Stability Analysis - Mechanics of Structures: Variational and Computational Methods - page 673

axial force N 0

is

v T N 0 N T N dx v N T EI N dx − δ W = δ −

v T G T EI N u dx G N 0 G T dx G v N T

u

N u N u

= δ − (11.60)

Define for element i

G T k lin = N T EI N dx N T

u

EI N u dx G =

N 0 N 0 G T

(11.61)

k i geo =− N T N dx N u N u dx G =−

Matrix k lin is the same element stiffness matrix that was discussed in Chapters 4 and 5. This
stiffness matrix is used in the static analysis of structures. If the same (static) shape function
N is used in forming k i geo as in forming k lin , then k i geo , which is called the geometric stiffness
matrix, is said to be consistent. The matrix k i geo is also known as the initial stress stiffness
matrix, stability coefficient matrix, or simply the stress stiffness matrix .
In this section, we will study the development of the element matrices k lin and k i geo
and applications to simple beam models. The assembly into system matrices of these ele-
ment matrices will be discussed in Section 11.4, along with stability analyses of complex
structures.
One of the most common assumed displacements is a trial function formed of Hermitian
polynomials. The Hermitian polynomial shape functions of Chapter 4, Eq. (4.47b) w =

w a H 1 + θ a H 2 + w b H 3 + θ b H 4 led to the exact element stiffness matrix for the static response
in Chapter 4. As mentioned above, the use of the same (static response) polynomial to form
the stability matrix k i geo

provides a “consistent” matrix. Introduce (Chapter 4, Eq. (4.47a))

1000
0

w

a
θ

− 100

a

2 3 ]

w = Nv = [1 ξξ ξ (11.62)

− 3231

2

w

b

− 1 − 2 − 1 θ

b

N u G v

with

2 ] 1

N u =

[012 ξ 3 ξ

1

N u =

[0026 ξ ]

2

into Eq. (11.61). This gives the stiffness matrix k lin

[ w a θ a w b θ b ]

12 − 6 − 12 − 6

EI

− 64 62

k lin =

(11.63)

− 26 26

−

3

62 64