The Finite Element Method - Mechanics of Structures: Variational and Computational Methods - page 346

Suppose there is only surface loading so that integral II need not be considered. The
surface integral III applies for the element borders, as well as the free surfaces to which
loads are applied. Inner or interface boundaries lie between two adjoining elements. Care
must be taken in carrying out the surface integral to see that the integration along the
element interfaces, which seem to cover the same surface twice, leads to the proper single
contribution to the virtual work. No question arises for problems in which the loading is
applied only on “outer” boundaries.
In terms of M elements, the virtual work relationship can be expressed as

i

M

V δ T σ dV

u T p dS
III

− δ W = −

S p δ

= 0 (6.21)

I

i = 1

The variables in this expression are to be written in terms of displacements.
Recall from Chapter 2 that the principle of virtual work requires that the displacements,
in this case the assumed displacement field, must be kinematically admissible. Thus, th e
displacements must satisfy = Du in V and the displacement boundary conditions u = u

on S u .

Kinematics
The strains are obtained from the displacement function u using the kinematic relation

= Du (6.22)

where, for the i th element,

u x
u y

Nv i

u = = (6.23)

For this in-plane case [Chapter 1, Eq. (1.24)], the operator matrix is

∂ x 0

D = D u = 0 ∂

y

∂ y ∂ x

where, since x = a ξ ,y = b η ,

∂ x = ∂
∂

1
a

∂
∂ξ =

1
a ∂ ξ x =

(6.24)

= ∂
∂

1
b

∂
∂η =

1
b ∂ η

∂

y =

y

The strains in terms of the trial functions and nodal unknowns appear as

u x

∂ 0

x x

= =

DNv i Bv i

∂

0 = (6.25)

y
γ

y

∂ ∂ u y

xy y x