SPLINES 131segment between two consecutive ducks to be a thin simply supported beam across which the
bending moment varies linearly. Applying the linearized Euler-Bernoulli beam equation for small
deformation
EI EIdy
dxχ = = + Ax B2
2 (5.1)whereEI is the flexural rigidity of the beam, χ the curvature, y the vertical deflection and A and B are
known constants. Solving for deflection yields
y Ax
EIBx
EI
= Cx C
6
+
2
+ +32
12 (5.2)whereC 1 and C 2 are unknown constants. Let l be the length of the beam segment. As the segment is
simply supported, at x = 0 and l,y = 0. Thus
C 2 = 0C Al
EIBl(^1) EI
2
= –
6
- 2
Hence y
Al x x
EI
Blx x
EI
=(– )
6
+(– )
232 2
(5.3)This is a cubic equation in 0≤x≤l. For continuity of the second derivative at junction points x = 0
andl, it is required from Eq. (5.1) that the bending moment, Ax + B compares with the neighboring
segments at those points. This is ensured by the equilibrium condition and thus the resulting deflection
curve inclusive of all segments is inherently a C^2 continuous curve. A cubic spline, therefore, is a
curve for which the second derivative is continuous throughout in the interval of definition. Note that
Eq. (5.1) represents the strong form of the equilibrium condition. Alternatively, the weak form in
terms of the strain energy stored in the beam may be written as
Minimize: Strain Energy =^1
2
;^2
∫
EIχ dx y = 0 at x = 0 and l (5.4)Eq. (5.4) provides an alternative description of a spline, that is, the resulting physical spline is a
smooth curve for which the strain energy or the mean squared curvature is a minimum. The general
mathematical definition of a spline, however, can be extended as:
Annth order (n– 1 degree) spline is a curve which is Cn–2 continuous in the domain of
definition, that is, the (n– 2)th derivative of the curve exists everywhere in the above domain.Wooden stripDucks
Figure 5.1 Schematic of the draughtsman’s approach and the simply supported beam model