Unit 4 Membrane Stress in Shells of Revolution
回转壳的薄膜应力
A shell of revolution is the form swept out by a line or curve rotated about an axis (A solid of revolution is formed by rotating an area about an axis). Most process vesls are made up from shells of revolution; cylindrical and conical ctions; and hemispherical, ellipsoidal and torispherical heads; Fig.1.13.
回转壳是由一条直线或曲线绕着一根轴旋转形成的曲面（回转体是由一个面绕着一轴旋转而成的）。大多数过程容器是由回转壳组成的：圆柱和圆锥形的；半球形的；椭球形的和准球形的；如图1.13.
Fig.1.13 Fig.1.14
The walls of thin vesls can be considered to be “membranes”; supporting loads without significant bending or shear stress; similar to the walls of a balloon.
薄容器的器壁可以看成是“薄膜”；假设载荷没有明显挠度和剪应力；就像气球的薄膜。
The analysis of the membrane stress induced in shells of revolution by internal pressure gives a basis for determining the minimum wall thickness required for vesl shells. The actual thickness required will also depend on the stress arising from the other loads to which the vesl is subjected.
对受内压回转体的薄膜应力分析为确定容器壳体最小壁厚奠定了基础。实际所需壁厚也依赖于容器上所受其他载荷引起的应力。
Consider the shell of revolution of general shape shown in Fig.1.14 under a loading that is rotationally symmetric; that is, the load per unit area (pressure) on the shell is constant round the circumference, but not necessarily the same from top to bottom.
如图1.14，研究的是回转壳一般模型在旋转轴对称的载荷作用下的情行；也就是，单位圆周面积上的载荷是一定的，但不都是从上到下。
Let P=pressure, 令：P=压力
t=thickness of shell, t=壁厚
=the meridional (longitudinal) stress, the stress acting along a meridian, =经向应力
=the circumferential or tangential stress, the stress acting along parallel circles (often called the hoop stress), =周向应力
=the meridional radius of curvature,r1=经向曲率半径
=circumferential radius of curvature.r2=环向曲率半径
Note: the vesl has a double curvature; the values of and are determined by the shape.
注意：容器有两个曲率；r1 和r2值由模型决定。
Consider the forces acting on the element defined by the points a. b c, d. Then the normal component (component acting at right angels to the surface) of the force on the element
假设作用在元件上的力由a、b、c、d四点确定。那么元件上的法向力（垂直于表面的分力）
This force is resisted by the normal component of the forces associated with the membrane stress in the walls of vesl （given by, force=stressarea）
通过与容器器壁的薄膜应力联系的法向力来抵抗这个力（即，力=应力*面积）
Equating the forces and simplifying, and noting that in the limit, and, gives:
假设这些力是相等的并简化，取极限, 和，有
(1.12)
An expression for the meridional stress can be obtained by considering the equilibrium of the forces acting about any circumferential line, Fig 1.14. The vertical component of the pressure force
经向应力的一个表达式可由作用在周向线上力的平衡得到，如图1.14.压力的垂直分量
This is balanced by the vertical component of the force due to the meridional stress acting in the ring of the wall of the vesl
=2σ1tл(r2sinθ) sinθ
这与作用在容器壁上的经向力的垂直分力是平衡的
=2σ1tл(r2sinθ) sinθ
Equating the forces gives:令这两个力相等有：
(1.13)
Equations (1.12) and (1.13) are completely general for any shell of revolution.
方程（1.12）和（1.13）对所有的回转壳都适用。
Cylinder(Fig.1.15a)圆柱体（图1.15a）
A cylinder is swpet out by the rotation of a line parallel to the axis of revolution, so:
圆柱体是由一条平行于轴的直线绕着轴旋转形成的，于是：
where D is the cylinder diameter.这里D是圆柱底面直径。
Substitution in equations (1.12) and (1.13) gives:
代入方程（1.12）和（1.13）得：
(1.14)
Sphere(Fig.1.15b)球（图1.15b）
hence:于是:
(1.15)
Cone(Fig.1.15c)圆锥（图1.15c）
