CIVIL ENGINEERING FORMULAS

68 CHAPTER TWO

and (2.23)

at (2.24)

By Castigliano,

(2.25)

Eccentrically Curved Beams

These beams (Fig. 2.23) are bounded by arcs having different centers of cur-
vature. In addition, it is possible for either radius to be the larger one. The
one in which the section depth shortens as the central section is approached
may be called the arch beam. When the central section is the largest, the
beam is of the crescent type.
Crescent Idenotes the beam of larger outside radius and crescent IIof
larger inside radius. The stress at the central sectionof such beams may be
found from SKMC/I. In the case of rectangular cross section, the equation
becomesS 6 KM/bh^2 , where Mis the bending moment, bis the width of the
beam section, and hits height. The stress factors,Kfor the inner boundary,
established from photoelastic data, are given in Table 2.5. The outside radius
is denoted by Roand the inside by Ri. The geometry of crescent beams is such
that the stress can be larger in off-center sections. The stress at the central

B^ x

FR^3

4 EI

(^) B (^) y

FR^3

2 EI

32.5

tan^1

2

(^) xtan^1 

FR^3

2 EI



4 EI

FR^3 

(^) B

FR^3

2 EIB

1 

2

4

FIGURE 2.22 Quadrant with fixed

end.

y

FXB

A

R

θ

FIGURE 2.23 Eccentrically curved beams.

Ro

Ri

Ri

Ro

Ro

Ri

Ri

Ro