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

Chapter 6 Finite Element Method for Plane stress and plane strain problems 第六章有限单元法解平面问题 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 1 Chapter 6 Finite Element Method for Plane Stress and Plane Strain Problems 第六章 有限单元法解平面问题

References参考书 徐芝纶,弹性力学简明教程第六章。高等教育 出版社。 华东水利学院,弹性力学问题的有限单元法, 水利电力出版社。 卓家寿,弹性力学中的有限元法,高等教育出 版社。 O C. Zien kiewicz. The Finite Element Method. Third Edition. 51. 818 766 K.C. Rockey and so on, The Finite Element Method. Second Edition. 51. 818. R682-2 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 2 References 参考书 • 徐芝纶，弹性力学简明教程第六章。高等教育 出版社。 • 华东水利学院， 弹性力学问题的有限单元法， 水利电力出版社。 • 卓家寿， 弹性力学中的有限元法，高等教育出 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, R682-2

Introduction-1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 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。 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 3 Introduction-1 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。 • 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

Introduction 2 The availability of automatic digital computers from 1950 onwards contributed to the rapid development of matrix methods during this period. 从1950以后数字计算机的出现使矩阵位移法 迅速发展。 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 4 Introduction-2 • The availability of automatic digital computers from 1950 onwards contributed to the rapid development of matrix methods during this period. 从 1950以后 数字计算机的出现使矩阵位移法 迅速发展

Introduction-3 The finite element method was developed rapidly from 1960 onwards and known in China from 1970 onwards 从1960以后有限元法迅速发展。1970以后传 入我国。 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 5 Introduction-3 • The finite element method was developed rapidly from 1960 onwards and known in China from 1970 onwards. • 从 1960以后 有限元法迅速发展。 1970以后 传 入我国

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] {}