Page 122 - Civil Engineering Formulas

BEAM FORMULAS 65

l l
2 a 2
6
2P P

R 1 a
R 2

Shear

Moment
FIGURE 2.20 Two moving loads of unequal weight.

When several wheel loads constituting a system are on a beam or beams, the
several wheels must be examined in turn to determine which causes the greatest
moment. The position for the greatest moment that can occur under a given
wheel is, as stated earlier, when the center of the span bisects the distance
between the wheel in question and the resultant of all loads then on the span.
The position for maximum shear at the support is when one wheel is passing off
the span.

CURVED BEAMS

The application of the flexure formula for a straight beam to the case of a
curved beam results in error. When all “fibers” of a member have the same
center of curvature, the concentric or common type of curved beam exists
(Fig. 2.21). Such a beam is defined by the Winkler-Bach theory. The stress at
a point y units from the centroidal axis is

M y
S 1 (2.18)
AR Z (R y)
M is the bending moment, positive when it increases curvature; y is posi-
tive when measured toward the convex side; A is the cross-sectional area;