Principle of Virtual Work
35 Derivation of the principle of virtual work starts with the assumption of that forces are in equilibrium and satisfaction of the static boundary conditions.
The Equation of equilibrium (Eq. 6.26) which is rewritten as dffx dx da
dTx dy
yy + VT* + , dy dx where b representing the body force. In matrix form, this can be rewritten as d dx
Note that this equation can be generalized to 3D.
37 The surface r of the solid can be decomposed into two parts r and T„ where tractions and displacements are respectively specified.
d ' 
axx  
d 
ayy  
dx 
1 Txy  
t = t on rt Natural B.C. u = û on r„ Essential B.C. Equations 13.35 and 13.36b constitute a statically admissible stress field. 38 The principle of virtual work (or more specifically of virtual displacement) can be stated as A deformable system is in equilibrium if the sum of the external virtual work and the internal virtual work is zero for virtual displacements ¿u which are kinematically admissible. The major governing equations are summarized IQ JQ JTt 39 Note that the principle is independent of material properties, and that the primary unknowns are the displacements. ■ Example 131: Tapered Cantiliver Beam, Virtual Displacement Analyse the problem shown in Fig. 13.2, by the virtual displacement method. Solution: 1. For this flexural problem, we must apply the expression of the virtual internal strain energy as derived for beams in Eq. 13.25. And the solutions must be expressed in terms of the displacements which in turn must satisfy the essential boundary conditions. Figure 13.2: Tapered Cantilivered Beam Analysed by the Vitual Displacement Method The approximate solutions proposed to this problem are 2. These equations do indeed satisfy the essential B.C. (i.e kinematic), but for them to also satisfy equilibrium they must satisfy the principle of virtual work. 3. Using the virtual displacement method we evaluate the displacements v2 from three different combination of virtual and actual displacement:
Where actual and virtual values for the two assumed displacement fields are given below.

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