Elastic Shear Buckling Coefficients for Oblique Plates

Mitsui, Kazuya; Ikarashi, Kikuo

doi:https://doi.org/10.1155/2022/9532380

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2022 | Article ID 9532380 | https://doi.org/10.1155/2022/9532380

Elastic Shear Buckling Coefficients for Oblique Plates

Kazuya Mitsui¹and Kikuo Ikarashi¹

Academic Editor: Federico Guarracino

Received12 Jul 2022

Accepted18 Aug 2022

Published15 Sept 2022

Abstract

Oblique plates are utilized in a variety of applications in steel structures such as panel zones. As shear stresses generally act upon panel zones, it is important to evaluate the shear buckling resistance of panel zones, especially as thinner steel plates have been used in recent years. In contrast to the stability problem of rectangular plates, which has been extensively studied, the elastic buckling strengths of oblique plates are not entirely clear, even under basic stress conditions such as uniform shear stress. This study investigated the elastic shear buckling stress of oblique plates. The main goals were to examine the buckling behavior of simply and clamped supported oblique plates and propose convenient and reliable design equations to predict the elastic shear buckling coefficients. By adopting the energy method based on Timoshenko’s plate buckling theory, the influence of the geometry of the oblique plates and the direction of the shear stress were investigated. Furthermore, highly convenient design equations for elastic shear buckling coefficients were proposed. The accuracy of the proposed equation was assessed using extensive parametric studies based on the finite element method using MSC Marc software.

1. Introduction

Parallelogram-shaped thin plate elements are frequently utilized in steel structures, such as steel moment-resisting frames, I-girders, and aircrafts. For example, steel structures, such as factories and warehouses, which require column-free space inside the building, utilize a one-directional moment-resisting frame system in Japan, in which an L-shaped connection is formed at the beam-to-column connection at the roof level, as shown in Figure 1 (where the beam members are connected to the column members in an L-shape). The variety of panel zone shapes further complicates their buckling behavior, as the shape of the L-shaped connection varies from rectangular to parallelogram, as shown in Figure 1 (depending on the roof slope and the type of beam-to-column connection). In addition, in recent years, the cross-sectional shape has become larger and thinner in steel structural buildings, modifications implemented from the viewpoints of economy and livability, respectively. Hence, it is becoming increasingly important to evaluate the elastic shear buckling strength that occurs inside panel zones. However, in contrast to the stability problem of rectangular plates, which has been extensively studied, the elastic buckling stresses of the oblique plate are not entirely clear, even under basic stress conditions such as uniform shear stress. As the linear stability problem of a plate is determined by the boundary conditions, including the shape of the plate, it is necessary to clarify the effects of these factors to elucidate the elastic buckling stress of the plate.

The linear stability problems of a plate with basic shapes, such as rectangles, under uniform shear stresses were reported by Southwell and Skan [1]. Timoshenko’s work [2] using the energy method and practical design equations based on research results was also presented in the specification for the stability design of steel structures in Japan [3]. Numerous studies applying these methods have been reported [4–12]. For example, Way [4] used a double Fourier sine series to express the buckling deformation of simply supported rectangular plates stiffened by two transverse stiffeners under shear stresses and derived the shear buckling strength. The rigidity required for the stiffeners to maintain the original geometry was also obtained. Wang [5] examined the case of three (or four transverse) stiffeners with simply supported rectangular plates and stated that the moment of inertia of stiffeners was dominant in determining the elastic shear buckling strength and that the torsional rigidity was comparatively small. Fourier sine series were also used as buckling displacement functions in Wang’s study [5]. Cook and Rockey [6] expressed the displacement function using a Fourier series with a mixture of sine and cosine waves and presented solutions for buckling under shear of infinitely long rectangular plates (with clamped edges stiffened by transverse stiffeners). Cook and Rockey [7, 8] also obtained the shear buckling strength of infinitely long rectangular plates (stiffened by stiffeners in the longitudinal and transverse directions) and the shear buckling strength for mixed simply and fixed support boundary conditions (using the mixture Fourier series). Ikarashi et al. [9] theoretically described the stress state inside a panel zone using Airy’s stress function and obtained the elastic shear buckling strength of a rectangular panel zone by setting the buckling displacement (using a double Fourier cosine series). A method for determining the ultimate resistance of an L-shaped connection was also presented based on the elastic shear buckling strength and tension field model [10]. Hence, from the aforementioned analysis, simple design equations are available for the shear buckling strength of rectangular plates and developmental studies have been conducted.

Studies on the buckling of oblique plates have primarily been conducted in aeronautical engineering. Wittrick [13–15], for example, used the upper bound solutions with the Raleigh–Ritz method to determine the elastic buckling strength of oblique plates with clamped supported edges subjected to uniform compressive or shear forces. For shear stress conditions, the buckling displacement function proposed by Iguchi [16] was used to determine the elastic shear buckling coefficients. The elastic buckling coefficients varied depending on the direction of shear stress. However, it was difficult to obtain a larger elastic buckling strength because a function to describe the buckling mode could not be determined. Yoshimura and Iwata [17] derived the elastic buckling strength of oblique plates with simply supported edges subjected to uniform compressive (or shear) stresses using the energy method. They also proposed a calculation method to obtain the elastic buckling strength by solving a fifth-order square matrix. Kennedy and Prabhakara [18] expressed the buckling displacement function in terms of a double Fourier sine series satisfying the boundary conditions at the edges of oblique plates. They also determined the elastic buckling strength of a simply supported oblique plate using the energy method. Ashton [19] obtained the elastic buckling strength of oblique plates with clamped supported edges subjected to combined stresses using the Ritz method. In addition to these studies, other methods for calculating the elastic buckling strength based on energy methods, such as the Galerkin and Raleigh–Ritz methods, have also been presented [20–25]. However, these calculation methods were not convenient: they provided tables or graphical charts of the elastic buckling coefficients at specific angles or aspect ratios or required complex matrix equations to be solved. Currently, there is no simple method for calculating the elastic buckling strength of oblique plates with arbitrary shapes.

Therefore, the purpose of this study was to evaluate the elastic shear buckling strength of simply and clamped supported oblique plates under uniform shear stresses using shear buckling coefficients and propose design equations with a uniform and simple expression. In Section 2, the displacement function that reproduced shear buckling behavior was presented, and the energy method based on Timoshenko’s [2] plate buckling theory was described. In Section 3, the relationship between the elastic shear buckling coefficient obtained based on the energy method and the parallelogram geometry was explored. In Section 4, convenient and reliable design equations for the elastic shear buckling coefficient for the oblique plate were developed, reflecting the findings obtained in the previous sections.

2. Outline of Theoretical Analysis Modeling and Energy Method Introduction

This study examines the elastic shear buckling strength of an oblique plate under uniform shear stress based on the energy method. The theoretical analysis model of an oblique plate of uniform thickness and its coordinate system is illustrated in Figure 2, where a denotes the length of the oblique plates (long side), b denotes the height (short side), t denotes the thickness, and θ denotes the angle of obliquity. Uniform shear stress acts at an arbitrary point on the oblique plate; the positive stress direction is defined as one tending to decrease the obliquity of the plate, and the negative stress direction is defined as one tending to increase the obliquity of the plate [14]. In this study, the discussion is mainly focused on aspect ratios greater than 1.0. However, for oblique plates under uniform shear stress, even aspect ratios less than 1.0 can be regarded as that of oblique plates with aspect ratios greater than 1.0, by switching the length and height. That is, all aspect ratios can be covered by considering an aspect ratio range greater than 1.0. The boundary conditions under consideration are simply and clamped supported edges. By setting the displacement functions using the Fourier series to satisfy the boundary conditions on the edges, the displacement functions for oblique plates with simply and clamped supported edges are expressed by the following equations:

Displacement function for simply supported plates is as follows:

Displacement function for clamped supported plates is as follows:where m and n are natural numbers representing the number of terms in the series and c_mn is an undetermined coefficient. Figure 3 illustrates the buckling displacement determined using the proposed function (equations (1) and (2)) for m = n = 1. Previous studies [13–25] on the buckling of oblique plates defined a coordinate system along the oblique edges, whereas the present study employed a Cartesian coordinate system to handle the stresses easily. On all edges of the oblique plate, both displacement functions are zero displacement ( = 0); the displacement function for simply supported edges allows rotation on the edges ( ≠ 0, ≠ 0) and the displacement function for clamped supported edges restrains the rotation on the edges ( = 0, = 0).

(a)

(b)

The elastic shear buckling stresses were obtained based on selected displacement functions. Based on Timoshenko’s plate buckling theory [2], the incremental strain energy ΔU for an oblique plate can be expressed as follows:where D denotes the flexural rigidity of the plate, E denotes the modulus of elasticity, ν denotes Poisson’s ratio, and A is the integral range enclosed by the plate. As shown in Figure 2, assuming that a uniform shear stress acts on the oblique plate, the work done by the external force ΔT can be written as follows:where τ_cr is the elastic shear buckling stress and k_τ is the shear buckling coefficient. σ₀ is defined as the base stress. The potential energy Π obtained by performing an integral calculation in the Cartesian coordinate system (assuming that the sum of the incremental strain energy is equal to the sum of the incremental work done by the external force), is as follows:

Based on the principle of minimum potential energy, the buckling condition equation (equation (6)) can be obtained by differentiating equation (5) with respect to the auxiliary variable c_mn (m = 1, 2, 3, …, M; n = 1, 2, 3, …, N) as follows:

The following buckling conditions can be obtained by differentiating equation (5) by all auxiliary variables c_mn and formulating a system of linear equations as follows:where B is a matrix composed of equation (3), C is a matrix composed of equation (7), and {c_mn} is the coefficient matrix expressed by equation (9). The shear buckling stress τ_cr = k_τσ₀ can be obtained by determining the lowest eigenvalues in each shear stress direction when equation (8) has nontrivial solutions; that is, when {c_mn} in equation (8) has a non-{0} solution. The buckling displacement corresponding to the shear buckling stresses in each direction is obtained as an eigenvector corresponding to the eigenvalues.

3. Relationship between Elastic Shear Buckling Stresses and Geometry of Oblique Plates

3.1. Validation of Energy Methods

In this section, the effects of the geometry and boundary conditions of the oblique plates on the elastic shear buckling stress were examined using equation (8), which was developed using the energy method. First, the validity of the elastic shear buckling stresses determined using the energy method was confirmed by comparing them with the results of the finite element method eigenvalue analysis. The finite element program MSC Marc version 2019 [26, 27] was used to create a numerical model of the oblique plates, as shown in Figure 4. The finite element (FE) model was constructed using Element 139 (four-node shell elements with six degrees of freedom at each node, with translations and rotations about three reference axes) [27], which was suitable for analyzing thin-shell structures. A mesh of 2 × 2 mm was selected based on a convergence study. For the material properties, the modulus of elasticity, E, was set to 205 GPa and Poisson’s ratio ν was set to 0.3. The FE model was a rectangular shape composed of a testing section and auxiliary plates. Shear loads were introduced to the nodes on the auxiliary plate edges, as shown in Figure 4, to apply uniform shear stress to the testing section. The out-of-plane displacement of the auxiliary plate was restrained and only the test section was allowed to deform in the out-of-plane direction. Therefore, shear buckling deformation occurred only in the test section. The test section and auxiliary plate were connected by RBE2 links [27]. The RBE2 link is a rigid link, to which arbitrary degrees of freedom can be selected from six degrees of freedom and applied. In this FE model, the nodes of the test section and the auxiliary plate nodes were rigidly connected by the RBE2 links for displacements in the x, y, and z directions, and in-plane shear forces applied to the auxiliary plate were transmitted to the test section. In the simply supported model, the rotational degrees of freedom for all the edges of the oblique plate were released. In the clamped supported model, the rotational degrees of freedom for all the edges of the oblique plate were restrained. As analytical variables, the height, b, was fixed at 100 mm, thickness, t, was fixed at 1.0, aspect ratio, a/b, was varied in the range of 0.1 to 5.0, and angle, θ, was varied in the range of 0 to 45 degrees.

Figure 5 shows a graphical comparison between the present theoretical results based on the energy method and the finite element analysis (FEA) results. The buckling modes of the first eigenmode in the positive and negative directions at θ = 30° are shown in Figure 5. As described by Wittrick [14, 15], the shear buckling stress of the oblique plate varied with the direction of the shear stress; therefore, the line type was changed for each direction of the shear stress and the color was changed for each angle. Note that a (1600 × a/b) × (1600 × a/b) square matrix was created with the x-axis series term m set to 40 times the aspect ratio and the y-axis series term n set to 40, which was larger than the 8 × 8 square matrix reported by Wittrick [14]. This was done to ensure sufficient convergence of the buckling coefficient obtained from the determinant. Regardless of the boundary conditions, the theoretical results based on the energy method corresponded well with the results of the FEA. In addition, the shapes of the 1st buckling modes obtained from the FEA and the energy method presented in Section 3.2 were almost identical. The relative error between the FEA results and the negative direction, that is, the smaller the elastic shear buckling coefficient, was small. For the same results, the relative error in the positive direction, that is, the larger the elastic shear buckling coefficient, tended to be slightly larger. The maximum error was 4.6% in the positive direction and 3.5% in the negative direction. As shown in Figure 5, reliable elastic shear buckling coefficients were obtained using the energy method.

(a)

(b)

Figure 6 compares the shear buckling coefficients obtained by the method presented in this study with those obtained by Wittrick [14], Kennedy and Prabhakara [18], and Durvasula [21]. Kennedy and Prabhakara [18] obtained the shear buckling coefficients in the positive and negative directions for simply supported parallelograms with angles of 15°, 30°, 45°, and 60° with aspect ratios of 0.5, 1.0, 1.5, and 2.0, respectively. Durvasula [21] also obtained shear buckling coefficients in the positive and negative directions for simply supported parallelograms with angles of 15°, 30°, 45°, and 60° with aspect ratios of 0.5, 1.0, 1.5, and 2.0, respectively. Wittrick [14] obtained a shear buckling coefficient in the negative direction for clamped supported parallelograms with an angle of 45° and an aspect ratio of approximately 0.8 to 2.4. The analytical results presented by Wittrick [14] and Kennedy and Prabhakara [18] were almost the same as those presented in this study, although the displacement function and coordinate system settings were different from those in this study. However, Wittrick [14] did not present larger critical shear buckling coefficients and Kennedy and Prabhakara [18] underestimated the elastic shear buckling coefficient in the negative direction. It was observed that the analysis results presented by Durvasula [21] overestimated the elastic shear buckling strength; such discrepancies were attributed to the failure in satisfying all the boundary conditions and an insufficient matrix size.

(a)

(b)

3.2. Effect of Geometry of Oblique Plate

Figure 7 shows the variation in the buckling coefficient k_τ for elastic shear buckling with respect to the change in the aspect ratio obtained from the energy method. The difference in the obliquity angle θ of the panel zone is described according to the line color. Equation (10) is the design equation for the elastic shear buckling stress of rectangular flat plates, and equations (11) and (12) are the design equations for the elastic shear buckling coefficients for each boundary condition [3, 28, 29]:

(a)

(b)

For simply supported plates,

For clamped supported plates,

Equations (10)–(12) can be used to calculate the shear buckling stress by considering a as the long side of the rectangle and b as the short side of the rectangle. Figure 8 illustrates the buckling modes obtained from the energy method corresponding to Figure 7. For θ = 0°, the curves of the elastic shear buckling coefficient coincided regardless of the direction of the shear stress. For θ ≠ 0°, it was confirmed that the elastic shear buckling coefficient was affected by the direction of shear stress in the range of small aspect ratios. In addition, the elastic shear buckling coefficient varied depending on the direction of shear stress and angle. However, the convergence value of the elastic shear buckling coefficient was not affected by the direction of the shear stress and was consistent with the convergence value for a rectangular plate at θ = 0°. As shown in Figure 8, as the aspect ratio a/b increased, the shear buckling deformation became approximately the same shape as the buckling deformation that occurred in a rectangular shape and could be approximated as a rectangle. The difference in the boundary conditions for the same angle and aspect ratio confirmed that the clamped supported plate was less affected by the angle than the simply supported plate. The elastic shear buckling coefficient k_τ was larger for positive shear stress and smaller for negative shear stress as compared to that of θ = 0°. This trend was more significant for larger angles. The differences in the shear buckling coefficient in the positive and negative directions were due to the buckling half-wavelength that formed in the panel. As shown in the deformation diagram in Figure 8, the positive direction led to a shorter buckling half-wavelength than that in the negative direction, resulting in a larger shear buckling coefficient.

(a)

(b)

(c)

(d)

(e)

4. Design Equations for Oblique Plates

For θ = 0°, as shown in Figure 7, the correspondence between equations (11) and (12), and the theoretical analysis results for both boundary conditions show a correlation when the aspect ratio a/b is greater than 1.0. As mentioned earlier, under uniform shear stress, it was synonymous to obtaining the shear buckling coefficient for a/b ≥ 1.0 and a/b ≤ 1.0 by switching the aspect ratio. The current design equations shown in equations (11) and (12) also addressed this by switching the height and length according to the aspect ratio. This study proposed convenient design equations for shear buckling coefficients of oblique plates by taking advantage of the fact that the convergence value of the shear buckling coefficients (oblique plates) coincided with that of a rectangular plate.

As shown in Figure 7, in the large aspect ratio range, the elastic shear buckling coefficient converged to 5.34 or 8.98, which were the convergence value of equations (11) and (12) for each boundary condition (regardless of the direction of the shear stresses and angle). However, the shear buckling coefficient at a/b = 1.0 varied with the direction of the shear stress and angle. To conveniently approximate the range of a/b ≥ 1.0, it was sufficient to propose approximate equations that passed through the value at a/b = 1.0, which varied with the panel angle and converged to the convergent values of equations (11) and (12) for each boundary condition.

Figure 9 shows the values at a/b = 1.0, which varied with the panel angle. The current design equations (equations (11) and (12)) were curves that were inversely proportional to the aspect ratio; thus, the value at a/b = 1.0, which varied with the direction of the shear stresses and angle, determined the slope of this curve. In other words, approximate equations could be developed by approximating the slope of these curves for each direction of the shear stresses and boundary conditions as a function of the angle. From an engineering viewpoint, this study proposed equations (13) and (14) as design equations for oblique plates subjected to uniform shear stress, utilizing the shear buckling coefficient design equations (equations (11) and (12)) for a rectangular plate element with θ = 0°.

For simply supported plates (larger elastic shear buckling strength for 0° ≤ θ ≤ 45°),

For simply supported plates (smaller elastic shear buckling strength for 0° ≤ θ ≤ 45°),

For clamped supported plates (larger elastic shear buckling strength for 0° ≤ θ ≤ 45°),

For clamped supported plates (smaller elastic shear buckling strength for 0° ≤ θ ≤ 45°),

Note that a is the long side of a rectangle or parallelogram and b is the short side.

Figure 10 shows a comparison of k_τ values from the theoretical analysis results based on the energy method and proposed design equations. k_τ represents the value obtained by the energy method and k_{τ, cal} represents the values obtained by the proposed design equation. The variables of the parametric investigation were the aspect ratio a/b, angle of obliquity θ, and direction of the shear stress and boundary conditions, as listed in Table 1. It was observed that the proposed design equations could well predict the k_τ values. For simply supported edges, the average ratios of the theoretical analysis to the predicted values were both 0.999 for the positive and negative directions, with standard deviations of 0.034 and 0.028, respectively. For the clamped supported edges, the average ratios of the theoretical analysis to the predicted values were 0.981 and 0.990 for the positive and negative directions, with standard deviations of 0.028 and 0.019, respectively. Therefore, the proposed design equations can be used to predict the elastic shear buckling coefficients of oblique plates.

(a)

(b)

5. Summary and Conclusions

This study provided a thorough investigation of the elastic shear buckling behavior of simply and clamped supported oblique plates based on the energy method. In this study, buckling displacement functions were proposed for a parallelogram in the Cartesian coordinate system. It was confirmed that the shear buckling stress of the oblique plate varied with the direction of shear stress based on the energy method and FEA. Convenient design equations for the elastic shear buckling coefficient for oblique plates were proposed. The primary findings of this study are summarized as follows:(1)Previous studies defined a local coordinate system along the inclined edge and used displacement functions according to the coordinate system, whereas this study proposed displacement functions for simply and clamped supported parallelograms using a Cartesian coordinate system.(2)The shear buckling stress of the oblique plate varied with the direction of the shear stress. Compared to the shear buckling coefficient of a rectangle with the same aspect ratio, the shear buckling coefficient of an oblique plate was larger than that of a rectangle when the direction of the shear stress was positive and smaller than that of a rectangle when the shear stress was negative. However, as the aspect ratio increased, shear buckling deformation was not formed and the parallelogram was approximately regarded as a rectangle; therefore, the convergence value of the shear buckling coefficient was not affected by the direction of shear stress and was consistent with the convergence value for a rectangular plate at θ = 0°.(3)Convenient design equations for the shear buckling coefficient of oblique plates were proposed by taking advantage of the fact that the convergence value of the shear buckling coefficient of the oblique plate coincided with the convergence value of the shear buckling coefficient of a rectangle. The proposed design formula could be used to evaluate the elastic shear buckling coefficient of a parallelogram with sufficient engineering accuracy.

Data Availability

All information and data are given in the text completely.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2022 Kazuya Mitsui and Kikuo Ikarashi. 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.

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