Response of flexure–torsion coupled composite thin-walled beams with closed cross-sections to random loads

A study of the flexure–torsion coupled random response of the composite beams with solid or thin-walled closed-sections subjected to various types of concentrated and distributed random excitations is dealt with in this paper. The effects of flexure–torsion coupling, shear deformation and rotary inertia are included in the present formulations. The random excitations are assumed to be stationary, ergodic and Gaussian. Analytical expressions for the displacement response of the composite beams are obtained by using normal mode superposition method combined with frequency response function method. The present method can produce the effective solutions for the composite Timoshenko beams with circumferentially antisymmetric (CAS) configuration and more general beam assemblages of connected beams. The influences of flexure–torsion coupling, shear deformation and rotary inertia on the random response of an appropriately chosen composite beam from the literature are demonstrated and discussed.     

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介绍了求解复杂截面闭口薄壁杆件扭转问题的网络理论解法,以普朗特应力函 数解法为基础,通过与电路问题比拟,借鉴了电路理论中的网络理论解法. 本文方法对解决复 杂截面闭口薄壁杆件扭转的工程问题有一定的应用价值. 在教学上,可以起到拓宽学生思路 的作用.     

A new finite element of spatial thin-walled beams

Based on the theories of Timoshenko＇s beams and Vlasov＇s thin-walled members, a new spatial thin-walled beam element with an interior node is developed. By independently interpolating bending angles and warp, factors such as transverse shear deformation, torsional shear deformation and their Coupling, coupling of flexure and torsion, and second shear stress are considered. According to the generalized variational theory of Hellinger-Reissner, the element stiffness matrix is derived. Examples show that the developed model is accurate and can be applied in the finite element analysis of thinwalled structures.     

开口薄壁梁的扭转理论与应用

以Vlasov薄壁构件理论为基础, 推导了开口薄壁构件一阶扭转理论. 该理论考虑了翘曲剪应力对截面转角的影响, 截面的转角分为自由翘曲转角和约束剪切转角, 在约束扭转中, St.Venant扭矩仅仅与自由翘曲转角有关, 而翘曲扭矩仅与约束剪切转角有关. 利用半逆解方法求出了约束扭转中薄壁构件的St.Venant扭矩表达公式; 依据能量方法, 建立了约束剪切转角和翘曲扭矩之间的关系, 并提出了翘曲剪切系数概念, 给出了一阶扭转理论的微分方程. 为了有效求解微分方程, 给出了求解微分方程的初参数法方程和相应的影响函数矩阵; 当St.Venant扭矩可以忽略时, 得到与一阶弯曲理论(Timoshenko梁理论)相似的一阶扭转理论简化形式. 最后利用算例证明了一阶扭转理论和简化理论的有效性.      

薄壁杆侧向稳定分析有限杆元模型

根据位移变分原理,建立有限杆元模型,对薄壁杆进行侧向稳定分析。在考虑截面扭转、翘曲同时,特别考虑了反映剪力滞后现象的杆壁中面上剪应交的影响。推导出薄壁杆侧向屈曲能量方程可适用求解常用边界条件,任意棱形截面形状的薄壁杆的屈曲特征值问题。     

Torsion of moderately thick hollow tubes with polygonal shapes

A simple formulation is presented for torsion analysis of hollow tubes with polygonal shapes. Thicknesses of segments of cross section can be different. Governing equations in term of Prandtl’s stress function are used to derive the formulas. The derived formulas are so simple that computations can be carried out with a pocket calculator. Several examples are presented to show the accuracy and efficiency of the formulation. The obtained results are verified by accurate finite element solutions. It will be seen that the derived formulas can be useful for analysis of thin-walled and moderately thick-walled hollow tubes.     

相变复合波在伪弹性TiNi合金薄壁管中的传播

相变可以改变材料的性质,从而严重影响波在介质中传播的结构。采用考虑静水压力和偏应力联合作用的增量型相变本构模型,研究了在拉(压)-扭联合作用下半无限长TiNi合金薄壁管内相变复合波的传播规律。基于广义特征理论分析了相变复合波的特征波速及简单波解的基本性质。利用数值方法研究了两种典型情况下管内相变耦合波传播的规律,管内传播的应力路径和波的结构与初始状态及加载幅值有关,展现出和普通弹塑性材料截然不同的性质。     

An analogue technique for determining torsion and flexure functions of uniform beams

This paper describes an extremely simple, and yet accurate technique, based on electric analogy, for determining torsion and flexure functions of beams of uniform section, acted upon by terminal loads. Equations governing the torsion and flexure of such beams have been expressed to state the problem as a Neumann-type boundary-value problem; thus, the problem reduces to finding a function (or functions), which is harmonic within the cross section of the beam, and whose normal derivatives at the boundary of the section are prescribed. Prescribed current densities are introduced at the boundary, and consequent voltages are obtained as the analogue of required functions. Instrumentation is very simple and, once what has been described here as the “influence matrix” of voltages is obtained, both torsion and flexure functions can be obtained simply by multiplying the “influence matrix” by vectors of appropriate normal derivatives at the boundary. None of the usual drawbacks of membrane analogy arise here. Use and accuracy of the technique have been demonstrated by a number of examples.     

外套有/无约束条件下波纹管扭转特性的实验研究

波纹管是一种具有极高扭转刚度，但易于失稳的弹性薄壁壳体。由于波纹管通常在很小的扭转角度下就会发生失稳，限制了其在传递扭矩方面的用途。通过实验研究发现，波纹管扭转屈曲时其材料远未到达强度极限，使得材料的利用率较低。波纹管失稳以后依然可以在一定范围内承受增大的扭矩，其在弹性范围内的扭转角度远大于屈曲发生时的扭转角度。为提高波纹管在后屈曲阶段的承载能力，采用具有合适内径的刚性外套来约束其横向弯曲变形。通过实验研究与有限元分析相结合的方法，比较了后屈曲阶段外套在提高波纹管承载能力方面的作用，两者得到了一致的结论，外套约束对波纹管后屈曲阶段的承载增加作用明显。     

Local buckling of wide-flange thin-walled anisotropic composite beams

The present contribution deals with the onset of local buckling of compressively loaded thin-walled beams with open I, C, Z, T and L-cross-sections made of laminated composite materials. The method employs a discrete plate analysis approach in the course of which each structural subelement of interest—which presently is the flange—of the thin-walled cross-section is considered as a separate composite plate with elastic rotational restraints at those edges where an adjacent substructural element is located. While in many investigations the lamination schemes of webs and flanges are considered to be purely orthotropic, in the present paper the laminate layups are allowed to be of an arbitrary non-orthotropic nature, which also allows for the analysis of laminates with inherent bending–torsion coupling. The analysis of the buckling loads of the flanges of thin-walled composite beams is performed using the Ritz-method for which some especially adjusted displacement shape functions are employed. For the case of pure orthotropy, a novel closed-form solution is described. The accuracy of the employed approaches is established by comparison with accompanying finite element simulations of thin-walled composite beams. It is revealed that the presented methodology is highly efficient in terms of computational effort and yet performs with satisfying accuracy, which makes it very attractive for actual practical applications whenever the local stability behaviour of wide-flange thin-walled composite beams is to be considered.     

New universal relations for nonlinear isotropic elastic materials

A nonlinear isotropic elastic block is subjected to a homogeneous deformation consisting of simple shear superposed on triaxial extension. Two new relations are established for this deformation which are valid for all nonlinear elastic isotropic materials, and hence are universal relations. The first is a relation between the stretch ratios in the plane of shear and the amount of shear when the deformation is supported only by shear tractions. The second relation is established for a thin-walled cylinder under combined extension, inflation and torsion. Each material element of the cylinder undergoes the same local homogeneous deformation of shear superposed on triaxial extension. The properties of this deformation are used to establish a relation between pressure, twisting moment, angle of twist and current dimensions when no axial force is applied to the cylinder. It is shown that these relations also apply for a mixture of a nonlinear isotropic solid and a fluid.     

Experimental method of determining the uniaxial-compression characteristics of thin-walled structures

An analytical treatment of the stability of thin-walled cylinders demonstrates that this geometry is free from the normally observed buckling instabilities when the external pressure is applied as a ‘winding’ load. Overwrapping vessels with elastomeric fiber under high tension produces an equilibrium state that is identical with simple external pressure. This state differs from the case of constant external pressure in that it is stable and has no buckling instabilities. An analysis is presented that demonstrates this experimentally observed observation and shows that the system will be stable independent of its material properties (i.e., elastic, plastic, viscoelastic, etc.). Consequently, this technique can be used to subject thin-walled cylindrical structures to large enough external pressure to cause fracture or homogeneous yielding. Simple experiments on a metal and polymer sample demonstrate the technique.     

THIRD-ORDER GENERALISED BEAM THEORY AND CALCULATION METHOD

一阶广义梁理论描述通过运用加入弯曲、扭转和畸变函数的普通非耦合微分方程组解决棱柱状结构行为.二阶广义梁理论,是添加上偏离力效果的微分方程. 通过引入纵向膜弯矩和膜剪应变虚功到广义梁理论系统当中,完全展开的三阶广义梁方程组将以一串大型离散迭代函数且能转化为可用于数值分析的若干切线刚度矩阵形式出现. 通过膜应力派生出三阶分项^ijrkv_σ和^ijrkv_τ并结合先进数值技术寻求全解,三阶广义梁理论提供了一种严谨和高效的数值工具用于调查薄壁结构后屈曲大变形行为.     

基于高阶剪切变形理论的四边形求积元板单元及其应用

近年来由各类新型复合材料或功能梯度材料构成的板结构在工程领域得到了广泛应用,其显著特点是材料性能沿板厚变化.为合理考虑横向剪切应变,许多学者基于Reddy高阶剪切变形理论,构建了不同的有限元单元对该类板结构进行分析,但其中满足$C^{1}$连续条件的单元相对较少.本文基于Reddy高阶剪切变形理论,采用求积元方法,建立了$C^{1}$连续的四边形板单元.利用该单元对均质材料、复合材料、功能梯度材料构成的等厚度矩形板、变厚度矩形板及等厚度斜板的线弹性弯曲和自由振动问题进行了计算分析,并与现有文献中的相应计算结果进行了对比.研究表明:基于高阶剪切变形理论的四边形求积元板单元具有较高的计算效率和良好的适应性,文中各类材料构成的等变厚度矩形板及等厚度斜板均只需1个单元即可得到理想的计算结果.对于等/变厚度矩形板,可仅使用9$\times$9个积分点,而对于等厚度斜板,随着斜角的增大,所需积分点的数目逐渐增多至15$\times $15.该四边形求积元板单元可进一步用于新型复合材料板的非线性分析.      

Plasticity with non-linear kinematic hardening: modelling and shakedown analysis by the bipotential approach

The class of generalised standard materials is not relevant to model the non-associative constitutive equations. The possible generalisation of Fenchel's inequality proposed by de Saxcé allows the recovery of flow rule normality for non-associative behaviours. The normality rule is written in the weak form of an implicit relation. This leads to the introduction of the class of implicit standard materials. This formulation is applied to constitutive equations involving non-linear kinematic hardening, indispensable to describe accurately and realistically the cyclic plasticity of metallic materials. For these plastic flow rules shakedown bound theorems can be extended; an analytical example of the shakedown of a thin-walled tube under constant traction and alternate cyclic torsion is considered and the obtained solution is proved to be exact.     

The application of graph theory to the limit analysis for torsion of thin-walled bars with complex cross-sections

Graph theory is employed in this paper as a means to establish the topological model of complex thin-walled cross-sections. On this basis, the upper and lower bound theorems of the plastic limit analysis are applied to the analysis of the plastic limit shear flows on the cross-section of thin-walled bars under St. Venant torsion. Corresponding mathematical programming problems are formulated and their duality is shown. After solving the linear programming problem corresponding to the lower bound theorem, the limit torsional moment of a thin-walled cross-section can be calculated according to the shear stress distribution in the limit state. The formula for calculating the limit torsional moment is given. Furthermore, the limit state of thin-walled cross-sections under St. Venant torsion is also discussed and the concept of the limit tree is introduced. A computer program has been developed by the author. Results calculated by the program for typical complex cross-sections are given.     

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用有限元法分析了开口薄壁圆弧曲梁的非线性弹性屈曲，并通过对有限元计 算结果的非线性回归分析，给出了开口薄壁曲梁极限承载力的实用估算公式.     

三阶广义梁理论及其计算方法

一阶广义梁理论描述通过运用加入弯曲、扭转和畸变函数的普通非耦合微分方程组解决棱柱状结构行为.二阶广义梁理论,是添加上偏离力效果的微分方程. 通过引入纵向膜弯矩和膜剪应变虚功到广义梁理论系统当中,完全展开的三阶广义梁方程组将以一串大型离散迭代函数且能转化为可用于数值分析的若干切线刚度矩阵形式出现. 通过膜应力派生出三阶分项ijrkv_σ和ijrkv_τ并结合先进数值技术寻求全解,三阶广义梁理论提供了一种严谨和高效的数值工具用于调查薄壁结构后屈曲大变形行为.      

周期性多孔材料等效剪切模量与尺寸效应研究

提出了以圆筒扭转力学模型为基础, 预测周期性多孔材料等效剪切模量及其研究尺寸效 应的一种简单有效计算方法. 以方形孔和圆形孔两种典型多孔材料为例进行了数值计算求解; 同时, 建立了几何参数随体胞尺寸缩放因子$n$的解析关系, 证明了两种构型的周期性多孔材 料的等效剪切模量均随尺寸缩放因子$n$的增大而减小. 当$n \to \infty $时, 即体胞尺寸相对整体结构无限小时, 多孔材料的等效剪切模量趋近收敛于一个恒定值; 当体胞的材料体分比增大时, 多孔材料等效剪切模量也随之增大. 此外, 依据周 期性多孔材料的结构对称特性, 使用体胞子结构有限元计算模型进行等效剪切模量及其尺寸 效应的预测, 极大地提高了计算效率.     

薄壁杆系结构的梁元分析模型

本文导出了用于薄壁杆系结构弹性分析的薄壁梁元分析模型，在空间梁元分析模型^[3]的基础上，采用了一种改进的位移模式，考察了薄壁杆件可能发生的拉压，剪切，弯曲，扭转和翘曲等各变形形式以及它们的耦合效应，得出了相应的单元形函数，同时从工程应变的定义出发，采用Taylor级数展开的方法，建立了单元的五阶近似正交变表达式，并建立了相应的薄壁单元刚度方程，从而得出了局部坐标系下单元刚度矩阵的显式，根据本文所导出的薄壁梁元分析模型，编制了相应的结构计算程序，通过算例验证了本文所推导的单元刚度矩阵，同时通过与传统空间梁元计算模型计算结果的比较，阐述了截面翘曲对薄壁杆系结构的影响。