CIVIL ENGINEERING FORMULAS

66 CHAPTER TWO

Ris the radius of the centroidal axis; Z is a cross-section propertydefined

by

(2.19)

Analyticalexpressions for Zof certain sections are given in Table 2.4. Zcan

also be found by graphicalintegration methods (see any advanced strength

book). The neutral surfaceshifts toward the center of curvature, or inside fiber,

an amount equal to eZR/(Z1). The Winkler-Bach theory, though practi-

cally satisfactory, disregards radial stresses as well as lateral deformations and

assumes pure bending. The maximum stressoccurring on the inside fiber is

SMhi/AeRi, whereas that on the outside fiber is SMho/AeRo.

Thedeflectionin curved beams can be computed by means of the moment-

area theory.

The resultant deflection is then equal to in the direction

defined by Deflections can also be found conveniently by use

ofCastigliano’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 Ris 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 Bis

By moment area,

yRsin xR(1cos ) (2.20)

dsRd MFRsin (2.21)

B^ x (2.22)

FR^3

4 EI

(^) B (^) y

FR^3

2 EI

tan y/ (^) x.
0  (^2) x ^2 y

Z

1

A



y

Ry

dA

F

ho

a

F

e

hi

Ri R

F

Ro

+y

Neutral

surface

F

FIGURE 2.21 Curved beam.