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)