5.3
Several assumptions are made in this method.
1) Plane sections remain plane
2) r/t = 10 with t being uniform and constant
3) The applied pressure, p, is the gage pressure (note that p is the
difference between the absolute pressure and the atmospheric pressure)
4) Material is linear-elastic, isotropic and homogeneous.
5) Stress distributions throughout the wall thickness will not vary
6) Element of interest is remote from the end of the cylinder and other
geometric discontinuities.
7) Working fluid has negligible weight
Cylindrical Vessels: A cylindrical pressure with wall thickness, t, and inner radius,
r, is considered, (see Figure 12.1). A gauge pressure , p, exists within the vessel by the
working fluid (gas or liquid). For an element sufficiently removed from the ends of the
cylinder and oriented as shown in Figure 12.1, two types of normal stresses are
generated: hoop, s h , and axial, s a , that both exhibit tension of the material.
Combination of axial and bending stresses acting on a member simultaneously, such as occurs in the top chord (compression + bending) or bottom chord (tension + bending) of a truss.
In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body. Because the loaded deformable body is assumed to behave as a continuum, these internal forces are distributed continuously within the volume of the material body, and result in deformation of the body s shape. Beyond certain limits of material strength, this can lead to a permanent shape change or structural failure.
However, models of continuum mechanics which explicitly express force as a variable generally fail to merge and describe deformation of matter and solid bodies, because the attributes of matter and solids are three dimensional. Classical models of continuum mechanics assume an average force and fail to properly incorporate "geometrical factors", which are important to describe stress distribution and accumulation of energy during the continuum.
The dimension of stress is that of pressure, and therefore the SI unit for stress is the pascal (symbol Pa), which is equivalent to one newton (force) per square meter (unit area), that is N/m2. In Imperial units, stress is measured in pound-force per square inch, which is abbreviated as psi.
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Cylindrical or spherical pressure vessels (e.g., hydraulic cylinders, gun barrels,
pipes, boilers and tanks) are commonly used in industry to carry both liquid s and gases
under pressure. When the pressure vessel is exposed to this pressure, the material
comprising the vessel is subjected to pressure loading, and hence stresses, from all
directions. The normal stresses resulting from this pressure are functions of the radius of
the element under consideration, the shape of the pressure vessel (i.e., open ended
cylinder, closed end cylinder, or sphere) as well as the applied pressure.
Two types of analysis are commonly applied to pressure vessels. The most
common method is based on a simple mechanics approach and is applicable to “thin
wall” pressure vessels which by definition have a ratio of inner radius, r, to wall thickness,
t, of r/t=10. The second method is based on elasticity solution and is always applicable
regardless of the r/t ratio and can be referred to as the solution for “thick wall” pressure
vessels. Both types of analysis are discussed here, although for most engineering
applications, the thin wall pressure vessel can be used.
Thin-Walled Pressure Vessels