beam

CHAPTER 2
BEAM
FORMULAS
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Source: CIVIL ENGINEERING FORMULAS

In analyzing beams of various types, the geometric proper-
ties 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deflec-
tion, ft (m); aspacing, ft (m); bspacing, ft (m); E
modulus of elasticity, lb/in2(kPa); Imoment of inertia,
in4(dm4); 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 com-
ponents. Formulas for analysis are given in the diagram.
Reactions of a continuous beam can be found by using the
formulas in Fig. 2.5. Fixed-end moment formulas for
beams of constant moment of inertia (prismatic beams) for
16 CHAPTER TWO
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BEAM FORMULAS

17
FIGURE 2.1 Geometric properties of sections.
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BEAM FORMULAS

18
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BEAM FORMULAS

19
FIGURE 2.1 (Continued) Geometric properties of sections.
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BEAM FORMULAS