Basic Equations: Differential Form - Mechanics of Structures: Variational and Computational Methods - page 25

FIGURE 1.11
Surface force conditions for stress resultants n x ,n y ,
n xy = n yx for a thin element lying in the xy plane.

FIGURE 1.12
The surface S of a solid is considered to be made up of two surfaces S u and S p . The symbol S u designates regions
with known displacements, whereas S p denotes everything else, including those portions of the surface where
applied forces occur.

Note that A T is of the same form as D T of Eq. (1.53) in that the derivatives in D T correspond
to the projection directions in A T . Equations (1.57), which are surface stress conditions for
a point on the boundary, are often referred to as Cauchy's formula.
In particular, for the thin, flat element used for plane stress and strain, the surface condi-
tion takes the form (Fig. 1.11)

p a
p t

n x
n y
n xy

sin 2 cos 2 α α 2 sin α cos α

= (1.59)

sin 2 cos 2 − sin α cos α sin α cos α − α + α

where n x ,n y , and n xy = n yx are stress resultants for the thin element, and p a ,p t are the

tractions normal and tangential to the boundary.
If surface force s (per unit area) are applied externally they are referred to as prescribed
surface tractions p . Suppose the whole surface of the body is designated by S and that
those portions of the surface with prescribed tractions are designated as S p (Fig. 1.12). Let
the remainder o f the surface, i.e., S − S p , be denoted as S u to indicate where prescribed

displacements u appear.
Equilibrium requires that the resultant stress p be equal to the applied surface tractions
on S p :

p = p on S p (1.60)