ELASTIC BUCKLING OF ARCHES BY FINITE-ELEMENT METHOD

A procedure for the computation of the elastic buckling load of arches is presented. The arch is represented by beam finite elements curved in one plane but deformable in three dimensional space. The curved axis of the element is represented by a fourth-order polynomial. The displacement functions are approximated by cubic polynomials. The expressions for the generalized strains include the linear and quadratic terms of the displacements. By using these functions the expression for the strain energy of an element is derived. This expression consists of three parts: the quadratic, cubic, and quartic terms. Proper differentiation of these expressions yields the linear stiffness matrix (K) and the incremental stiffness matrices (N1 and N2) of the element.;Assuming that the system is elastic and conservative, the equilibrium equation is obtained from the first variation of the potential energy. This represents a set of nonlinear algebraic equations. The equation governing the linear incremental behavior is obtained from the second variation of the potential energy. A basis for obtaining the critical load of a structural system is the vanishing of the load increment vector corresponding to a change in the displacement vector. To avoid dealing with nonlinear equations, an estimate of the buckling load is obtained by assuming that the displacement increase linearly with the applied load until buckling occurs. This leads to a quadratic eigenvalue problem for the buckling loads and their associated buckling modes. Assuming that at buckling the displacements are sufficiently small the quadratic eigenproblem reduces to a linear one.;The quadratic eigenproblem is solved by the determinant search method in conjunction with the modified regula falsi iteration technique. Inverse vector iteration is used for the solution of the linear problem.;Eigenvalue problems are also formulated for the case of tilted loads (for example, due to the horizontal rigidity of the deck of an arch bridge). In addition, the buckling problem involving interactions between horizontal transverse loading and vertical in-plane loading is formulated.;A computer program was prepared for the implementation of the linear equilibrium solution and the buckling load solutions. Numerical results were obtained involving arch ribs with in-plane and out-of-plane behavior. The influence of the number of elements on the accuracy of the results was investigated by considering both linear equilibrium problems and buckling problems. The types of buckling problems considered are: in-plane, out-of-plane, tilted loads, and the effect of out-of-plane horizontal loads on the in-plane buckling load. Good agreement was indicated by comparisons of the first three types of problems with existing analytical solutions based on the classical buckling theory. Results for the last type of problems indicated that while a small out-of-plane horizontal load may have little effect on the in-plane buckling load, the latter decreases rapidly with increases in the horizontal load.