Principles of Virtual Work: Integral Form of the Basic Equations - Mechanics of Structures: Variational and Computational Methods - page 92

FIGURE 2.5
Statically indeterminate structure.

EXAMPLE 2.4 Truss
Find the displacement of point a and the forces in the members of the statically indetermi-
nate structure of Fig. 2.5.
For the principle of virtual work, apply a kinematically admissible virtual displacement
and use Eq. (2.54) to find a condition relating the unknowns. Choose a virtua l d isplacement
of δ V of point a in the vertical direction. The external virtual work is δ W e = P δ V . Then the

first equation in Eq. (2.54) becomes
i (σ δ

i

LA ) − P δ V = 0 (1)

where i 1 , 2 , 3 identifies the number of each truss element as shown in Fig. 2.5. Suppose

that the geometry of the undeformed truss is valid for the deformed configuration. The
elongation of bar 1 is

=

δ V , and the kinematically admissible displacement of bar 2 or 3 is

δ V cos α . Then the virtual strain in bar 1 is δ V / L ,andinbar2or3,itis

V cos 2 δ = δ

V cos α α

α = δ

(2)

L / cos L

The stresses can be expressed in terms of the unknown displacement V . Since the strains
are

V
L ,

cos α

= = ( α) =

V cos

bar3 , Hooke's law gives the stresses

bar1 bar2

L

EV cos 2

EV
L

α

σ bar1 = , σ bar2 = σ bar3 = (3)

L

where E is the modulus of elasticity for each of these bars. Substitution of the virtual strains
(2) and the stresses (3) into the principle of virtual work (1) yields
EV
L

δ LA bar1

2 EV cos 2

δ L

cos a

A bar2

V
L

α V cos 2 α

+ − P δ V = 0 (4)

L L

Factoring out δ V , one sees that for an arbitrary virtual displacement δ V

2 EAV cos 3

EAV
L

α

+ − P = 0 (5)

L

= = .

for A bar1 A bar2 A