Mathematical Problems in Engineering
Volume 2020 (2020), Article ID 3097062, 18 pages
https://doi.org/10.1155/2020/3097062

Research Article

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

Jiaxing Ma,¹ Tao Wang,¹ Yinhui Wang,¹ and Kikuo Ikarashi²

¹School of Civil Engineering and Architecture, NingboTech University, Ningbo 315100, China
²Department 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

Copyright © 2020 Jiaxing Ma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Numerical analyses and theoretic analyses are presented to study the elastic buckling of H-section beam web under combined bending and shear force. Results show that the buckling stress of a single web with clamped edges gives a good agreement with the buckling stress of an H-section beam web when the local buckling of the beam is dominated by the web buckling. Based on theoretic analyses, a parametric study is conducted to simplify the calculation of buckling coefficients. The parameters involved are clarified first, and the improved equations for the buckling coefficient and buckling stress are suggested. By applying the proposed method, the web buckling slenderness ratio is defined. It is verified that the web buckling slenderness ratio has a strong correlation with the normalized ultimate strength of H-section beams when the buckling of the beams is dominated by web buckling. Finally, a design equation is proposed for the ultimate strength of H-section beams.

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.

Figure 1: The moment diagram of a steel frame.

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

where Mb is the left end bending moment, β is moment gradient (0 ≤ β ≤ 2), and the right end bending moment is (1 − β)Mb. The relationship between Mb and shear load Qs is expressed as

(a)

(b)

Figure 2: Load condition of H-section beam.

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 Qs and moment (1 − β)Mb are acting on the center point of the rigid plane in which Qs and Mb 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.

Figure 3: FEA model.

The load condition of a beam end (Figure 1) is close to a cantilever beam with β = 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 β = 1) are collected as shown in Table 1. According to the data, it can be confirmed that, in most cases, 4L/D ≤ 6 and L/B ≤ 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, , and tf = 16) and Figure 5 (L/D = 1400/350 = 4, L/B = 1400/175 = 8, , and tf = 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 ≈ 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.

Table 1: Test data.

Number
L (mm)
D (mm)
B (mm)
(mm)
tf (mm)
swy (MPa)
sfy (MPa)
(b/tf)eq

τmax (test)
Load (M. stands for monotonic load; C. stands for cyclic load)

Reported by Fukuchi and Ogura [15]

1
1000
223.4
269.6
5.8
8.8
297
270
0.594
0.305
1.08
M.

2
1000
223.7
269.5
5.6
8.9
297
270
0.591
0.323
1.07
C.

3
1000
223.3
269.6
5.7
8.9
297
270
0.589
0.314
1.08
C.

4
1000
223.6
269.5
5.8
8.9
297
270
0.588
0.307
1.08
C.

5
1000
223.4
270
5.8
8.8
297
270
0.595
0.306
1.07
C.

6
1000
223.4
288.1
8.8
12
322
262
0.451
0.190
1.24
M.

7
1000
224.2
287.8
8.8
12
322
262
0.451
0.191
1.22
C.

8
1000
223.9
287.8
8.9
12.1
322
262
0.447
0.188
1.19
C.

9
1000
224.3
287.7
8.9
11.9
270
257
0.446
0.184
1.17
C.

10
1000
224.5
287.8
8.9
11.9
270
257
0.447
0.184
1.21
C.

11
1000
222.7
240.6
8.8
11.9
322
262
0.387
0.177
1.4
M.

12
1000
222.7
240.6
8.8
11.9
322
262
0.387
0.177
1.26
C.

13
1000
223.7
240.1
9.1
12.1
322
262
0.380
0.170
1.26
C.

14
1000
225.1
239.6
8.9
12
270
257
0.376
0.173
1.29
C.

15
1000
223.8
240
8.8
11.9
270
257
0.379
0.174
1.34
C.

16
1000
222.7
192.7
8.9
12
322
262
0.318
0.162
1.47
M.

17
1000
224.5
191.7
9.1
11.9
322
262
0.319
0.158
1.31
C.

18
1000
223.8
190.8
8.8
11.7
322
262
0.324
0.164
1.34
C.

19
1000
224
191.9
8.7
12
270
257
0.312
0.164
1.39
C.

20
1000
225.2
192.1
8.9
11.9
270
257
0.313
0.160
1.43
C.

Reported by Kato et.al. [16]

21
1040
198
144
6
9
291
291
0.349
0.211
1.36
M.

22
1040
318
144
6
9
291
291
0.421
0.351
1.3
M.

23
1040
438
144
6
9
291
291
0.510
0.490
1.19
M.

24
1300
198
180
6
9
291
291
0.416
0.211
1.23
M.

25
1300
318
180
6
9
291
291
0.478
0.351
1.22
M.

26
2600
318
180
6
9
291
291
0.478
0.313
1.17
M.

27
1300
438
180
6
9
291
291
0.558
0.491
1.1
M.

28
1570
198
216
6
9
291
291
0.485
0.211
1.16
M.

29
3130
198
216
6
9
291
291
0.485
0.191
1.03
M.

30
1570
318
216
6
9
291
291
0.539
0.351
1.1
M.

31
3130
318
216
6
9
291
291
0.539
0.314
1.07
M.

32
1830
198
252
6
9
291
291
0.556
0.211
1.06
M.

33
1830
318
252
6
9
291
291
0.604
0.351
1.08
M.

34
2090
198
288
6
9
291
291
0.628
0.211
1.0
M.

35
2090
318
288
6
9
291
291
0.671
0.351
1.05
M.

36
1040
198
144
6
9
523
523
0.468
0.283
1.2
M.

37
1040
318
144
6
9
523
523
0.565
0.470
1.07
M.

38
1300
198
180
6
9
523
523
0.558
0.283
1.05
M.

39
2600
198
180
6
9
523
523
0.558
0.255
0.99
M.

40
1300
318
180
6
9
523
523
0.641
0.471
1.04
M.

41
2600
318
180
6
9
523
523
0.641
0.420
1.0
M.

42
1570
198
216
6
9
523
523
0.651
0.283
0.98
M.

43
1570
318
216
6
9
523
523
0.723
0.470
0.98
M.

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

44
900
300
125
3.2
9
274
268
0.562
0.739
1.09
M.

45
900
300
125
6
9
297
268
0.376
0.321
1.43
M.

46
1200
300
125
3.2
9
274
268
0.562
0.663
1.15
M.

47
1200
300
125
4.5
9
286
268
0.443
0.426
1.29
M.

48
1200
300
125
6
9
297
268
0.376
0.300
1.28
M.

49
1200
300
125
9
9
268
268
0.307
0.185
1.3
M.

50
1500
300
125
3.2
9
274
268
0.562
0.616
1.09
M.

51
1500
300
125
6
9
297
268
0.376
0.289
1.23
M.

52
1400
450
125
4.5
9
516
268
0.793
0.700
0.93
M.

53
1400
450
125
4.5
12
516
259
0.764
0.724
0.96
M.

54
1400
450
125
4.5
15
516
268
0.747
0.765
0.98
M.

55
1250
250
125
5.8
8.5
395
374
0.418
0.298
1.22
M.

56
1950
250
125
5.6
8.3
395
374
0.430
0.295
1.18
M.

57
1200
300
100
9
9
384
384
0.321
0.216
1.41
C.

58
2100
350
150
6
16
499
449
0.463
0.456
1.37
M.

59
2100
350
150
6
12
499
506
0.521
0.464
1.33
M.

60
2100
350
150
6
9
499
488
0.589
0.448
1.3
M.

61
2400
400
150
6
16
499
449
0.521
0.513
1.33
M.

62
2400
400
150
6
12
499
506
0.574
0.524
1.3
M.

63
1800
450
200
6
12
415
368
0.611
0.579
1.06
M.

64
1800
450
200
6
16
415
377
0.558
0.628
1.14
M.

65
1800
450
200
6
19
415
375
0.532
0.658
1.16
M.

66
1800
450
200
9
16
383
377
0.412
0.367
1.3
M.

67
1800
450
150
6
12
415
368
0.565
0.539
1.11
M.

68
2500
450
200
6
16
415
377
0.558
0.566
1.18
M.

69
1100
450
150
6
12
415
368
0.565
0.618
1.14
M.

70
1600
400
150
6
16
398
376
0.467
0.520
1.3
M.

71
1600
400
200
6
16
398
376
0.500
0.575
1.15
M.

72
1600
400
150
6
22
398
357
0.432
0.550
1.14
M.

73
1200
400
200
6
12
398
382
0.562
0.591
1.1
M.

74
1200
400
200
6
16
398
376
0.500
0.644
1.07
M.

Reported by Fujikawa and Fujiwara [18]

75
900
241
150.2
4.48
8.6
344
272
0.451
0.399
1.39
C.

76
899
248
150.4
4.56
12.16
344
293
0.391
0.458
1.22
C.

77
1125
241
149.7
4.49
9.27
344
548
0.524
0.506
1.11
C.

78
1260
331
149.9
4.6
8.57
344
272
0.541
0.485
1.29
C.

79
1258
338
150.4
4.38
12.13
344
293
0.515
0.596
1.26
C.

80
1266
330
150.2
4.3
9.21
344
548
0.627
0.716
1.05
C.

81
450
242
150.1
4.32
8.61
344
272
0.460
0.532
1.16
C.

82
462
247
150.5
4.33
12.18
344
293
0.403
0.526
1.24
C.

83
450
241
150.5
4.31
9.23
344
548
0.535
0.528
1.2
C.

84
629
330
151
4.34
8.63
344
272
0.561
0.666
1.16
C.

85
630
339
150.5
4.35
12.15
344
293
0.519
0.739
1.06
C.

86
629
332
151.7
4.37
9.28
344
548
0.624
0.736
1.02
C.

Reported by Fujiwara and Kato [19]

87
1496
244
151
4.5
8.99
372
282
0.457
0.353
1.25
C.

88
1492
250
150.4
4.5
11.99
372
309
0.414
0.397
1.28
C.

89
1497
244
150
4.4
8.7
372
368
0.501
0.402
1.3
C.

90
1499
244
149.6
4.4
8.83
372
547
0.556
0.479
1.08
C.

Reported by Yoda et al. [20]

91
1100
270.5
108.5
4.23
5.57
306
310
0.529
0.400
1.18
M.

92
1350
314.4
108.3
4.23
5.57
306
310
0.574
0.455
1.15
M.

93
1600
359.7
108.8
4.23
5.57
306
310
0.626
0.514
1.13
M.

94
1850
405.8
108
4.23
5.57
306
310
0.678
0.575
1.03
M.

95
2100
449.6
108.9
4.23
5.57
306
310
0.732
0.634
1.0
M.

96
2350
494.6
108.2
4.23
5.57
306
310
0.786
0.696
0.91
M.

97
1400
269.6
144.3
4.23
5.57
306
310
0.624
0.403
1.09
M.

98
1650
315.5
143.8
4.23
5.57
306
310
0.664
0.464
1.03
M.

99
1950
359.5
144.8
4.23
5.57
306
310
0.709
0.522
1.03
M.

100
2250
405.1
144.3
4.23
5.57
306
310
0.755
0.583
1.0
M.

101
2550
449.2
144.5
4.23
5.57
306
310
0.803
0.643
0.93
M.

102
2900
494.9
144.1
4.23
5.57
306
310
0.854
0.706
0.93
M.

103
1550
270.4
180.3
4.23
5.57
306
310
0.730
0.414
0.99
M.

104
1850
315
180.3
4.23
5.57
306
310
0.764
0.471
0.97
M.

105
2200
360.5
179.8
4.23
5.57
306
310
0.801
0.530
0.92
M.

106
2550
404.8
180.4
4.23
5.57
306
310
0.844
0.590
0.87
M.

107
2850
449.2
180.9
4.23
5.57
306
310
0.888
0.651
0.84
M.

108
3200
494.5
180.3
4.23
5.57
306
310
0.934
0.713
0.8
M.

109
1799
479.6
108.1
3.23
5.89
286
309
0.917
0.907
0.92
M.

110
1849
576.4
107.9
3.23
5.89
286
309
1.080
1.083
0.82
M.

111
2549
479.4
144.1
3.23
5.89
286
309
0.969
0.911
0.87
M.

112
2603
576
143.6
3.23
5.89
286
309
1.124
1.089
0.8
M.

113
3309
479.4
179.6
3.23
5.89
286
309
1.031
0.916
0.86
M.

114
3448
575.8
180.2
3.23
5.89
286
309
1.179
1.094
0.71
M.

115
1100
269.5
108.5
4.23
5.57
306
310
0.528
0.398
1.18
M.

116
1600
359.5
108.2
4.23
5.57
306
310
0.624
0.513
1.14
M.

117
2101
449.3
108.1
4.23
5.57
306
310
0.730
0.634
1.02
M.

118
1399
270.4
144.5
4.23
5.57
306
310
0.625
0.405
1.08
M.

119
1950
360.3
144
4.23
5.57
306
310
0.708
0.522
1.08
M.

120
2551
451.1
144
4.23
5.57
306
310
0.804
0.645
1.0
M.

121
1550
270.5
180.3
4.23
5.57
306
310
0.730
0.414
1.08
M.

122
2199
359.6
179
4.23
5.57
306
310
0.798
0.529
0.96
M.

123
2850
449.5
179.5
4.23
5.57
306
310
0.885
0.651
0.9
M.

Reported by Konomi et al. [21]

124
1850
450
200
9
12
290
305
0.425
0.310
1.37
C.

125
1850
450
200
9
12
290
305
0.425
0.310
1.35
C.

126
1850
450
200
9
12
290
305
0.425
0.310
1.33
C.

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

127
1800
506
201
11
19
367
357
0.357
0.325
1.43
C.

128
1800
506
201
11
19
367
357
0.357
0.325
1.35
C.

129
1800
506
201
11
19
367
357
0.357
0.325
1.38
C.

130
1800
506
201
11
19
367
357
0.357
0.325
1.35
C.

Reported by Ito et al. [25]

131
1200
300
130
6
12
403
358
0.391
0.367
1.45
C.

132
1200
300
130
6
12
403
358
0.391
0.367
1.51
C.

133
1200
300
130
6
12
403
358
0.391
0.367
1.39
C.

134
1200
300
130
6
12
403
358
0.391
0.367
1.46
C.

Reported by Minami et al. [29]

135
2150
488
300
11
18
387
387
0.457
0.342
1.56
C.

136
2150
488
300
11
18
387
387
0.457
0.342
1.55
C.

137
2150
488
300
11
18
387
387
0.457
0.342
1.42
C.

Reported by Wang et al. [30]

138
1400
350
175
3.2
12
281
306
0.653
1.006
0.73
C.

139
1400
350
175
4.5
12
267
306
0.496
0.604
1.13
C.

140
1400
350
175
4.5
16
267
266
0.444
0.622
1.07
C.

141
1400
350
175
4.5
19
267
257
0.423
0.666
0.96
C.

142
1400
350
175
6
9
332
291
0.505
0.384
1.15
C.

143
1400
350
175
6
12
332
306
0.443
0.417
1.21
C.

144
1400
350
175
6
16
332
266
0.387
0.423
1.26
C.

145
1400
350
175
9
12
291
306
0.353
0.245
1.32
C.

146
1050
350
175
4.5
12
267
306
0.496
0.688
0.95
C.

147
1050
350
175
6
12
332
306
0.443
0.461
1.25
C.

Reported by Kimura [14]

148
1000
202
150
8.2
11.9
323
283
0.270
0.154
1.45
M.

149
1000
202
150
8.1
12.1
323
283
0.267
0.157
1.67
C.

150
1000
201
150
8.25
12
323
283
0.268
0.152
1.53
C.

151
1000
201
101
8.25
12
323
283
0.205
0.140
1.3
M.

152
1000
202
100
8.25
12.1
323
283
0.204
0.140
1.42
C.

153
1000
201
200
8.2
11.8
323
283
0.342
0.167
1.26
M.

154
1000
201
201
8.2
11.8
323
283
0.344
0.168
1.4
C.

155
800
202
150
8.25
12
323
283
0.268
0.164
1.42
M.

156
800
202
150
8.25
11.9
323
283
0.270
0.164
1.52
C.

157
1200
201
150
8.35
11.9
323
283
0.269
0.143
1.41
M.

158
1200
201
151
8.25
12
323
283
0.269
0.146
1.51
C.

(a)

(b)

(c)

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.

(a)

(b)

(c)

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.

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 σ(x,y) in the web can be expressed as

where σb is the maximum value of bending normal stress. Mb can be expressed as

where is the section area of web and Af = 2btf is the section area of flange. The shear stress is assumed to be uniformly distributed, the shear force Qs is expressed as

where τs is the web shear stress. According to the above expression, the ratio of τs to σb can be approximately expressed as

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

where U is the strain energy and Vb and Vs are the external work due to bending and shear force, respectively. U, Vb, and Vs can be expressed as follows:

where W is the out-of-plane displacement function and is the flexural rigidity:

where E is Young’s modulus and υ is the Poisson ratio (υ = 0.3). σcrw and τcrw in equations (9) and (10) are the critical value of σb and τs, which are defined as

where kbw and ksw denote the buckling coefficients due to σcrw, τcrw, and kbw/ksw = α (equation (6)). The out-of-plane displacement W is expressed by a double Fourier series function as follows:

where emn is the series coefficient and fm(x) and are the Fourier series functions. Assuming the edges of web are clamped, the functions fm(x) and can be expressed as

Figure 6: Boundary conditions of web.

Figure 7: Stress distribution of web.

Here, W, fm(x), and in above equations are replaced with ω, µm(x), and , respectively, as follows:

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

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

To solve the buckling coefficients kbw and ksw in equation (20), a generalized eigenvalue analysis is required. The involved parameters to be input are the aspect ratio , moment gradient β, and stress ratio (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, β = 1, d = D = 400 mm (due to the shell elements, d = D), , b = 150 mm, and variable dimension of tf in the range of 5 mm ≤ tf ≤ 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 ksw of H-section beams with clamped flanges (Case 1) obtained by FEA is close to the coefficient ksw 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 ksw of H-section beams with tf > 10 mm obtained by FEA is close to the coefficient ksw of single webs as well. However, when tf < 10 mm, the ksw of H-section beams are smaller than ksw of single webs due to the effect of flange buckling.

Figure 8: ksw versus tf by different methods.

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 em,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.

(a)

(b)

(c)

Figure 9: Buckling mode of H-section beam with different tf. (a) L = 2400, β = 1, d = 400, , b = 150, and tf = 12. (b) L = 2400, β = 1, d = 400, , b = 150, and tf = 10. (c) L = 2400, β = 1, d = 400, , b = 150, and tf = 6.

(a)

(b)

(c)

Figure 10: Buckling mode of web with different α. (a) , β = 1, and α = 0.4028. (b) , β = 1, and α = 0.3403. (c) , β = 1, and α = 0.2153.

As shown in Figure 9(a), the buckling mode of an H-section beam (with tf = 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 (=2400/400 = 6, β = 1, and α = 0.4028), and the shear buckling coefficients are close to each other (8.64 ≈ 8.84).

As shown in Figure 9(b), even with a slightly buckled flange, the local buckling of an H-section beam (with tf = 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 (=6, β = 1, and α = 0.3403), and the shear buckling coefficients are close to each other (8.22 ≈ 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 tf = 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 (= 6, β = 1, and α = 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:

where

As indicated by Suzuki and Ikarashi [12], the proposed equations are only applicable when 1 < ß ≤ 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 kbw = 0 (without considering the effect of bending stress); the critical conditional expression equation (20) can be written as follows:

According to equations (17), (19), and (24), the shear buckling coefficient ksw0 is related to aspect ratio only. Result (Figure 11) shows that ksw0 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 is larger than 1, the analyzed result corresponds to the approximate equation suggested by Moheit [3] as follows:

Figure 11: ksw0 versus curves.

3.3. Buckling Coefficient of Web under Unequal Bending Moment

Let ksw = 0 (without considering the effect of shear stress); the critical conditional expression equation (20) can be written as follows:

According to equations (17), (18), and (26), the bending buckling coefficient kbw0 is related to and β. According to Bijlaard’ research [4], for an infinitely long plate under uniform bending (β = 0), the buckling coefficient kbw0 was 39.6. In this study, the analyzed results (Figure 12) show that kbw0 converges to 39.6, which shows a good agreement with Bijlaard’ research [4]. kbw0 is only slightly larger than the lower limit 39.6 when ≥1. As shown in Figure 13, when β > 0, the larger value of β is, the higher value of kbw0 is. By changing the abscissa of Figure 13 into , Figure 14 can be obtained. It is shown that the curves of kbw0 versus with various β almost overlap with each other, and they can be approximated by the following equation:

Figure 12: kbw0 versus curves with β = 0.

Figure 13: versus curves with various β.

Figure 14: versus curves with various β.

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, versus curves and versus curves with β = 2 and various are analyzed based on the critical conditional expression equation (20). As shown in Figure 15, versus curves obtained by considering the combined bending and shear stress are lower than versus curve obtained by considering the shear stress only. When is small enough, all versus curves converge to versus curve. As shown in Figure 16, versus curves obtained by considering the combined bending and shear stress are lower than versus curve obtained by considering the bending stress only. When is large enough, all versus curves converge to versus curve.

Figure 15: versus curves with β = 2 and various .

Figure 16: versus curves with β = 2 and various .

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 and β = 2 and different aspect ratio (, 8, and 12) are shown in Figure 17; the symbols of versus and versus are shown in Figures 15 and 16, respectively. The web with 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 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 presents intermediate bucking mode when the effects of shear and bending stress are comparable (Figure 17(b)). For all webs, the buckling coefficients are always smaller than and the buckling coefficients are always smaller than .

(a)

(b)

(c)

Figure 17: Different types of web buckling modes. (a) Shear type web buckling mode (, , and β = 2). (b) Intermediate type web buckling mode (, , and β = 2). (c) Bending type web buckling mode (, , and β = 2).

As above, and can be regarded as the upper limits of and , respectively. By taking the ordinate as and taking the abscissa as , Figures 15 and 16 can be expressed by Figure 18. It is shown that all the interaction curves (with β = 2 and various ) overlap with each other, and they can be evaluated by an approximate equation as follows:

Figure 18: versus curves with β = 2 and various .

Figure 19 shows a large number of analytical data with various , β, and in the range of 1 ≤  ≤ 40, 0 ≤ ß ≤ 2, and 0.3 ≤  ≤ 2.5; generally, results of versus distribute around the curve of equation (28). Therefore, equation (28) can be regarded as an interaction formula to calculate and .

Figure 19: versus plots with various β and .

3.5. Improved Equations

Taking ksw/kbw = a and substituting equations (25) and (27) into the interaction formula equation (28), the approximate equations for kbw and ksw are obtained as follows:

where α 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 and β are shown in Figures 20–23. It is found that equation (21) gives good agreements with the FEA results when β = 2 and β = 1, as shown in Figures 20 and 21. However, when β = 0.5 and β = 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 ≤ β ≤ 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 ≤ ß ≤ 2), which are valuable for further studies.

Figure 20: ksw versus with β = 2.

Figure 21: ksw versus with β = 1.

Figure 22: ksw versus with β = 0.5.

Figure 23: ksw versus with β = 0.1.

4. Design Equation for the Ultimate Strength

In the following research, 158 sets of experimental data [14–30] of H-section cantilever beams (with β = 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 τmax is defined as

where Mu and Qu are the ultimate bending strength and shear strength. Mp is the full plastic bending moment, Qp is the shear strength in the full plastic bending state, and is the yield shear strength. Mp, Qp, and are expressed as follows:

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]:

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

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

Figure 24: τmax versus (b/tf)eq.

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

Figure 25: τmax versus (b/tf)eq of flange buckling dominant beams.

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 τmax and web buckling slenderness ratio , in which is defined as follows:

where τcrw is the shear buckling stress which is calculated by using equations (13) and (30); Mcrw is the bending buckling moment which is calculated by using equations (12), (29), and (39):

By changing the abscissa of Figure 24 into , Figure 26 can be obtained. By comparison, the data dispersion in Figure 26 is smaller than that in Figure 24 when σcrw ≤ 1.5σcrf, and it is reasonable to suppose that the local buckling is dominated by web buckling when σcrw ≤ 1.5σcrf. Therefore, the new defined web buckling slenderness ratio () can be regarded as a major parameter to evaluate the normalized ultimate strength (τmax) 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:

Figure 26: τmax versus .

τe is the normalized elastic buckling strength. The following asymptotic equation (41) is attempted to evaluate τmax, in which τmax converges to equation (40) with a large value of , and τmax converges to 1 with a small value of :

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

As shown in Figure 26, equation (42) takes the lower limit in the range of σcrw ≤ 1.5σcrf. However, it has been indicated [14] that τmax may be larger than 1 for an inelastic buckling H-section beam. To avoid underestimations, equation (43) is attempted:

Figure 27 shows the relationship between τmax and in the range of σcrw ≤ 1.5σcrf only. By taking the maximum value of equations (42) and (43), (44) is obtained:

Figure 27: τmax versus of web buckling dominant beams.

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 σcrw ≤ 1.5σcrf.

Moreover, according to the test data (Figure 27), the normalized ultimate strength τmax 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 kbw0 of web under unequal bending stress is related to the aspect ratio and moment gradient β, and when  > 1, kbw0 can be approximately expressed by equation (27). The shear buckling coefficients ksw0 (equation (25)) and bending buckling coefficient kbw0 (equation (27)) can be regarded as the upper limits of the buckling coefficients ksw and kbw 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 τmax has a strong correlation with the equivalent width-thickness ratio (b/tf)eq in the range of σcrw > 1.5σcrf, whereas τmax has a strong correlation with the web buckling slenderness ratio in the range of σcrw ≤ 1.5σcrf. It is reasonable to assume that the local buckling is dominated by the flange buckling when σcrw > 1.5σcrf, whereas it is dominated by the web buckling when σcrw ≤ 1.5σcrf.(4)The distribution of the normalized ultimate strength τmax obtained by cyclic tests is slightly higher than that obtained by monotonic tests, when the local buckling is dominated by flange buckling. However, τmax 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

λw: | Aspect ratio of web, λw = L/d (Figure 2)
β: | Moment gradient (Figure 2)
d/tw: | Depth-thickness ratio of web (Figure 2)
b/tf: | Width-thickness ratio of flange (Figure 2)
Aw = dtw, Af = Btf: | Section area of web and flange (Figure 2)
: | Flexure rigidity of web (equation (11))
E: | Young’s modulus, E = 2.05 × 105 MPa
υ: | Poisson ratio υ = 0.3
Mb: | Bending moment on the left end (Figure 2)
Mb: | Full plastic bending moment (equation (32))
Mcrw: | Bending buckling moment (equation (39))
Qs: | Shear load (Figure 2)
QP: | Shear strength in the full plastic bending state (equation (33))
wQP: | Yield shear strength (equation (34))
σb: | Maximum value of bending normal stress (Figure 7)
τs: | Uniformly distributed shear stress (Figure 7)
α: | Ratio of τs to σb (equation (6))
σcrw: | Bending buckling stress (equation (12)), critical value of σb
τcrw: | Shear buckling stress (equation (13)), critical value of τs
kbw, ksw: | Buckling coefficient due to σcrw and τcrw
kbw0: | Buckling coefficient due to σb in the case of τs = 0
ksw0: | Buckling coefficient due to τs in the case of σb = 0
σcrf: | Buckling stress of flange (equation (35))
σwy, σfy: | Yield stress of web and flange
Mu, Qu: | Ultimate bending strength and shear strength
τmax: | Normalized ultimate strength (equation (31))
τe: | Normalized elastic buckling strength (equation (40))
(b/tf)eq: | Equivalent width-thickness ratio (equation (36))
Sw: | Web buckling slenderness ratio (equation (38)).

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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