Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

10.2 moment–curvature Relationship
When a straight beam is loaded and the action is elastic, the longitudinal centroidal axis
of the beam becomes a curve, which is termed the elastic curve. The relationship between
the internal bending moment and the curvature of the elastic curve was developed in
Section 8.4. Equation 8.6 summarized the moment–curvature relationship:
v
O
+ρ
+M +M
–M
–M
ρ –
FIGURE 10.2 Radius of
curvature ρ related to sign of M.
O
x
1
κ = = M ρ EI
This equation relates the radius of curvature ρ of the neutral surface of the beam to the
internal bending moment M (about the z axis), the elastic modulus of the material E, and
the moment of inertia of the cross-sectional area, I z . Since E and I z are always positive, the
sign of ρ is consistent with the sign of the bending moment. As shown in Figure 10.2, a
positive bending moment M creates a radius of curvature ρ that extends above the beam—
that is, in the positive v direction. When M is negative, ρ extends below the beam in a
negative v direction.
z
10.3 The Differential Equation of the Elastic curve
The relationship between the bending moment and the radius of curvature is applicable
when the bending moment M is constant for a flexural component. For most beams, however,
the bending moment varies along its span and a more general expression is required
to express the deflection v as a function of the coordinate x.
From calculus, the curvature is defined as
2 2
1 d v/
dx
κ = =
ρ [1 + ( dv/ dx)]
For typical beams, the slope dv/dx is very small and its square can be neglected in comparison
to unity. This approximation simplifies the curvature expression to
and Equation (8.6) becomes
1
2
κ = = dv
2
ρ dx
2 3/2
EI d 2
v
2
dx
=
Mx ( )
(10.1)
This is the differential equation of the elastic curve for a beam. In general, the bending
moment M will be a function of position x along the beam’s span.
The differential equation of the elastic curve can also be obtained from the geometry of
the deflected beam, as shown in Figure 10.3. The deflection v at point A on the elastic curve
is shown in Figure 10.3a. Point A is located at a distance x from the origin. A second point, B,
is located at a distance x + dx from the origin, and it has deflection v + dv.
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