Page 203 - Standard Handbook Of Petroleum & Natural Gas Engineering

188 General Engineering and Science

direction of the shear stress is indicated by two subscripts, the first of which indicates
the direction normal to the plane in which the load is applied, and the second of
which indicates the direction of the load. Note that for static equilibrium to exist,
zxy = zyx, zu = z=, and zYz = T~~.
If a multidimensional state of stress exists, the Poisson’s ratio effect causes the tensile
and compressive strains to be dependent upon each of the components of stress.

1
E, = -[ox - V(OY +o,)] (2-78)
E
1
E
E, = -[oy - v(0, + (T,)] (2-79)

1
E, = -[oz - v(0, + (T,)] (2-80)
E
Likewise the stresses may be written in terms of the components of strain.

(2-81)

(T= E ((1 - V)E, + V(E, + 4 (2-82)
(1 + v)(l- 2v)
E
(T= [ (1 - V)E, + V(E, + E, )] (2-83)
(1 + v)(l- 2v)
The components of the shear stress all obey Equation 2-74.
While the foregoing discussion of stress and strain is based on a Cartesian coordinate
system, any orthogonal coordinate system may be used.

Elementary Loading Configurations
Torsion of a Cylinder

Consider a uniform cylindrical bar or tube to which some balanced torque T is
applied (Figure 2-28). The bar will be subject to a torsional stress, or shear stress z,,
which increases with the radial position within the bar.

zze = Tr/J (2-84)
where r is the radial distance from the z axis, and J is the polar moment of inertia.
The polar moment of inertia for a hollow cylinder with an internal radius ri and an
external radius, ro, is

R
J = -(r: - r: 1 (2-85)
2