(新)弹性力学弹性力学 solution of plane problems in polar coordinates2——精品.pptVIP

* * 第四章 平面问题的极坐标解答 4—1 Differential Equations of Equilibrium in Polar Coordinates 4—2 Geometrical Equations and Physical Equations in Polar Coordinates 4—3 Stress Function and Compatibility Equations in Polar Coordinates 4—9 Effect of circular holes on stress distribution 4. Solution of Plane Problems in Polar Coordinates 4—4 Coordinates Transformation of Stress Components 4—5 Axisymmetrical Stresses and Cooresponding Displacements 4—6 Hollow Cylinder Subjected to Uniform Pressures 4—11 Concentrated Normal Load on a Straight Boundary 4.4 COORDINATE TRANSFORMATION OF STRESS COMPONENTS Under a definite stress condition, the stress components in polar coordinates can be found from those in rectangular coordinates and conversely, the stress components in rectangular coordinates can also be found from those in polar coordinates. Now, suppose the stress components in rectangular coordinates, ?x, ?y and ?xy, are known and it is required to find those in the polar coordinates, ?r, ?? and ?r?. ?x ?xy ?y ?yx ?r ?r? A x y o a c b ? ? Take a triangular element A，with side ab in the y direction, side ac in the x direction and side cb in the ? direction. The thickness of the element is taken as unity. Let the length of cb be ds, then ab=ds·cos? ac=ds·sin? According to the equilibrium of element A in the r direction, we can write out the equilibrium equation as： Similarly, we can write out the equilibrium equation for the element A and obtain： ?x B x y o ? ? ?y ?yx ?xy ?? ?r? Taking another triangular element B, and writing out the equilibrium equation , we can obtain Thus, the formulas for evaluating the stress components in polar coordinates from those in rectangular coordinates are： The formulas for evaluating the stress components in rectangular coordinates from those in polar coordinates are： The students are advised to derive those. 4.5 AXISYMMETRIAL STRESSES AND COORESPONDING DISPLACEMENTS Now we apply the inverse method and assume the str