Summary

60 Summary of Virtual work methods, Table 13.2.

Starts with

Ends with

In terms of virtual

Solve for

Virtual Work U

Complimentary Virtual Work U*

SAS

SAS KAD

Displacement/strains Forces/Stresses

Displacement Displacement

KAD: Kinematically Admissible Dispacements SAS: Statically Admissible Stresses

Table 13.2: Comparison of Virtual Work and Complementary Virtual Work

61 A summary of the various methods introduced in this chapter is shown in Fig. 13.7.

62 The duality between the two variational principles is highlighted by Fig. 13.8, where beginning with kinematically admissible displacements, the principle of virtual work provides statically admissible solutions. Similarly, for statically admissible stresses, the principle of complementary virtual work leads to kinematically admissible solutions.

63 Finally, Table 13.3 summarizes some of the major equations associated with one dimensional rod elements.

U

Virtual Displacement 5U

Virtual Force SU *

General

Linear

General

Linear

Axial

S

Vc EIz dx

7 ÖS

S

W

Virtual Displacement SW

Virtual Force SW *

^i 2

ViPiSAi

ViSPiAi

M

Vi 2 Midi

ViMiSdi

ViSMidi

w

w(x)v(x)dx c

J c

Sw(x)v(x)dx c

Table 13.3: Summary of Variational Terms Associated with One Dimensional Elements

Table 13.3: Summary of Variational Terms Associated with One Dimensional Elements

Natural B.C.

Natural B.C.

The Rayleigh Ritz Principle
Figure 13.7: Summary of Variational Methods
Figure 13.8: Duality of Variational Principles
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