河海大学：《弹性力学及有限元》课程PPT教学课件（双语版）第六章 限单元法解平面问题 Finite Element Method for Plane Stress and Plane Strain Problems

Introduction-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,须人为划分为单 元 徐汉忠第一版2000/7 弹性力学第六章有限元

徐汉忠第一版2000/7 弹性力学第六章有限元 6 Introduction-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，须人为划分为单 元

Introduction -5 The artificial elements. which are termed ' finite elements'or discrete elements, are usually chosen to be either rectangular or triangular in shape 单元通常取为三角形或矩形。 徐汉忠第一版2000/7 弹性力学第六章有限元

徐汉忠第一版2000/7 弹性力学第六章有限元 7 Introduction-5 • The artificial elements, which are termed ‘finite elements’ or discrete elements, are usually chosen to be either rectangular or triangular in shape. 单元通常取为三角形或矩形

6.1 Fundamental quantities and fundamental equations expressed by matrix 61基本量和基本方程的矩程表示 Body force体力:{p}=XYT Surface force面力:{p}=区XYT Displacement位移:{f}=uvT Strain应变: te=& E ryI Stress应力: to=lo oy txy l Geometrical equations Physical equations virtual work equations 徐汉忠第一版2000/7 弹性力学第六章有限元 8

徐汉忠第一版2000/7 弹性力学第六章有限元 8 6.1 Fundamental quantities and fundamental equations expressed by matrix 6.1 基本量和基本方程的矩程表示 Body force 体力： {p}=[X Y]T Surface force 面力： {p}=[X Y]T Displacement 位移： {f}=[u v]T Strain 应变： {}=[x y rxy ] T Stress 应力： {}= [x y xy ] T Geometrical equations Physical equations virtual work equations

Geometrical equation几何方程 Ou/ax a/Ox 0u {8} 88r av/a 0 a/Oyv=ILRf Ou/ay+ov/ax a/ay a/ax a/ax 0 (8)=Ey [Ll0 O/Oy ( f=lu vI o/ay a/ax {8}==[L{f 徐汉忠第一版2000/7 弹性力学第六章有限元

徐汉忠第一版2000/7 弹性力学第六章有限元 9 Geometrical Equation 几何方程 x u/x /x 0 u {}= y = v/y = 0 /y v =[L]{f} rxy u/y+v/x /y /x x /x 0 {}= y [L]= 0 /y {f} =[u v]T rxy /y /x {}==[L]{f}

Physical equation for Plane stress problem 平面应力问题的物理方程 Ex+ua =E/(1-2)6+ue1=E/(1-y2) r 00(1-μ)/2r 0 8 }={a,Dl=E(1-2p10 8 00(1-)/2 {o}=[Dl{e} 徐汉忠第一版20007 弹性力学第六章有限元

徐汉忠第一版2000/7 弹性力学第六章有限元 10 Physical Equation for Plane Stress Problem 平面应力问题的物理方程 x x+y 1  0 x y = E/(1-  2 ) y+x = E/(1-  2 )  1 0 y xy rxy(1-)/2 0 0 (1-)/2 rxy x 1  0 x {}= y [D] = E/(1-  2 )  1 0 {}= y xy 0 0 (1-)/2 rxy {}= [D] {}