Page 123 - Civil Engineering Formulas

66 CHAPTER TWO

F F

h o e
+y
a
h i

Neutral
surface R
R i
R o
F F
FIGURE 2.21 Curved beam.

R is the radius of the centroidal axis; Z is a cross-section property defined
by
Z y
1
A R y dA (2.19)

Analytical expressions for Z of certain sections are given in Table 2.4. Z can
also be found by graphical integration methods (see any advanced strength
book). The neutral surface shifts toward the center of curvature, or inside fiber,
an amount equal to e ZR/(Z 1). The Winkler-Bach theory, though practi-
cally satisfactory, disregards radial stresses as well as lateral deformations and
assumes pure bending. The maximum stress occurring on the inside fiber is
S Mh i /AeR i , whereas that on the outside fiber is S Mh o /AeR o .
The deflection in curved beams can be computed by means of the moment-
area theory.
2 2
The resultant deflection is then equal to 0 x y in the direction
defined by tan y / x . Deflections can also be found conveniently by use
of Castigliano’s theorem. It states that in an elastic system the displacement in
the direction of a force (or couple) and due to that force (or couple) is the partial
derivative of the strain energy with respect to the force (or couple).
A quadrant of radius R is fixed at one end as shown in Fig. 2.22. The force F
is applied in the radial direction at free-end B. Then, the deflection of B is
By moment area,

y R sin x R(1 cos ) (2.20)
ds Rd M FR sin (2.21)

FR 3 FR 3
B x B y (2.22)
4EI 2EI