Complex equations of flexible circular ring shells overall-bending in a meridian plane and general solution for the slender ring shells

ＩｎｔｒｏｄｕｃｔｉｏｎＥｑｕａｔｉｏｎｓｆｏｒｃｉｒｃｕｌａｒｒｉｎｇｓｈｅｌｌｓａｒｅｄｉｆｆｉｃｕｌｔｔｏｓｏｌｖｅ．Ｔｈｅｒｅｓｅａｒｃｈｏｆｔｈｉｓｐｒｏｂｌｅｍｓｔａｒｔｅｄａｔｔｈｅｂｅｇｉｎｎｉｎｇｏｆｔｈｉｓｃｅｎｔｕｒｙ．Ｉｎｔｈｅｌａｔｅ１９７０ｓａｎｄｅａｒｌｙ１９８０ｓ,Ｗ．Ｚ．Ｃｈｉｅｎ（１９７９,１９８０,１９８１）［１～３］ｒｅｂｕｉｌｔｔｈｅｃｏｍｐｌｅｘｅｑｕａｔｉｏｎｓｏｆａｘｉｓ＿ｓｙｍｍｅｔｒｉｃａｌｌｙｌｏａｄｅｄｒｉｎｇｓｈｅｌｌｓｐｒｅｓｅｎｔｅｄ…     

用初参数法解C型波纹管在子午面内整体弯曲

波纹管是一种呈波纹状的管形薄壁结构,作为弹性敏感元件、密封元件和位移补偿器,在现代工业、国防和民用设备中的应用日益广泛.波纹管的计算,特别是在非轴对称受力和变形方面亟待完善.本文讨论了C型波纹管在子午面内整体弯曲的应力分布和端面角位移.从柔性圆环壳在子午面内整体弯曲的复变量方程出发,化边值问题为初值问题,用S.Gill法,得到了该波纹管的数值解,适用于0＜α＜1的范围.通过3个例子,将本文的数值解和已有的细环壳一般解(解析解)作了比较.结果表明,两者基本一致,并且,当环壳较细(α较小)时,两种计算结果更为接近.     

FINITE ELEMENT DISPLACEMENT PERTURBATION METHOD FOR GEOMETRIC NONLINEAR BEHAVIORS OF SHELLS OF REVOLUTION OVERALL BEDING IN A MERIDIONAL PLANE AND APPLICATION TO BELLOW (Ⅱ)

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｅｑｕａｔｉｏｎｓａｎｄｔｈｅｓｏｌｕｔｉｏｎｐｒｏｃｅｄｕｒｅｓｏｆｔｈｅｆｉｎｉｔｅ＿ｅｌｅｍｅｎｔ＿ｄｉｓｐｌａｃｅｍｅｎｔ＿ｐｅｒｔｕｒｂａｔｉｏｎｍｅｔｈｏｄ (ＦＥＤＰＭ)ｆｏｒｔｈｅｇｅｏｍｅｔｒｉｃｎｏｎｌｉｎｅａｒｂｅｈａｖｉｏｒｓｏｆｓｈｅｌｌｓｏｆｒｅｖｏｌｕｔｉｏｎｓｕｂｊｅｃｔｅｄｔｏｐｕｒｅｂｅｎｄｉｎｇｍｏｍｅｎｔｓｏｒｌａｔｅｒａｌｆｏｒｃｅｓｉｎｏｎｅｏｆｔｈｅｉｒｍｅｒｉｄｉｏｎａｌｐｌａｎｅｓｗｅｒｅｐｒｅｓｅｎｔｅｄｉｎ (Ⅰ ) .…     

FINITE ELEMENT DISPLACEMENT PERTURBATION METHOD FOR GEOMETRIC NONLINEAR BEHAVIORS OF SHELLS OF REVOLUTION OVERALL BEDING IN A MERIDIONAL PLANE AND APPLICATION TO BELLOW (Ⅱ)

The finite-element-displacement-perturbation method (FEDPM)for the geometric nonlinear behaviors of shells of revolution subjected to pure bending moments or lateral forces in one of their meridional planes (Ⅰ) was employed to calculate the stress distributions and the stiffness of the bellows. Firstly, by applying the first-order perturbation solution (the linear solution)of the FEDPM to the bellows, the obtained results were compared with those of the general solution and the initial parameter integration solution proposed by the present authors earlier, as well as of the experiments and the FEA by others.It is shown that the FEDPM is with good precision and reliability, and as it was pointed out in (Ⅰ) the abrupt changes of the meridian curvature of bellows would not affect the use of the usual straight element. Then the nonlinear behaviors of the bellows were discussed. As expected, the nonlinear effects mainly come from the bellows ring plate,and the wider the ring plate is, the stronger the nonlinear effects are. Contrarily, the vanishing of the ring plate, like the C-shaped bellows, the nonlinear effects almost vanish. In addition, when the pure bending moments act on the bellows, each convolution has the same stress distributions calculated by the linear solution and other linear theories, but by the present nonlinear solution they vary with respect to the convolutions of the bellows. Yet for most bellows, the linear solutions are valid in practice.     

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.     

薄板理论的正交关系及其变分原理

利用平面弹性与板弯曲的相似性理论，将弹性力学新正交关系中构造对偶向量的思路推广到 各向同性薄板弹性弯曲问题，由混合变量求解法直接得到对偶微分方程并推导了对应的变分 原理. 所导出的对偶微分矩阵具有主对角子矩阵为零矩阵的特点. 发现了两个独立的、对称 的正交关系，利用薄板弹性弯曲理论的积分形式证明了这种正交关系的成立. 在恰当选择对 偶向量后，弹性力学的新正交关系可以推广到各向同性薄板弹性弯曲理论.     

GENERAL SOLUTION OF THE OVERALL BENDING OF FLEXIBLE CIRCULAR RING SHELLS WITH MODERATELY SLENDER RATIO AND APPLICATIONS TO THE BELLOWS (Ⅱ)- CALCULATION FOR OMEGA-SHAPED BELLOWS

ＩｎｔｒｏｄｕｃｔｉｏｎＢｅｌｌｏｗｓ[1,2 ]ｉｓａｓｈｅｌｌｏｆｒｅｖｏｌｕｔｉｏｎｗｉｔｈｗｒｉｎｋｌｅｄｍｅｒｉｄｉａｎ ,ｕｓｕａｌｌｙｎａｍｅｄａｆｔｅｒｉｔｓｍｅｒｉｄｉｏｎａｌｓｈａｐｅｉｎａｐｅｒｉｏｄｏｒｉｔｓｐｒｏｆｉｌｅｏｆａｃｏｎｖｏｌｕｔｉｏｎ ,ｓｕｃｈａｓΩ＿ｓｈａｐｅｄｂｅｌｌｏｗｓ,Ｃ＿ｓｈａｐｅｄｂｅｌｌｏｗｓ,Ｕ＿ｓｈａｐｅｄｂｅｌｌｏｗｓａｎｄｓｏｏｎ .Ｔｈｅｙａｒｅｗｉｄｅｌｙｕｓｅｄｉｎｍａｎｙｆｉｅｌｄｓ,ｆｏｒｅｘａｍｐｌｅ ,ｔｈｅｅｌａｓｔｉｃｓｅｎｓｅ…     

微结构非局部效应的多重性模型及在微梁弯曲中的应用

基于Eringen非局部弹性理论,直接利用逐次逼近法推导了非局部应力场的精确表达,该精确的非局部应力可具体表示为一个无穷级数的形式. 然后以微梁的横向弯曲和纯弯曲变形为例,建立平衡方程并求解及分析了挠度受非局部效应的影响. 结果表明：根据所取非局部小尺度参数大小的不同,非局部微梁的弯曲挠度可低于也可以高于经典力学下的挠度,非局部效应的增大可提高亦可降低结构的抗弯刚度. 本文结果证明了Wang以及Lim等人分别提出的两种相反的非局部模型的各自正确性. 同时首次发现,弯曲挠度随着非局部效应的增大而上下波动且存在若干跳跃点,挠度是非局部小尺度参数的非单调函数,研究同时给出了一种确定材料非局部常数的建议途径.     

GENERAL SOLUTION OF THE OVERALL BENDING OF FLEXIBLE CIRCULAR RING SHELLS WITH MODERATELY SLENDER RATIO AND APPLICATIONS TO THE BELLOWS (Ⅱ)- CALCULATION FOR OMEGA-SHAPED BELLOWS

is one of the applications of (Ⅰ), in which the angular stiffness, the lateral stiffness and the corresponding stress distributions of Omega-shaped bellows were calculated, and the present results were compared with those of the other theories and experiments. It is shown that the non-homogeneous solution of (Ⅰ) can solve the pure bending problem of the bellows by itself, and be more effective than by the theory of slender ring shells; but if a lateral slide of the bellows support exists the non-homogeneous solution will no longer entirely satisfy the boundary conditions of the problem, in this case the homogeneous solution of (Ⅰ) should be included, that is to say, the full solution of (Ⅰ) can meet all the requirements.     

GENERAL SOLUTION OF THE OVERALL BENDING OF FLEXIBLE CIRCULAR RING SHELLS WITH MODERATELY SLENDER RATIO AND APPLICATIONS TO THE BELLOWS (Ⅰ)-GOVERNING EQUATION AND GENERAL SOLUTION

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｔｈｅｏｒｙｏｆｆｌｅｘｉｂｌｅｓｈｅｌｌｓ,ｅｓｔａｂｌｉｓｈｅｄｂｙＥ .Ｌ .Ａｘｅｌｒａｄ[1,2 ]( 1 976,1 987) ,ｉｓａｓｉｍｐｌｉｆｉｅｄｔｈｅｏｒｙｆｏｒｔｈｉｎｓｈｅｌｌｓ,ｉｎｗｈｉｃｈｓｈｅｌｌｓｏｆｒｅｖｏｌｕｔｉｏｎａｒｅｆｏｒｍｕｌａｔｅｄｉｎｍｅｒｉｄｉｏｎａｌｄｉｒｅｃｔｉｏｎｂｙｂｅｎｄｉｎｇｔｈｅｏｒｙｏｆｔｈｅｓｈｅｌｌｓａｎｄｂｙｔｈｅｍｅｍｂｒａｎｅｔｈｅｏｒｙｉｎｔｈｅｃｉｒｃｕｍｆｅｒｅｎｔｉａｌｄｉｒｅｃｔｉｏｎ .ＴｈｅＡｘｅｌｒ…     

旋转薄壳转点频段的轴对称振动解

用渐近方法求解了含转点频段的旋转薄壳自由振动方程,重新定义了第一和第二类广义相关函数,求得了一致有效解.该解的新颖之处在于具有对称的耦合结构,即转点一侧的奇异无矩解在另一侧含有快变弯曲解成分,同时转点一侧的1个弯曲解在另一侧含有慢变无矩解成分.数值计算表明,所得一致有效解与有限元计算结果吻合.     

GENERAL SOLUTION OF THE OVERALL BENDING OF FLEXIBLE CIRCULAR RING SHELLS WITH MODERATELY SLENDER RATIO AND APPLICATIONS TO THE BELLOWS (Ⅰ)-GOVERNING EQUATION AND GENERAL SOLUTION

The overall bending of circular ring shells subjected to bending moments and lateral forces is discussed. The derivation of the equations was based upon the theory of flexible shells generalized by E.L. Axelrad and the assumption of the moderately slender ratio less than 1/3 (i.e., ratio between curvature radius of the meridian and distance from the meridional curvature center to the axis of revolution). The present general solution is an analytical one convergent in the whole domain of the shell and with the necessary integral constants for the boundary value problems. It can be used to calculate the stresses and displacements of the related bellows. The whole work is arranged into four parts: (Ⅰ) Governing equation and general solution; (Ⅱ) Calculation for Omega-shaped bellows; (Ⅲ) Calculation for C-shaped bellows; (Ⅳ) Calculation for U-shaped bellows. This paper is the first part.     

Pressure distribution in an elastomer confined by a long thin-walled flexible hollow cylinder under the assumption of a hydrostatic stress state

The stress distribution in a pressurized elastomer confined by a hollow cylinder is of interest in various applications of material testing and manufacturing. A relatively accurate closed form solution for the pressure distribution inside an elastomer confined by a rigid hollow cylinder was presented by Yu et al. (2001). But in many practical applications the assumption of a rigid hollow cylinder is not appropriate, because the cylinder deformations have a significant influence on the stresses inside the elastomer. Thus in this paper a solution for an elastomer confined by a deformable hollow cylinder is derived. Both axial and radial deformations of the hollow cylinder are taken into account, while the bending stiffness of the cylinder wall is neglected, i.e. the cylinder wall is treated according to the membrane theory. The accuracy of the proposed closed form solution is verified by a parametric finite element simulation.     

GENERAL SOLUTION OF THE OVERALL BENDING OF FLEXIBLE CIRCULAR RING SHELLS WITH MODERATELY SLENDER RATIO AND APPLICATIONS TO THE BELLOWS (Ⅳ)-CALCULATION FOR U-SHAPED BELLOWS

ＩｎｔｒｏｄｕｃｔｉｏｎＵ＿ｓｈａｐｅｄｂｅｌｌｏｗｓａｒｅｕｓｕａｌｌｙｓｅｅｎｉｎｔｈｅｐｉｐｅｌｉｎｅｓｙｓｔｅｍ (Ｆｉｇ .1) ,ｗｈｉｃｈａｒｅｕｓｅｄｔｏｃｏｍｐｅｎｓａｔｅｆｏｒｔｈｅａｘｉａｌｍｏｖｅｍｅｎｔ,ｔｈｅｒｏｔａｔｉｏｎａｎｄｔｈｅｄｅｆｌｅｃｔｉｏｎｏｆｔｈｅｐｉｐｅｌｉｎｅｓｄｕｅｔｏｔｈｅｔｈｅｒｍａｌａｃｔｉｏｎ ,ｔｈｅｂａｓｅｕｎｅｑｕａｌｓｅｔｔｌｅｍｅｎｔｓａｎｄｔｈｅｉｎｓｔａｌｌａｔｉｏｎｅｒｒｏｒｓ.Ｔｈｅａｘｉｓｙｍｍｅｔｒｉｃａｌｄｅｆｏｒｍａｔｉｏ…     

GENERAL SOLUTION OF THE OVERALL BENDING OF FLEXIBLE CIRCULAR RING SHELLS WITH MODERATELY SLENDER RATIO AND APPLICATIONS TO THE BELLOWS (Ⅳ)-CALCULATION FOR U-SHAPED BELLOWS

This is one of the applications of Part (I), in which the angular stiffness, and the corresponding stress distributions of U-shaped bellows were discussed. The bellows was divided into protruding sections, concave sections and ring plates for the calculation that the general solution (I) with its reduced form to ring plates were used respectively, but the continuity of the surface stresses and the meridian rotations at each joint of the sections were entirely satisfied. The resent results were compared with those of the slender ring shell solution proposed earlier by the authors, the standards of the Expansion Joint Manufacturers Association (EJMA), the experiment and the finite element method. It is shown that the governing equation and the general solution (I) are very effective. Contributed by HUANG Qian Biography: ZHU Wei-ping (1962-)     

Bending and buckling of superconducting partial toroidal field coils

Based on the theory of curved beams and the Biot–Savart law, a new theoretical model for magnetoelastic bending and buckling of superconducting toroidal field coils is developed and corroborated numerically. In contrast to the existing models, this model incorporates in-plane deformation of the coil. A semi-analytical approach is used to obtain the solution to the coupled problem. In order to validate the model and associated solution method, the experiment of Miya et al., 1982 is modeled. The theoretical predictions of the critical current of coils with none or one pin support are shown to be in excellent agreement with the published experimental data. It is also shown that the in-plane deformation has a significant influence on the critical current.     

Bending stresses in out-of-round uni- and multi-axial internally pressurized strength specimens

A derivation is presented for the bending stresses in internally pressurized slightly elliptical tubular or ring specimens used in the testing of uni- and multiaxial strength. The solution obtained shows that the bending stresses are directly proportional to Young's modulus of elasticity of the material being tested. A numerical example applicable to strength testing of aluminum oxide demonstrates, that a slight out-of-round-ness can result in bending stresses which are an appreciable percentage of the observed values of the fracture strength. It is concluded that strength testing by means of the internal pressurization of tubular or ring specimens requires close dimensional control in specimen manufacture.     

计及表面能的应变梯度理论本构参数识别

本文提出利用静态位移信息对一种计及表面能的应变梯度理论本构参数进行识别的求解策略.基于Vardoulakis和Sulem的计及表面能的简单线性应变梯度理论,文献[13]给出了伯努力-欧拉梁弯曲问题的正演解析模型,本文将其反演归结为两个带有不等式约束的非线性规划问题.在此基础上,采用黄金分割一维搜索方法进行求解,给出了数值验证,讨论了信息误差对反演结果的影响.结果显示,这种方法可以用来对应变梯度理论本构参数进行识别,即使在体积和表面能常数非常小的情况下,仍然能够得到满意的结果.     

波纹管在内压作用下柱失稳临界压力的计算

 讨论了波纹管在内压作用下的柱失稳问题，将波纹管等效成圆 柱壳，证明了该柱壳在内压作用下失稳的临界压力与在轴向均布压力 作用下失稳的临界压力相等，利用作者对波纹管整体弯曲问题的研究 成果确定等效柱壳的抗弯刚度，给出了波纹管在内压作用下柱失稳临 界压力的计算公式. 由对前人实验的观察分析得出波纹管呈弹性柱失 稳的条件为其长细比大于1，在此条件下本文的计算结果和前人的实 验结果相一致.     

Buckling life estimation of circular tubes under cyclic bending

In this paper, an experimental investigation of the degradation and buckling of circular tubes subjected to cyclic bending is discussed. The machinery specimens (with different diameter-to-thickness ratios but the same inside diameter) and method of testing (cyclic bending) in this study were the same as the ones used by Lee et al. (Lee, K.L., Pan, W.F., Kuo, J.N., 2001. The influence of the diameter-to-thickness ratio on the stability of circular tubes under cyclic bending. International Journal of Solids and Structures 38, 2401–2413.) for 316L stainless steel circular tubes. The experimental investigation was extended to different outside and inside diameters of the same circular tubes subjected to cyclic bending. Based on the experimental findings, the empirical formulation proposed by Lee et al. (2001) was modified so that it can now be used to simulate the relationship between the prescribed curvature and the number of cycles necessary to produce buckling. In addition, it was found that the experimental curve of the ovalization and the number of cycles necessary to produce buckling could be divided into three stages – an initial, secondary and tertiary stage. An empirical relationship, similar to the Bailey–Norton creep formulation, was proposed for simulating the aforementioned curve for the initial and secondary stages in this study. The derived empirical relationship was in good agreement with the experimental data.