Abstract
This paper presents analytical solutions for the problem of skewed thick plates under transverse load on a Winkler foundation, which has not been reported in the literature. The thick plate solution is obtained by using a framework of an oblique coordinate system. First, the governing differential equation in that system is derived, and the solution is obtained using deflection and rotation as derivatives of the potential function developed here. This method is applicable for arbitrary loading conditions, boundary conditions, and materials. The solution technique is applied to two illustrative application examples, and the results are compared with numerical solutions. The two approaches yielded results in good agreement.
1. Introduction
Plates resting on an elastic foundation are important structural elements that are used in a wide range of applications including foundation structures and concrete pavements of roads. In the past few decades, there have been efforts to analytically investigate the behavior of plates resting on an elastic foundation. Most of the work has been performed on rectangular plates and circular plates. For example, Timoshenko and Woinowsky-Krieger [1] provided an analytical solution for rectangular thin plates, and Celep [2] presented the behavior of circular thin plates on an elastic foundation.
Due to its mathematical complexity, there are very few studies which have considered skewed plates on an elastic foundation. Kennedy [3] was the first to develop an analytical solution as polynomial series for skewed isotropic thin plates with four edges clamped on an elastic foundation. Ng and Das [4] and Chell et al. [5] developed an analytical solution as trigonometric series for skewed sandwich thin plates on an elastic foundation. Though these studies are informative, they have limitations in terms of loading and boundary conditions. All of the solutions developed by the above researchers can handle only uniformly distributed load, and a designated boundary condition. The solutions by Kennedy [3] and Ng and Das [4] are limited to clamped plates, and that of Chell et al. [5] is limited to simply supported plates.
In addition, there is another issue; these methods are applicable only to thin plates of which the thickness-to-side ratio is less than 1/20 because Kirchhoff plate theory has been employed to develop the governing equation in these studies. Though the Kirchhoff theory is widely used in plate analysis, it suffers from under predicting deflections when thick plates are analyzed because it neglects the effect of the transverse shear deformation [6]. To resolve this issue, the Mindlin theory [7, 8], which relaxes the perpendicular restriction for transverse normals, was developed so that the effect of transverse shear deformation could be considered. Several studies which have employed the Mindlin theory for analysis are worth mentioning. For example, Kobayashi and Sonoda [9] developed an analytical solution for rectangular thick plates with two opposite edges, simply supported. In their research, Levy-type trigonometric series solutions were derived. Liew et al. [10] and Liu [11] solved the rectangular thick plate problem by the differential quadrature method and offered a number of numerical results under several boundary conditions. Though there are some studies dealing with rectangular thick plates, there has been no research analyzing skewed thick plates.
In the present manuscript, analytical solutions for skewed thick plates on an elastic foundation with arbitrary boundary and loading conditions are newly developed based on the Mindlin theory. The Winkler foundation, which is the simplest model to represent the elastic foundation, is employed here. The present method is new in the following two aspects. First of all, the present method has developed analytical solutions based on the Mindlin theory for skewed thick plates, and such solutions have never previously been developed in the literature. Second, the present method allows arbitrary loading conditions, boundary conditions, and materials.
We first show the derivation of a governing equation in a framework of an oblique coordinate system, and then it is solved using trigonometric series. The governing equation in this system is derived and the solution is obtained using the deflection and rotation as derivatives of a potential function. The solution technique is applied to an illustrative application example with different skew angles and modulus of foundations, and the results are compared with the commercial finite element package ANSYS [12]. The present approach is in reasonable agreement with the FEM solution.
2. Governing Equation under the Oblique Coordinate System
When a plate’s boundary profile is a parallelogram, the oblique Cartesian coordinate system can be advantageous. We first present the concept of the oblique coordinate system and then show the governing differential equation of skewed thick plates on a Winkler foundation based on the Mindlin theory. The authors have developed an analytical solution for skewed thick plates without an elastic foundation in [13], and the formulas developed in this chapter have utilized the relationships developed in the research. Figure 1 shows the oblique coordinate system spanned by the X and Y axes, along with the rectangular coordinate system x and y, with angle XOY denoted as skew angle a. Parallelogram ABCD in Figure 1 represents the skewed plate of interest, and the edge lengths CD and AD are 2a and 2b, respectively.
The relationship between the rectangular and oblique coordinate systems can be written as follows [14, 15]: where and are, respectively, the rotations normal to the X and Y axes. The corresponding relationships for stress, strain, moment, and shear force under the two coordinate systems are also available in [14]. They are readily derivable according to the principles of statics and continuum mechanics. Their corresponding quantities in the oblique system are derived in this research effort. The resulting flexural stiffness matrix under the oblique coordinate system is related to its counterpart in the rectangular system as follows [16]: The flexural stiffness matrices relate the moments to the curvatures in the respective coordinate systems. For example, in the rectangular coordinate system for isotropic material is [1]: where is Young’s modulus, is Poisson’s ratio, and is the thickness of the plate. Note that (2.2) is also applicable for other, more complex situations, such as orthotropic or anisotropic materials. By using the flexural stiffness matrices, the moment-strain relationship of the rectangular and oblique coordinate systems can be described as in the following (2.4) and (2.5), respectively [6]: In this derivation, the Mindlin theory is applied, which assumes that strains are linearly distributed along the thickness direction in the plate’s cross-section.
Similarly, the extensional stiffness matrix in the oblique coordinate system is related to that of the rectangular system as follows [13]: The extensional stiffness matrix relates the shear forces to the shear strains. For example, for isotropic material is By using the extensional stiffness matrices, the relationships between the shear force and deflection, rotation angle of the rectangular and oblique coordinate system can be described as in the following (2.8) and (2.9), respectively, [6]: where is the transverse deformation perpendicular to the plane of the plate, and is the shear correction factor to account for nonuniform transverse shear distribution. Hereafter, the components in and are referred to using their respective elements to and to as follows: where the diagonal components of relate the moments to the curvatures in the same directions. The off-diagonal terms relate the same moments to the curvatures in other directions due to the Poisson’s effect and coordinate system obliquity. Similarly, the diagonal components of relate the shear forces to the shear strains in the same directions, and off-diagonal terms to the shear strains in other directions due to obliquity.
The following (2.11) to (2.13) are equilibrium conditions of the skewed plates shown in Figure 1.
Equilibrium of force in the direction (perpendicular to the plane of the plate) Equilibrium of moments along the X-axis Equilibrium of moments along the Y-axis where in (2.11) is the load normal to the upper surface of the plate, and k is the modulus of the foundation. By substituting (2.5), (2.9), and (2.10) into (2.11) to (2.13), the following (2.14) to (2.16) are obtained in the oblique system: To make the solution process simpler, a new function is introduced below to represent the condition of the skewed thick plate. We assume that consists of terms up to the 4th derivative, and and up to the 3rd derivative of , with respect to the spatial variables X and Y. The following relations in (2.17) to (2.19) are obtained to satisfy (2.15) and (2.16). has only even derivatives and and have only odd derivatives though it is assumed that , , and may have both even and odd derivatives. This is because the coefficients of odd derivatives of and the coefficients of even derivatives of and should be 0 to satisfy (2.14) to (2.16). By substituting these relations into (2.14), the governing equation of the Mindlin skewed thick plate is then formulated as a 6th-order partial differential equation as follows: where is a linear differential operator in the oblique coordinate system
3. Analytical Solution in Series Form
In this section, a general solution to the governing differential equation (2.11) is developed as the sum of a fundamental (homogeneous) and a particular (nonhomogeneous) solution, detailed separately, as follows.
3.1. Homogeneous Solution
The homogeneous solution is the solution to (2.20) for , obtained as a trigonometric series in (3.1) below where is the imaginary unit, and , , , and trigonometric functions are as follows: where the bar on denotes the conjugate of . , , , , and , and are the eigenvalues to be obtained by satisfying . For example, is derived by solving the following equation: The trigonometric function in (3.1) has unknowns , and , with being the number of the trigonometric terms needed for convergence. In addition, as for the coefficients of real parts (, and ), () terms are considered so that 12 more unknowns (, and ) can be determined. Therefore, the homogeneous solution has unknowns, and these will be determined according to the boundary conditions discussed below.
3.2. Particular Solution
For a particular solution in the series form, the transverse load in (2.20) is expanded to a trigonometric series as follows: Equation (3.4) can express any transverse load, such as a uniformly distributed load, a concentrated load, a line load, or a patch load. Therefore, the particular solution for (2.20) can be written in a series form as where and are to be determined to satisfy (2.20) and (3.4), and is the number of the trigonometric terms needed for convergence. The general solution for is derived as the sum of the homogeneous solution and the particular solution as Since no unknowns exist in the particular solution, the total number of unknowns in the general solution is still , as in the homogeneous solution.
4. Determination of Unknown Constants for Series Solution
Using the Mindlin theory, the boundary conditions for various edges are given below for determining the unknown constants in the homogeneous solution. The normal and tangential directions to the edge are denoted here using the subscripts and , respectively. The moments on the edges are noted using these subscripts in a way that is consistent with the directions of the stresses thereby induced. Namely, is for the moment causing normal stresses and is the torsional moment inducing shear stresses. Note that the Kirchhoff theory treats SS1 and SS2 in (4.2) and (4.3) as having the same boundary condition. The difference between them is explained graphically in Figure 2. The boundary condition of SS1 restricts the tangential rotation by supporting two points in the cross-section, thereby generating a nonzero torsional moment. In contrast, the boundary condition of SS2 supports the plate only at one point in the cross-section, allowing a tangential rotation and generating no twisting moment. The boundary conditions in (4.1) to (4.4) can be unified as follows: where to represent the left hand side of (4.1) to (4.4). to are expanded as a Fourier series as follows for the solution method pursued in this paper: where coefficients , , and are Fourier coefficients for the boundary condition . Because of the need for truncation, terms are kept for each of the 12 boundary conditions so that a total of equations are established for the same number of Fourier coefficients that have the unknowns included to be solved for.
5. Application Examples
In this section, two application examples are presented using the developed analytical solution for skewed thick plates. They are also compared with solutions published in [10] and the FEM analysis result obtained using the commercial package ANSYS 11. In the analysis by ANSYS, 2D 8-node quadrilateral plate elements (SHELL99) appropriate for thick plate analysis are used for the skewed plates with various skewed angles. The numbers of nodes and elements are 4961 and 1600, respectively. The convergence figure of transverse displacements along with the number of nodes is provided for both of the application examples. In addition, the effect on convergence of the number of terms and in the fundamental and particular solutions is studied. In the following examples, the shear correction factor is taken to be 5/6 since this factor is commonly used in plate analyses.
5.1. Isotropic Clamped Skewed Thick Plate under Uniformly Distributed Load
Isotropic skewed thick plates are analyzed here. As an external force, uniformly distributed load is applied, and, as skewed angles, , 60°, and 90° are employed. The geometrical properties used in this research are , and . The clamped boundary condition is used for all edges. The modulus of foundation is expressed in a dimensionless form as . D is the bending stiffness and is expressed as for the isotropic plate. , 5, and 7 are employed here.
As a first step, convergence of the finite element model is checked. Figure 3 shows the convergence figure of transverse displacements w at the center of the plates when is 30°. It can be concluded that the finite element model converges adequately when the number of elements = 1600. Similar results are confirmed for other .
Then, the numbers of terms in the series solution and are determined. The deflection at the center of the plates for and differs by less than 0.1% from that of as in Figure 4 when and . It can be concluded that the solution is already convergent while truncated at , and similar results are observed for other and . Therefore, and are employed in this example.
Figures 5, 6, and 7 give a comparison between the present method and FEM analysis using ANSYS for the deflection for , 60°, and 90° along line HF defined in Figure 1 and on the top of the plate. Note that the deflection is expressed in a dimensionless form as . For the plate of , results in the literature [10] are also shown in the figure. In addition, Table 1 also shows the comparison results between the present solution, FEM, and literature at the center of the plate. The results show that our analytical and numerical solutions agree with each other very well.
5.2. Orthotropic Skewed Thick Plate under Concentrated Load
Orthotropic thick skewed plates are analyzed in this example, with the following material and geometrical properties: , , , , , , , where and are Young’s modulus along the and directions, is the major Poisson’s ratio, and , , and are the shear modulus in the , , and planes. These values determine , , , and in (2.2) and (2.6). The external transverse force is a concentrated force of applied at . Plates with skew angle , 60°, and 90° and the modulus of foundation of , 5, and 7 are analyzed here, where is expressed in a dimensionless form as , and is the bending stiffness which is expressed as . To show the applicability of this method for the arbitrary boundary condition, the following complex boundary condition is applied in this example: AB in Figure 1 is soft, simply supported condition (SS1), BC is clamped (C), CD is free (F), and DA is a hard, simply supported condition (SS2).
As with the previous example, the convergence of the finite element model should be checked. Figure 8 shows the convergence study of transverse displacements at the center of the plates when is 30°. It can be concluded that the finite element model converges adequately when the number of elements = 1600. Similar results are confirmed for other .
In addition, the numbers of terms including and in (3.1) and (3.5) need to be determined. Figure 9 shows the deflection at the point where a concentrated load is applied for skew angle and , as one of the cases considered, for various and values. It is seen that the deflection at is well converged. Similar results are observed for other cases. Therefore is employed here and is also used as the reference for comparison.
For this example, because no previous work in the literature has been found reporting a similar experience, only FEM analysis results by ANSYS are employed for comparison with our analytical solution results. Table 2 shows comparison results of the deflection at the center of the plate . The deflection is expressed in a dimensionless form as .
Figures 10, 11, and 12 indicate the results for deflection along line HF in Figure 1. The results indicate that the response behavior for this case is much more complex than the examples above, due to nonsymmetric loading and boundary conditions. Due to the oblique coordinate system, the load at has different relative relations to the interested responses on , along with different skew angles. For example, this causes the peak responses of the deflection to move towards the center of the plate with the skew angle decreasing from 90 to 30 degrees, and it makes the deflection of larger than that of .
6. Conclusions
The governing differential equation of skewed thick plates on an elastic foundation in an oblique coordinate system is formulated for the first time in this paper. This work allows derivation of an analytical solution for any boundary condition, loading condition, and material, also for the first time reported. All response quantities including shear forces, moments, stresses, strains, deflections, and rotation angles can be readily derived from the proposed potential function . The two illustrative examples show that the analytical solutions are in good agreement with those reported in literature and numerical solutions by FEM. Further validation of the analytical solution developed here will be provided by an experimental test.
It is also worth noting that the approach to the governing differential equation and its analytical solution developed in this study can be used for further studies including, but not limited to, continuous plate analysis and dynamic analysis, for which only numerical solutions exist. These studies are in progress now and may contribute additional understanding of the behavior of a structure on an elastic foundation.