Advances in Civil Engineering

Volume 2019, Article ID 1560171, 12 pages

https://doi.org/10.1155/2019/1560171

## Deformation and Mechanical Behaviors of SCSF and CCCF Rectangular Thin Plates Loaded by Hydrostatic Pressure

^{1}School of Civil Engineering and Architecture, Xi’an University of Technology, 5 South Jinhua Road, Xi’an, Shaanxi 710048, China^{2}State Key Laboratory of Eco-Hydraulics in Northwest Arid Region, Xi’an University of Technology, No. 5 South Jinhua Road, Xi’an, Shaanxi 710048, China

Correspondence should be addressed to Faning Dang; nc.ude.tuax.liam@nfgnad

Received 28 December 2018; Accepted 6 February 2019; Published 7 March 2019

Academic Editor: Harry Far

Copyright © 2019 Jun Gao 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

Elastic rectangular thin plate problems are very important both in theoretical research and engineering applications. Based on this, the flexural deformation functions of the rectangular thin plates with two opposite edges simply supported, one edge clamped and one edge free (SCSF) and three edges clamped and one edge free (CCCF), loaded by hydrostatic pressure are determined by single trigonometric series. And the flexural deformation functions are solved via the principle of minimum potential energy. Next, the internal force and stress functions of rectangular thin plates with two boundary conditions are obtained based on the small deflection bending theory of thin plates. The dimensionless deflection, dimensionless internal force, and dimensionless stress functions of the rectangular thin plates are established as well. The analytic solution in this paper is validated by the finite element method. Finally, the influence of aspect ratio *λ* and Poisson’s ratio *μ* on the deformation and mechanical behaviors of the rectangle thin plates is analyzed in this paper. This research can provide references for the plane water gate problem in seaports and channels.

#### 1. Introduction

Bending of rectangular thin plates has been heavily researched and gained great achievements. Numerical and analytical methods are two research methods that are often used for the analysis of thin plate problems. It is well known that many effective numerical methods have been developed in recent years. Representative methods include the finite element method (FEM) [1, 2], the finite difference method [3–5], the finite strip method [6], the meshless method [7], the spline element method [8], etc. These numerical methods normally meet the engineering requirements with acceptable errors and are greatly applied in practice. Meanwhile, analytical solutions are regarded as the benchmarks for verification of various numerical methods and have been investigated by many researchers. The problem of a rectangular thin plate is first given by Dixon [9]. The solution of double trigonometric series of a rectangular thin plate with four edges simply supported (SSSS) under arbitrary loading has been proposed by Navier (Navier’s solution). The solution of single trigonometric series of a rectangular thin plate with two opposite edges simply supported and other two opposite edges free (SFSF) under transverse loading has been given by Levy (Levy’s solution). Different methods are used to analyze the rectangular thin plate problem under different boundary conditions and loading, such as the Fourier series method, the Rayleigh–Ritz method, the superposition method, the semi-inverse method, the symplectic geometry method, the integral-transform method, etc. The bending problem of a square plate with two adjacent edges clamped and the others either simply supported or free (CCSS or CCFF) under uniform loading has been investigated by Huang and Conway [10]. Many exact solutions for the bending problem of elastic rectangular thin plates have been obtained by Timoshenko and Woinwsky-Krieger [11]. The bending problem of a rectangular thin plate with two opposite edges simply supported has been analyzed by Hutchinson [12]. The buckling problem of clamped rectangular plates with different aspect ratios has been solved by El-Bayoumy using extended Kantorovich method [13]. The problem of an isotropic rectangular thin plate with four edges clamped has been given by Imrak and Gerdemeli [14]. The accurate solution for a rectangular thin plate with two adjacent edges clamped and the others free (CCFF) is proposed by Chang [15], and the solution is yielded by superposing six known solutions. The symplectic geometry method is employed by Lim et al. [16] to investigate the bending problem of a rectangular thin plate with two opposite edges simply supported and the others free (SFSF). Moreover, the symplectic geometry method is also employed by Zhong and Li [17] and Liu and Li [18] to solve the deflection function and bending moment of a rectangular thin plate with four edges clamped (CCCC) under arbitrary loading. The analytic bending solutions of free rectangular thin plates resting on elastic foundations are obtained by Li et al. [19] via a new accurate symplectic superposition method. Moreover, Li et al. [20] extend the approach to the free vibration problems of the same plates and obtain the analytic solutions which cannot be obtained by the conventional symplectic approach. Besides, the analytic bending solutions of rectangular thin plates with a corner point-supported, its adjacent corner free, and their opposite edge clamped or simply supported are obtained by Li et al. [21] via the superposition method in the symplectic space. The method of symplectic geometry is more reasonable than traditional semi-inverse solution. Khan et al. [22] employed the variation method to obtain a higher approximate solution for a rectangular thin plate with four edges simply supported (SSSS) under uniform loading. Based on this, the bending problem of rectangular thin plates has also been investigated by some other researchers under different boundary conditions and loading [23–26]. However, it is still difficult to obtain the exact solution through solving the differential equation for the bending problem of the rectangular thin plate with certain boundary conditions. Thus, many exact solutions for the bending of thin plates are obtained with simple boundary conditions and transverse loading, such as the bending of thin plates with four edges clamped or simply supported, two opposite edges clamped or simply supported, three edges clamped or simply supported under uniform loading, and transverse loading. Most approximate solutions have been obtained for the bending problem of rectangular thin plates with relatively complicated boundary conditions and transverse loading.

In this paper, the flexural deformation functions of two types of rectangular thin plates (two opposite edges simply supported, one edge clamped and one edge free (SCSF) and three edges clamped and one edge free (CCCF)) loaded by hydrostatic pressure are established with single trigonometric series. The flexural deformation functions are solved using the principle of minimum potential energy. Internal force and stress functions of the rectangular thin plates under the two boundary conditions are obtained using the small deflection bending theory of thin plates. The dimensionless deflection, dimensionless internal force, and dimensionless stress functions of rectangular thin plates under the two boundary conditions are established in this paper. Moreover, the influence of aspect ratio and Poisson’s ratio on the deformation and mechanical characteristics of rectangular thin plates under the two boundary conditions is analyzed in this paper. This research can provide references for the plane water gate problem of seaports and channels.

#### 2. Deflection and Internal Force Function of the SCSF Rectangular Thin Plate

##### 2.1. Bending Equation and Boundary Condition of the SCSF Rectangular Thin Plate

The hydrostatic pressure is loaded on the surface of the rectangular thin plate. The width is *a* along the *x* axis. The height is *b* along the *y* axis. The thickness is *δ* along the *z* axis. The dimensions and load condition of the rectangular thin plate are shown in Figure 1.