Principles of Virtual Work: Integral Form of the Basic Equations - Mechanics of Structures: Variational and Computational Methods - page 115

The corresponding complementary hybrid functional is obtained by assuming that the
stresses σ satisfy equilibrium in V . Then from Eq. (2.101),

∗ H =− U 0 (σ ) p T u dS u T

dV + + dS

Additional term for the
boundary

( p − p ) (2.104)

V S u S p

Complementary virtual work D

δ p = 0on S p

All of the principles considered thus far involve the variation of δ u or δ σ or both. An

interesting form, requiring the variation of e , is obtained if the material law is

introduced explicitly as an additional condition. The four global generalized forms can be
extended by including the term

δ u , δ σ ,or δ

− δ

ij

(σ − E ijkl ) dV (2.105)

ij kl

In the literature, the resulting functional is sometimes referred to as the Hu 9 - Washizu 10
functional [Washizu, 1982]. As an example of how this modifies the generalized principles,
consider the form CB = C + B . This first two terms of form CB in Table 2.3 would be altered.

U 0 =

Introduce (Eq. 2.14) − U 0 − σ

ij , with the strains ij being treated as independent ij

unknowns. Then

V σ

U 0 (σ ) − dV +

V (

u i, j σ ) dV = U 0 () dV −

V σ

ij dV

+

ij u i, j dV

ij ij

V V

= U 0 () dV +

V σ ij (

u i, j − ij ) dV (2.106)

V

or rewrite the first term of form CB of Table 2.4 as
[

u D T

0 0 u

σ u T σ T T ]

dV

δ δ δ D u 0 − I (2.107)

0 − I E

where I is a unit diagonal matrix.
The generalized principles AB, BC, CD and AD can be further modified by replacing the
variation in the stresses by a variation in the strains with the help of the material law, i.e.,
by making

δ σ = E δ e [Wunderlich, 1972]. In this case, the first two terms of form CB in

Table 2.3 become

V (

U 0 () −

ij E ijkl

+ u i, j E ijkl ) dV =

V ( −

U 0 () + u i, j E ijkl ) dV (2.108)

kl kl kl

or the first term of form CB is altered to

e T ] 0

u dV

u D T E

u T [ δ δ (2.109)

ED u − E

V

9 Hai-Chang Hu was born in Zhejiang Province, China, in 1928 and graduated from the Department of Civil
Engineering, Zhejiang University, in 1950. He proposed the Hu-Washizu principle in 1954. Much of his career was
spent at positions in the Chinese Academy of Sciences, the Chinese Academy of Space Technology, and Beijing
University.
10 Kyuichoro Washizu (1924-1981) graduated in 1942 from Tokyo University where he subsequently became a
professor of Aerospace Engineering. He contributed to such areas as variational methods and plasticity. Research
with T. Pian at MIT led to a 1955 report that proposed the Hu-Washizu principle. At the time of his death he was
a professor of the Faculty of Engineering Science, Osaka University.