Structural Analysis Methods I: Beam Elements - Mechanics of Structures: Variational and Computational Methods - page 205

FIGURE 4.10
Bar undergoing axial extension.

EXAMPLE 4.1 Axial Deformation
Derive the stiffness matrix for a bar of cross-sectional area A undergoing axial deformation.
The axial end displacements are given by u a and u b as shown in Fig. 4.10. Choose the
polynomial shape function

x ]

u 1

u = N u u = [1 (1)

u 2

In terms of the displacements at ends a and b,

= = +

u a u 1 , b u 1 u 2 (2)

or in matrix notation,

u a
u b

10
1

u 1

= N u

= v u = (3)

u 2

Then

1

0

= N − 1
u

= N − 1
u

u v = Gv , G = (4)

− 1 / 1 /

and u ] v .

The internal virtual work term for axial motion takes the form

= N u Gv = Nv = [1 − x / x /

0 δ

u k D udx − δ W i = (5)

To express this in discrete form,

v T G T N u

δ u = δ( N u Gv ) = N u G δ v = δ (6)

From Chapter 2, Example 2.7,

k D = x dEAd x (7)

The element stiffness matrix k i is given by

G T
0

k i N u k D N u dx G = (8)

The term d x N u would be equal to [0 1] and

0
1

N u

x d

=

With the integral

EA
0

0
1

[0

EA 00
0

N u k D N u dx = 1] dx = (9)

0