Basic Equations: Differential Form - Mechanics of Structures: Variational and Computational Methods - page 23

( p Vx = 0 , p Vy = 0 ) , stresses defined as

2 2 2

= ∂

ψ

= ∂

ψ

=− ∂

ψ

σ σ τ (1.55)

x y xy

∂ y 2 ∂ x 2 ∂ x ∂ y

identically satisfy the equilibrium equations of Eq. (1.50a). Here ψ is the appropriate stress

function, also called the Airy 13

stress function . Observe that in this case all stresses σ x , σ y ,

and τ xy in the body are derivable from a single stress function.

A variety of other stress functions can be chosen which are similar to the Airy stress
function in that they are selected such that the equilibrium conditions are satisfied. For the
general three-dimensional case, the equations of equilibrium of Eq. (1.51) can be satisfied
if six stress functions ψ =

ψ x ψ y ψ z ψ xy ψ xz ψ yz ] T

[ are defined such that

p Vx dx

p Vy dy

p Vz dz
0
0
0

D 1 ψ − σ 0 σ = D 1 ψ − =

(1.56a)

where D 1 is defined in Eq. (1.28). The six functions in ψ are a combination of Maxwell's 14
and Morera's stress functions. The first three components of the vector ψ belong to
Maxwell's stress function and the final three compose Morera's stress function. These
functions, which are not independent, can be expressed as three stress functions φ . The
relationship between stresses and stress functions takes the form
σ = D φ − σ 0

(1.56b)

where D is the same operator matrix D as in the kinematical relations of Eq. (1.21).

1.5 Surface Forces and Boundary Conditions
The stress components on the surface, i.e., the boundary, of a body must be in equilibrium
with the forces applied to the surface. The equilibrium conditions are obtained by consider-
ing the state of stress at a point on the surface. Suppose a small element lies on the surface of
a body (Fig. 1.10) with unit normal vector a (positive outward) defining its orientation with

13 George Biddel Airy (1801-1892) was an English mathematician with a primary interest in astronomy. He was a
professor at Cambridge University. Although most of his efforts were directed to astronomical work, he showed
an interest in the application of mathematics to structural mechanics problems. He proposed the stress function
named after him while trying to solve a beam problem of rectangular cross-section. He chose a polynomial form
for , with the coefficients selected such that the boundary conditions were satisfied. His solution was incomplete

as the

ψ

he used did not satisfy compatibility requirements.
14 James Clerk Maxwell (1831-1879) was a Scottish mathematician who was interested in photoelasticity and
analytical solid mechanics. He did his work in developing the science of photoelasticity, along with solving
numerous problems of the torsion of bars and cylinders and the bending of beams and plates, before he was
nineteen. Then he began his studies at Cambridge University. He remained at Cambridge after his 1855 graduation
and broadened his interests to include electricity, magnetism, and the kinetic theory of gases. In 1865 he retired to
write the celebrated Treatise on Electricity and Magnetism . He returned to Cambridge in 1871 where he developed
the Cavendish Laboratory. His reciprocal theorem (Chapter 3, Section 3.3) appeared eight years before the more
general theorem of Betti.

ψ