Stability of a highly elastic rectangular plate with clamped-free edges under uniaxial compression

Abstract

The symmetrical buckling modes of a rectangular Kirchhoff plate with two clamped and two free parallel faces (CFCF-plate) under the action of a distributed compressive load applied to the clamped faces have been studied. The function of plate deflections due to loss of stability is represented by two hyperbolic-trigonometric series with indefinite coefficients which are found when all conditions of the boundary value problem are exactly satisfied. The problem is reduced to solving a homogeneous infinite system of linear algebraic equations with respect to one sequence of uncertain coefficients which contain the desired critical load as a parameter. To obtain nontrivial solutions, the determinant of the system must be equal to zero. This eigenvalue problem has countless solutions. It is proposed to find non-trivial solutions of the system using the method of successive approximations with enumeration of the load parameter. Using computer calculations, the first four critical loads (including the Euler load) were found applied to the clamped parallel faces of a square plate and giving symmetrical forms of buckling. The influence on the accuracy of calculations of the number of terms retained in the series and the number of iterations is studied. 3D images of the found buckling modes are presented. A comparison with known solutions is provided. The results obtained can be used in the design of various flat rectangular elements in microelectronics and nanotechnology.

Keywords

• rectangular plate
• cfcf-plate
• critical loads
• hyperbolic-trigonometric series