Problem 13.10 The curvature of a slender column subject to an axial load P (Fig. P13.10) can be
modeled by
d’y
dx 2
where
+ply=0
p=
EI
where E = the modulus of elasticity, and / = the moment of inertia of the cross section about its
neutral axis.
This model can be converted into an eigenvalue problem by substituting a centered finite-
difference approximation for the second derivative to give
where 7 = a node located at a position along the rod’s interior, and Ax = the spacing between
nodes. This equation can be expressed as
Vor —Q=-AC py, +, =0
Writing this equation for a series of interior nodes along the axis of the column yields a
homogeneous system of equations. For example, if the column is divided into five segments
(i.e., four interior nodes), the result is
Q2-A*p?) = 0 0 »
=1 2-Ax*p?) -1 0 "| _o
0 =1 Q2-A*p?) =) Ys
0 0 -1 C-A pH)».
An axially loaded wooden column has the following characteristics: E£ = 10x10? Pa, I=1.25%10-
m*, and L=3 m. For the five-segment, four-node representation:
(a) Implement the polynomial method with MATLAB to determine the eigenvalues for this
system.
(b) Use the MATLAB eig function to determine the eigenvalues and eigenvectors.
(¢) Use the power method to determine the largest eigenvalue and its co
esponding eigenvector.