The bending of elastic circular ring of non-homogeneous and variable cross section under the actions of arbitrary loads

On the basis of the stepped reduction method suggested in [1], we investigate the problem of the bending of elastic circular ring of non-homogeneous and variable cross section under the actions of arbitrary loads. The general solution of this problem is obtained so that it can be used for the calculations of strength and rigidity of practical problems such as arch, tunnel etc. In order to examine results of this paper and explain the application of this new method, an example is brought out at the end of this paper. Circular ring and arch are commonly used structures in engineering. Timoshenko, S.^[2], Barber, J. R.^[3], Tsumura Rimitsu^[4] et al. have studied these problems of bending, but, so far as we know, it has been solely restricted to the general solution of homogeneous uniform cross section ring. The only known solution for the problems with variable cross section ones has been solely restricted to the solution of special case of flexural rigidity in linear function of coordinates. On account of fundamental equations of the non-homogeneous variable cross section problem being variable coefficients, it is very difficult to solve them. In this paper, we use the stepped reduction method suggested in [1] to transform the variable coefficient differential equation into equivalent constant coefficient one. After introducing virtual internal forces, we obtain general solution of an elastic circular ring with non-homogeneity and variable cross section under the actions of arbitrary loads.     

Optimal elastic design of beams

According to the principle of minimum complementary energy a mathematical statement of optimal strength design problem for elastic beams is formulated in this research, which is an extremum problem of functionals with equality and inequality constraints. Further the application of the Lagrangian multiplier method yields the necessary conditions for extrema. A set of relations that must be satisfied for the optimal solution follows afterwards. This set of relations can be used to verify the optimality of a uniform strength design or any feasible elastic design. An iterative numerical method to find the optimal solution when the uniform strength design is not optimal is also presented in this paper.Project supported by the Science and Technics Fund of the Chinese National Educational Committee.     

Nonlinear stability of spherical shallow shell under compound loads

Using a semi-analytical method,the nonlinear stability of a spherical shallow shellunder centrally distributed and concentrated loads is investigated in this paper.The longermanual calculation has been avoided when finding the approximate solution,and the P-Wmcharacteristic relation can be given analytically.     

非均匀变截面弹性圆环在任意载荷下的弯曲问题

本文在等刚度弹性圆环的初参数公式的基础上,利用[2]提出的阶梯折算法,进一步研究非均匀变截面弹性圆环的弯曲,得到了这类问题的通解,应当指出,这组通解对非均匀变截面圆柱拱的相应问题也是适用的.为验证所得的公式并说明这种方法的应用,文末给出了示例并进行了求解,圆环、圆拱是工程上经常采用的结构,它们的弯曲,Timoshenko,S.[5],Barber,J.R.[3],Roark,R J[4],津村利光[6]等曾作过很多研究.然而,迄今只求得了均匀材料、等截面圆环的通解。对变截面问题,仅仅求得了抗弯刚度是坐标的线性函数这一特殊情况的解.由于非均匀变截面问题常常导出变系数微分方程,它们的求解遇到很大的数学困难.本文通过阶梯折算法把非均匀变截面弹性圆环弯曲问题的变系数微分方程转化成一等效的等刚度圆环弯曲的常系数微分方程.为保证内力连续,引入虚拟内力,并以[1]导出的初参数公式为影响函数,通过积分构造出了非齐次解,从而求得了非均匀变截面弹性圆环弯曲问题的通解.     

Nonlinear dynamical stability analysis of the circular three-dimensional frame

The three-dimensional frame is simplified into flat plate by the method of quasiplate. The nonlinear relationships between the surface strain and the midst plane displacement are established. According to the thin plate nonlinear dynamical theory, the nonlinear dynamical equations of three-dimensional frame in the orthogonal coordinates system are obtained. Then the equations are translated into the axial symmetry nonlinear dynamical equations in the polar coordinates system. Some dimensionless quantities different from the plate of uniform thickness are introduced under the boundary conditions of fixed edges, then these fundamental equations are simplified with these dimensionless quantities. A cubic nonlinear vibration equation is obtained with the method of Galerkin. The stability and bifurcation of the circular three-dimensional frame are studied under the condition of without outer motivation. The contingent chaotic vibration of the three-dimensional frame is studied with the method of Melnikov. Some phase figures of contingent chaotic vibration are plotted with digital artificial method.     

非线性弹性元件—碟形弹簧,膜片弹簧的力学与数学模型的研究

本文提出了研究碟形弹簧和膜片弹簧的新的力学模型。用梁的大变形理论与一般轴对称壳体Reissnef有限变形理论,建立了符合力学模型和实验资料的新的数学模型——非线性边值问题。应用Green函数方法,将边值问题化为等价的非线性积分方程组,便于理论研究和求解。     

扁薄锥壳非对称大变形问题

本文用双参数摄动法研究了扁锥壳非对称大变形问题，求得了在线性载荷作用下的扁锥壳变形的三次近似解析解并绘出了摄动点的挠度与载荷的特征曲线·应用本文方法还可对其它板壳的非轴对称大变形问题进行讨论·本文通过算例对平板及不同初挠度的扁锥壳大挠度变形进行了讨论·     

Nonlinear vibration of thin shallow conic shells under combined action of peripheral moment and transverse loads

IntroductionTheproblemofnonlinearvibrationisveryimportantnotonlyintheorybutalsoinapplication .Theconicshellplaysanimportantroleinarchitecture ,navigation ,spaceflightandwasusedastheelasticcomponentoftheinstrumentwidely .Thenonlinearvibrationofshellisacomplicatedproblemanditisdifficulttosolveinmechanicsandmathematics .Itismoredifficultiftheactionofouterloads (staticload ,thermalload ,magnetic)istakenintoaccount[1～7].Thereisdifferentinterestineventhenonlinearvibrationundertheactionofstaticload…     

Free vibration of nonlinear deformation ring-and stringer-stiffened nonuniform cylindrical shell

Cylindrical shells stiffened with rings and stringers are used in many structural applications such as pipes conveying fluids or gases and acrospace. In this paper, the general solution is obtained for free vibration of nonlinear deformation ring-and stringer-stiffened cylindrical shell with arbitrary boundary condition by step reduction method^[1]. Finally, it is only necessary to solve a nonlinear algebraic equation. This equation is expressed as an analytic form. Its convergence is proved. Three numerical examples are given at the end of the paper which indicate that satisfactory results can be obtained by step reduction method. Project Supported by the Science and Technic Fund of the National Education Committee.     

扁薄球壳非对称大变形问题

本文用修正选代法研究了扁球壳非对称大变形问题，求得了在线性液体载荷作用下的扁球壳变形的二次近似解析解并绘出了摄动点的挠度与载荷的特征曲线族．应用本文方法还可对其他板壳的非轴对称大变形问题进行讨论．本文通过算例对平板及不同初挠度的扁球壳大挠度变形进行了讨论．