A novel weak-form plane frame quadrature element and its application to bridge analysis

A novel weak-form plane frame quadrature element (WPFQE) is developed. A complex structure can be divided into several non-homogeneous (or homogeneous) elements with large size. The Chebyshev–Lobatto differential quadrature dealing with differentiation and the Gauss–Lobatto quadrature dealing with integration have the same integration points including the ends of each element. Based on the energy variational principle, the diagonal mass matrix and positive definite real symmetric stiffness matrix for the element can be obtained. The unknown quantities at the internal quadrature points of the element can be represented by those at the endpoints of the element based on the principle of static equal effect anterior to the development of global matrices, which significantly reduces the sizes of modified element matrices while maintaining sufficient accuracy, thereby obtaining the small-sized global matrices. Assembly of elements can be performed efficiently just as that in the finite element (FE) method. In the case study, the static and dynamic performances of a three-span high-pier rigid-frame bridge with variable cross-section are studied by applying the proposed method. The results indicate that the orders of structural matrices and calculation time obtained by the WPFQE method are much smaller than those of the FE method.