Page 30 - Civil Engineering Formulas

CHAPTER 2
BEAM

FORMULAS

In analyzing beams of various types, the geometric properties of a variety of
cross-sectional areas are used. Figure 2.1 gives equations for computing area A,
moment of inertia I, section modulus or the ratio S I/c, where c distance
from the neutral axis to the outermost fiber of the beam or other member. Units
used are inches and millimeters and their powers. The formulas in Fig. 2.1 are
valid for both USCS and SI units.
Handy formulas for some dozen different types of beams are given in Fig. 2.2.
In Fig. 2.2, both USCS and SI units can be used in any of the formulas that are
applicable to both steel and wooden beams. Note that W load, lb (kN); L
length, ft (m); R reaction, lb (kN); V shear, lb (kN); M bending moment,
lb ft (N m); D deflection, ft (m); a spacing, ft (m); b spacing, ft (m);
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E modulus of elasticity, lb/in (kPa); I moment of inertia, in (dm );
less than; greater than.
Figure 2.3 gives the elastic-curve equations for a variety of prismatic beams.
In these equations the load is given as P, lb (kN). Spacing is given as k, ft (m)
and c, ft (m).
CONTINUOUS BEAMS

Continuous beams and frames are statically indeterminate. Bending moments in
these beams are functions of the geometry, moments of inertia, loads, spans,
and modulus of elasticity of individual members. Figure 2.4 shows how any
span of a continuous beam can be treated as a single beam, with the moment
diagram decomposed into basic components. Formulas for analysis are given in
the diagram. Reactions of a continuous beam can be found by using the formu-
las in Fig. 2.5. Fixed-end moment formulas for beams of constant moment of
inertia (prismatic beams) for several common types of loading are given in Fig. 2.6.
Curves (Fig. 2.7) can be used to speed computation of fixed-end moments in
prismatic beams. Before the curves in Fig. 2.7 can be used, the characteristics
of the loading must be computed by using the formulas in Fig. 2.8. These
include xL, the location of the center of gravity of the loading with respect
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to one of the loads; G
b n P n /W, where b n L is the distance from each
load P n to the center of gravity of the loading (taken positive to the right); and
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