Superposition Principle For Composite Cantilever Tube As Well As Its Shear Effect

In this paper,a complete stress field is employed for the general inclined-wound composite cantilever tube and the corresponding complete displacement field is derived.The superposition is then proposed for the inclined-wound composite cantilever tube so that the shear effect is discovered.Due to the fact that the complete stress field as well as its complete displacement field include sufficient information and necessary unknown functions fur the analysis of compositse cantilever tubes,and the superposition not only dramatically simplifies the procedures of analysis but also provide more intuitive understanding for the influence of different loading,they are the key to find out the shear effect.The main work of this paper includes:(1)Complete stress field as well as its complete displacement fieldAccording to the loadings of inclined-wound composite cantilever tube,i.e.the moment loading proportional to the axis coordinates and the shear loading unvaried along the axis,a comp lete stress field for the inclined-wound composite cantilever tube is employed.All the components of this stress field include both part ? and part ?.The part? stresses are related to the moment loading so they are proportional to the axis coordinates while the part ? stresses are related to the shear loading so they are unvaried along the axis.In addition,the necessity of the construction of this stress field is proved in theory,namely,all the stress components must include both part ? and part ?.Based on,the general formula of the complete displacement field is derived for inclined-wound composite cantilever tube,in which the part ? displacement corresponds to the part ?stress as well as the part ? strain while the part ? displacement to the part ? stress as well as the part ? strain.Since the complete stress field as well as its corresponding complete displacement field includes sufficient information and necessary unknown functions for the composite cantilever tube,they can be used to describe more effectively the tube.(2)Superposition PrincipleThe superposition principle for the analysis of composite cantilever tube is proposed by carefully examining the governing equations of composite cantilever tube,including equilibrium equation,physical equation and geometric equation,free boundary condition on inner surface and on outer surface,continuity condition between layers and loading condition over cross section.The cantilever tube problem is divided into a bending-like sub-problem and a shear-like sub-problem.The bending-like sub-problem corresponds to stress ? as well as its displacement ? so that it is related to moment loading while the shear-like sub-problem corresponds to stress ? as well as its displacement ? so that it is related to shear loading.However,the additional body-like force constructed by stress gradient is necessary for the shear-like sub-problem,so the stress ? as well as its displacement ? is affected by stress ? as well as its displacement ?.With the help of superposition principle,the complex composite cantilever tube problem is divided into two simpler sub-problems,which not only the analysis cost is greatly reduced,but also the understanding for the clearer influence between different loading is provided.(3)Shear effectBy using the unified parameter method(Zhang&etc.,2014)the solution for the pure bending problem of composite tube subjected to moment gradient can be obtained.This solution is the gradient of stress I as well as of its local displacement ?,namely,is the stress gradient as well as its local displacement of the composite cantilever tube.The body-like force constructed by the stress gradient is substituted into the equilibrium equation of shear-like problem.According to its inhomogeneous terms,the circumferential triangular function for stress ? can be derived.Since these triangular functions are just opposite to the circumferential triangular function for stress I,the symmetrical axis for the complete stress field over the cross section rotates at some angle even though with good symmetry.At the same time,the local displacement gradient is substituted into the geometric equation of shear-like problem.According to its inhomogeneous terms,the circumferential trigonometric function for the gradient of the local displacement ? can be obtained.This gradient is substituted into the corresponding displacement II and the deviation of the axis of the cantilever tube after deformation is discovered.This phenomenon of the rotation of the stress symmetry axis over cross section and the deviation of the axis of the cantilever tube after deformation is the comprehensive effect from the shear load ing and moment load ing in the inclined-wound composite cantilever tube,which is the special shear effect of the inclined-wound composite cantilever tube.(4)Numerical examplesSeveral numerical examples of the simulation by using the fine finite element method have been provided for a series of cormposite cantilever tubes,including the single-layer special composite cantilever tubes[0°]and[90°],the single-layer general inclined-wound composite cantilever tubes[0°],[45°],and[75°]as well as the multi-layer composite cantilever tubes[0/90]and[900/450]and etc.The numerical results of special composite cantilever tubes[0°]and[90°]are essentially in agreement with those by Lekhnitskii,verifying the feasibility of the fine finite element numerical simulation method in this paper.In addition,the numerical results of the other single-layer as well as the multi-layer composite cantilever tubes demonstrate the Validation of the special shear effect discovered in this paper.