Complimentary Virtual Work
3-D Problems
θ1
P1
M1
f
u1
u
fl
S (surface on which traction acts)
w
u
T(
n)
δ Wvc = ∫ δ T( n ) ⋅ u dS + ∫ δ f ⋅ u dV + ∫ δ fl wds + ∑ δ Pu
i i + ∑ δ M iθ i
S
V
i
l
i
for multiple loads and moments
As in the 1-D case can show
3
3
δ W = ∫ ∑∑ eijδσ ij dV = δ U c
c
v
V i =1 j =1
Principle of Complimentary Virtual Work
If the principle of complementary virtual work is satisfied for all
variations of the stresses, where those variations satisfy local
equilibrium, then the compatibility equations will also be satisfied
δ Wvc = δ U c
for all possible stresses
satisfying local equilibrium
compatibility of
the stresses
(and strains)
Determine the stresses in the three elastic bars by the principle
of complimentary virtual work. All bars have the same E, A
bar 1
θ
Lsinθ
L
θ
bar2
L
Q
bar 3
θ
σ3A
θ
σ2A
σ1 A
P
P
Q
u
Equilibrium Eqs:
σ 2 A sin θ − σ 1 A sin θ + Q = 0
σ 1 A cos θ + σ 2 A cos θ + σ 3 A − P = 0
Complimentary Strain Energy for a linear elastic bar of length l
Alσ 2
U =U =
2E
c
v
Variations must satisfy
δσ 2 A sin θ − δσ 1 A sin θ + δ Q = 0
δσ 1 A cos θ + δσ 2 A cos θ + δσ 3 A − δ P = 0
Alσδσ
= A∆δσ
and for a single bar δ U =
E
c
so from
δ Wvc = δ U c = uδ P + vδ Q
uδ P
A∆1δσ + A∆ 2δσ + A∆ 3δσ 3 = (δσ 1uA cos θ + δσ 2uA cos θ + δσ 3u )
− ( vδσ 2 A sin θ − vδσ 1 A sin θ )
vδ Q
From the principle of complimentary virtual work we found
A∆1δσ 1 + A∆ 2δσ 2 + A∆ 3δσ 3 = δσ 1uA cos θ + δσ 2uA cos θ + δσ 3 Au
−vδσ 2 A sin θ + vδσ 1 A sin θ
satisfying this for all
δσ 1 , δσ 2 , δσ 3
gives
∆1 = u cos θ + v sin θ
∆ 2 = u cos θ − v sin θ
∆3 = u
so these are the compatible bar elongations (compatible with the two
displacements u, v, at the applied loads) . We see
∆1 + ∆ 2 = 2∆ 3 cos θ
∆1 + ∆ 2 = 2∆ 3 cos θ
This equation is the compatibility equation for our problem. It shows that
the three elongations in the bar must be related in order that the three
bars all give consistent end displacements ( u, v). We could also write
this equation in terms of compatibility of the strains in the bars
e1 L + e2 L = 2e3 L sin θ cos θ
Using Hooke's law, in terms of the stresses we find
2σ 3 L sin θ
+
=
cos θ
E
E
E
σ 1L σ 2 L
Thus, we have solved the problem by the method of complimentary virtual
work since from equilibrium
σ 2 A sin θ − σ 1 A sin θ + Q = 0
σ 1 A cos θ + σ 2 A cos θ + σ 3 A − P = 0
and from compatibility
σ1L σ 2 L
E
+
E
=
2σ 3 L sin θ
cos θ
E
we have three equations in three unknowns