the growth to some finite value. To capture post-buckling behavior one needs to
drop the assumption of small rotations, introduced through formulating Hooke’s
law as M(x) = EIy′′. The correct expression is M = EIj with j being the curvature,
j = y′′/(1 + (y ′)
2 )
3/2 . Thus, the linear approximation j = y′′ is only valid for small
rotations, (y′)
2
( 1. Still, linear theory is sufficient to predict the onset of buckling
and the initial post-buckling behavior.
We observe from the example that:
• The constituents of the EVP (differential equation + boundary conditions) are
both linear and homogeneous. Many, but not all, EVPs possess this property.
• For every value of k the EVP has the trivial solution u(x)=0. All EVPs possess
this property.
• For some values of k the EVP has additional nontrivial solutions. This is not a
property of all EVPs (cf. the following example).
2.4.2 The Paradox of Follower-Loading
The clamped-free column in Fig. 2.4 is loaded by a special type of force, known as
a follower-load. It acts along tangents to the free tip of the deformed column.
The differential Eq. (2.6) for the previous example still applies. However, at
x = 0 the geometric boundary condition y(0) = 0 in (2.7) is replaced by the static
condition M′ = 0, that is (by Hooke’s law) y′′′(0) = 0.
The general solution (2.8) applies here too, since the differential equation of the
EVP is unchanged. Requiring this solution to satisfy y′′(0) = y′′′(0) = 0 one obtains
that either c 1 = c 2 = 0, or that c = 0. Substituting the remaining conditions y(l) = y′
(l) = 0, one obtain for the case c 1 = c 2 = 0 that c 3 = c 4 = 0, implying y(x) = 0. For
the case c = 0 one finds that c 3 = c 2 + c 4 = 0, giving also y(x) = 0. So, in either
case there are no nontrivial solutions to this EVP.
Fig. 2.3 Buckling modes of the clamped-hinged column
Fig. 2.4 Follower-loaded column
56
2 Eigenvalue Problems of Vibrations and Stability
drop the assumption of small rotations, introduced through formulating Hooke’s
law as M(x) = EIy′′. The correct expression is M = EIj with j being the curvature,
j = y′′/(1 + (y ′)
2 )
3/2 . Thus, the linear approximation j = y′′ is only valid for small
rotations, (y′)
2
( 1. Still, linear theory is sufficient to predict the onset of buckling
and the initial post-buckling behavior.
We observe from the example that:
• The constituents of the EVP (differential equation + boundary conditions) are
both linear and homogeneous. Many, but not all, EVPs possess this property.
• For every value of k the EVP has the trivial solution u(x)=0. All EVPs possess
this property.
• For some values of k the EVP has additional nontrivial solutions. This is not a
property of all EVPs (cf. the following example).
2.4.2 The Paradox of Follower-Loading
The clamped-free column in Fig. 2.4 is loaded by a special type of force, known as
a follower-load. It acts along tangents to the free tip of the deformed column.
The differential Eq. (2.6) for the previous example still applies. However, at
x = 0 the geometric boundary condition y(0) = 0 in (2.7) is replaced by the static
condition M′ = 0, that is (by Hooke’s law) y′′′(0) = 0.
The general solution (2.8) applies here too, since the differential equation of the
EVP is unchanged. Requiring this solution to satisfy y′′(0) = y′′′(0) = 0 one obtains
that either c 1 = c 2 = 0, or that c = 0. Substituting the remaining conditions y(l) = y′
(l) = 0, one obtain for the case c 1 = c 2 = 0 that c 3 = c 4 = 0, implying y(x) = 0. For
the case c = 0 one finds that c 3 = c 2 + c 4 = 0, giving also y(x) = 0. So, in either
case there are no nontrivial solutions to this EVP.
Fig. 2.3 Buckling modes of the clamped-hinged column
Fig. 2.4 Follower-loaded column
56
2 Eigenvalue Problems of Vibrations and Stability