Page 333 - Mechanics of Microelectromechanical Systems

320 Chapter 5

As Fig. 5.55 indicates, the bending moment is:

such that substitution of Eq. (5.149) into Eq. (5.148) results in:

where:

From basic differential calculus, it is known that the solution to the
homogeneous differential equation (5.150) is of the form:

where A and B are integration constants that are determined from the
boundary conditions. When x = 0, the deflection at that point is
and Eq. (5.152) gives A = 0. Similarly, when x = 1, which, after
substitution into Eq. (5.152), gives the non-trivial solution:

Equation (5.153) is equivalent to:

which, combined to Eq. (5.151), gives the equation of the forces that produce
buckling as:

Out of the set of forces that are obtained when n = 1, 2, 3,..., the critical
buckling load is the smallest one, corresponding to n = 1, and therefore:

Boundary conditions that are different from the ones of Fig. 5.55 are also
possible in other buckling-related problems, as shown in Fig. 5.56. The