Jerzy Rakowski

The exact thick arch finite element

Abstract The exact stiffness matrix is derived for a curved beam element with constant curvature. The plane two-node six degree-of-freedom element is considered in which effects of flexural, axial and shear deformations are taken into account. The analytical shape functions describing radial and tangential displacements as well as cross-section rotations are found in the algebraic–trigonometric form. They contain the coupled influences of shear and membrane effects. Based on these shape functions, using the strain energy formula, the stiffness matrix for shear flexible and compressible arch element is formulated. Obviously, this element is completely free of shear and membrane locking effects. The advantage of the elaborated element is its applicability to any combination of geometrical properties of the arch structure, e.g. the depth–length ratio of element. In presented numerical examples the shear and membrane influences on the displacements for various cases of boundary conditions and loading are investigated. The results coincide exactly with the analytical ones obtained for continuous arches.

AN EFFICIENT CURVED BEAM FINITE ELEMENT

The plane two-node curved beam finite element with six degrees of freedom is considered. Knowing the set of 18 exact shape functions their approximation is derived using the expansion of the trigonometric functions in the power series. Unlike the ones commonly used in the FEM analysis the functions suggested by the authors have the coefficients dependent on the geometrical and physical properties of the element. From the strain energy formula the stiffness matrix of the element is determined. It is very simple and can be split into components responsible for bending, shear and axial forces influences on the displacements. The proposed element is totally free of the shear and membrane locking effects. It can be referred to the shear-flexible (parameter d) and compressible (parameter e) systems. Neglecting d or e yields the finite elements in all necessary combinations, i.e. curved Euler–Bernoulli beam or curved Timoshenko beam with or without the membrane effect. Applying the elaborated element in the calculations a very good convergence to the analytical results can be obtained even with a very coarse mesh without the commonly adopted corrections as reduced or selective integration or introduction of the stabilization matrices, additional constraints, etc., for the small depth–length ratio.

DIFFERENCE EQUATION METHOD FOR STATIC PROBLEM OF INFINITE DISCRETE STRUCTURES

Abstract The static problem of infinite, regular systems is considered. The structures are treated as discrete ones. The equilibrium conditions are derived using the FEM formulation. The set of infinite number of equilibrium conditions is replaced by one equivalent difference equation and solved in analytical form.

Solution of stability problem of infinite plate strips

The aim of the paper is to solve a stability problem of infinite plate strips using the finite strip method (FSM). Contrary to well-known solutions for 2D continuous systems the authors present an idea for solving a stability problem of infinite discrete plates.