Page 126 - Civil Engineering Formulas

68 CHAPTER TWO

FR 3
2
and B 1 (2.23)
2EI B 4
FR 3 4EI 3
1
at x tan 2EI
FR (2.24)
2
tan 1

32.5
By Castigliano,

FR 3 FR 3
B x B y (2.25)
4EI 2EI

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 I denotes the beam of larger outside radius and crescent II of
larger inside radius. The stress at the central section of such beams may be
found from S KMC/I. In the case of rectangular cross section, the equation
2
becomes S 6KM/bh , where M is the bending moment, b is the width of the
beam section, and h its height. The stress factors, K for the inner boundary,
established from photoelastic data, are given in Table 2.5. The outside radius
is denoted by R o and the inside by R i . The geometry of crescent beams is such
that the stress can be larger in off-center sections. The stress at the central

R o R i
A
R i
R o
y

R
R o R o
θ
R i
F B X R i
FIGURE 2.22 Quadrant with fixed FIGURE 2.23 Eccentrically curved beams.
end.