Vibration of plate and shell structures using triangular finite elements

Abstract

Mass and stiffness matrices are obtained for a general-triangle element and for a right-angled-triangle element. Both bending and in-plane actions are considered, although no coupling is assumed, and the matrices relating to bending actions are obtained independently of those relating to in-plane actions. Coupling is introduced, unless all the elements lie in a single plane, when the transformation from local to global co-ordinates is made. In deriving the stiffness matrices assumptions have been made about the form of the stress components within, and on the boundaries of, the element, together with assumptions about the form of the displacement components on the boundary of the element only. The commonly made assumptions in the derivation of stiffness matrices relate to the form of the displacement components not only on the boundary but throughout the element. In order to derive satisfactory mass matrices it is necessary to assume the form of the displacement components throughout the element. For the right-angled-triangle mass matrices these displacement components have been assumed independently of the assumed boundary displacements needed for the stiffness matrix. For the general-triangle mass matrix, however, the displacements throughout the element have been made consistent with the boundary displacements which were needed for the stiffness matrix. Numerical results are given for the first few natural frequencies of a square simply supported slab and a square encastré slab. Comparison with accepted values shows that the finite-element values are accurate, and convergent as the element size is reduced. For the same number of elements it is indicated that general triangles give a more accurate solution than right-angled triangles, probably because of the more satisfactory derivation of the mass matrix for the general triangle. This advantage is offset, however, by the greater computation time required by general triangles. Calculated and experimental frequency values are also given for a single-curvature arch dam of constant thickness. Mode shapes are not given in any of the numerical solutions although they are produced as an integral part of the computer programme.