1Chapter 6 Finite Element Method for Plane Stress and Plane Strain Problems第六章第六章 有限单元法解平面问题有限单元法解平面问题2References 参考书参考书 徐芝纶，徐芝纶，弹性力学简明教程第六章。弹性力学简明教程第六章。高等教育高等教育 出版社。出版社。 华东水利学院华东水利学院， 弹性力学问题的有限单元法弹性力学问题的有限单元法， 水利电力出版社。水利电力出版社。 卓家寿，卓家寿， 弹性力学中的有限元法弹性力学中的有限元法，高等教育出，高等教育出 版社。版社。 O.C. Zienkiewicz, The Finite Element Method,Third Edition, 51.818, Z66 K.C. Rockey and so on ,The Finite Element Method, Second Edition, 51.818,R682- 23Introduction- 1 导引导引- 1 The finite element method is an extension of the analysis techniques (matrix method) of ordinary framed structures. 有限元法是刚架结构分析技术的扩充。有限元法是刚架结构分析技术的扩充。 The finite element method was pioneered in the aircraft industry where there was an urgent need for accurate analysis of complex airframes. 有限元法首先应用于飞机工业。有限元法首先应用于飞机工业。4Introduction- 2 The availability of automatic digital computers from 1950 onwards contributed to the rapid development of matrix methods during this period. 从从 1950以后 数字计算机的出现使矩阵位移法以后 数字计算机的出现使矩阵位移法 迅速发展。迅速发展。5Introduction- 3 The finite element method was developed rapidly from 1960 onwards and known in China from 1970 onwards. 从从 1960以后 有限元法迅速发展。以后 有限元法迅速发展。 1970以后 传以后 传 入我国。入我国。6Introduction- 4 In a continuum structure , a corresponding natural subdivision does not exist so that the continuum has to be artificially divided into a number of elements before the matrix method of analysis can be applied.连续结构不存在自然的单元，连续结构不存在自然的单元，须人为划须人为划分分为为单元单元7Introduction- 5 The artificial elements, which are termed finite elements or discrete elements, are usually chosen to be either rectangular or triangular in shape.单元单元通常取为三角形或通常取为三角形或矩矩形形。86.1 Fundamental quantities and fundamental equations expressed by matrix 6.1 基本量和基本方程的矩程表示基本量和基本方程的矩程表示Body force 体体力力：p=X YT Surface force 面力面力： p=X YT Displacement 位移位移： f=u vT Strain 应应变变：=x yrxyT Stress 应力应力：= x y xyT Geometrical equations Physical equations virtual work equations9Geometrical Equation 几何方几何方程程x u/x /x 0 u = y= v/y = 0 /y v =Lf rxyu/y+v/x /y /x x /x 0 = yL= 0 /y f =u vT rxy/y /x =Lf10Physical Equation for Plane Stress Problem 平面应力问题的物理方程平面应力问题的物理方程xx+y1 0 x y= E/(1-2) y+x= E/(1-2) 1 0 y xyrxy(1- )/2 0 0 (1- )/2 rxyx1 0 x = yD = E/(1-2) 1 0 = y xy0 0 (1- )/2 rxy= D 11Virtual Work Equation 虚功方程虚功方程状态状态1： p=X YT p=X YT = x y xyT 状态状态2： f*=u* v*T *=L f* 虚功方虚功方程程 ： f*Tpdx dy t+ f*Tpds t = *Tdx dy t 注：注： f*Tp = u* v* X =X u*+Y v* Y *T =x *y*rxy* x = x x*+ y y*+ xyrxy* y xy126.2 Basic Concepts about Finite Element Method 6.2 有限单元法的有限单元法的概念概念有限单元法的计算有限单元法的计算模型模型 1.The continuum structure is idealized as a structure consisting of a number of individual elements connected only at nodal points. 连续的结构连续的结构理想化为仅由理想化为仅由在结在结点相点相连的连的 单元单元组成组成。13 2.Displacement boundary: place a bar support at the node where displacement is zero. 位移位移边界：边界：结结点点位移位移为零处为零处，设置设置连连杆杆. 3. The system of external loads acting on the actual structure has to be replaced by an equivalent system of forces concentrated at the element nodes.This can be done by using the principle of virtual work and equating the work done by the actual loads to the work done by the equivalent nodal loads. 外外力力按静按静力等力等效效的的原则原则移移置到置到结结点上点上141516补充：关于离散补充：关于离散 About Discretization In reality elements are connected together along their common boundaries. Here it is assumed that these elements are only interconnected at their nodes.实际：实际：单元单元间相间相连连- - - - 假定：只假定：只结结点相点相连连17关于离散关于离散- 2 However,in the finite element method, the individual elements are constrained to deform in specific patterns.然然而而，单元，单元变形按指定模式变形按指定模式. 18关于离散关于离散- - 3 Hence, although continuity is only specified at the nodal points, the choice of a suitable pattern of deflection for the elements can lead to the satisfaction of some,if not all,of the continuity requirements along the sides of adjacent elements. 位移位移模式模式使使相相连单元位移连续连单元位移连续得某些满得某些满 足足19关于离散关于离散- - 4 Hence, as stated by Clough, finite elements are not merely pieces cut from the original structure, but are special types of elastic elements constrained to deform in specific patterns such that the overall continuity of the assemblage tends to be maintained206.3 Displacement pattern and convergence criteria 6.3 位移模式和收敛性位移模式和收敛性 Fig. 1 shows the typical triangular element with nodes ijm numbered in an anti- clockwise order. Y m图图1为一典型为一典型的的三角形三角形单元单元，i 结结点点 ijm 逆钟向编号逆钟向编号- - x正向到正向到j y正向正向