Direct Variational and Weighted Residual Methods: Classical Trial Function Methods - Mechanics of Structures: Variational and Computational Methods - page 466

The terms in brackets are the virtual work expressions

] 0

[ − δ V w − V δw − δ M θ − M δθ

on that portion of the boundary where the displacement boundary conditions occur, i.e.,
on S u .

N u

Introduce the trial solution w( x ) = w , and note that

w T N u

δw = N u δ w = δ

Then

w T EI L
0

N T
u

L

N u N u − N T

u

N u − N u N u

on S u

N u +

N T
u

N u dx = δ − δ W + w

0

p z dx

L

N u

− = 0

0

Let

EI N u N u L N u − N u

R = (7.70)

0

which corresponds to the term [ − V δw − M δθ ] 0 on S u . Then

w T

EI L
0

L

!
" =

N T
u

N u dx R T = δ N u p z dx δ W + ( R + ) w − 0 (7.71a)

0

k u p u

Thus, the free parameters w can be found from

R T

+ ( + ) − =

[ k u R ] w p u 0 (7.71b)

EXAMPLE 7.9 Beam with Linearly Varying Loading
Consider again the beam of Fig. 7.1. This time we will employ a trial function that does not
necessarily satisfy the displacement boundary conditions.
We begin with a polynomial trial solution in the form

w

1
w 2
w 3
w 4

2 3 4 ]

w( x ) = N u w = [ ξξ ξ ξ (1)

Although it is not necessary that any boundary conditions be satisfied, note that the con-
dition w( 0 ) = 0isfulfilled by (1). With d ξ = dx / L the needed derivatives become

1
L [1

N u =

2 3 ]

2 ξ 3 ξ 4 ξ

1
L 2 [026

N u =

2 ]

ξ 12 ξ (2)

1
L 3 [00624

N u = ξ ]

which are obtained using

d N u
dx

d N u
d

d ξ

dx

N u =

=

ξ