Elastic Mechanics Equations And Its Simplification Of Bi-curvature Beam Based On Tensor Theory

Formerly,researching on bi-curvature beam was based on linear analysis,but,linear model cannot be found in large deformation problem.That makes the results which conclude are incorrect when the nonlinear deformation is ignored.Digger deeper,the finite element analytical of geometry nonlinear must be taken out on bi-curvature beam.In order to simplify the calculation,quadratic term and higher terms are neglected;one term has been taken in it only for the curvature.Moreover,alternative curvature beam uses straight beam usually in the nonlinear finite element analysis of bi-curvature.However,this method did not consider the coupling effect among the deformation of axial,bend and torsion.In the accuracy of engineering limiting,straight beam element alternate curvature beam element in approximate solution.Though,with the developing of compound material used in structure engineering,even the span of the structures is bigger.The question of huge deformation and stability are appearing in nonlinear feature of curvature beam.That makes the theory of curvature should be taken to solve the question.For the sake of researching in the material behavior of bi-curvature beam,the curvature beam element must be established to reflect the characteristic of the beam.For the curvature beam,exist of the initial curvature,the deformation carried by internal force is coupling.So,the neutral layer cannot be confirmed easily.This thesis applies tensor theory to build arbitrary section centric axis of curvature beam.Furthermore,under the condition of the small rotation,the warping before and after deformation in the cross section of the curvature beam can be ignored.Plain section and straight normal assumption which assume the cross section of the bi-curvature after deformation is still maintaining plain.In this paper,the basic mechanics equations of bi-curvature beam derivation are totally based on plain section and straight normal assumption.The main work in this thesis has several aspects as following:(1)Arbitrary geometry equations of bi-curvature beam in the plain section and straight normal assumption are given based on tensor theory in this paper,including the contact of curvature and angle,angle and displacement,strain and displacement of bi-curvature beam.The Green strain tensor is deduced.Finally,according to the coordinate transformation relationship,the strain-displacement relationship of bi-curvature beam is got.(2)Got the differential balance equation of bi-curvature beam under the force of distribution and the force coupling of distribution.The expression between the sectional internal force and displacement of cross section is derived based on differential balance equation under the boundary conditions.For the internal force-displacement equation of bi-curvature beam in general case and ignore the impact of higher curvature,the elastic mechanics equations that without considering shear deformation of bi-curvature beam are deduced.(3)The unit load method for computing the displacements of curvature beam is put forward.The main vector and the main moment on arbitrary section are derived.The displacements of arbitrary point for curvature beam are calculated and compared with close circle of helical spring.The universal expressions are given.