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 13.36-b 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 13-1: 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:

Solution

Total

Virtual

1

Eqn. 13.40

Eqn. 13.41

2

Eqn. 13.40

Eqn. 13.40

3

Eqn. 13.41

Eqn. 13.41

Where actual and virtual values for the two assumed displacement fields are given below.

Trigonometric (Eqn. 13.40)

Polynomial (Eqn. 13.41)

v

(1 _cos nx) V2

3( f)2 - 2( D3

v2

Sv

(1 - cos ff) 5v2

31 f )2 -21 f )3

Sv2

v"

-L cos nf v2

( L2 - —) v2

Sv"

4ZT2 cos nx SV2

Sv EIzv dx

J 0 P2SV2

Solution 1:

rL n2

3nEI1 P2SV2

fnx\

v2

^ 6

v2lJ

vi2 -

10

16"

n

n2

cos which yields:

Solution 2:

which yields:

Solution 3:

which yields:

P2L3 2.648EI1

32L3 P2SV2

--V2SV2

P2L3 2.57 EI 1

9 EI

P2SV2

V2SV2

P2L3 9 EI

0 0

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