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 (strain-displacement):

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

Figure 13.3: Tapered Cantilevered Beam Analysed by the Virtual Force Method

/ £ij Saij dQ

— UiStidT =







Sffijj =0 in



Sti = 0 on



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 13-2: Tapered Cantilivered Beam; Virtual Force

"Exact" solution of previous problem using principle of virtual work with virtual force.


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:


2P2 L EIX 2P2L EIi 2P2L EI1

P2L3 2.5887Eh



+1 -1


  • Katja
    What is complementary virtual work?
    2 years ago
  • robert
    How does complementary virtual work are differ from virtual work.?
    2 years ago
  • medhane
    What is virtual work design of concrete structure?
    2 years ago
  • frank
    What is complementary elasticity?
    2 years ago

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