《弹性力学》（双语版） 第九章 扭转

Elasticity Chanter g torsion

1 Elasticity

弹单性生力学 第九转

2

Chapter 9 Torsion 89-1 The Torsion of Equal Section Pole 89-2 The Torsion of Elliptic Section Pole 89-3 Membrane assimilation 89-4 The Torsion of Rectangular Section Pole 89-5 The Torsion of Ringent Thin Cliff Pole

3 Chapter 9 Torsion §9-1 The Torsion of Equal Section Pole §9-2 The Torsion of Elliptic Section Pole §9-3 Membrane assimilation §9-4 The Torsion of Rectangular Section Pole §9-5 The Torsion of Ringent Thin Cliff Pole

第九章扭转 §9-1等截面直杆的扭转 □§9-2椭圆截面杆的扭转 「§9-3薄膜比拟 □§9-4矩形截面杆的扭转 §9-5开口薄壁杆件的扭转

4 第九章 扭 转 §9-1 等截面直杆的扭转 §9-2 椭圆截面杆的扭转 §9-3 薄膜比拟 §9-4 矩形截面杆的扭转 §9-5 开口薄壁杆件的扭转

Material mechanics has solved the torsion problems of round section pole, but it cant be used to analyze the torsion problems of non-round section pole. For the torsion of any section pole, it is a relatively simple spatial problem. According to the characteristic of the problem, this chapter first gives the differential functions and boundary conditions, which the stress function should satisfy of solving the torsion problems. Then, in order to solve the torsion problems of relatively complex section pole, we introduction the method of membrane assimilation

5 Material mechanics has solved the torsion problems of round section pole, but it can’t be used to analyze the torsion problems of non-round section pole. For the torsion of any section pole, it is a relatively simple spatial problem. According to the characteristic of the problem, this chapter first gives the differential functions and boundary conditions, which the stress function should satisfy of solving the torsion problems. Then , in order to solve the torsion problems of relatively complex section pole, we introduction the method of membrane assimilation

材料力学解决了圆截面直杆的扭转问题,但对非 圆截面杄的扭转问题却无法分析。对于任意截面杆的」 扭转,这本是一个较简单的空间问题,根据问题的特 点,本章首先给出了求解扭转问题的应力函数所应满 足的微分方程和边界条件。其次,为了求解相对复杂 截面杆的扭转问题,我们介绍了薄膜比拟方法

6 材料力学解决了圆截面直杆的扭转问题，但对非 圆截面杆的扭转问题却无法分析。对于任意截面杆的 扭转，这本是一个较简单的空间问题，根据问题的特 点，本章首先给出了求解扭转问题的应力函数所应满 足的微分方程和边界条件。其次，为了求解相对复杂 截面杆的扭转问题，我们介绍了薄膜比拟方法

89-1 The Torsion of Equal Section Pole 1. Stress Function A equal section straight pole, ignoring the body force, is under the action of torsion M at its two end planes. Take one end as the xy plane, as shown in fig. The other stress components are zero except for the shear stress txx, tzy o.=0.=0.=7=0 Substitute the stress components and body forces X-r-=Z-0 into the equations of equilibrium, we get

7 §9-1 The Torsion of Equal Section Pole 1. Stress Function A equal section straight pole, ignoring the body force, is under the action of torsion M at its two end planes. Take one end as the xy plane,as shown in fig. The other stress components are zero except for the shear stress τzx、τzy = = = = 0 x y z x y     Substitute the stress components and body forces X=Y=Z=0 into the equations of equilibrium, we get x M M o y z

§9-1等微面直杆的扭转 应力函数 设有等截面直杆,体力不计, 在两端平面内受扭矩M作用。取杆 的一端平面为x面,图示。横截 面上除了切应力了 以外, 其余的应力分量为零 .=0.=0==0 将应力分量及体力￥2=0代入平 衡方程。得 8

8 §9-1 等截面直杆的扭转 一 应力函数 设有等截面直杆，体力不计， 在两端平面内受扭矩M作用。取杆 的一端平面为 xy面，图示。横截 面上除了切应力τzx、τzy以外， 其余的应力分量为零 = = = = 0 x y z x y     将应力分量及体力X=Y=Z=0代入平 衡方程，得 x M M o y z

Annotation: the differential equations r 0 =0.x+ r =0 of equilibrium for spatial problems are a From the first two equations. we ++x+Y= 0 know,rx、τ are functions of onyx az do at aT and y, they have nothing to do with z. +二+Y=0 From the third formula aT +Z=0 a x二 According to the theory of differential equations, there must exist a function op(x y), from it T三 00 T-_op(a) a ax The function (p(x,y) is called stress function of torsion problems 9

9 ( ) ( ) x z y z x y       = − From the first two equations, we know,τzx、τzy are functions of only x and y, they have nothing to do with z. From the third formula: = 0, = 0, + = 0 z z x y z x z y x z y z                  + =   +   +   + =   +   +   + =   +   +   0 0 0 Z z x y Y y z x X x y z xz yz z y zy xy x yx zx          Annotation : the differential equations of equilibrium for spatial problems are: According to the theory of differential equations, there must exist a function (x,y), from it , y z x    = x zy   = −   The function (x,y) is called stress function of torsion problems. (a)

扭装 Ot aT 注:空间问题平衡微分方程 =0,=0,+-=0 a y ++x+Y= 根据前两方程可见,x、a只是 0 az x和y的函数,与z无关,由第三式 do at aT +二+Y=0 (x2)=,(-y) ao ar ar +Z=0 az Ox ay 根据微分方程理论,一定存在一 个函数φ(x,y),使得 X (a) a 函数(x,y)称为扭转问题的应力函 数。 10

10 ( ) ( ) x z y z x y       = − 根据前两方程可见，τzx、τzy只是 x和y的函数，与z无关，由第三式 = 0, = 0, + = 0 z z x y z x z y x z y z                  + =   +   +   + =   +   +   + =   +   +   0 0 0 Z z x y Y y z x X x y z xz yz z y zy xy x yx zx          注：空间问题平衡微分方程 根据微分方程理论，一定存在一 个函数(x,y)，使得 , y z x    = x zy   = −   函数(x,y)称为扭转问题的应力函 数。 (a)