Page 30 - Mechanics of Microelectromechanical Systems
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1. Stiffness basics 17
Figure 1.11 Beam element under the action of distributed load, shear force and bending
moment
Equations (1.53) are valid for small-deformations only and under the
assumptions that plane cross-sections remain plane after deformations, and
that cross-sections remain perpendicular to the neutral axis (as mentioned,
the normal stresses are zero at the neutral axis). The latter two assumptions
define what is known as the Euler-Bernoulli beam model, which is
recognized to be valid for long beams where the length is at least 5-7 times
larger than the largest cross-sectional dimension. For relatively-short beams,
shearing effects become important, and the regular bending deformations are
augmented by the addition of shearing deformations, according to a model
known as Timoshenko’s beam model. In this case, the cross-sections are no
longer perpendicular to the neutral axis in the deformed state, and the
deformations are described by the following equations:
as shown, for instance, by Reddy [5] or Pilkey [6].
The strain energy stored in a beam that is acted upon by a bending
moment over its length is:
