Principles of Virtual Work: Integral Form of the Basic Equations - Mechanics of Structures: Variational and Computational Methods - page 130

FIGURE P2.15
A bar structure. Each bar has the same modulus of elasticity and cross-sectional area.

10 7

0 . 6 .

2.16 Resolve Problem 2.15 if the material is nonlinearly elastic with σ = 20 ×

2.17 Solve Problem 2.15 using the principle of stationary potential energy.
2.18 Use the principle of stationary potential energy to find the equation of equili br ium
for a flexible string of length L with tension N and transverse loading intensity p z ( x ) .

be the transverse displacement.
2.19 An elastic string of length L , modul us E , and cross-sectional area A is fixed at its ends
and subjected to a transverse force p z .

Let w( x )

1 2

The strain is given by x = u ,x +

2 w

,x , where

u is the axial component of displacement and is the transverse displacement. Use

the principle of stationary potential energy to show that the governing equations are

w

EA u ,x +

d
dx

1
2 w

2
,x

= 0 and u | x = 0 = u | x = L = 0

EA u ,x +

w ,x

d
dx

1
2 w

2
,x

+ p z = 0 w | x = 0 = w | x = L = 0

Hint:

1

2 EA u ,x +

2 dx

1
2

1
2 w

2
,x

= σ x x dV − p z w dx = − p z w

L L

δ = 0 leads to the governing equations.