1 / 34

Theory of Elasticity 弹性力学

Theory of Elasticity 弹性力学. Chapter 8 Two-Dimensional Solution 平面问题的直角坐标求解. Theory of Elasticity. Chapter. Page. Content （内容）. Introduction （概述） Mathematical Preliminaries （数学基础） Stress and Equilibrium （应力与平衡） Displacements and Strains （位移与应变）

keith-norris
Download Presentation

Theory of Elasticity 弹性力学

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Theory of Elasticity弹性力学 Chapter 8 Two-Dimensional Solution 平面问题的直角坐标求解

  2. Theory of Elasticity Chapter Page Content（内容） • Introduction（概述） • Mathematical Preliminaries （数学基础） • Stress and Equilibrium（应力与平衡） • Displacements and Strains （位移与应变） • Material Behavior- Linear Elastic Solids（弹性应力应变关系) • Formulation and Solution Strategies（弹性力学问题求解） • Two-Dimensional Formulation （平面问题基本理论） • Two-Dimensional Solution （平面问题的直角坐标求解） • Two-Dimensional Solution （平面问题的极坐标求解） • Three-Dimensional Problems（三维空间问题） • Bending of Thin Plates （薄板弯曲） • Plastic deformation – Introduction（塑性力学基础） • Introduction to Finite Element Mechod（有限元方法介绍） 1 1

  3. Theory of Elasticity Page Chapter Two-Dimensional Solution in Cartesian Coordinate • 8.1 Cartesian Coordinate Solutions Using Polynomials(直角坐标下的多项式解答) • 8.2 Uniaxial Tension of a Beam(梁的单轴拉伸问题) • 8.3 Bending of a Beam by Uniform Transverse Loading(受均匀横向载荷的梁弯曲问题) 2 8

  4. Theory of Elasticity Page Chapter 8.1 Cartesian Coordinate Solutions Using Polynomials （直角坐标下的多项式解答） 3 D 15 unknowns including 3 displacements, 6 strains, and 6 stresses. 2 D 1 unknowns 3 8

  5. Theory of Elasticity Page Chapter 8.1 Cartesian Coordinate Solutions Using Polynomials（直角坐标下的多项式解答） General Solution Strategies(求解方法) 1、Direct Method(直接法) Direct integration of the field equations(直接积分场方程) Or stress and/or displacement formulations (得到应力/位移方程.) 4 8

  6. Theory of Elasticity Page Chapter 8.1 Cartesian Coordinate Solutions Using Polynomials（直角坐标下的多项式解答） Example: Stretching of Prismatic Bar Under Its Own Weight (受自重的等截面杆) The equilibrium equations reduce to (平衡方程简化为) 4 8

  7. Theory of Elasticity Page Chapter 8.1 Cartesian Coordinate Solutions Using Polynomials（直角坐标下的多项式解答） Example: boundary condition σz |z=0=0 using Hooke’s law 5 8

  8. Theory of Elasticity Page Chapter 8.1 Cartesian Coordinate Solutions Using Polynomials（直角坐标下的多项式解答） Example: integrating the strain-displacement relations: with boundary conditions of zero displacement and rotation at point A(积分应变-位移方程,加上A点位移和转动都为0) 5 8

  9. Theory of Elasticity Page Chapter 8.1 Cartesian Coordinate Solutions Using Polynomials（直角坐标下的多项式解答） 2、Inverse Method(逆解法) particular displacements or stresses are selected that satisfy the basic field equations. A search is then conducted to identify a specific problem that would be solved by this solution field.(选择满足相容方程的应力函数,再根据应力边条和几何边条找出能用所选取的应力函数解决的问题.) φ Boundary Conditions geometry 6 8

  10. Theory of Elasticity Page Chapter 8.1 Cartesian Coordinate Solutions Using Polynomials（直角坐标下的多项式解答） 3、Semi-Inverse Method(半逆解法) part of the displacement and/or stress field is specified, and the other remaining portion is determined by the fundamental field equations (normally using direct integration) and the boundary conditions.(假定部分或全部应力分量为某种形式的双调和函数,从而导出应力函数,再考察由这个应力函数得到的应力分量是否满足全部边界条件) σ（part) φ σ B. C.? 7 8

  11. Theory of Elasticity Page Chapter 8.1 Cartesian Coordinate Solutions Using Polynomials（直角坐标下的多项式解答） Inverse Method in terms of Polynomials:(多项式解法) 1:one-order Polynomials (一次多项式) Assume Fx=Fy=0 One-order Polynomials is fit for zero body force, zero stress state.(适用于零体力,零面力情况) 8 8

  12. Theory of Elasticity Page Chapter 8.1 Cartesian Coordinate Solutions Using Polynomials（直角坐标下的多项式解答） 2:two-order Polynomials (二次多项式) Assume Fx=Fy=0 two-order Polynomials is fit for a uniformity distribution of stress(适用于均匀应力分布) 8 8

  13. Theory of Elasticity Page Chapter 8.1 Cartesian Coordinate Solutions Using Polynomials（直角坐标下的多项式解答） 3:general states(推广到无穷阶) 9 8

  14. 满足双调各方程: Theory of Elasticity Page Chapter 8.1 Cartesian Coordinate Solutions Using Polynomials（直角坐标下的多项式解答） 8.1 specifies one constant in terms of the other two leaving two constants to be determined by the boundary conditions.(由上式8.1再加上边界条件 就可求出所有系数) 9 8

  15. 满足双调各方程: Theory of Elasticity Page Chapter 8.1 Cartesian Coordinate Solutions Using Polynomials（直角坐标下的多项式解答） the general relation that must be satisfied to ensure that the polynomial grouping is biharmonic(对于任意阶多项式要满足双调和方程) 10 8

  16. Theory of Elasticity Page Chapter 8.2 Uniaxial Tension of a Beam(单轴拉伸梁) Problem: plane stress case Saint Venant approximationto the more general case with nonuniformly distributed tensile forces at the ends x =±l. (由圣维南原理可知对于在x =±l处拉力分布不均但静力等效的情况也适用) Solution( inverse method)(逆解法): The boundary conditions (边界条件): 11 8

  17. Theory of Elasticity Page Chapter 8.2 Uniaxial Tension of a Beam(单轴拉伸梁) 求应力函数Φ constant stresses on each of the beam’s boundaries: polynomial is biharmonic Boundary condition Therefore, this problem is given by: 12 8

  18. Theory of Elasticity Page Chapter 8.2 Uniaxial Tension of a Beam(单轴拉伸梁) 求u ,ε Integral(积分)

  19. Theory of Elasticity Page Chapter 8.2 Uniaxial Tension of a Beam(单轴拉伸梁) inverse method(逆解法) Physical Equations e Geometrical Equations u 14 8

  20. q 1 ql ql h/2 z h/2 x y l l y Theory of Elasticity Page Chapter 8.3 Bending of a Beam by Uniform Transverse Loading(受均匀横向载荷的梁弯曲问题) Airy stress function(艾里应力函数) stress field(应力场) Boundary Conditions(边界条件) compare with elementary strength of materials(和材力结果相比) 15 8

  21. q 1 ql ql h/2 z h/2 x y l l y Theory of Elasticity to be determined Page Chapter 8.3 Bending of a Beam by Uniform Transverse Loading(受均匀横向载荷的梁弯曲问题) ∵ q =const. 2) Format of Stress function plane stress conditions (semi-inverse method 半逆解法) 1, Airy stress function Integrate above: 1) Stress components Stress Component Force M (主要由弯矩引起；) Q (主要由剪力引起；) q (由 q 引起) 16 8

  22. Theory of Elasticity Page Chapter 8.3 Bending of a Beam by Uniform Transverse Loading(受均匀横向载荷的梁弯曲问题) 3）satisfy the biharmonic equation 17 8

  23. q 1 ql ql h/2 z h/2 x y l l y Theory of Elasticity Page Chapter 8.3 Bending of a Beam by Uniform Transverse Loading(受均匀横向载荷的梁弯曲问题) 关于 x 的二次方程，且要求 －l≤ x ≤ l 内方程均成立。 此处略去了f1(y)中的常数项 18 8

  24. Theory of Elasticity Page Chapter 8.3 Bending of a Beam by Uniform Transverse Loading(受均匀横向载荷的梁弯曲问题) 9 unknown coefficients 2, stress field 19 8

  25. q ql ql x l l y Theory of Elasticity Page Chapter 8.3 Bending of a Beam by Uniform Transverse Loading(受均匀横向载荷的梁弯曲问题) 3, Boundary Conditions 1) Symmetry Condition —— x 的偶函数 —— x的奇函数 20 8

  26. q ql ql x l l y Theory of Elasticity Page Chapter 8.3 Bending of a Beam by Uniform Transverse Loading(受均匀横向载荷的梁弯曲问题) 2) Boundary Conditions a) Top and bottom(上下面，主要) 21 8

  27. q ql ql x l l y Theory of Elasticity Page Chapter 8.3 Bending of a Beam by Uniform Transverse Loading(受均匀横向载荷的梁弯曲问题) b) End conditions(左右边界，次要) Using Saint-Venant’s Principle 轴力 N = 0； statically equivalent force 弯矩 M = 0； 剪力 Q = －ql； 22 8

  28. Theory of Elasticity Page Chapter 8.3 Bending of a Beam by Uniform Transverse Loading(受均匀横向载荷的梁弯曲问题) 轴力 N = 0； 弯矩 M = 0； 剪力 Q = －ql； 自动满足 23 8

  29. q ql ql x l l y Theory of Elasticity Page Chapter 8.3 Bending of a Beam by Uniform Transverse Loading(受均匀横向载荷的梁弯曲问题) 4, compare with elementary strength of materials 24 8

  30. Theory of Elasticity Page Chapter 8.3 Bending of a Beam by Uniform Transverse Loading(受均匀横向载荷的梁弯曲问题) 4, compare with elementary strength of materials 第一项与材力结果相同，为主要项。 第二项为修正项。当 h / l<<1，该项误差很小，可略；当 h / l较大时，须修正。 为梁各层纤维间的挤压应力，材力中不考虑。 与材力中相同。 25 8

  31. q ql ql x l l y Theory of Elasticity Page Chapter 8.3 Bending of a Beam by Uniform Transverse Loading(受均匀横向载荷的梁弯曲问题) 26 8

  32. Theory of Elasticity Page Chapter Vocabulary(词汇) Polynomials 多项式 Uniaxial 单轴的 Beam 梁 Uniform 均匀的 Inverse Method逆解法 Semi-Inverse Method半逆解法 Biharmonic 双调和的 Airy stress function艾里应力函数 27 8

  33. o x b q ρg Theory of Elasticity y 图1 o x α ρg y 图2 Page Chapter Homework 1:设有矩形截面的长竖柱,密度为ρ,在一边侧面上受均布剪力q,如图1,试求应力分量. 提示:可假设σx=0,或假设τxy=f(x),或假设σy如材料力学中偏心受压公式所示.上端边界条件如不能精确满足,可应用圣维南原理. 2:设图2中的三角形悬臂梁只受重力作用,而梁的密度为ρ,试用纯三次式的应力函数求解. 27 8

  34. y o b/2 b/2 ρ2g ρ1g Theory of Elasticity x 图3 Page Chapter Homework 3:挡水墙的密度为ρ1,厚度为b,图3,水的密度为ρ2,试求应力分量. (提示:可假设σy=xf(y)上端的边界条件如不能精确满足,可应用圣维南原理,求出近似的解答) 27 8

More Related