Using Elasticity Theory To Exactly Solve The Composite Cantilever Tube Under Transverse Force

The composite thick-walled multilayer tube of cylindrically orthotropic materials has been proved to be very useful for many structures as the main load-bearing structure components.In this paper,the composite cantilever tube under transverse force has been solved without ignoring any stress component.According to decomposing the problem in two times,the original problem has been simplified in a great extent.Besides,the characteristic of stress distribution and deformation,i.e.,the phenomenon of transverse displacement has also been analyzed.The main work of this paper includes:(1)The compatibility equations are derived at first.Then the composite cantilever tube is decomposed into composite tube Ⅰ*which only imposes proportional moment on cross section and composite tube Ⅱ*which only imposes constant force on cross section.Obviously,composite tube Ⅰ*and composite tube Ⅱ*cannot exist in reality.However,we can design composite tube Ⅰ to solve composite tube Ⅰ*,which is equivalent to pure bending tube when the gradient of moment is seen as a kind of loading,as well as composite tube Ⅱ with additional terms like body force and initial deformation to solve composite tube Ⅱ*.(2)Composite tube Ⅰ can be directly solved by pure bending formula.However,composite tube Ⅱ is difficult to separate variables directly because the non-homogeneous terms of the governing differential equations contain both the parameters for main deformation and the parameters for secondary deformation.So,it needs to further decompose it into composite tube Ⅱ(Ⅰ)and composite tube Ⅱ(Ⅱ)according to the characteristics of the function.The depth decomposition belongs to the superposition principle,in which the composite tube Ⅱ(Ⅱ)is a pure bending tube in the secondary direction which can also be solved directly by pure bending formula,while the composite tube Ⅱ(Ⅰ)has the same homogeneous equation as the pure bending problem.So we only need to apply the method of undetermined coefficients to find its special solution.According to the free boundary conditions,continuity conditions and alternative loading conditions of the composite cantilever tube,the combination coefficients and deformation parameters in the general solution can be determined.Then the exact solution in elasticity theory is obtained.(3)In this paper,the characteristics of stresses and displacements are also discussed in quantity.Because the accurate solution has been obtained in this paper,the phase angle of the peak stresses,which corresponds to the circumferential variable,is determined by the location of the radius.Therefore,it indicates that all the stress components in the cross section are no longer symmetrical.Besides,the shear effect,i.e.the displacements of cantilever tube in secondary direction,is also quantitatively analyzed in this paper.It is strictly proved that there is no shear effect in the special composite cantilevered tube,which clearly point out that this phenomenon only happened in the composite cantilever tube with layers including inclined winding angle.(4)The theory in this paper is used to analyze several simple composite cantilever tubes including[0°]、[90°]、[90°/0°]、[45°]、[75°]and[75°/45°],in which the result of composite cantilever tube[0°]is compared with Lekhnitskii’s theory and the results of others are compared with fine finite element methods.The good agreement implies the accuracy and effectiveness of our method.Besides,the alternative winding composite tubes[75°/-75°]and[(75°/-75°)40]which is commonly used in engineering are also analyzed.The property of symmetry is also discussed.