《弹性力学》（双语版） 第四章 平面问题的极坐标解答

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‖弹性力

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POtLSOUTIOFORP Chapter 4 Polar Solutions For Planar Problems 84-1 Differential Equations of Equilibrium in Polar Coordinates 84-2 Geometric and Physical Formulas in Polar Coordinates D$4-3 Stress Functions and Consistent Equations in Polar coordinates 84-4 Coordinates Transforms for Stress Components 84-5 Axially Symmetric Stress and Its Displacement >84-6 Circular Ring or Cylinder under Uniform Loading pressure Tunnel

3 Chapter 4 Polar Solutions For Planar Problems §4-1 Differential Equations of Equilibrium in Polar Coordinates §4-5 Axially Symmetric Stress and Its Displacement §4-2 Geometric and Physical Formulas in Polar Coordinates §4-3 Stress Functions and Consistent Equations in Polar Coordinates §4-4 Coordinates Transforms for Stress Components §4-6 Circular Ring or Cylinder under Uniform Loading Pressure Tunnel

第四章平面问题的极坐标解答 §4-1极坐标中的平衡微分方程 §4-2极坐标中的几何方程及物理方程 s4-3极坐标中的应力函数与相容方程 §4-4应力分量的坐标变换式 s4-5轴对称应力和相应的位移 §4-6圆环或圆筒受均布压力压力隧洞

4 第四章 平面问题的极坐标解答 §4-1 极坐标中的平衡微分方程 §4-4 应力分量的坐标变换式 §4-3 极坐标中的应力函数与相容方程 §4-2 极坐标中的几何方程及物理方程 §4-5 轴对称应力和相应的位移 §4-6 圆环或圆筒受均布压力压力隧洞

POtLSOUTIOFORP Chapter 4 Polar Solutions For Planar problems 84-7 Pure Bending of Curved Beam 84-8 Stress and Displacement of Disc in Uniform Rotation 84-9 Stress Concentration Close by Circular Aperture 84-10 Force on the Top and the Faces of Wedge >84-11 Normal Concentrated Forces on the Boundary for Semi-plane body Exercise

5 §4-7 Pure Bending of Curved Beam Chapter 4 Polar Solutions For Planar Problems §4-11 Normal Concentrated Forces on the Boundary for Semi-plane Body Exercise §4-9 Stress Concentration Close by Circular Aperture §4-10 Force on the Top and the Faces of Wedge §4-8 Stress and Displacement of Disc in Uniform Rotation

第四章平面问题的极坐标解答 §4-7曲梁的纯弯曲 s4-8圆盘在匀速转动中的应力及位移 s49圆孔的孔边应力集中 §4-10楔形体在楔顶或楔面受力 §4-11半平面体在边界上受法向集中力 习题课

6 §4-9 圆孔的孔边应力集中 §4-7 曲梁的纯弯曲 §4-8 圆盘在匀速转动中的应力及位移 §4-10 楔形体在楔顶或楔面受力 §4-11 半平面体在边界上受法向集中力 习题课 第四章 平面问题的极坐标解答

84-1 Differential Equations of Equilibrium in Polar Coordinates Dealing with elasticity problems, what form of coordinate system we choose, which can t affect on describing problem essence, but relate to the level of difficulty on solving problem directly. If coordinate is suitable, it can simplify the problem considerably o For example, for circular wedged and sector and so on, solved by using polar coordinates are more convenient than using rectangular coordinates Considering an differential field PacB in the plate 7

7 §4-1 Differential Equations of Equilibrium in Polar Coordinates Dealing with elasticity problems, what form of coordinate system we choose，which can’t affect on describing problem essence, but relate to the level of difficulty on solving problem directly。If coordinate is suitable，it can simplify the problem considerably。For example, for circular、wedged and sector and so on，solved by using polar coordinates are more convenient than using rectangular coordinates. Considering an differential field PACB in the plate

§4-1极坐标中的平衡微分方程 在处理弹性力学问题时,选择什么形式的坐标系统,虽 不会影响对问题本质的描绘,但将直接关系到解决问题的难 易程度。如坐标选得合适,可使问题大为简化。例如对于圆 形、楔形、扇形等物体,采用极坐标求解比用直角坐标方便 的多。 考虑平面上的一个微分体PACB

8 §4-1 极坐标中的平衡微分方程 在处理弹性力学问题时，选择什么形式的坐标系统，虽 不会影响对问题本质的描绘，但将直接关系到解决问题的难 易程度。如坐标选得合适，可使问题大为简化。例如对于圆 形、楔形、扇形等物体，采用极坐标求解比用直角坐标方便 的多。 考虑平面上的一个微分体PACB

POtLSOUTIOFORP normal stress in the r direction is called radial normal stress denoted by o. normal stress in the 0 direction is called tangential normal stress ss denoted by oes shear stress is denoted by I-o, stipulation of sign of each stress component are similar to ones in rectangular coordinates. Body force components of radial and hoop are denoted by k and Ke respectively. Fig 4-1 Considering equilibrium of an unit O叉 element, there have three equilibrium equations fro+.gp dr dr ∑F=0.∑F=0.∑M=0 o+-rdr ar Fig 4-1 9

9 Considering equilibrium of an unit element,there have three equilibrium equations: r r r r d       r r r r r d              d          d r r     d r r  dr K Kr y x o P A B C Fig.4－1 Fr  0,F  0, M  0 normal stress in the direction is called radial normal stress denoted by ;normal stress in the direction is called tangential normal stress denoted by ;shear stress is denoted by ,stipulation of sign of each stress component are similar to ones in rectangular coordinates.Body force components of radial and hoop are denoted by and ,respectively. Fig.4-1.  r     r Kr K r

沿r方向的正应力称为径向正应力, x 用表沿方的正应力称为 P 环向正应力,用表,剪应力 100 用表,各应力分量的正负号 T 的规定和直角坐标中一样。径向及 环向的体力分量分别用及K,表。 dr .+ 示。如图4-1。 考虑图示单元体的平衡,有三个平衡方程: 图4-1 ∑F=0.∑F=0∑M=0 10

10 图4－1 沿 方向的正应力称为径向正应力， 用 表示沿 方向的正应力称为 环向正应力，用 表示，剪应力 用 表示，各应力分量的正负号 的规定和直角坐标中一样。径向及 环向的体力分量分别用 及 表 示。如图4-1。 r  r     r Kr K r r r r d       r r r r r d              d          d r r     d r r  dr K Kr y x o P A B C 考虑图示单元体的平衡，有三个平衡方程： Fr  0,F  0, M  0