Principles of Virtual Work: Integral Form of the Basic Equations - Mechanics of Structures: Variational and Computational Methods - page 90

The expanded operator matrix k D

can be formed using the expressions for D and E from

Chapter 1, Eqs. (1.22) and (1.34).

∂ 00

x

0 ∂ 0

y

00 ∂

z
··· ··· ···
∂

D u =

∂ 0

y x

∂ ∂

0

z x

∂ ∂

0

z y

− ν v v .

1 0 0 0

.

ν − ν ν

1 0 0 0

.

ν ν 1 − ν 0 0 0

2 G

E = ··· ··· ··· ··· ··· ··· ···

1 − 2 ν

. (

− ν)/

0 0 0 1 2 2 0 0

.

0 0 0 0 ( 1 − 2 ν)/ 2 0

.

0 0 0 0 0 ( 1 − 2 ν)/ 2

E

G =

2 ( 1 + ν)

It is then found that

V δ

− δ W = [ u x u y u z ]

. .

2 G ( 1 − ν) 2 G ν 2 G ν

∂

ν ∂

∂

ν ∂

∂

ν ∂

x x x y x z

1 − 2 1 − 2 1 − 2

. + ∂ . +

+ ∂ G ∂ + ∂ G ∂

y G

∂ ∂ G ∂

y y z z x z x

............ ... ............ ... ............

. .

u x
u y
u z

p Vx
p Vy
p Vz

2 G ν 2 G ( 1 − ν) 2 G ν

∂

ν ∂ x

∂

ν ∂ y

∂

ν ∂ z

y y y

−

×

1 − 2 1 − 2 1 − 2

dV

. + x ∂ . + z ∂

+ x ∂ G ∂ y G ∂ x + z ∂ G ∂ z G ∂ y

............ ... ............ ... ............

. .

2 G ν 2 G ν 2 G ( 1 − ν)

∂

ν ∂

∂

ν ∂

∂

ν ∂

z x z y z z

1 − 2 1 − 2 1 − 2

. + . +

+ ∂ G ∂ ∂ G ∂ ∂ G ∂ + ∂ G ∂

x z y z x x y y

k D

p x
p y
p z

dS

−

S p δ

[ u x u y u z ] = 0 (2.59)

= ∂ x , = ∂ y , = ∂ z

∂ ∂ ∂

z

∂ ∂ ∂

where are partial derivatives acting on the preceding

variable, i.e., u x ,u y , or u z . In Eq. (2.59), the only unknowns are the displacements. As will
be seen later, this form of the principle of virtual work embodies the displacement method
of structural analysis. The introduction of the operator matrix k D is very useful for a unified
description of several different problem types. It reveals the structure of a corresponding

and x , y , z

x y