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

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

(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)

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

■ 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 )

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

■ 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

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