9.2 Principle of Virtual Displacement
533
B 1
B 2
B n − 1
B n
x
y
A
δx
δy
L 1
L 2
L n − 1
L n
α 1
α 2
α n − 1
α n
β
P
Fig. 9.3 The forces and displacements of a truss structure
In this structure only node A is not constrained, let its displacement components
be u and v. These are the real displacement components of the node A produced by
the load P. The strain of the i-th bar caused by these displacement components is
ε i =
u cos α i
L i
+
v sin α i
L i
(3)
The stress of the i-th bar caused by the displacement u and v
σ i = E i ε i = E i
u cos α i
L i
+
v sin α i
L i
(4)
where, E i is the modulus of elasticity.
In the structure only the load P does the virtual work. The virtual displacement
Eq. (9.2.29) is applied to the structure, we obtain
P cos βδx =
n
i=1
˚
V
σ i δε i dV =
n
i=1
σ i δε i L i A i
(5)
Substituting Eqs. (2) and (4) into the Eq. (5), we give
P cos βδx =
n
i=1
E i
u cos α i
L i
+
v sin α i
L i
δx cos α i
L i
L i A i
(6)