扇形截面杆扭转问题的差分线法

导出了扇形截面杆扭转问题偏微分方程的差分线法常微分方程组, 并解析求解了该方程组, 得到了扭转应力函数的半解析解, 计算了扭转应力及扭转刚度. 计算过程中, 用追赶法计算 常微分方程组的特解, 用公式计算三对角矩阵的特征值与特征向量, 利用实对阵矩阵的特征 向量相互正交的特性避免矩阵求逆计算, 利用复化梯形公式计算扭转刚度. 整个求解过程在 角度方向离散微分方程和用复化梯形公式进行面积积分时引入了误差, 其他求解过程是精确 的. 计算结果与已有结果进行了对比, 显示了算法的正确性. 该算法对工程中扇形截面扭 转杆的设计有一定的实用价值.     

螺旋细杆的均匀扭转振动

讨论螺旋细杆的特殊形式扭转振动，即均匀扭转振动．以非圆截面杆和有原始曲率的圆截面杆为研究对象．杆作均匀扭转振动时各截面有相同的扭角变化规律，且杆中心线的几何形状不受振动过程的影响．研究表明，扭振来源于杆截面的非对称性及杆的原始曲率．杆的扭振规律与单摆运动相似，其动力学方程存在精确解．圆环杆的均匀扭振为螺旋杆的倾角为零时的特例．     

TRIAL FUNCTION METHOD FOR THE LONGITUDINAL VIBRATION OF RODS WITH VARIABLE CROSS-SECTIONS

给出了一种试探函数法,并研究了变截面杆的纵振动问题. 先给出振动控制方程的特殊函数形式的试探解,然后要求此解满足控制方程,反过来确定了控制方程各种可能的系数函数（即截面变化函数）并得到了控制方程的精确解. 作为例子,给出了一种变截面杆在3 种边界条件下的频率方程,计算出了固有频率. 研究表明,试探函数法简单、直接,适合于研究变截面杆的纵振动问题. 对于杆扭转振动、薄膜振动以及管中波传播等问题,该方法同样有推广应用价值.     

极坐标系下扇形截面杆扭转问题的有限差分法

导出了极坐标系下扇形截面杆扭转问题的差分格式；结合Mathcad编程用逐次超松弛迭代法求出了扭转应力函数差分值；采用复化二维辛普生求积公式计算了抗扭刚度；求出了相应的扭转应力的差分解；给出了张角为2π时裂纹尖端附近的第三型应力强度因子．     

矩形截面杆扭转的康脱洛维奇解法

本文利用康脱洛维奇变分解法来计算矩形截面杆的扭转问题，所得到的计算结果形式简单，应用方便，与文［１］的精确解很接近     

非圆截面杆中的非线性扭转波及其行波解

研究了非圆截面杆中非线性扭转波的传播特性.由于非圆截面杆的扭转运动会伴随有横截面的翘曲,这种翘曲运动将引起扭转波的弥散.如果同时考虑有限扭转变形和翘曲弥散的共同作用,将会得到非线性扭转波的方程.在相平面上,对非线性扭转波动方程进行定性分析,结果表明,在一定条件下方程存在同宿轨道或异宿轨道,分别相应于方程的孤波解或冲击波解.本文利用Jacobi椭圆函数展开法,对该非线性方程进行求解,得到了非线性波动方程的三类准确周期解及相应的孤波解和冲击波解,讨论了这些解存在的必要条件.这些条件与定性分析的结果相一致.     

圆截面弹性细杆的平面振动

基于Kirchhoff理论讨论圆截面弹性细杆的平面振动．以杆中心线的Frenet坐标系为参考系建立动力学方程．杆作平面运动时，其扭转振动与弯曲振动解耦．讨论任意形状杆的扭转振动和轴向受压直杆在无扭转条件下的弯曲振动，证明直杆平衡的静态Lyapunov稳定性与欧拉稳定性条件为动态稳定性的必要条件．考虑轴向力和截面转动惯性效应的影响，导出弯曲振动的固有频率．     

轴线存在应变时弹性杆力学的两个概念

讨论Kirchhoff弹性杆力学向精确Cosserat弹性杆推广中的两个概念: 轴 线切向量对截面法向量是怎样偏离以及本构方程中应变矢和弯扭度的基准问题. 从单元体的 剪应变出发, 导出了截面法矢、轴线切矢以及剪应变矢三者关系, 即Cosserat弹性杆的变形 几何关系;从Hook定律出发, 论证了在一次近似下本构方程中的截面弯扭度和形心应变矢都 以原始弧坐标为基准.     

半椭圆和半圆环截面弯曲中心的分析解

本文利用对称截面悬臂梁在力作用下的弯曲解,推导出半椭圆截面和半圆环截面弯曲中心的分析解。在推导过程中算出了半椭圆截面扭转刚度的精确解,并在椭圆长短轴相等和圆环内半径趋于零的两种情况,分别给出半圆环截面弯曲中心的分析解。     

强化材料弹塑性扭转问题的渐近解法

本文对强化规律以一幂级数表示的材料,提出了等截面柱形杆的强化弹塑性扭转问题的渐近摄动解法。求原问题的渐近解被变为解一系列有齐次边界条件的Poisson方程。以圆形截面杆的扭转作为例题,验证了本文方法的可靠性。还举了一个椭圆形截面杆强化弹塑性扭转问题,获得了它的渐近解。     

The warping functions of regular prismatic bars under torsion evaluated by caustics

Reflection of a bundle of coherent light on the warped cross section of a prismatic bar submitted to torsion forms a caustic on a receiver plane. From the mathematical expression of this curve and the theory of reflected caustics, it is possible to evaluate accurately the warping function of the cross section. Using this idea, it was possible to study the torsion problem in prismatic bars with sections which were equilateral triangles and squares. It was observed that the shape of the caustic is an hypocycloid curve with three or four cusps respectively. By evaluating the warping function by using elements from the respective caustics it was possible to find out that, for the triangular cross section, the expression for the warping function coincided exactly with the expression given by the exact solution of the problem. For the square cross section, a closed-form solution for its warping function was readily derived, to which the series approximation solution differed only by a few percent at maximum for the shear stresses. Since the method can be readily extended to any canonical polygonic cross section, it constitutes a general solution for the torsion of prismatic bars, which approximates their exact deformations better than the solutions based on the Saint-Vénant assumptions.     

Torsion of prismatic bars under pressure linearly varying along the generator

The problems on the limit state of prismatic bars in torsion is studied under the assumption that the bar is under pressure linearly varying along the generator. The bar stress-strain state is determined and the characteristic field is constructed.     

Torsion prismatischer Stäbe nach der Singularitätenmethode

übersicht Das St. Venantsche Torsionsproblem wird auf eine Fredholmsche Integralgleichung 2. Art zurückgeführt. Als unbekannte Funktion tritt dabei eine Singularit?tenverteilung auf. Die Integralgleichung wird für prismatische St?be numerisch gel?st. Ist die Singularit?tenverteilung bekannt, so k?nnen Spannungs- und Verformungszustand berechnet werden. Um die Brauchbarkeit des Verfahrens zu zeigen, werden in einigen Beispielen die Ergebnisse mit denen der exakten L?sungen verglichen.

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Summary The St. Venant's torsion problem is reduced to a Fredholm integral equation of the second kind with a distribution of singularities as unknown function. This equation is solved numerically for prismatic bars. The state of tension and displacement can be computed if the distribution of singularities is given. To illustrate the usefullness of the procedure some examples are given and their results are compared with those of the exact solution.

     

Application of an improved boundary element method on the problem of elastic torsion

An improved boundary clement method has been used in analyzing and calculating the problems of the torsion of a prismatic bar with elliptical cross-section. In this paper the calculated results correspond with the values of boundary element method. However, the quantity of data required by the improved boundary element method is much less than that required by boundary element method, and the calculating time will be greatly reduced. Therefore, the procedure of this paper is an economical and efficient numerical computational way for solving Poisson equation problem.     

Eddy current analogy of torsion problem

In as paper, an eddycurrent analogy and a brief sketch of required equipment are presented. Values oftorsional rigidity and shearing stresses of a prismatic bar under free torsion can be obtained experimentally to a high degree of accuracy in an instant with this equipment whether the cross-section is bounded bv a single boundary or multi-connected boundaries. The error is les than two per cent generally, as shown in Table 3- ’Hits new analogy can be used extensively to solve various physical problems expressed by Poisson’s (or Laplace’s) equation with constant boundary condition.     

Zur Lösung des torsionsproblems bei verbundstäben. nach der integralgleichungsmethode von Muschelischwili

Übersicht Die meisten bekannt gewordenen nicht-trivialen Beispiele exakter Lösungen des Torsionsproblems zylindrischer Verbundstäbe wurden durch Anwendung der konformen Abbildung erhalten, die oft mit der schwerfälligen Rücktransformation der Ergebnisse in die Originalebene verbunden ist. In einigen Fällen kann jedoch durch direkte Anwendung der Integralgleichungsmethode von Muschelischwili eine bessere, streng analytische Lösung des Problems gelingen. Eine neue Lösung eines solchen Beispieles, nämlich die Torsion eines Kreiszylinders, der mit einem exzentrischen kreiszylindrischen Kern aus anderem Material verbunden ist, wird hier vorgestellt.

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On the solution of the torsion problem of composite bars by the integral equation method of Muskhelishvili

Summary The majority of known non-trivial examples of exact solutions to the torsion problem of composite cylindrical bars were obtained by application of the conformal mapping which is often connected with cumbersome retransformation of the results to the physical domain. In several cases, however, a better exact analytic solution of the problem can be found by direct application of the integral equation method of Muskhelishvili. A new solution of such an example, i.e. the torsion of a circular cylinder reinforced by a eccentric circular core made of different material, is presented here.

     

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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.     

Meshless methods for the inverse problem related to the determination of elastoplastic properties from the torsional experiment

The problem of determining the elastoplastic properties of a prismatic bar from the given experimental relation between the torsional moment M and the angle of twist per unit length of the rod’s length θ is investigated as an inverse problem. The proposed method to solve the inverse problem is based on the solution of some sequences of the direct problem by applying the Levenberg-Marquardt iteration method. In the direct problem, these properties are known, and the torsional moment is calculated as a function of the angle of twist from the solution of a non-linear boundary value problem. This non-linear problem results from the Saint-Venant displacement assumption, the Ramberg–Osgood constitutive equation, and the deformation theory of plasticity for the stress–strain relation. To solve the direct problem in each iteration step, the Kansa method is used for the circular cross section of the rod, or the method of fundamental solutions (MFS) and the method of particular solutions (MPS) are used for the prismatic cross section of the rod. The non-linear torsion problem in the plastic region is solved using the Picard iteration.     

A duality theorem for plastic torsion

Limit analysis of prismatic torsion bars was the earliest attempt to apply plasticity theory to a continuum. The simplicity of the problem made it feasible to use the two-dimensional Prandtl stress function, defined for the elastic torsion problems, for the plastic stress distributions as well. The gradient of the stress functions for plastic torsion has a constant magnitude, and hence a function of this type assumes the profile of a sand hill. This sand hill analogy of Nadai (1950, The Theory of Flow and Fracture of Solids. McGraw-Hill, U.K.) gave a visual sense of possible nonsmoothness of such stress functions and thus discontinuous stress fields. Many stress functions of plastic torsion for relatively simple cross-sections have been constructed graphically. However, collapse modes in terms of warping functions were much less reported. In this paper, we shall establish a duality theorem which relates the correct stress function to the correct warping function, thus providing the means to obtain complete static and kinematic solutions. This dual variational principle leads naturally to a general numerical algorithm which guarantees convergence and accuracy. In this paper, we shall only present three exact solutions to verify the theorem, to demonstrate the possible non-smooth feature of the solutions and to reiterate this effective dual variational approach to limit analysis in general.