Semi-analytical Analysis Of Static Problem Of Composite Structures With Variable Cross-section

Beam and column are the most common structure in engineering application. A lot of structures can be simply considered as beams or columns in many fields such as aerospace, mechanical, civil engineering and so on, the cross-section of which remain the same during the structures. However,with the development of society, these structures can not meet the various requirements of the structure in the engineering application,so the variable cross-section structure is gradually be used. The stepped structure is a common variable cross-section structure and is widely used in different fields which strength and rigidity present a step change. The mechanical analysis of isometric structure is easy because of its simple structure shape. However, due to the irregular geometry of the stepped structure, the corresponding mechanical analysis becomes complicated, so the semi-analytical method is used to obtain the approximate solution. The state space differential quadrature method is an efficient method to solve the problem. In this paper, the static problem of composite structures with variable cross-section is analyzed with this method. The main contents are as follows:Firstly, The semi-analytical analysis of the static problem of the rectangular plane of non - uniform boundary is carried out. Based on the two-dimensional elastic mechanics theory, the state equation of the structure is obtained, and then the control equation is obtained according to the end bearing conditions and the boundary conditions of the upper and lower surfaces, and then the equation can be solved. The horizontal displacement and deflection of the upper and lower surfaces as well as the normal stress and shear stress of the upper surface are calculated and then compared with those calculated by the finite element method.Secondly, the semi-analytical analysis of the static problem of the variable -section ladder - type combined rectangular domain is carried out. The upper and lower part of the structure are selected as the research object respectively, and then the state equation are obtained. Then the respective control equation can be obtained according to the end bearing condition and the loading conditions of the upper and lower surfaces. Under the continuous condition of the upper and lower structures, the whole control equation of the structure can be obtained and the problem can be solved. The stress distribution of the contact part of the upper and lower structures are calculated,besides, the distribution of stress and displacement in the horizontal and the thickness direction in the upper structure are calculated.Then, based on the two - dimensional elasticity theory in polar coordinate system,the static problem of non - uniform boundary plane annular domain is analyzed semi -analytically. The control equation is obtained by SS-DQM and the solution is solved.The displacement of the upper and lower surfaces of the structure and the normal stress and shear stress of the upper surface of the structure are calculated when the annular domain is degraded to a rectangular domain,and the results are compared with those of the rectangular field in the paper. Then, the deformation and stress of the annular domain under the displacement of both ends are also calculated and analyzed.Finally, The semi-analytical analysis of the static problem of the variable -section ladder - type combined annular domain is carried out. The upper and lower part of the structure is selected respectively as the object of study, and their control equations are obtained in the polar coordinate system. Then it can be solved according to the upper and lower structure of the continuous conditions. The stress of the contact of the upper and lower structures are calculated, besides, the stress and displacement distribution in the upper structure of the composite structure with variable ladder-type cross-section are also calculated.