Page 209 - Standard Handbook Of Petroleum & Natural Gas Engineering

194 General Engineering and Science

Pr
(T =(T =- (2-97)
a 2t
For a cylindrical vessel, the radius of curvature in the axial direction is infinite,
and the stress in the direction of the circumference, called the hoop stress, is

Pr
fl=- (2-98)
t
The stress in the axial direction in a cylindrical vessel is found by taking a cross-
section perpendicular to the longitudinal axis and imposing the conditions of static
equilibrium. This yields

Pr
(T =- (2-99)
2t
Prediction of Failure
For most practical purposes, the onset of plastic deformation constitutes failure. In an
axially loaded part, the yield point is known from testing (see Tables 2-15 through 2-18),
and failure prediction is no problem. However, it is often necessary to use uniaxial tensile
data to predict yielding due to a multidimensional state of stress. Many failure theories
have been developed for this purpose. For elastoplastic materials (steel, aluminum, brass,
etc.), the maximum distortion energy theory or urn Misa theory is in general application.
With this theory the components of stress are combined into a single effective stress,
denoted as G~, which can be compared to known data for uniaxial yielding. The ratio of
the measure yield stress to the effective stress is known as the factor of safety.

1
(T, = { [ ( (T, - (T, )2 + ( 6, - 0. )' + ( (T, - 6, )* + 6 ( T:, + zfZ + T: I]} 1/2 (2-1 00)

For brittle materials such as glass or cast iron, the maximum shear-stress theory is
usually applied.

Example 2-21
A cylindrical steel pressure vessel (AIS1 SAE 1035, cold rolled) with a wall thickness
of 0.1 in. and an inside diameter of 1 ft is subject to an internal pressure of 1,000 psia
and a torque of 10,000 ft-lb (see Figure 2-30). What is the effective stress at point A in
the wall? What is the factor of safety in this design?
Hoop stress:

(1,000 psi)(6 in.)
(To = = 60,000 psi
(0.1 in.)
Axial stress:
(1,000 psi)(6 in.)
OZ = = 30,000 psi
2(0.1 in.)

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