The exact thick arch finite element
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 algebraictrigonometric 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 depthlength 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.
thick arch finite element stiffness matrix shape function