## Virtual Work

23 We define the virtual work done by the load on a body during a small, admissible (continuous and satisfying the boundary conditions) change in displacements.

Internal Virtual Work |
JQ |
(13.19) |

SW S |
/ t-SudT+ / bSudQ |
(13.20) |

Jrt JQ |

where all the terms have been previously defined and b is the body force vector.

24 Note that the virtual quantity (displacement or force) is one that we will approximate/guess as long as it meets some admissibility requirements.

### 13.1.3.1 Internal Virtual Work

25 Next we shall derive a displacement based expression of 5U for each type of one dimensional structural member. It should be noted that the Virtual Force method would yield analogous ones but based on forces rather than displacements.

26 Two sets of solutions will be given, the first one is independent of the material stress strain relations, and the other assumes a linear elastic stress strain relation.

Elastic Systems In this set of formulation, we derive expressions of the virtual strain energies which are independent of the material constitutive laws. Thus 5U will be left in terms of forces and displacements.

Axial Members:

aSedQ

Flexural Members:

ja y

dAdx

OJ A

Linear Elastic Systems Should we have a linear elastic material (a = Ee) then: Axial Members:

Flexural Members:

J oxSexdû

d2v T7 T

dxV EIz d2V E

dAdx d2(Sv)y dx2 y

13.1.3.2 External Virtual Work 5W

27 For concentrated forces (and moments):

where: SAi = virtual displacement.

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