the buckling coefficients of a single web with clamped edges gave good agreements with those of an H-section beam web when the rotations of flanges were clamped [12]. However, relevant studies are rather limited towards the condition when flanges are not clamped.

2.3. Elastic Buckling of an H-Section Beam and a Single Web. When the out-of-plane displacement of flanges is constrained, the flange buckling will be prevented to allow web buckling to occur only. For a normal H-section beam, the lower flange (in compression) is not constrained, and the web buckling and flange buckling may occur simultaneously. In the following finite element analysis (FEA), two cases of boundary conditions are considered: Case 1: the out-of-plane displacement of flanges which is constrained (clamped flanges); Case 2: flanges without any constraint (free flanges).

The elastic buckling analysis results of H-section beams obtained by FEA and the elastic buckling analysis results of single webs obtained by theoretical method are shown in Figure 8. The H-section beams with constant dimension of L = 2400 mm, [beta] = 1, d = D = 400 mm (due to the shell elements, d = D), [t.sub.w] = 4 mm, b = 150 mm, and variable dimension of [t.sub.f] in the range of 5 mm [less than or equal to] [t.sub.f] [less than or equal to]15 mm are analyzed by FEA, where L/D = 2400/400 = 6 and L/B = 2400/ 300 = 8 < 16. The single webs with the same shape and stress condition (using equation (6)) are analyzed as well by the theoretical energy method presented above. It is shown that the shear buckling coefficient [k.sub.sw] of H-section beams with clamped flanges (Case 1) obtained by FEA is close to the coefficient [k.sub.sw] of single webs obtained by theoretical energy method, which corresponds to the previous research [12]. When the flanges are not clamped (Case 2: free flanges), the coefficient [k.sub.sw] of H-section beams with [t.sub.f] > 10 mm obtained by FEA is close to the coefficient [k.sub.sw] of single webs as well. However, when [t.sub.f] < 10 mm, the [k.sub.sw] of H-section beams are smaller than [k.sub.sw] of single webs due to the effect of flange buckling.

For comparison, three typical buckling modes of H-section beams obtained by FEA with free flanges are given in Figure 9, and three typical buckling modes of single webs obtained by theoretical analysis are given in Figure 10. The theoretical buckling mode is expressed by equation (14), where [e.sub.m, n] (m = 1, 2, ..., 20; n = 1, 2, ..., 10) is the first-order eigenvector. By applying the Wolfram Mathematica program, the web buckling modes expressed by equation (14) can be drawn as shown in Figure 10.

As shown in Figure 9(a), the buckling mode of an H-section beam (with [t.sub.f] = 12 mm and free flanges) is controlled by web buckling, in which the web buckling mode is basically the same as the buckling mode (Figure 10(a)) of a single web with identical shape and stress condition ([[lambda].sub.w] = 2400/400 = 6, [beta] = 1, and [alpha] = 0.4028), and the shear buckling coefficients are close to each other (8.64 [approximately equal to] 8.84).

As shown in Figure 9(b), even with a slightly buckled flange, the local buckling of an H-section beam (with [t.sub.f] = 10 mm and free flanges) is dominated by web buckling, and the buckling mode of the beam web is similar to the buckling mode (Figure 10(b)) of a single web with identical shape and stress condition ([[lambda].sub.w] = 6, [beta] = 1, and [ alpha] = 0.3403), and the shear buckling coefficients are close to each other (8.22 [approximately equal to] 8.67), which corroborate that the edge condition of a web can be regarded as clamped condition when the local buckling of an H-section beam is dominated by web buckling.

However, when the local buckling of an H-section beam (with [t.sub.f] = 6 mm and free flanges) is dominated by flange buckling, the buckling mode (Figure 9(c)) of the beam web is different from a single web buckling mode (Figure 10(c)) with identical shape and stress condition ([[lambda].sub.w] = 6, [beta] = 1, and [alpha] = 0.2153).

Through comparisons, it is concluded that the buckling of an H-section beam with free flanges is dominated by web buckling when the flange buckling does not occur or slightly occur, and the boundary condition of the longer web edges is close to clamped condition. Therefore, to calculate the elastic local buckling stress of an H-section beam dominated by web buckling, the analytic model can be simplified as a single web with the boundary condition as shown in Figure 6 and with the stress condition as shown in Figure 7. The presented theoretical analytic method with high accuracy and small amount of computations is valuable for studying the buckling stress of an H-section beam web.

3. Parametric Study

3.1. Previous Study. Based on theoretic analyses, Suzuki and Ikarashi [12] proposed approximate equations for the buckling coefficients of web with clamped edges as follows:

[mathematical expression not reproducible] (21)

[k.sub.bw] = [k.sub.sw]/[alpha], (22)

where

[eta] = 1/6 + [A.sub.f]/[A.sub.w], [[eta].sup.*] = [eta][beta]/2, [LAMBDA] = 2.1[[eta].sup.*] + 1.7, R = 107[[eta].sup.*] - 20.92.1[[eta].sup.*] - 7.22, [alpha] = [eta][beta]/[[lambda].sub.w]. (23)

As indicated by Suzuki and Ikarashi [12], the proposed equations are only applicable when 1 < [beta] [less than or equal to] 2, and the equations are too complex to be applied for practical uses. These defects should be annihilated. In this study, a new method based on parametric studies is proposed to simplify the calculation method and to expand the application range. The first study is to formulate the equation under simple stress condition such as pure shear, uniform bending, and unequal bending. The interaction between the buckling coefficients of shear and bending is then studied to suggest an approximate formula for the calculation of buckling stress.

3.2. Buckling Coefficient of Web under Pure Shear Force. Let [k.sub.bw] = 0 (without considering the effect of bending stress); the critical conditional expression equation (20) can be written as follows:

[partial derivative] [PI]/[partial derivative][e.sub.i,j] = [partial derivative]U - [partial derivative] [V.sub.s]/[partial derivative][e.sub.i,j] = 0 (i = 1,2, ..., M, j = 1, 2, ..., N). (24)

According to equations (17), (19), and (24), the shear buckling coefficient [k.sub.sw0] is related to aspect ratio [[lambda].sub.w] only. Result (Figure 11) shows that [k.sub.sw0] converges to 8.98 in the case of the infinitely long web, which agrees well with previous research [2]. For the finite length web, when [[lambda].sub.w] is larger than 1, the analyzed result corresponds to the approximate equation suggested by Moheit [3] as follows:

[k.sub.sw0] = 8.98 + 5.6/[[lambda].sup.2.sub.w]. (25)

3.3. Buckling Coefficient of Web under Unequal Bending Moment. Let [k.sub.sw] = 0 (without considering the effect of shear stress); the critical conditional expression equation (20) can be written as follows:

[partial derivative][PI]/[partial derivative][e.sub.i,j] = [partial derivative]U -[partial derivative] [V.sub.b]/[partial derivative][e.sub.i,j] = 0 (i = 1,2, ..., M, j = 1, 2, ..., N). (26)

According to equations (17), (18), and (26), the bending buckling coefficient [k.sub.bw0] is related to [[lambda].sub.w] and [beta]. According to Bijlaard' research [4], for an infinitely long plate under uniform bending ([beta] = 0), the buckling coefficient [k.sub.bw0] was 39.6. In this study, the analyzed results (Figure 12) show that [k.sub.bw0] converges to 39.6, which shows a good agreement with Bijlaard' research [4]. [k.sub.bw0] is only slightly larger than the lower limit 39.6 when [[lambda].sub.w] [greater than or equal to] 1. As shown in Figure 13, when [beta] > 0, the larger value of [beta] is, the higher value of [k.sub.bw0] is. By changing the abscissa of Figure 13 into [beta]/ [[lambda].sub.w], Figure 14 can be obtained. It is shown that the curves of [k.sub.bw0] versus [beta]/[[lambda].sub.w] with various [beta] almost overlap with each other, and they can be approximated by the following equation:

[k.sub.bw0] = 39.6 + 40[beta]/[[lambda].sub.w]. (27)

3.4. Interaction Curve of the Buckling Coefficients under Combined Bending and Shear Force. For an H-section beam web, the combined bending and shear stress must be considered. To investigate the interaction between bending and shear stress, [k.sub.sw] versus [[lambda].sub.w] curves and [k.sub.bw] versus [[lambda].sub.w] curves with [beta] = 2 and various [A.sub.f]/[A.sub.w] are analyzed based on the critical conditional expression equation (20). As shown in Figure 15, [k.sub.sw] versus [[lambda].sub.w] curves obtained by considering the combined bending and shear stress are lower than [k.sub.sw0] versus [[lambda].sub.w] curve obtained by considering the shear stress only. When [[lambda].sub.w] is small enough, all [k.sub.sw] versus [[lambda].sub.w] curves converge to [k.sub.sw0] versus [[lambda].sub.w] curve. As shown in Figure 16, [k.sub.bw] versus [[lambda].sub.w] curves obtained by considering the combined bending and shear stress are lower than [k.sub.bw0] versus [[lambda].sub.w] curve obtained by considering the bending stress only. When [[lambda].sub.w] is large enough, all [k.sub.bw] versus [[lambda].sub.w] curves converge to [k.sub.bw0] versus [[lambda].sub.w] curve.

The beams may present various buckling modes due to the different configurations [32, 33]. According to the theoretical analyzed results in this study, the elastic local buckling modes of a single web can be roughly divided into three types (shear type, bending type, and intermediate type). Three web buckling modes with [A.sub.f]/[A.sub.w] = 1 and [beta] = 2 and different aspect ratio ([[lambda].sub.w] = 6, 8, and 12) are shown in Figure 17; the symbols of [k.sub.sw] versus [[lambda].sub.w] and [k.sub.bw] versus [[lambda].sub.w] are shown in Figures 15 and 16, respectively. The web with [[lambda].sub.w] = 6 presents shear type buckling mode (Figure 17(a)) due to the relatively large shear stress, in which the similar shapes of buckling waves are observed along the length direction. The web with [[lambda].sub.w] = 12 presents bending type buckling mode (Figure 17(c)) due to the relatively large bending stress, in which the buckling waves concentrate close to the web ends. The web with [[lambda].sub.w] = 8 presents intermediate bucking mode when the effects of shear and bending stress are comparable (Figure 17(b)). For all webs, the buckling coefficients [k.sub.sw] are always smaller than [k.sub.sw0] and the buckling coefficients [k.sub.bw] are always smaller than [k.sub.bw0].

As above, [k.sub.sw0] and [k.sub.bw0] can be regarded as the upper limits of [k.sub.sw] and [k.sub.bw], respectively. By taking the ordinate as [k.sub.sw]/[k.sub.sw0] and taking the abscissa as [k.sub.bw]/[k.sub.bw0], Figures 15 and 16 can be expressed by Figure 18. It is shown that all the interaction curves (with [beta] = 2 and various [A.sub.f]/[A.sub.w]) overlap with each other, and they can be evaluated by an approximate equation as follows:

[([k.sub.bw]/[k.sub.bw0]).sup.2.5] + [([k.sub.sw]/[k.sub.sw0]).sup.2.5] = 1. (28)

Figure 19 shows a large number of analytical data with various [lambda.sub.w], [beta], and [A.sub.f]/[A.sub.w] in the range of 1 [less than or equal to] [lambda.sub.w] [less than or equal to] 40, 0 [less than or equal to] [beta] [less than or equal to] 2, and 0.3 [less than or equal to] [A.sub.f]/[A.sub.w] [less than or equal to] 2.5; generally, results of [k.sub.bw]/[k.sub.bwO] versus [k.sub.sw]/[k.sub.sw0] distribute around the curve of equation (28). Therefore, equation (28) can be regarded as an interaction formula to calculate [k.sub.bw] and [k.sub.sw].

3.5. Improved Equations. Taking [k.sub.sw]/[k.sub.bw] = a and substituting equations (25) and (27) into the interaction formula equation (28), the approximate equations for [k.sub.bw] and [k.sub.sw] are obtained as follows:

[mathematical expression not reproducible] (29)

[k.sub.sw] = [k.sub.bw] [alpha], (30)

where [alpha] is the stress ratio as shown in equation (6).

The proposed equations (equations (29) and (30)) are far simpler than the previous ones (equation (21) and (22)). For verification and comparison, the proposed curves and FEA results with various cases of [A.sub.f]/[A.sub.w] and [beta] are shown in Figures 20-23. It is found that equation (21) gives good agreements with the FEA results when [beta] = 2 and [beta] = 1, as shown in Figures 20 and 21. However, when [beta] = 0.5 and [beta] = 0.1, equation (21) does not agree with FEA as shown in Figures 22 and 23. As indicated previously, the equations (equations (21) and (22)) can only apply to the range of 1 [less than or equal to] [beta] [less than or equal to] 2. This defect is overridden in this study. As shown in Figures 20-23, equation (30) gives good agreements with the FEA results for all cases. The proposed equations with high accuracies are applicable for the full range (0 [less than or equal to] [beta] [less than or equal to] < 2), which are valuable for further studies.

4. Design Equation for the Ultimate Strength

In the following research, 158 sets of experimental data [14-30] of H-section cantilever beams (with [beta] = 1) as shown in Table 1 are collected to investigate the ultimate strength.

All the beams are welded H-section nonscallop beams. Here, the normalized ultimate strength [[tau].sub.max] is defined as

[mathematical expression not reproducible] (31)

where [M.sub.u] and [Q.sub.u] are the ultimate bending strength and shear strength. [M.sub.p] is the full plastic bending moment, [Q.sub.p] is the shear strength in the full plastic bending state, and [sub.w][Q.sub.p] is the yield shear strength. [M.sub.p], [Q.sub.p], and [sub.w][Q.sub.p] are expressed as follows:

[mathematical expression not reproducible] (32)

[Q.sub.p] = [M.sub.p][beta]/L, (33)

[mathematical expression not reproducible] (34)

4.1. Equivalent Width-Thickness Ratio and Previous Design Equation. When the width-thickness ratio of flange is large, the buckling of an H-section beam may be dominated by flange buckling. The elastic buckling stress of a long flange (assuming no restraint from the web) under compressive force is expressed as follows [5]:

[mathematical expression not reproducible] (35)

According to previous studies [13,14], the equivalent width-thickness ratio [(b/[t.sub.f]).sub.eq] can be written as follows:

[mathematical expression not reproducible] (36)

By taking the ordinate as the normalized ultimate strength [[tau].sub.max] (obtained from test data) and taking the abscissa as the equivalent width-thickness ratio [(b/[t.sub.f]).sub.eq], Figure 24 can be obtained. There are 94 test data in the range of [[sigma].sub.crw] > 1.5 [[sigma].sub.crf] and 64 test data in the range of [[sigma].sub.crw] [less than or equal to] 1.5 [[sigma].sub.crf], in which [[sigma].sub.crw] is calculated by using equations (12) and (29) and [[sigma].sub.crf] is calculated by equation (35). It is shown that [[tau].sub.max] has a strong correlation with [(b/[t.sub.f]).sub.eq], and it is reasonable to suppose that the local buckling is dominated by flange buckling when [[sigma].sub.crw] > 1.5 [[sigma].sub.crf]. The calculation of [[tau].sub.max] has been suggested by Kimura [14] for the flange buckling dominant H-section beams, and [[tau].sub.max] of the beams under monotonic loads can be expressed as follows:

[[tau].sub.max] = 1.5 - 0.57 [(b/[t.sub.f]).sub.eq] - 0.01L/D. (37)

By taking the ordinate as [[tau].sub.max] + 0.01 L/D and the abscissa as [(b/[t.sub.f]).sub.eq] in the range of [[sigma].sub.crw] > 1.5 [[sigma].sub.crf], Figure 25 can be obtained, in which equation (37) gives a good agreement with test results. Moreover, the distribution of [[tau].sub.max] obtained by cyclic tests is slightly higher than that obtained by monotonic tests as indicated by Kimura [14]. However, the correlation between [[tau].sub.max] and [(b/[t.sub.f]).sub.eq] is not strong when [[sigma].sub.crw] [less than or equal to] 1.5[[sigma].sub.[bar.crf]], due to the effect of web buckling as shown in Figure 24. As demonstrated in equation (36), [(b/[t.sub.f]).sub.eq] contains the width-thickness ratio of flange and the depth-thickness ratio of web only. Thus, it is insufficient to regard [(b/[t.sub.f]).sub.eq] as the major parameter to evaluate the ultimate strength when the buckling is dominated by web buckling.

4.2. Web Buckling Slenderness Ratio and New Design Equation. To evaluate the ultimate strength of a web buckling dominant H-section beam, not only the depth-thickness ratio but also other parameters such as aspect ratio, bending gradient, and section areas should be considered. To avoid complex calculations, a direct strength method based on the calculation of elastic buckling stress is employed to investigate the relationship between the normalized ultimate strength [[tau].sub.max] and web buckling slenderness ratio [S.sub.w], in which [S.sub.w] is defined as follows:

[mathematical expression not reproducible] (38)

where [[tau].sub.crw] is the shear buckling stress which is calculated by using equations (13) and (30); [M.sub.crw] is the bending buckling moment which is calculated by using equations (12), (29), and (39):

[mathematical expression not reproducible] (39)

By changing the abscissa of Figure 24 into [S.sub.w], Figure 26 can be obtained. By comparison, the data dispersion in Figure 26 is smaller than that in Figure 24 when [[sigma].sub.crw] [less than or equal to] 1.5[[sigma].sub.crw], and it is reasonable to suppose that the local buckling is dominated by web buckling when [[sigma].sub.crw] [less than or equal to] 1.5[[sigma].sub.crf]. Therefore, the new defined web buckling slenderness ratio ([S.sub.w]) can be regarded as a major parameter to evaluate the normalized ultimate strength ([[tau].sub.max]) of a web buckling dominant H-section beam. However, the test results are lower than the Euler curve equation (40), due to the inelastic buckling:

[[tau].sub.e] = 1/[S.sup.2.sub.w]. (40)

[[tau].sub.e] is the normalized elastic buckling strength. The following asymptotic equation (41) is attempted to evaluate [[tau].sub.max], in which [[tau].sub.max] converges to equation (40) with a large value of [S.sub.w], and [[tau].sub.max] converges to 1 with a small value of [S.sub.w]:

1/[[tau].sup.2.sub.e] + 1/[[tau].sup.2.sub.max]. (41)

By substituting equation (40) into equation (41), (41) can be written as

[[tau].sub.max] = 1/[square root of ([S.sup.4.sub.w] + 1)] (42)

As shown in Figure 26, equation (42) takes the lower limit in the range of [[sigma].sub.crw] [less than or equal to] 1.5 [[sigma].sub.crf]. However, it has been indicated [14] that [[tau].sub.max] may be larger than 1 for an inelastic buckling H-section beam. To avoid underestimations, equation (43) is attempted:

[[tau].sub.max]= 1.35 - [S.sup.2.sub.w]. (43)

Figure 27 shows the relationship between [[tau].sub.max] and [S.sub.w] in the range of [[sigma].sub.crw] [less than or equal to] 1.5 [[sigma].sub.crf] only. By taking the maximum value of equations (42) and (43), (44) is obtained:

[mathematical expression not reproducible] (44)

It is shown that proposed equation (44) takes the lower limit of test results, and the upper limit is about its 125%. This means that the deviation caused by initial geometric imperfection and residual stress is lower than 25% in the tests. The proposed design equation (44) produces good predictions for the test results of the ultimate strengths of the web buckling dominant H-section beams, and the application range is [[sigma].sub.crw] [less than or equal to] 1.5 [[sigma].sub.crf].

Moreover, according to the test data (Figure 27), the normalized ultimate strength [[tau].sub.max] is not affected by the loading program (including monotonic loading and various kinds of cyclic loading) when the buckling is dominated by web buckling. For both monotonic tests and cyclic tests, the dispersion is small.

The deviation in Figure 27 is about 25%, meaning that the beams with the same buckling slenderness ratio may have different normalized ultimate strengths with 25% deviation. In Section 2.1 (Figures 4 and 5), the FEA results have shown that the influence of the geometric imperfections with D/800 ~ D/200 only causes approximately 5% deviation, which is far smaller than 25%. In addition, other influences such as residual stresses, material characteristics, and the test methods may also cause deviations to a certain degree. However, these could not be the primary reason for the large deviation. As mentioned previously, the direct strength method is used in this study, which calculates the elastic buckling stress and buckling slenderness ratio to predict the normalized ultimate strength. The beams with the same buckling slenderness ratio do not necessarily mean they have the same normalized ultimate strength. The beams with different configurations may have different buckling behaviors and different normalized ultimate strengths, even though they have the same value of buckling slenderness ratio. Therefore, to improve the evaluation method of the normalized ultimate strength, further parameters and their influences should be studied to reduce the deviation.

5. Summary and Conclusions

Theoretic analysis by Ritz's energy method for the H-section beam under combined bending and shear force is presented. The theoretic analysis was verified against the FEA when the buckling of the beam is dominated by web buckling. A parametric study based on the stress separation concept is conducted to simplify the calculation method for buckling coefficients. The design equation based on direct strength method for the normalized ultimate strength of a web buckling dominant H-section beam is proposed. The conclusions are drawn as follows:

(1) Even when the flange is slightly buckled, the buckling mode and buckling stress of an H-section beam web show no obvious difference with a single web with clamped edges, when the local buckling of the beam is dominated by web buckling. The analytic model of a web buckling dominant H-section beam can be simplified by a single web with clamped edges.

(2) Without considering the effect of shear stress, the bending buckling coefficient [k.sub.bw0] of web under unequal bending stress is related to the aspect ratio [[lambda].sub.w] and moment gradient [beta], and when [[lambda].sub.w] > 1, [k.sub.bw0] can be approximately expressed by equation (27). The shear buckling coefficients [k.sub.sw0] (equation (25)) and bending buckling coefficient [k.sub.bw0] (equation (27)) can be regarded as the upper limits of the buckling coefficients [k.sub.sw] and [k.sub.bw] of web under combined bending and shear stress, respectively. The interaction curve of the buckling coefficients can be expressed by equation (28), and the approximate equations for calculating the buckling coefficients (equations (29) and (30)) are proposed.

(3) According to a number of tests, it is shown that the normalized ultimate strength [[tau].sub.max] has a strong correlation with the equivalent width-thickness ratio [(b/[t.sub.f]).sub.eq] in the range of [[sigma].sub.crw] > 1.5 [[sigma].sub.crf], whereas [[tau].sub.max] has a strong correlation with the web buckling slenderness ratio [S.sub.w] in the range of [[sigma].sub.crw] [greater than or equal to] 1.5 [[sigma].sub.crf]. It is reasonable to assume that the local buckling is dominated by the flange buckling when [[sigma].sub.crw] > 1.5 [[sigma].sub.crf], whereas it is dominated by the web buckling when [[sigma].sub.crw] [less than or equal to] 1.5 [[sigma].sub.crf].

(4) The distribution of the normalized ultimate strength [[tau].sub.max] obtained by cyclic tests is slightly higher than that obtained by monotonic tests, when the local buckling is dominated by flange buckling. However, Tmax is not affected by the loading program when the buckling is dominated by web buckling.

(5) A new design equation (equation (44)) to evaluate the ultimate strength of a web buckling dominant H-section beam is proposed, which showed sound agreements with test results.
Nomenclature

[[lambda].sub.w]:     Aspect ratio of web, [[lambda].sub.w] = L/d
                      (Figure 2)

[beta]:               Moment gradient (Figure 2)

d/[t.sub.w]:          Depth-thickness ratio of web (Figure 2)

b/[t.sub.f]:          Width-thickness ratio of flange (Figure 2)

[A.sub.w] = dtw,      Section area of web and flange (Figure 2)
[A.sub.f] = Btf:

[D.sub.w]:            Flexure rigidity of web (equation (11))

E:                    Young's modulus, E = 2.05 * 105 MPa

v:                    Poisson ratio u = 0.3

[M.sub.b]:            Bending moment on the left end (Figure 2)

[M.sub.b]:            Full plastic bending moment (equation (32))

[M.sub.crw]:          Bending buckling moment (equation (39))

[Q.sub.s]:            Shear load (Figure 2)

[Q.sub.p]:            Shear strength in the full plastic bending
                      state (equation (33))

wQP:                  Yield shear strength (equation (34))

[[sigma].sub.b]:      Maximum value of bending normal stress
                      (Figure 7)

[[tau].sub.s]:        Uniformly distributed shear stress (Figure 7)

[alpha]:              Ratio of [[tau].sub.s] to [sigma]b (equation (6))

[[sigma].sub.crw]:    Bending buckling stress (equation (12)),
                      critical value of [[sigma].sub.b]

[[tau].sub.crw]:      Shear buckling stress (equation (13)), critical
                      value of [[tau].sub.s]

[k.sub.bw],           Buckling coefficient due to [[sigma].sub.crw]
[k.sub.sw]:           and [[tau].sub.crw]

[k.sub.bw0]:          Buckling coefficient due to [[sigma].sub.crw] in
                      the case of [[tau].sub.s] = 0

[k.sub.sw0]:          Buckling coefficient due to [[tau].sub.s] in the
                      case of [[sigma].sub.b] = 0

[[sigma].sub.ocrf]:   Buckling stress of flange (equation (35))

[[sigma].sub.wy],     Yield stress of web and flange
[[sigma].sub.fy]:

[M.sub.u],            Ultimate bending strength and shear
[Q.sub.u]:            strength

[[tau].sub.max]:      Normalized ultimate strength (equation
                      (31))

[[tau].sub.e]:        Normalized elastic buckling strength
                      (equation (40))

[(b/[t.sub.f])        Equivalent width-thickness ratio (equation
.sub.eq]:             (36))

[S.sub.w]:            Web buckling slenderness ratio (equation
                      (38)).


https://doi.org/ 10.1155/2020/3097062

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of Zhejiang Province, China (Grant no. LQ19E080008).

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Jiaxing Ma, (1) Tao Wang [ID], (1) Yinhui Wang, (1) and Kikuo Ikarashi (2)

(1) School of Civil Engineering and Architecture, NingboTech University, Ningbo 315100, China

(2) Department of Architecture and Building Engineering, Tokyo Institute of Technology, Tokyo 1528550, Japan

Correspondence should be addressed to Tao Wang; wangtao@nit.net.cn

Received 11 June 2020; Accepted 3 August 2020; Published 19 August 2020

Academic Editor: Francesco Cannizzaro

Caption: Figure 1: The moment diagram of a steel frame.

Caption: Figure 2: Load condition of H-section beam.

Caption: Figure 3: FEA model.

Caption: Figure 4: Comparison between FEA and test [30]. (a) Load versus deflection response. (b) Web buckling collapse mode by test [30]. (c) Web buckling collapse mode by FEA with IMP = D/400.

Caption: Figure 5: Comparison between FEA and test [30]. (a)Load versus deflection response. (b) Flange buckling collapse mode by test [30]. (c) Flange buckling collapse mode by FEA with IMP = D/400.

Caption: Figure 6: Boundary conditions of web.

Caption: Figure 7: Stress distribution of web.

Caption: Figure 8: [k.sub.sw] versus [t.sub.f] by different methods.

Caption: Figure 9: Buckling mode of H-section beam with different [t.sub.f] (a) L = 2400, [beta] = 1, d = 400, t.sub.w] = 4, b = 150, and [t.sub.f] = 12. (b) L = 2400, [beta] = 1, d = 400, [t.sub.w] = 4, b = 150, and [t.sub.f] = 10. (c) L = 2400, [beta] = 1, d = 400, [t.sub.w] = 4, b = 150, and [t.sub.f] = 6.

Caption: Figure 10: Buckling mode of web with different [alpha]. (a) [[lambda].sub.w] = 6, [beta] = 1, and [alpha] = 0.4028. (b) [[lambda].sub.w] = 6, [beta] = 1, and [alpha] = 0.3403. (c) [[lambda].sub.w] = 6, [beta] = 1, and [alpha] = 0.2153.

Caption: Figure 11: [k.sub.sw0] versus [[lambda].sub.w] curves.

Caption: Figure 12: [k.sub.bw0] versus [[lambda].sub.w] curves with [beta] = 0.

Caption: Figure 13: [k.sub.bw0] versus [[lambda].sub.w] curves with various [beta].

Caption: Figure 14: [k.sub.bw0] versus [beta]/[[lambda].sub.w] curves with various [beta].

Caption: Figure 15: [k.sub.sw] versus [[lambda].sub.w] curves with [beta] = 2 and various [A.sub.f]/[A.sub.w].

Caption: Figure 16: [k.sub.bw] versus [[lambda].sub.w] curves with [beta] = 2 and various [A.sub.f]/[.sub.w].

Caption: Figure 17: Different types of web buckling modes. (a) Shear type web buckling mode ([[lambda].sub.w] = 6, [a.sub.f]/[A.sub.w] = 1, and [beta] = 2). (b) Intermediate type web buckling mode ([[lambda].sub.w] = 8, [A.sub.f]/[A.sub.w] = 1, and [beta] = 2). (c) Bending type web buckling mode ([[lambda].sub.w] = 12, [A.sub.f]/[A.sub.w] = 1, and [beta] = 2).

Caption: Figure 18: [k.sub.bw]/[k.sub.bw0] versus [k.sub.sw]/ [k.sub.sw0] curves with [beta] = 2 and various [A.sub.f]/[A.sub.w].

Caption: Figure 19: [k.sub.bw]/[k.sub.bw0] versus [k.sub.sw]/[k.sub.sw0] plots with various [beta] and [A.sub.f]/[A.sub.w].

Caption: Figure 20: [k.sub.sw] versus [[lambda].sub.w] with [beta] = 2.

Caption: Figure 21: [k.sub.sw] versus [[lambda].sub.w] with [beta] = 1.

Caption: Figure 22: [k.sub.sw] versus [[lambda].sub.w] with [beta] = 0.5.

Caption: Figure 23: [k.sub.sw] versus [[lambda].sub.w] with [beta] = 0.1.

Caption: Figure 24: [[tau].sub.max] versus [(b/[t.sub.f]).sub.eq].

Caption: Figure 25: [[tau].sub.max] versus [(b/[t.sub.f]).sub.eq] of flange buckling dominant beams.

Caption: Figure 26: [[tau].sub.max] versus [S.sub.w].

Caption: Figure 27: [[tau].sub.max] versus [S.sub.w] of web buckling dominant beams.
Table 1: Test data.

Number     L      D        B     [t.sub.w]   [t.sub.f]  [s.sub.wy]
         (mm)   (mm)     (mm)      (mm)      (mm)        (MPa)

Reported by Fukuchi and Ogura [15]

1        1000    223.4   269.6      5.8       8.8        297
2        1000    223.7   269.5      5.6       8.9        297
3        1000    223.3   269.6      5.7       8.9        297
4        1000    223.6   269.5      5.8       8.9        297
5        1000    223.4    270       5.8       8.8        297
6        1000    223.4   288.1      8.8       12         322
7        1000    224.2   287.8      8.8       12         322
8        1000    223.9   287.8      8.9      12.1        322
9        1000    224.3   287.7      8.9      11.9        270
10       1000    224.5   287.8      8.9      11.9        270
11       1000    222.7   240.6      8.8      11.9        322
12       1000    222.7   240.6      8.8      11.9        322
13       1000    223.7   240.1      9.1      12.1        322
14       1000    225.1   239.6      8.9       12         270
15       1000    223.8    240       8.8      11.9        270
16       1000    222.7   192.7      8.9       12         322
17       1000    224.5   191.7      9.1      11.9        322
18       1000    223.8   190.8      8.8      11.7        322
19       1000     224    191.9      8.7       12         270
20       1000    225.2   192.1      8.9      11.9        270

Reported by Kato et.al. [16]

21       1040     198     144        6        9          291
22       1040     318     144        6        9          291
23       1040     438     144        6        9          291
24       1300     198     180        6        9          291
25       1300     318     180        6        9          291
26       2600     318     180        6        9          291
27       1300     438     180        6        9          291
28       1570     198     216        6        9          291
29       3130     198     216        6        9          291
30       1570     318     216        6        9          291
31       3130     318     216        6        9          291
32       1830     198     252        6        9          291
33       1830     318     252        6        9          291
34       2090     198     288        6        9          291
35       2090     318     288        6        9          291
36       1040     198     144        6        9          523
37       1040     318     144        6        9          523
38       1300     198     180        6        9          523
39       2600     198     180        6        9          523
40       1300     318     180        6        9          523
41       2600     318     180        6        9          523
42       1570     198     216        6        9          523
43       1570     318     216        6        9          523

Reported by Suzuki et al. [17, 24, 26-28]

44        900     300     125       3.2       9          274
45        900     300     125        6        9          297
46       1200     300     125       3.2       9          274
47       1200     300     125       4.5       9          286
48       1200     300     125        6        9          297
49       1200     300     125        9        9          268
50       1500     300     125       3.2       9          274
51       1500     300     125        6        9          297
52       1400     450     125       4.5       9          516
53       1400     450     125       4.5       12         516
54       1400     450     125       4.5       15         516
55       1250     250     125       5.8      8.5         395
56       1950     250     125       5.6      8.3         395
57       1200     300     100        9        9          384
58       2100     350     150        6        16         499
59       2100     350     150        6        12         499
60       2100     350     150        6        9          499
61       2400     400     150        6        16         499
62       2400     400     150        6        12         499
63       1800     450     200        6        12         415
64       1800     450     200        6        16         415
65       1800     450     200        6        19         415
66       1800     450     200        9        16         383
67       1800     450     150        6        12         415
68       2500     450     200        6        16         415
69       1100     450     150        6        12         415
70       1600     400     150        6        16         398
71       1600     400     200        6        16         398
72       1600     400     150        6        22         398
73       1200     400     200        6        12         398
74       1200     400     200        6        16         398

Reported by Fujikawa and Fujiwara [18]

75        900     241    150.2      4.48      8.6        344
76        899     248    150.4      4.56     12.16       344
77       1125     241    149.7      4.49      9.27       344
78       1260     331    149.9      4.6       8.57       344
79       1258     338    150.       4.38     12.13       344
80       1266     330    150.2      4.3       9.21       344
81        450     242    150.1      4.32      8.61       344
82        462     247    150.5      4.33     12.18       344
83        450     241    150.5      4.31      9.23       344
84        629     330     151       4.34      8.63       344
85        630     339    150.5      4.35     12.15       344
86        629     332    151.7      4.37      9.28       344

Reported by Fujiwara and Kato [19]

87       1496     244     151       4.5       8.99       372
88       1492     250    150.4      4.5      11.99       372
89       1497     244     150       4.4       8.7        372
90       1499     244    149.6      4.4       8.83       372

Reported by Yoda et al. [20]

91       1100    27.05   108.5      4.23      5.57       306
92       1350    314.4   108.3      4.23      5.57       306
93       1600    359.7   108.8      4.23      5.57       306
94       1850    405.8    108       4.23      5.57       306
95       2100    449.6   108.9      4.23      5.57       306
96       2350    494.6   108.2      4.23      5.57       306
97       1400    269.6   144.3      4.23      5.57       306
98       1650    315.5   143.8      4.23      5.57       306
99       1950    359.5   144.8      4.23      5.57       306
100      2250    405.1   144.3      4.23      5.57       306
101      2550    449.2   144.5      4.23      5.57       306
102      2900    494.9   144.1      4.23      5.57       306
103      1550    270.4   180.3      4.23      5.57       306
104      1850     315    180.3      4.23      5.57       306
105      2200    360.5   179.8      4.23      5.57       306
106      2550    404.8   180.4      4.23      5.57       306
107      2850    449.2   180.9      4.23      5.57       306
108      3200    494.5   180.3      4.23      5.57       306
109      1799    479.6   108.1      3.23      5.89       286
110      1849    576.4   107.9      3.23      5.89       286
111      2549    479.4   144.1      3.23      5.89       286
112      2603     576    143.6      3.23      5.89       286
113      3309    479.4   179.6      3.23      5.89       286
114      3448    575.8   180.2      3.23      5.89       286
115      1100    269.5   108.5      4.23      5.57       306
116      1600    359.5   108.2      4.23      5.57       306
117      2101    449.3   108.1      4.23      5.57       306
118      1399    270.4   144.5      4.23      5.57       306
119      1950    360.3    144       4.23      5.57       306
120      2551    451.1    144       4.23      5.57       306
121      1550    270.5   180.3      4.23      5.57       306
122      2199    359.6    179       4.23      5.57       306
123      2850    449.5   179.5      4.23      5.57       306

Reported by Konomi et al. [21]

124      1850    450      200        9        12         290
125      1850    450      200        9        12         290
126      1850    450      200        9        12         290

Reported by Makishi and Yamamoto et al. [22, 23]

127      1800    506      201        11       19         367
128      1800    506      201        11       19         367
129      1800    506      201        11       19         367
130      1800    506      201        11       19         367

Reported by Ito 131 1200 et al. [25]

131      1200    300     130         6        12         403
132      1200    300      130        6        12         403
133      1200    300      130        6        12         403
134      1200    300      130        6        12         403

Reported by Minami et al. [29]

135      2150    488      300        11       18         387
136      2150    488      300        11       18         387
137      2150    488      300        11       18         387

Reported by Wang et al. [30]

138      1400    350      175       3.2       12         281
139      1400    350      175       4.5       12         267
140      1400    350      175       4.5       16         267
141      1400    350      175       4.5       19         267
142      1400    350      175        6        9          332
143      1400    350      175        6        12         332
144      1400    350      175        6        16         332
145      1400    350      175        9        12         291
146      1050    350      175       4.5       12         267
147      1050    350      175        6        12         332

Reported by Kimura [14]

148      1000    202      150       8.2      11.9        323
149      1000    202      150       8.1      12.1        323
150      1000    201      150       8.25      12         323
151      1000    201      101       8.25      12         323
152      1000    202      100       8.25     12.1        323
153      1000    201      200       8.2      11.8        323
154      1000    201      201       8.2      11.8        323
155       800    202      150       8.25      12         323
156       800    202      150       8.25     11.9        323
157      1200    201      150       8.35     11.9        323
158      1200    201      151       8.25      12         323

Number   [s.sub.fy]   [(b/[t.sub.f])   [S.sub.w]   [[tau].sub.max]
          (MPa)       .sub.eq]                        (test)

Reported by Fukuchi and Ogura [15]

1            270          0.594          0.305         1.08
2            270          0.591          0.323         1.07
3            270          0.589          0.314         1.08
4            270          0.588          0.307         1.08
5            270          0.595          0.306         1.07
6            262          0.451          0.190         1.24
7            262          0.451          0.191         1.22
8            262          0.447          0.188         1.19
9            257          0.446          0.184         1.17
10           257          0.447          0.184         1.21
11           262          0.387          0.177         1.4
12           262          0.387          0.177         1.26
13           262          0.380          0.170         1.26
14           257          0.376          0.173         1.29
15           257          0.379          0.174         1.34
16           262          0.318          0.162         1.47
17           262          0.319          0.158         1.31
18           262          0.324          0.164         1.34
19           257          0.312          0.164         1.39
20           257          0.313          0.160         1.43

Reported by Kato et.al. [16]

21           291          0.349          0.211         1.36
22           291          0.421          0.351         1.3
23           291          0.510          0.490         1.19
24           291          0.416          0.211         1.23
25           291          0.478          0.351         1.22
26           291          0.478          0.313         1.17
27           291          0.558          0.491         1.1
28           291          0.485          0.211         1.16
29           291          0.485          0.191         1.03
30           291          0.539          0.351         1.1
31           291          0.539          0.314         1.07
32           291          0.556          0.211         1.06
33           291          0.604          0.351         1.08
34           291          0.628          0.211         1.0
35           291          0.671          0.351         1.05
36           523          0.468          0.283         1.2
37           523          0.565          0.470         1.07
38           523          0.558          0.283         1.05
39           523          0.558          0.255         0.99
40           523          0.641          0.471         1.04
41           523          0.641          0.420         1.0
42           523          0.651          0.283         0.98
43           523          0.723          0.470         0.98

Reported by Suzuki et al. [17, 24, 26-28]

44           268          0.562          0.739         1.09
45           268          0.376          0.321         1.43
46           268          0.562          0.663         1.15
47           268          0.443          0.426         1.29
48           268          0.376          0.300         1.28
49           268          0.307          0.185         1.3
50           268          0.562          0.616         1.09
51           268          0.376          0.289         1.23
52           268          0.793          0.700         0.93
53           259          0.764          0.724         0.96
54           268          0.747          0.765         0.98
55           374          0.418          0.298         1.22
56           374          0.430          0.295         1.18
57           384          0.321          0.216         1.41
58           449          0.463          0.456         1.37
59           506          0.521          0.464         1.33
60           488          0.589          0.448         1.3
61           449          0.521          0.513         1.33
62           506          0.574          0.524         1.3
63           368          0.611          0.579         1.06
64           377          0.558          0.628         1.14
65           375          0.532          0.658         1.16
66           377          0.412          0.367         1.3
67           368          0.565          0.539         1.11
68           377          0.558          0.566         1.18
69           368          0.565          0.618         1.14
70           376          0.467          0.520         1.3
71           376          0.500          0.575         1.15
72           357          0.432          0.550         1.14
73           382          0.562          0.591         1.1
74           376          0.500          0.644         1.07

Reported by Fujikawa and Fujiwara [18]

75           272          0.451          0.399         1.39
76           293          0.391          0.458         1.22
77           548          0.524          0.506         1.11
78           272          0.541          0.485         1.29
79           293          0.515          0.596         1.26
80           548          0.627          0.716         1.05
81           272          0.460          0.532         1.16
82           293          0.403          0.526         1.24
83           548          0.535          0.528         1.2
84           272          0.561          0.666         1.16
85           293          0.519          0.739         1.06
86           548          0.624          0.736         1.02

Reported by Fujiwara and Kato [19]

87           282          0.457          0.353         1.25
88           309          0.414          0.397         1.28
89           368          0.501          0.402         1.3
90           547          0.556          0.479         1.08

Reported by Yoda et al. [20]

91           310          0.529          0.400         1.18
92           310          0.574          0.455         1.15
93           310          0.626          0.514         1.13
94           310          0.678          0.575         1.03
95           310          0.732          0.634         1.0
96           310          0.786          0.696         0.91
97           310          0.624          0.403         1.09
98           310          0.664          0.464         1.03
99           310          0.709          0.522         1.03
100          310          0.755          0.583         1.0
101          310          0.803          0.643         0.93
102          310          0.854          0.706         0.93
103          310          0.730          0.414         0.99
104          310          0.764          0.471         0.97
105          310          0.801          0.530         0.92
106          310          0.844          0.590         0.87
107          310          0.888          0.651         0.84
108          310          0.934          0.713         0.8
109          309          0.917          0.907         0.92
110          309          1.080          1.083         0.82
111          309          0.969          0.911         0.87
112          309          1.124          1.089         0.8
113          309          1.031          0.916         0.86
114          309          1.179          1.094         0.71
115          310          0.528          0.398         1.18
116          310          0.624          0.513         1.14
117          310          0.730          0.634         1.02
118          310          0.625          0.405         1.08
119          310          0.708          0.522         1.08
120          310          0.804          0.645         1.0
121          310          0.730          0.414         1.08
122          310          0.798          0.529         0.96
123          310          0.885          0.651         0.9

Reported by Konomi et al. [21]

124          305          0.425          0.310         1.37
125          305          0.425          0.310         1.35
126          305          0.425          0.310         1.33

Reported by Makishi and Yamamoto et al. [22, 23]

127          357          0.357          0.325         1.43
128          357          0.357          0.325         1.35
129          357          0.357          0.325         1.38
130          357          0.357          0.325         1.35

Reported by Ito 131 1200 et al. [25]

131          358          0.391          0.367         1.45
132          358          0.391          0.367         1.51
133          358          0.391          0.367         1.39
134          358          0.391          0.367         1.46

Reported by Minami et al. [29]

135          387          0.457          0.342         1.56
136          387          0.457          0.342         1.55
137          387          0.457          0.342         1.42

Reported by Wang et al. [30]

138          306          0.653          1.006         0.73
139          306          0.496          0.604         1.13
140          266          0.444          0.622         1.07
141          257          0.423          0.666         0.96
142          291          0.505          0.384         1.15
143          306          0.443          0.417         1.21
144          266          0.387          0.423         1.26
145          306          0.353          0.245         1.32
146          306          0.496          0.688         0.95
147          306          0.443          0.461         1.25

Reported by Kimura [14]

148          283          0.270          0.154         1.45
149          283          0.267          0.157         1.67
150          283          0.268          0.152         1.53
151          283          0.205          0.140         1.3
152          283          0.204          0.140         1.42
153          283          0.342          0.167         1.26
154          283          0.344          0.168         1.4
155          283          0.268          0.164         1.42
156          283          0.270          0.164         1.52
157          283          0.269          0.143         1.41
158          283          0.269          0.146         1.51

Number   Load (M. stands for monotonic load;
         C, stands for cyclic load)

Reported by Fukuchi and Ogura [15]

1                 M.
2                 C.
3                 C.
4                 C.
5                 C.
6                 M.
7                 C.
8                 C.
9                 C.
10                C.
11                M.
12                C.
13                C.
14                C.
15                C.
16                M.
17                C.
18                C.
19                C.
20                C.

Reported by Kato et.al. [16]

21                M.
22                M.
23                M.
24                M.
25                M.
26                M.
27                M.
28                M.
29                M.
30                M.
31                M.
32                M.
33                M.
34                M.
35                M.
36                M.
37                M.
38                M.
39                M.
40                M.
41                M.
42                M.
43                M.

Reported by Suzuki et al. [17, 24, 26-28]

44                M.
45                M.
46                M.
47                M.
48                M.
49                M.
50                M.
51                M.
52                M.
53                M.
54                M.
55                M.
56                M.
57                C.
58                M.
59                M.
60                M.
61                M.
62                M.
63                M.
64                M.
65                M.
66                M.
67                M.
68                M.
69                M.
70                M.
71                M.
72                M.
73                M.
74                M.

Reported by Fujikawa and Fujiwara [18]

75                C.
76                C.
77                C.
78                C.
79                C.
80                C.
81                C.
82                C.
83                C.
84                C.
85                C.
86                C.

Reported by Fujiwara and Kato [19]

87                C.
88                C.
89                C.
90                C.

Reported by Yoda et al. [20]

91                M.
92                M.
93                M.
94                M.
95                M.
96                M.
97                M.
98                M.
99                M.
100               M.
101               M.
102               M.
103               M.
104               M.
105               M.
106               M.
107               M.
108               M.
109               M.
110               M.
111               M.
112               M.
113               M.
114               M.
115               M.
116               M.
117               M.
118               M.
119               M.
120               M.
121               M.
122               M.
123               M.

Reported by Konomi et al. [21]

124               C.
125               C.
126               C.

Reported by Makishi and Yamamoto et al. [22, 23]

127               C.
128               C.
129               C.
130               C.

Reported by Ito 131 1200 et al. [25

131               C.
132               C.
133               C.
134               C.

Reported by Minami et al. [29]

135               C.
136               C.
137               C.

Reported by Wang et al. [30]

138               C.
139               C.
140               C.
141               C.
142               C.
143               C.
144               C.
145               C.
146               C.
147               C.

Reported by Kimura [14]

148               M.
149               C.
150               C.
151               M.
152               C.
153               M.
154               C.
155               M.
156               C.
157               M.
158               C.
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Title Annotation:Research Article
Author:Ma, Jiaxing; Wang, Tao; Wang, Yinhui; Ikarashi, Kikuo
Publication:Mathematical Problems in Engineering
Article Type:Report
Geographic Code:9CHIN
Date:Aug 31, 2020
Words:11873
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