文档介绍：Chapter 9 Torsion
Elasticity
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第九章扭转
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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
Torsion
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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
扭转
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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 plex section pole, we introduction the method of membrane assimilation.
Torsion
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扭转
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。
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§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 ponents are zero except for the shear stress τzx、τzy
Substitute the ponents and body forces X=Y=Z=0 into the equations of equilibrium, we get
x
M
M
o
y
z
Torsion
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扭转
§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
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
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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:
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
The function x,y is called stress function of torsion problems.