Page 36 - Mechanics of Microelectromechanical Systems

1. Stiffness basics 23
The first theorem of Castigliano can easily be applied when the
deformation field is known in advance, but this proves to be difficult in
situations where the cross-section of the line element is variable. Instead, the
loads acting on a body can be known amounts, and the application of
Castigliano’s second (displacement) theorem is more feasible, especially in
cases where the material is linear, and therefore the strain and
complementary energies are equal (the force-displacement characteristic of
Fig. 1.15 is a line). The strain energy for a relatively-long line member that is
subject to complex load formed of axial force, torsion moment, shearing
force and bending moment can be written –see Den Hartog [7] or Cook and
Young [3] – in the form:

Equation (1.71) considered that the member’s cross-section has two principal
directions (it possesses two symmetry axes, and therefore a symmetry center)
and that bending moments and shearing forces act about these axes. Similarly,
the complementary energy can be expressed in terms of loading, and in the
case of a linear material this energy is:

which has been obtained by collecting individual strain energy terms from
axial, torsion, two-direction shearing and two-directional bending loads.

Example 1.5
Find the slope at the midspan of the beam shown in Fig. 1.16 by
considering that the beam is relatively long and is constructed of a material
with linear properties. An external moment loads the beam.

Solution
The beam is statically-indeterminate because there are four unknown
reactions (one at point 1, and 3 at point 3), and therefore an additional
equation needs to be written in order to complement the regular three
equations of static equilibrium. It can be seen that the specific boundary
condition at point 1 prevents the vertical (z) motion at that point, and
therefore: