The purpose of this paper is to provide a general solution of spatial warping curved beams under multiple loads based on the existed theory. The transverse shear deformation and torsion-related warping effects are taken into account. In natural coordinates, equilibrium, geometrical and constitutive equations are established in matrix form. The analytical expressions to the problems can be obtained by solving the matrix equations. The internal forces, stresses, strains, and displacements are calculated along the beam axis. By using these solutions, a plane curved beam subjected to uniform vertical loads and torsions is analyzed. The accuracy and the efficiency of the present theory are demonstrated by comparing between the results with the solution of C.P.Heins.
Curved beams have been applied in many engineering structures such as bridges, highway constructions, long span roof structures and machinery structures, etc. Therefore, many scientists have tried to analyze and solve problems of curved beams by different methods. The majority work in theoretical method aimed to the plane curved beams where deformations include out-of-plane bending and torsion. Analytical solutions of internal forces and deformations were derived (Heins, 1981; Yao, 1989) for plane curved beams with constant curvature under out-of-plane loads, based on traditional mechanics of structures, in which the coupling of flexure and torsion was taken into account. Further, Tufekci and Dogruer (2006) obtained the exact solution of out-of-plane problems of a plane arch with varying curvature and cross section. In order to investigate the reason of in-plane damage happened in curved bridges in recent years, Li, Liu and Zhao (2007) and Li and Zhao (in press) presented the exact solution of the displacement of the plane curved beams subjected to an in-plane load and a changed temperature. Along with the development of the study, some theoretical results for plane curved beams have been applied in the design of curved bridges.