Related Variational and Energy Principles - Mechanics of Structures: Variational and Computational Methods - page 141

is embodied in the equations

W i W e

− δ − δ = 0

V ij δσ ij dV

− u i δ p i dS = 0 (3.7)

S u

σ T dV p T u dS

V δ

−

S u δ

= 0

with statically admissible δσ

ij , i.e.,

δσ

ij that satisfy the equilibrium conditions and static

boundary conditions

D T σ + p V = 0 in V

p on S p
If the potential functions exist, the principle of complementary virtual work is specialized
as the principle of stationary complementary energy. This principle contends that among
the statically admissible states of stress, the actual state of stress which corresponds to
kinematically admissible deformations is the one for which the total complementary energy
∗ is stationary. This is expressed as
δ ∗ =

p =

∗ = U i + U e −→ 0or Stationary (3.8)

where

D T σ +

p V = 0 in V

p = p on S p

Other useful theorems can be derived from these principles.

3.2.1 Castigliano's Theorem, Part II
The complementary energy theorem in a form similar to Eq. (3.5) is Castigliano's theorem, part
II or Castigliano's second theorem . A general three-dimensional solid acted on by a system of
forces P 1 ,P 2 , V n at their

points of application will be used to illustrate this. Here, the forces can be regarded as re-
ac tions generated by prescribed displacements V k , which more precisely should be written
V k .

... ,P k , ... P n with corresponding displacements V 1 ,V 2 , ... ,V k , ...

That is, these are the displacements that would have to be applied to create the forces
P 1 ,P 2 , If the V k 's are considered as being independent of the P k 's, the total comple-

mentary potential energy can be expressed as

... P n .

n

∗ = U i + U e = U i − P k V k (3.9)

k = 1

The complementary energy is a function of the P k , which are sometimes referred to as the
generalized forces of the system. Recall that for the complementary virtual work theorems, the
variations are taken on the forces. According to the principle of stationary complementary
energy, δ ∗ = 0 . Therefore,

U i
∂

U i
∂

= ∂ + ∂ δ ∗ = 0

P 1 δ

P 1

P 2 δ

P 2 +···− V 1 δ P 1 − V 2 δ P 2 ···− V n δ P n

or

∂

P k

∂

V k P k

n n

U i
∂

U i
∂

P k δ

− δ =

P k −

δ =

P k V k 0

k = 1 k = 1