Stability Analysis - Mechanics of Structures: Variational and Computational Methods - page 659

If are the imposed curvature and strain respectively, due, for example, to a temper-

ature change, the material laws appear as

κ and ε

EI

d 2 w

dx 2 − κ

M = EI (κ − κ) = −

EA du

N = EA (ε − ε) =

dx − ε

(11.20)

Q = k s GA γ

Of course, the final expression is zero for Euler-Bernoulli beams. The shear force V follows
from Eqs. (11.12), (11.19), and (11.20).

N 0

V = Q − H θ ≈ Q − N θ = k s GA γ − ( EA ε)θ ≈− θ (11.21)

Governing Equations
These fundamental equations can be combined to form a set of first order or higher order
differential equations. The first order equations are:

Extension:

u
H

0 EA
00

u
H

ε
−

d
dx

= +

(11.22)

p x

Bending:

w
V
M

0 10 0

000 EI
0000
0

− w

V
M

0
κ
−

= +

d
dx

(11.23)

p z

H θ

H 10

0

d z
dx

= A z + P

In higher order form:
Extension:

EA du
dx

ε
dx

d
dx

EA d

=−

p x +

(11.24)

EA du H =

dx −

EA ε

Bending:

dx 2 EI d 2 dx 2

d 2 w

H d 2

w
dx 2 =

EI d 2

κ
dx 2 −

H d

θ

0
dx

− p z −

dx EI d 2

d w

dx 2 +

H d

dx −

EI d

κ
dx −

V =− H θ 0

(11.25)

EI d 2

w
dx 2 −

=− κ

M EI

dx

d

θ =−