1. 3.9 Linear models : boundary-value problems
■ Introduction
- Boundary conditions
: conditions specified on the unknown function
or on one of its derivatives
or on a linear combination of the unknown functions and one of its derivatives
at two (or more) different points
■ Deflection of a beam
axis of symmetry : a curve joining the centroids of all its cross sections
deflection curve (elastic curve) : the curve when a load is applied to the beam in a vertical plane
From the theory of elasticity
M(x) : the bending moment at a point x along the beam
w(x) : the load per unit length

2. The bending moment is proportional to the curvature of the elastic curve
E : Young’s modulus of elasticity of the material of the beam
I : the moment of inertia of a cross-section of the beam
 E & I are constants
When the deflection y(x) is small

3. (a) Embedded at both ends
Left : (no deflection, the slope is zero)
Right : (no deflection, the slope is zero)
(b) Cantilever beam
Left : (no deflection, zero slope)
Right : (zero bending moment,
zero shear force)
(c) Simply supported
Left : (no deflection, zero bending moment)
Right : (no deflection,
zero bending moment)

4. Example 1 Embedded beam
(Q) A beam of length L is embedded at both ends
A constant load w0 is uniformly distributed along its length, w(x)=w0
Find the deflection of the beam
(A)
Integrate the equation four times in succession

5. ■ Eigenvalues and eigenfunctions
We solve a two-point boundary value problem involving a linear DE that contains a parameter 
We seek the values of  for which the boundary value problem has nontrivial solutions
Example 2 Nontrivial solutions of a BVP
(Q) Solve
(A) We consider three cases : =0, <0, >0
(case I) For =0
(case II) For <0
(because )

6. (case III) For >0
If  If
Eigenvalues
Eigenfunctions

7. ■ Buckling of a thin vertical column
Leonhard Euler studied an eigenvalue problem in analyzing how a thin elastic column buckles under a compressive axial force
L : length of vertical column
y(x) : the deflection of the column when load P is applied to its top
E : Young’s modulus of elasticity
I : the moment of inertia of a cross section

8. Example 3 The Euler Load (Q) - The column is hinged at both ends
- Find the deflection of a thin vertical column of length L subjected to a constant axial load P
(A) BVP
y=0 is a perfectly good solution. Then, for what values of P does the BVP possess nontrivial solutions?
This is the case III in example 2.
Critical loads
Euler loads
First buckling mode