J. Cent. South Univ. (2019) 26: 1946-1957

DOI: https://doi.org/10.1007/s11717-019-4122-0

Lateral-torsional buckling of box beam with corrugated steel webs

FENG Yu-lin(冯玉林)¹, JIANG Li-zhong(蒋丽忠)^{1, 2},ZHOU Wang-bao(周旺保)^{1, 2}, HAN Jian-ping(韩建平)³

1. School of Civil Engineering, Central South University, Changsha 410075, China;

2. National Engineering Laboratory for High Speed Railway Construction, Central South University, Changsha 410075, China;

3. Key Laboratory of Disaster Prevention and Mitigation in Civil Engineering of Gansu Province, Lanzhou University of Technology, Lanzhou 730050, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: Corrugated steel web is folded along the longitudinal direction and has the mechanical properties such as axial compression stiffness corrugation effect, shear modulus corrugation effect, similar to that of an accordion. In order to study the lateral-torsional buckling of box beams with corrugated steel webs (BBCSW) under the action of bending moment load, the neutral equilibrium equation of BBCSW under the action of bending moment load is derived through the stationary value theory of total potential energy and further, along with taking Kollbrunner-Hajdin correction method and the mechanical properties of the corrugated web into consideration. The analytical calculation formula of lateral-torsional buckling critical bending moment of BBCSW is then obtained. The lateral-torsional buckling critical bending moment of 96 BBCSW test specimens with different geometry dimensions are then calculated using both the analytical calculation method and ANSYS finite element method. The results show that the analytical calculation results agree well with the numerical calculation results using ANSYS, thus proving the accuracy of the analytical calculation method and model simplification hypothesis proposed in this paper. Also, compared with the box beams with flat steel webs (BBFSW) with the same geometry dimensions as BBCSW, within the common range of web space-depth ratio and web span-depth ratio, BBCSW’s lateral-torsional buckling critical bending moment is larger than that of BBFSW. Moreover, the advantages of BBCSW’s stability are even more significant with the increase of web space-depth ratio and web depth-thickness ratio.

Key words: box beams with corrugated steel webs; lateral-torsional buckling; analytical solution; accordion effect; Kollbrunner-Hajdin method

Cite this article as: FENG Yu-lin, JIANG Li-zhong, ZHOU Wang-bao, HAN Jian-ping. Lateral-torsional buckling of box beam with corrugated steel webs [J]. Journal of Central South University, 2019, 26(7): 1946-1957. DOI: https://doi.org/10.1007/s11717-019-4122-0.

1 Introduction

Corrugated web is folded along the longitudinal direction and has the mechanical properties such as axial compression stiffness corrugation effect, shear modulus corrugation effect and so on. With further study on the mechanical properties of box beams with corrugated steel webs, it has been widely used in architecture, highway, railway, and urban rail transits [1-3].

Compared to box beams with flat steel webs, the out-of-plane stiffness of beams with corrugated steel webs is relatively high, and hence cross-sections with corrugated steel webs are considered to be less prone to distortion [4]. On the other hand, the low axial stiffness of corrugated webs for BBCSW enables different torsional behavior compared to conventional concrete or steel box beam [5]. The beam generally has a narrow and high form of cross section, and there is a considerable difference between the inertia moments on the two principal inertia axes. If there is no lateral bracing in the middle of beam span, or the lateral bracing has a greater distance, when the transverse load or bending moment reach a certain value, the beam cross section may display lateral displacement and torsion, causing the loss of carrying capacity, known as the beam’s lateral- torsional buckling. So far, there has not been much research on lateral-torsional buckling of BBCSW under the action of bending moment load.

In the following, the contributions of research studies on the understanding of the behavior of web-corrugated beams are briefly presented. Most of the studies used analytical and numerical methods on I-beams with corrugated web, with only few researchers on BBCSW. DENAN et al [6] conducted both experimental and numerical study on lateral-torsional buckling behavior of steel I-beams with trapezoidal corrugated web. It was found that the steel beam with trapezoidal corrugated web section shows higher resistance to lateral-torsional buckling than a section with flat web. Based on the study of the lateral-torsional behavior of beams tested by DENAN [6], LINDNER [7] developed a formula to calculate the warping constant of web-corrugated beams, letting the remaining cross-sectional constants equal to those used for beams with flat webs. Based on the results of previous studies, MOON et al [8] described the bending and pure torsional rigidities of I-beams with corrugated web. Derivation of the section properties of these types of beam led to the determination of the lateral-torsional buckling strength, followed by discussion on the effect of web corrugation shape on the lateral-torsional buckling strength. MOON et al [9] proposed approximate methods for calculating the shear center of I-beams with corrugated webs under uniform bending, which enables a rapid computation of the lateral-torsional buckling strength of the member. The work of MOON focused on the lateral-torsional buckling strength of I-beams with corrugated steel webs subjected to linear moment gradient using finite element analysis, and proposed a simple equation for the moment gradient correction factor. The study results showed that the buckling behavior of an I-beam with corrugated steel webs depended on the number of periods of the corrugation [10]. SAYED-AHMED [11] calculated the lateral- torsional buckling strength of I-beams with corrugated webs and found that their resistance to lateral-torsional buckling is higher than the resistance of plate beams with flat webs, by up to 37%. IBRAHIM [12] investigated the lateral- torsional buckling strength of plate beams with corrugated webs using tubular rectangular flanges and the elastic strength of plate beams with flat web using the same formula but with the modified warping constant suggested by LINDNER [7]. It was concluded that combination of corrugated webs and tubular flanges may result in lateral-torsional buckling strength higher by up to 46% compared to conventional plate beams with flat web and the same cross-sectional area. IBRAHIM [13] further investigated the lateral-torsional buckling strength of unsymmetrical I-beams with corrugated web subjected to uniform pure flexure moment, where the pure torsional constant and shear modulus were based on previous work. LIM et al [14] investigated the elastic lateral-torsional buckling of I-beams under linear moment gradient and obtained the elastic critical buckling moments independently by using the Bubnov–Galerkin method and the finite element method. PIMENTA et al [15] presented reliability-based design recommendations for I-beams with sinusoidally-shaped webs for the limit state of lateral-torsional buckling. KAZEMI NIA KORRANI [16] investigated the stiffness requirement of lateral restraints that are intended to provide restraining of simply supported I-beams with corrugated webs in order to increase their lateral buckling moment strength under pure bending. TARAS et al [17] developed analytical Ayrton-Perry-type formulae for the lateral-torsional and torsional buckling modes of members with doubly symmetrical open I-sections. The similarities and differences to the flexural buckling mode were highlighted. OLIVEIRA et al [18] developed a finite element model using the commercial software ANSYS to determine the elastic critical moments of continuous composite steel-concrete beams with corrugated sinusoidal web, which were compared to numerical data from the literature for validation purposes. NGUYEN et al [19, 20] proposed new general formulae for the cross-sectional properties, as well as for the determination of the moment modification factors of web corrugated beams under moment gradients, they further presented the results of theoretical and finite element analyses focusing on the lateral- torsional buckling of an I-beam with trapezoidal web corrugations under uniform moment and the elastic lateral-torsional buckling strength [21, 22].

In the existing literature, studies on the lateral- torsional buckling of plate beams with corrugated webs mainly have focused on I-beams with corrugated webs, with little on the lateral-torsional buckling of BBCSW. Furthermore, the existing studies are mostly based on finite element numerical calculations. On the basis of Kollbrunner-Hajdin correction method [23] and the mechanical properties of the corrugated web, this paper establishes the differential equation of BBCSW’s constraint torsion and proposes the analytical calculation formula of the lateral- torsional buckling critical bending moment of BBCSW. Through 96 examples of BBCSW with different geometries of cross sections, the analytical solution calculation method in this paper is then compared with the finite element numerical method, and the accuracy of the analytical solution in this paper is verified. Also, compared with the BBFSW with the same geometry dimensions as BBCSW test specimens, within the common range of web space-depth ratio and web depth-thickness ratio, BBCSW’s lateral-torsional buckling critical bending moment are larger than BBFSW’s, moreover, the advantages of BBCSW’s stability are even more significant with the increase of web space-depth ratio and web depth-thickness ratio. The obtained formula displays a certain development compared with the former calculation theory of BBCSW’s lateral-torsional buckling critical bending moment and lays a theoretical foundation for further application of BBCSW in engineering.

2 Theoretical background

The 3D geometry of BBCSW is shown in Figure 1, and the fundamental assumptions of BBCSW are as follows:

1) As shown in Figure 2, corrugated steel web is folded along the longitudinal direction and has the mechanical properties such as axial compression stiffness corrugation effect, shear modulus corrugation effect, similar to that of an accordion. Under the axial loads, the corrugated steel webs display greater axial deformation, normally, the axial elastic modulus of corrugated steel webs is only a few hundredths or even a few thousandths of the steel elastic modulus E_s. Therefore, the axial stiffness of corrugated steel web is very small and can hardly resist the axial force [24]. The corrugated web’s axial stiffness can be ignored, that is, letting the value of corrugated web’s longitudinal effective elastic modulus as E_e=0 [25-27].

Figure 1 3D geometry of BBCSW

2) Due to the corrugated web’s fold structure, its effective shear modulus G_s decreases a certain extent compared with the steel’s shear modulus G_s and the corrugated web’s effective shear modulus value can be approximately expressed as G_e=G_s(d_w2+d_w3)/(d_w2+d_w3secθ_w) [28, 29], where d_w1, d_w2, d_w3, and h_w denote the length of inclined plate section, the length of corrugated web’s plate section, the projected length of inclined plate section on the horizontal line, and the corrugated web’s wave height, respectively. θ_w denotes the corrugated web’s folding angle, as shown in Figure 2.

3) With small deformation, the projection shape of the section after BBCSW’s torsion displayed on its original plate is identical to the original section’s shape, that is, it satisfies Vlasov’s rigid perimeter hypothesis.

4) The shearing stress of the thin walls of the BBCSW produced by constraint torsion is evenly distributed along the wall thickness.

Figure 2 Geometry of BBCSW:

3 Theoretical analysis of BBCSW’s torsion

3.1 BBCSW sectorial coordinates

BBCSW with both ends subjected to the effect of torque M_z is taken as the first state and that with both ends subjected to the effect of unit torque is taken as the second state. According to principle of virtual work, the following can be obtained:

                         (1)

where τ_B denotes BBCSW’s shear stress under the action of torque M_z; h denotes the distance from the bending center to the central line of thin walls of the sections; s denotes the natural coordinates along the central line of thin walls of the BBCSW cross sections; φ is BBCSW’s torsion angle under the action of torque M_z; L denotes the calculation length along BBCSW; G_i (i=1, 2, 3, 4) is the shear modulus of BBCSW’s lower flange, upper flange, left web, and right web, respectively; G₁=G₂=G_s; G₃=G₄=G_e.

Letting , the following can be obtained according to Eq. (1):

                            (2)

where t_i (i=1, 2, 3, 4) is the thickness of BBCSW’s lower flange, upper flange, left web, and right web, respectively; t₁=t₂=t_f; t₃=t₄=t_w.

According to Vlasov’s rigid perimeter hypothesis, the tangential displacement of any point on the central line of thin walls when BBCSW experiences torsion can be expressed as:

                                 (3)

The shear strain of BBCSW’s central line of the thin walls within the cross sections can be expressed as:

               (4)

where γ_szi (i=1, 2, 3, 4) is the shear strain of BBCSW’s lower flange, upper flange, left web, and right web, respectively; w is the warping displacement for BBCSW’s free torsion.

According to Eq. (4), the warping displacement for BBCSW’s free torsion can be obtained as follows:

         (5)

                        (6)

where ω denotes the sectorial coordinates of BBCSW’s cross section.

Substituting Eq. (2) into Eq. (6), the sectorial coordinates of BBCSW’s cross section can be obtained as follows:

                       (7)

3.2 Normal strain and shear strain of BBCSW’s constraint torsion

According to the Kollbrunner-Hajdin correction theory, the warping displacement of BBCSW’s constraint torsion can be expressed as follows:

                     (8)

where θ(z) is the warping function of constraint torsion.

Normal strain of BBCSW constraint torsion warping can then be expressed as follows:

                       (9)

From axial force equilibrium of warping, the following can be obtained:

         (10)

where E_i (i=1, 2, 3, 4) is the elastic modulus of BBCSW’s lower flange, upper flange, left web, and right web, respectively; E₁=E₂=E_s; E₃=E₄=E_e; σ_z denotes the axial positive stress of BBCSW’s constraint torsion.

From Eq. (10), in order to satisfy we only need to choose a starting point appropriate for s coordinate to make the following equation workable:

                (11)

where b_i (i=1, 2, 3, 4) refers to the length of BBCSW’s lower flange, upper flange, left web, and right web, respectively; b₁=b₂=b_f; b₃=b₄=b_w.

Thus, BBCSW’s warping normal strain and constraint torsion shear strain can be respectively simplified as:

                              (12)

                   (13)

3.3 Strain energy of BBCSW

As to the BBCSW component shown in Figure 2, this component is the bi-axially symmetric cross section involving x and y axes, so centroid O coincides with the bending center. Supposing in x-direction, displacement of BBCSW section’s bending center, i.e., original point O is u, and displacement of the bending center in y-direction is v, it can be determined from Vlasov’s rigid perimeter hypothesis that the displacement of any point B(x, y) on the cross section in the x-direction and y-direction is as follows:

,                   (14)

BBCSW’s strain energy is composed of the bending strain energy, warping strain energy, and constraint torsion shear strain energy, and can be expressed as follows:

                   (15)

                             (16)

                     (17)

 (18)

where A_i (i=1, 2, 3, 4) denotes the sectional area of BBCSW’s lower flange, upper flange, left web, and right web, respectively; A₁=A₂=A_f; A₃=A₄=A_w; I_fy and I_fx are the total inertia moment of BBCSW’s upper flange and lower flange cross section on the y axis and x axis, respectively; I_fω is the sum of sectorial inertia moments of upper flange and lower flange.

Substituting Eq. (13) into Eq. (15), the following can be obtained:

                       (19)

where

3.4 BBCSW’s external potential energy

BBCSW’s normal stress can be expressed as follows:

                (20)

                    (21)

where P denotes the axial force of the beam end; M_x and M_y denote the beam-end bending moment around the x axis and y axis, respectively.

Substituting Eq. (14) and Eq. (21) into Eq. (20), and the following can be obtained:

 (22)

where

;

;

.

BBCSW’s total potential energy can be expressed as:

                             (23)

Substituting Eq. (15) and Eq. (22) into Eq. (23), and according to the stationary value theory of total potential energy [30-32], the following can be obtained:

            (24)

            (25)

      (26)

Setting P=0, M_y=0, and M_x=-M₀, the buckling equilibrium differential equation of BBCSW in a pure bending state can be obtained as follows [33, 34]:

                      (27)

      (28)

3.5 Solution of differential equations

Using simply supported BBCSW as an example, the boundary conditions can be expressed as follows:

                       (29)

Supposing the displacement function is:

                          (30)

                          (31)

Letting and substituting Eqs. (30) and (31) into Eqs. (27) and (28), the following can be obtained:

                      (32)

             (33)

The characteristic equation corresponding to Eqs. (32) and (33) is as follows:

    (34)

Because the transverse section of BBCSW is bi-axial symmetric, β_y=0. Solving this equation and when n=1, BBCSW’s lateral-torsional buckling critical bending moment can be obtained as follows:

         (35)

4 Verification and application

4.1 Verification of above analytical calculation method

In order to verify the practicability of the above analytical calculation method, by selecting 96 BBCSW test specimens with different web space-depth ratio λ₁, web depth-thickness ratio λ₂ and web span-depth ratio λ₃ according to engineering experiences [35-38]. The lateral- torsional buckling critical bending moment of 96 BBCSW test specimens were calculated by employing the above analytical calculation method and ANSYS finite element method, the calculation results obtained by the two methods were then compared. The comparative results are shown in Tables 1-3 and Figures 3 and 4, where, M_F1 denotes the ANSYS calculation results of BBCSW’s lateral-torsional buckling critical bending moment; M_an denotes the analytical calculation results of BBCSW’s lateral-torsional buckling critical bending moment (kN·m); E_rr denotes the calculation error of the analytical calculation method in this paper (%).

Table 1 Comparison of lateral-torsional buckling critical bending moment (λ₁=0.35)

Table 2 Comparison of lateral-torsional buckling critical bending moment (λ₁=0.40)

Table 3 Comparison of lateral-torsional buckling critical bending moment (λ₁=0.45)

Figure 3 shows the results of one selected member. The SHELL43 element was used to conduct the simulation for the corrugated web and upper and lower flanges [39]. The simply supported boundary of the beam end used the method of restricting the vertical and transverse degrees of freedom of the beam-end joints for simulation and the axial translational degrees of freedom of one joint of the right end also was restricted so as to satisfy the requirement of BBCSW’s static determinacy, and allow the cross sections to rotate along the direction of the bending moment action as well.

From Tables 1-3 and Figure 4, as to the analytical calculation method and the ANSYS finite element numerical calculation method, in calculating the lateral-torsional buckling critical bending moment of 96 BBCSW test specimens with the cross sections of different geometry dimensions, the errors between the two methods are less than 4.17%. This indicates that the analytical calculation results in this paper agree well with the numerical calculation results using ANSYS, thus confirming the rationality of the analytical calculation method and model simplification hypothesis.

Figure 3 Lateral-torsional buckling mode of a test specimen

4.2 Application

In order to study the mechanical properties of BBCSW, the finite element software ANSYS was used to conduct the numerical simulation calculation on the lateral-torsional buckling critical bending moment of 96 BBFSW [24] with the same geometry dimensions as the BBCSW specimens. The lateral-torsional buckling critical bending moment of BBFSW are shown in Table 4 and the comparison of lateral-torsional buckling critical bending moment ratio M_F1/M_F2 are shown in Figure 5, where M_F2 denotes calculation results of the lateral-torsional buckling critical bending moment of BBFSW (kN·m).

From Tables 1-4 and Figure 5, it can be seen that, within the common range of λ₁ and λ₃, the ratio of non-dimensional ratio M_F1/M_F2 decreases with the increase of λ₃, increases with the increase of λ₁ and λ₂; within the common range of λ₁, λ₂ and λ₃, M_F1/M_F2 of all specimens were greater than 1, moreover, BBFSW’s modes of instability are all lateral-torsional buckling, and when λ₁=0.45, λ₂=200 and λ₃=8, the ratio of non-dimensional ratio M_F1/M_F2 reaches 7.83. This shows that BBCSW’s stability bearing capacity has significant stability property advantage compared with BBFSWs, and BBCSW can reduce or eliminate the use of cross diaphragms.

Figure 4 Precision analysis of analytical calculation results in this paper and ANSYS finite element numerical calculation results:

Table 4 Lateral-torsional buckling critical bending moment of BBFSW

Figure 5 Comparison of lateral-torsional buckling critical bending moment ratios determined from BBCSW and BBFSW:

5 Conclusions

1) Taking the Kollbrunner-Hajdin correction method and mechanical properties of corrugated web into consideration, the neutral equilibrium equation of BBCSW under the action of bending moment load is derived through the stationary value theory of total potential energy, along with obtaining the analytical calculation formula of the lateral-torsional buckling critical bending moment of the BBCSW. Using the analytical calculation method and ANSYS finite element method, we calculated the lateral-torsional critical buckling bending moment of 96 BBCSW test specimens with different geometry dimensions of cross sections. The analytical calculation results in this paper agree well with the numerical calculation results of ANSYS, thus confirming the suitability of the analytical calculation method and model simplification hypothesis.

2) Within the common range of λ₁ and λ₃, BBCSW’s lateral-torsional buckling critical bending moment are all larger than BBFSW’s, the advantage of BBCSW’s stability property are even more significant with the increase of λ₁ and λ₂, and BBCSW can reduce or eliminate the use of cross diaphragms.

3) The obtained formula displays a certain development compared with the former calculation theory of BBCSW’s lateral-torsional buckling critical bending moment and lays a theoretical foundation for further application of BBCSW in practical engineering.

Appendix

BGCSW’s total potential energy can be expressed as:

                              (1)

               (2)

The above equation can be written as:

             (3)

The integrand consists of four variables u, υ, θ, φ and their first and second derivatives. According to the principle of potential energy, is the sufficient and necessary condition that the elastic system is in neutral equilibrium, by using variational method, the following can be obtained:

  (4)

Because of the first variation δu, δυ and δθ are the arbitrary values that are not equal to zero, respectively, the following can be obtained:

              (5)

Thus, the following equation can be obtained:

             (6)

             (7)

                     (8)

                 (9)

And then, the equations can be written as:

            (10)

            (11)

      (12)

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(Edited by HE Yun-bin)

中文导读

波纹钢腹板箱形梁的侧扭屈曲分析

摘要：波纹钢腹板沿纵向呈褶皱状，具有轴压刚度波折效应及剪切模量波折效应等力学特性。为研究弯矩荷载作用下波纹钢腹板箱形梁(BBCSW)的侧扭屈曲，在考虑Kollbrunner-Hajdin修正方法及波纹钢腹板力学特性基础上，利用势能驻值原理推导出BBCSW在弯矩荷载作用下的中性平衡方程，并进一步获得BBCSW屈曲临界弯矩解析计算公式。利用本文提出的解析计算方法及ANSYS有限元方法对96个不同截面几何尺寸及计算长度的BBCSW模型的临界屈曲弯矩进行计算。结果表明：本文解析计算结果与ANSYS有限元数值计算结果吻合良好，论证了本文解析计算方法及模型简化假设的合理性；BBCSW与普通钢腹板箱形梁(BBFSW)在同等几何尺寸的情况下，BBCSW可取更小的腹板厚度而无需加劲肋，从而降低工程造价。在腹板距高比和腹板跨高比的共同范围内，BBCSW侧扭屈曲临界弯矩大于BBFSW，而且，随着腹板距高比和腹板跨高比的增加，BBCSW的稳定性优势更加显著。

关键词：波纹钢腹板箱形梁；侧扭屈曲；解析解；手风琴效应；Kollbrunner-Hajdin方法

Foundation item: Projects(51408449, 51778630) supported by the National Natural Science Foundation of China; Project(2018zzts189) supported by the Fundamental Research Funds for the Central Universities, China

Received date: 2018-02-14; Accepted date: 2019-05-17

Corresponding author: ZHOU Wang-bao, PhD, Professor; Tel: +86-13677300601; E-mail: zhouwangbao@163.com; ORCID: 0000- 0003-3338- 4971