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

When the beam is bent, points along the beam both deflect and rotate. The
angle of rotation θ of the elastic curve is the angle between the x axis and the
tangent to the elastic curve, as shown for point A in the enlarged view of Figure 10.3b.
Similarly, the angle of rotation at point B is θ + dθ, where dθ is the increase in
rotation angle between points A and B.
The slope of the elastic curve is the first derivative dv/dx of the deflection v.
From Figure 10.3b, the slope can also be defined as the vertical increment dv divided
by the horizontal increment dx between points A and B. Since dv and dx are
infinitesimally small, the first derivative dv/dx can be related to the rotation angle
θ by the tangent function:
dv
dx
= tan θ
(a)
Note that the slope dv/dx is positive when the tangent to the elastic curve slopes
upward to the right.
In Figure 10.3b, the distance along the elastic curve between points A and B
is denoted as ds, and from the definition of arclength, ds = ρ dθ. If the angle of
rotation θ is very small (as it would be for a beam with small deflections), then the
distance ds along the elastic curve in Figure 10.3b is essentially the same as the
increment dx along the x axis. Therefore, dx = ρ dθ, or
1 θ
= d ρ dx
Since tan θ ≈ θ for small angles, Equation (a) can be approximated as
Therefore, the beam angle of rotation θ (measured in radians) and the slope dv/dx
are equal if beam deflections are small.
Taking the derivative of Equation (c) with respect to x gives
dv
dx
≈ θ
(b)
(c)
v
x
v
x
ρ
393
THE dIFFERENTIAL EQuATION
OF THE ELASTIC CuRVE
O
ρ
A
ds
B
dθ
dx
(a) Elastic curve
A
dθ
ds
FIGURE 10.3 Geometry of deflected
beam.
v
dx
B
θ+
dθ
v + dv
(b) Enlarged region around point A
θ
v + dv
x
x
2
d v
2
dx
dθ
=
dx
(d)
From Equation (b), dθ/dx = 1/ρ. In addition, Equation (8.6) gives the relationship between
M and ρ. Combining these equations gives
2
d v
2
dx
dθ
1
= = =
dx ρ
M
EI
(e)
or
EI d 2
v
2
dx
=
Mx ( )
(10.1)
In general, the bending moment M will be a function of position x along the beam’s span.