Page 100 - Civil Engineering Formulas

BEAM FORMULAS 53

Shear at the end of a beam necessitates modification of the forms deter-
mined earlier. The area required to resist shear is P/S v in a cantilever and R/S v in
a simple beam. Dashed extensions in Figs. 2.15 and 2.16 show the changes nec-
essary to enable these cantilevers to resist shear. The waste in material and extra
cost in fabricating, however, make many of the forms impractical, except for
cast iron. Figure 2.17 shows some of the simple sections of uniform strength. In
none of these, however, is shear taken into account.

SAFE LOADS FOR BEAMS OFVARIOUSTYPES

Table 2.2 gives 32 formulas for computing the approximate safe loads on
steel beams of various cross sections for an allowable stress of 16,000 lb/in 2
(110.3 MPa). Use these formulas for quick estimation of the safe load for any
steel beam you are using in a design.
Table 2.3 gives coefficients for correcting values in Table 2.2 for various
methods of support and loading. When combined with Table 2.2, the two sets of
formulas provide useful time-saving means of making quick safe-load compu-
tations in both the office and the field.

ROLLING AND MOVING LOADS

Rolling and moving loads are loads that may change their position on a beam or
beams. Figure 2.18 shows a beam with two equal concentrated moving loads,
such as two wheels on a crane girder, or the wheels of a truck on a bridge.
Because the maximum moment occurs where the shear is zero, the shear diagram
shows that the maximum moment occurs under a wheel. Thus, with x a/2:
2x a
R 1 P 1 l l (2.13)

Pl a 2x a 4x 2
M 2 1 2 (2.14)
2 l l l l
2x a
R 2 P 1 (2.15)
l l
Pl a 2a 2 2x 3a 4x 2
M 1 1 2 (2.16)
2 l l 2 l l l
1
M 2 max when x 4 a
3
M 1 max when x 4 a
Pl a 2 P a 2
M max 1 l (2.17)
2 2l 2l 2