Finite element model of bending plate with considering shear deformations

Abstract

The arbitrary quadrangular finite element of plate is proposed. Finite element is based on piecewise constant approximations of moments and shear forces. The values of moments and shear forces at the finite element mesh nodes are used as unknown parameters. To construct the solution, the minimum additional energy principle is used. Using the possible displacements principle, algebraic equilibrium equations are formed for the finite element mesh nodes. The vertical displacement and rotation angles along the coordinate axes are considered as possible. The resulting equilibrium equations are included in the additional energy functional using Lagrange multipliers. Lagrange multipliers are vertical displacements and rotation angles values of the finite element mesh nodes. The use of piecewise constant approximations of moments and shear forces allows one to obtain a block-diagonal structure of the flexibility matrix. The solution is reduced to linear algebraic equations system for Lagrange multipliers. The proposed finite element allows one to consider shear deformations regardless of the plate thickness ratio to its dimensions. There is no “locking” effect when thin plates are calculating. Comparison of the calculating oblique plate results with the results of calculations using other programs is performed. It is shown that when the finite element mesh is refined, the displacements values tend to exact values from above.