Principle of Complementary Virtual Work
40 Derivation of the principle of complementary virtual work starts from the assumption of a kinematicaly admissible displacements and satisfaction of the essential boundary conditions.
41 Whereas we have previously used the vector notation for the principle of virtual work, we will now use the tensor notation for this derivation.
42 The kinematic condition (straindisplacement):
43 The essential boundary conditions are expressed as ui = u on r
44 The principle of virtual complementary work (or more specifically of virtual force) which can be stated as
A deformable system satisfies all kinematical requirements if the sum of the external complementary virtual work and the internal complementary virtual work is zero for all statically admissible virtual stresses Saj.
The major governing equations are summarized
/ £ij Saij dQ 
— UiStidT = 
0 
(13.52) 
Jq 
Jru  
SW* 
SW*  
Sffijj =0 in 
Q 
(13.53)  
Sti = 0 on 
rt 
(13.54) 
45 Note that the principle is independent of material properties, and that the primary unknowns are the stresses.
46 Expressions for the complimentary virtual work in beams are given in Table 13.3
Example 132: Tapered Cantilivered Beam; Virtual Force
"Exact" solution of previous problem using principle of virtual work with virtual force.
Internal
SPA External
Note: This represents the internal virtual strain energy and external virtual work written in terms of forces and should be compared with the similar expression derived in Eq. 13.25 written in terms of displacements:
'dx2 dx2
Here: SM and SP are the virtual forces, and ¡M and A are the actual displacements. See Fig. 13.3
L1J 0 21 P2 2L
L x2
From Mathematica we note that:
Thus substituting a = L and b = 1 into Eqn. 13.58, we obtain:
Similarly:
2P2 L EIX 2P2L EIi 2P2L EI1
P2L3 2.5887Eh
~EhJo
.721EI1
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