Chapter 2 Theory of Plane Problems 第二章:平面问题的理论 2.1 Plane stress and plane strain 平面应力问题与平面应变问题 徐汉忠第二版2001/10 弹性力学第二章
徐汉忠第二版2001/10 弹性力学第二章 1 Chapter 2 Theory of Plane Problems 第二章：平面问题的理论 2.1 Plane stress and plane strain 平面应力问题与平面应变问题
Spatial problems and plane problems 空间问题转化为平面问题。 a spatial problem- a plane problem the body has a particular shape particular external forces. 当物体的形状特殊,外力分布特殊,空间问题 转化为平面问题 Plane problems: plane stress problems and plane strain problems 平面问题:平面应力问题与平面应变问题 徐汉忠第二版2001/10 弹性力学第二章 2
徐汉忠第二版2001/10 弹性力学第二章 2 Spatial problems and plane problems 空间问题转化为平面问题。 • a spatial problem a plane problem the body has a particular shape. particular external forces. • 当物体的形状特殊，外力分布特殊，空间问题 转化为平面问题。 • Plane problems: plane stress problems and plane strain problems 平面问题: 平面应力问题与平面应变问题
A conditions for plane stress problem 平面应力问题的条件 Body-a very thin plate of uniform thickness t 很薄的等厚度薄板 External forces 1. The surface forces act on the edges only 面力仅作用板周边 2. The surface forces and body forces are parallel to the faces of the plate and distributed uniformly over the thickness 面力体力平行于板面且沿厚度无变化 徐汉忠第二版2001/10 弹性力学第二章
徐汉忠第二版2001/10 弹性力学第二章 3 A. conditions for plane stress problem 平面应力问题的条件 • Body--a very thin plate of uniform thickness t. 很薄的等厚度薄板 • External forces-- 1. The surface forces act on the edges only. 面力仅作用板周边 2. The surface forces and body forces are parallel to the faces of the plate and distributed uniformly over the thickness. 面力体力平行于板面且沿厚度无变化
t/2 t/2 z Fig 2.1.1 徐汉忠第二版2001/10 弹性力学第二章
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B Coordinate system for plane stress problem 平面应力问题的坐标系 X and y axes are in the middle plane and z axis is perpendicular to the middle pla ane x,y轴放在薄板的中面内,z垂直于中面 徐汉忠第二版2001/10 弹性力学第二章
徐汉忠第二版2001/10 弹性力学第二章 5 B. Coordinate system for plane stress problem 平面应力问题的坐标系 • x and y axes are in the middle plane and z axis is perpendicular to the middle plane. • x,y轴放在薄板的中面内，z垂直于中面
C Stresses for plane stress problem 平面应力问题的应力 Noting the absence of surface forces on the faces of the plate, we have板面无面力作用故: z=±t/2 since stress gradients through plate are small, 通过板的应力梯度小 ( oz tzx t)=any≈0 (o oy tx=0, They are functions of x and y only. the plate is said to be in a plane stress condition 板称为处于平面应力状态。 徐汉忠第二版2001/10 弹性力学第二章
徐汉忠第二版2001/10 弹性力学第二章 6 C. Stresses for plane stress problem 平面应力问题的应力 • Noting the absence of surface forces on the faces of the plate, we have板 面无面力作用故: (z zx zy)z=t/2=0 • since stress gradients through plate are small, 通过板的应力梯度小 (z zx zy)z=any ≈ 0 • (x y xy)≠0 ,They are functions of x and y only. • the plate is said to be in a plane stress condition 板称为处于平面应力状态
D. conditions for plane strain problem 平面应变问题的条件 Body--a cylindrical or prismatical body with infinite length无限长的柱形体 External forces 1. The surface forces are acting on the lateral surface 面力仅作用横向周边 2. The surface forces and body forces are parallel to any cross section of the body and not varying along the axial direction 面力体力平行于横截面且不沿长度变化。 徐汉忠第二版2001/10 弹性力学第二章
徐汉忠第二版2001/10 弹性力学第二章 7 D. conditions for plane strain problem 平面应变问题的条件 • Body--a cylindrical or prismatical body with infinite length 无限长的柱形体 • External forces-- 1. The surface forces are acting on the lateral surface. 面力仅作用横向周边 2. The surface forces and body forces are parallel to any cross section of the body and not varying along the axial direction. 面力体力平行于横截面且不沿长度变化
Fig.2.12 徐汉忠第二版2001/10 弹性力学第二章
徐汉忠第二版2001/10 弹性力学第二章 8
E Coordinate system for plane strain problem 平面应变问题的坐标系 X and y axes are in any cross section of the body, and z axis is perpendicular to the xy plane. xy轴放在任意的横截面内,z垂直于xy面 徐汉忠第二版2001/10 弹性力学第二章
徐汉忠第二版2001/10 弹性力学第二章 9 E. Coordinate system for plane strain problem 平面应变问题的坐标系 • x and y axes are in any cross section of the body, and z axis is perpendicular to the xy plane. • x,y轴放在任意的横截面内，z垂直于xy面
F Displacements for plane strain problem 平面应变问题的位移 Noting motion constrained in z direction, we have W=0 z向运动受限制,故:w=0 (u, v)*0, They are functions of x and y only uⅴ通常不为零,且只是xy的函数 Plane displacement problem 平面位移问题 徐汉忠第二版2001/10 弹性力学第二章
徐汉忠第二版2001/10 弹性力学第二章 10 F. Displacements for plane strain problem 平面应变问题的位移 • Noting motion constrained in z direction, we have ： w=0 z向运动受限制，故：w=0 • (u,v)≠0,They are functions of x and y only u v 通常不为零，且只是x y的函数。 • Plane displacement problem 平面位移问题