CLASSIFICATION OF VARIATIONAL PRINCIPLES IN ELASTICITY

In this paper, variational principels in elasticity are classified according to the differences in the constraints used in these principles. It is shown in a previous paper that the stress-strain relations are the constraint conditions in all these variational principles, and cannot be removed by the method of linear Lagrange multiplier. The other possible constraints are four of them: (1) equations of equilibrium,(2) strain-displacement relations, (3) boundary conditions of given external forces and (4) boundary conditions of given boundary displacements. In variational principles of elasticity, some of. them have only one kind of such constraints, some have two kinds or three kinds of constraints and at the most four kinds of constraints. Thus, we have altogether 15 kinds of possible variational principles. However, for every possible variational principle, either the strain energy density or the complementary energy density may be used. Hence, there arealtogether 30 classes of functional of variation     

再论弹性力学中的广义变分原理——就等价定理问题和胡海昌先生商榷

本文就胡海昌先生提出的等价定理的论争,申述个人的观点和论证,与胡海昌先生商榷。 本文主要论证了下列三点: (1)通过待定的拉格朗日乘子法证明了胡海昌-鹫津久一郎原理(下文简称胡鹫原理)的三类变量之间并不独立,应力应变关系仍然是应力和应变之间应该首先满足的变分约束条件。这个变分原理只是在形式上有应力、应变、位移三类变量,在实际上,这些变量中只有两类是独立的。(2)通过高阶拉格朗日乘子法,我们求得了比胡海昌鹫津久一郎原理的泛函更一般形式的具有三类变量的变分泛函,而且证明有无穷个这样的变分泛函。利用唯一性定理,我们证明了这些泛函的变量中必须满足应力应变关系这个条件。同样也证明了胡鹫原理并不是三类变量都独立的和没有任何约束条件的完全的变分原理,而是一个以应力应变关系为变分约束条件的变分原理。(3)在应力应变关系的变分约束条件下,我们证明了Hellinger-Reissner原理和胡鹫原理的等价定理。 本文的结论是:等价定理是正确的,并非象胡海昌先生所指的那样是“误解”。郭仲衡、戴天民、陈至达、刘殿魁、张其洁、邬瑞铎、奚肖风等通过各自的努力,在各种变分问题上论证了等价定理,都是正确的,没有什么“误解”,更没有“误入迷途”。胡海昌先生认为大家都有“误解”的原因,似乎在于     

THE FIRST ORDER APPROXIMATION OF NON-KIRCHHOFF-LOVE THEORYFOR ELASTIC CIRCULAR PLATE WITH FIXED BOUNDARY UNDERUNIFORM SURFACE LOADING (Ⅰ)

I.1ntroductionTheaxisymmetricproblemofthreedimensionalelasticcircularplatecanbetreatedasthreedimensionalaxisymmetricproblemofelasticity.Weconsideracircularplatewithauniformthicknessh,andsetupacircumferentialcoordinates(r,o)onitsmidd1esurfacewithabscissazp…     

弹性理论中广义变分原理的研究及其在有限元计算中的应用(续)

四大位移非线性弹性体的广义变分原理--完全的和不完全的我们也可以通过拉格朗日乘子法,导出大位移非线性弹性体的有关广义变分原理,设λ_(ij)和μ_i为待定的拉格朗日乘子,于是根据(49)式导出的无条件的广义变分泛函为 ...     

弹性理论中广义变分原理的研究及其在有限元计算中的应用

本文的目的在于说明怎样系统地建立各种广义变分原理,怎样合理地使用各种广义变分原理来改进有限元计算的成效。为了易于说明问题,本文只局限于弹性理论的各种广义变分原理,但其推广并不困难。本文指出,广义变分原理的泛函,可以系统地采用拉格朗日乘子法,把一般有条件的变分原理化为无条件的变分原理来唯一地决定的。拉格朗日乘子所代表的物理量,可以通过变分求极值或驻值的过程求得,从而消除了在建立广义变分原理的泛函时,人们经常陷入的象猜谜一样的困境。本文也指出:我们同样可以用拉格朗日乘子法把一般有多个条件的变分原理,化为条件个数较少的变分原理。我们称变分条件减少了的变分原理为各级不完全的广义变分原理。凡是把全部变分条件都消除了的变分原理,称为完全的广义变分原理,或简称广义变分原理;实际上是完全无条件的变分原理。本文建立了弹性小位移变形理论中的各级不完全的广义位能原理,和各级不完全的广义余能原理,包括从最小位能原理和最小余能原理分别导出的最完全的广义变分原理;并且证明了这两个弹性力学广义变分原理的泛函是等同的。在这些广义变分原理中,包括了Hellinger-Reissner(1950),胡海昌-鹫津久一郎(1955)的广义变分原理。本文也建立了弹性大位移变形理论中的位能原理和余能原理,并建立了有关位能余能的各级不完全的广义变分原理,包括以大位移变形的最小位能和最小余能原理分别导出的弹性力学广义变分原理,并且也证明了在大位移变形情况下,这两个弹性力学的广义变分原理也是等同的。本文除了列举广义变分原理在有限元法上的众所周知的应用外,还补充了三个比较重要的应用范围。     

宁波甬江大桥的大挠度非线性计算问题

研究了宁波甬江铁路大桥的大挠度非线性设计计算问题.提出了非线性方程的叠代近似算法,同时指出了如果两岸落差约5m,两岸跨度约100m计算,则桥梁中间的最大斜度将超过5%,这远远超过铁路设计允许的斜度.为此,提出了减小路轨斜度的设计方案,即铁路在两岸都有长度约1km斜度为0.5%的路基,使两岸的落差减小到约为原来落差的1/10.这样路轨在跨越甬江时,其挠度的斜度就会大大缩小,也在0.5%～0.6%之间.     

A RECTANGULAR ELEMENT OF THIN PLATES BASED UPON THE GENERALIZED VARIATIONAL PRINCIPLES

Based on generalized variational principles,an element called MR-12 was constructed for the static and dynamic analysis of thin plates with orthogonal anisotropy.Numerical results showed that this incompatible clement converges very rapidly and has good accuracy.It was demonstrated that generalized variational principles are useful and effective in founding incompatible element.Moreover,element MR-12 is easy for implementation since it does not differ very much from the common rectangular element R-12 of thin plate.     

FINITE ELEMENT METHOD COMBINED WITH DYNAMIC PHOTOELASTIC ANALYSIS TO DETERMINE DYNAMIC STRESS-INTENSITY FACTORS

The present paper is addressed to the finite element method combined with dynamicphotoelastic analysis of propagating cracks,that is,on the basis of[1]by Chien Wei-zang,finite elements which incorporate the propagating crack-tip singularity intrinsic to two-dimensional elasticity are employed.The relation between crack opening length and timestep obtained from dynamic photoelastic analysis is used as a definite condition for solvingthe dynamic equations and simulating the crack propagations as well.As an example,theimpact response of dynamic-bending-test.specimen is investigated and the dynamic stress-intensity factor obtained from the mentioned finite element analysis and dynamicphotoelasticity is in reasonable agreement with each other.     

DYNAMIC FINITE ELEMENT WITH DIAGONALIZED CONSISTENT MASS MATRIX AND ELASTIC-PLASTIC IMPACT CALCULATION

There are some common difficulties encountered in elastic-plas-tic impact codes such as EPIC,NONSAP etc.Most of thesecodes use the simple linear functions usually taken from static pro-blems to represent the displacement components.In such finite ele-ment formulation,the strain and stress components are constants inevery element.In the equations of motion,the stress components ingeneral appear in the form of their space derivatives.Thus,if weuse such form functions to represent the displacement components,the effect of internal stresses to the equations of motion vanishesidentically.The usual practice to overcome such difficulties isto establish as self-equilibrium system of internal forces actingon various nodal points by means of transforming equations of mo-tion into variational form of energy relation through the applica-tion of virtual displacement principle.The nodal acceleration.isthen calculated from the total force acting on this node from allthe neighbouring elements.The transformation of virt     

The Calculation of Semi-circular Corrugated Tube-An Application of the General Solution of Ring Shell

In this paper,the deformation and stress distribution in semi-circularcorrugated tube under axial force are calculated by means of the generalsolutions of circular ring shell given in previous paper[1].