《弹性力学》（双语版） 第十二章 薄板弯曲

Elasticity

1 Elasticity

弹单性力等 第十二氯言

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Chapter l2 Sheet bending D Summarization D8 12-1 Basic Hypothesis 812-2 Basic Functions D8 12-3 Internal Force of Cross Section 」§12-4 Boundary condition of sheet D8 12-5 Solution of Sheet Bending under Rectangular Coordinate [8 12-6 Axisymmetric Bending of Circular Sheet D8 12-7 Solution of Displacement of Sheet by Calculus of variation 3

3 Chapter12 Sheet bending Summarization § 12－1 Basic Hypothesis § 12－2 Basic Functions § 12－3 Internal Force of Cross Section § 12－4 Boundary Condition of Sheet § 12－5 Solution of Sheet Bending under Rectangular Coordinate § 12－6 Axisymmetric Bending of Circular Sheet § 12－7 Solution of Displacement of Sheet by Calculus of Variation

第十二章薄板弯曲 D]概述 >第一节基本假设 第二节基本方程 第三节横截面上的内力 D]第四节薄板的边界条件 ]第五节薄板弯曲的直角坐标求解 ]第六节圆形薄板的轴对称弯曲 >第七节变分法求薄板的位移

4 第十二章 薄板弯曲 概述 第一节 基本假设 第二节 基本方程 第三节 横截面上的内力 第四节 薄板的边界条件 第五节 薄板弯曲的直角坐标求解 第六节 圆形薄板的轴对称弯曲 第七节 变分法求薄板的位移

Summarization Sheet is different from thick board Generally if the ratio of the thickness of the board and the minimal dimension of the board face satisfies 80100b(58 We call the board sheet We call the plane halves 三==== the thickness of the board middle pl ane Choose the ordinate origin as a point of the middle plane and y axes of x and y in the middle plane, z perpendicular to it which are shown in fig

5 Summarization Sheet is different from thick board。Generally,if the ratio of the thickness of the board and the minimal dimension of the board face satisfies:             8 1 ~ 5 1 100 1 ~ 80 1 ＜ ＜ b t We call the board sheet. Choose the ordinate origin as a point of the middle plane, and axes of x and y in the middle plane, z perpendicular to it, which are shown in fig. . x y z o We call the plane halves the thickness of the board middle plane

薄城曲 概述 薄板区别于厚板。通常情况下,板的厚度t与板面的最 小尺寸b的比值满足如下条件: 80100b(58 则称为薄板。 我们把平分板厚度的平 三==== 面称为中面。 将坐标原点取于中面 内的一点,x和y轴在中面 内,z垂直轴向下,如图 所示

6 概述 薄板区别于厚板。通常情况下，板的厚度t与板面的最 小尺寸b的比值满足如下条件:             8 1 ~ 5 1 100 1 ~ 80 1 ＜ ＜ b t 则称为薄板。 将坐标原点取于中面 内的一点,x和y轴在中面 内，z 垂直轴向下，如图 所示。 x y z o 我们把平分板厚度的平 面称为中面

When sheet accepts common load, we always divide the load into two components. One is transverse load, which is perpendicular to middle plane, one is longitudinal load, which acts in middle plane. For the latter, we assume its distributing is even along the thickness of the sheet, and treat it as the plane stress problem. In this chapter, we just discuss the stress, Strain and displacement when sheet is bent because of transverse load

7 When sheet accepts common load, we always divide the load into two components. One is transverse load, which is perpendicular to middle plane, one is longitudinal load, which acts in middle plane. For the latter, we assume its distributing is even along the thickness of the sheet, and treat it as the plane stress problem. In this chapter, we just discuss the stress、strain and displacement when sheet is bent because of transverse load

等遗 当薄板受有一般载荷时,总可以把每一个载荷分解 为两个分量,一个是垂直于中面的横向载荷,另一个是 作用于中面之内的纵向载荷。对于纵向载荷,可认为它 沿薄板厚度均匀分布,按平面应力问题进行计算。本章 只讨论由于横向载荷使薄板发生小挠度弯曲所引起的应 力、应变和位移

8 当薄板受有一般载荷时，总可以把每一个载荷分解 为两个分量，一个是垂直于中面的横向载荷，另一个是 作用于中面之内的纵向载荷。对于纵向载荷，可认为它 沿薄板厚度均匀分布，按平面应力问题进行计算。本章 只讨论由于横向载荷使薄板发生小挠度弯曲所引起的应 力、应变和位移

8 12-1 Basic Hypothesis For small bending problem of sheet, we generally adopt these assumptions (1) fixity of the board thickness The normal strain E perpendicular to middle plane is very small, thus we can ignore it. Namely .0 From geometric equations, we have =0, thus we get w=w(x, y) Namely: any normal which is perpendicular to middle plane has the same bending (2) fixity of normal of the middle plane

9 § 12－1 Basic Hypothesis For small bending problem of sheet, we generally adopt these assumptions: （1）fixity of the board thickness Namely: any normal which is perpendicular to middle plane has the same bending. （2）fixity of normal of the middle plane w= w(x, y) The normal strain perpendicular to middle plane is very small, thus we can ignore it. Namely . From geometric equations, we have ,thus we get: = 0 z  = 0   z w z 

薄城曲 §12-1基本假设 薄板小挠度弯曲问题,通常采用如下假设: (1)板厚不变假设 垂直于中面方向的正应变E很小,可以忽略不计。 即E=0,由几何方程aW=0,从而有: w=w(x,y) 即:在垂直于中面的任一条法线上,各点都具有相同的 挠度。 (2)中面法线保持不变假设 10

10 § 12－1 基本假设 薄板小挠度弯曲问题，通常采用如下假设： （1）板厚不变假设 W =W(x, y) 即：在垂直于中面的任一条法线上，各点都具有相同的 挠度。 （2）中面法线保持不变假设 垂直于中面方向的正应变 很小，可以忽略不计。 即  z = 0 ，由几何方程得 = 0 ，从而有：   z W  z