Page 135 - Civil Engineering Formulas

76 CHAPTER TWO

multiply it by EI/wL 4 . To get T, divide the appropriate constant by EI/wL 4 .
In these equations,
natural frequency, rad/s
W beam weight, lb per linear ft (kg per linear m)
L beam length, ft (m)
2
E modulus of elasticity, lb/in (MPa)
4
4
I moment of inertia of beam cross section, in (mm )
T natural period, s
To determine the characteristic shapes and natural periods for beams with
variable cross section and mass, use the Rayleigh method. Convert the beam
into a lumped-mass system by dividing the span into elements and assuming
the mass of each element to be concentrated at its center. Also, compute all quanti-
ties, such as deflection and bending moment, at the center of each element. Start
with an assumed characteristic shape.

TORSION IN STRUCTURAL MEMBERS

Torsion in structural members occurs when forces or moments twist the beam
or column. For circular members, Hooke’s law gives the shear stress at any giv-
en radius, r. Table 2.7 shows the polar moment of inertia, J, and the maximum
shear for five different structural sections.

STRAIN ENERGY IN STRUCTURAL MEMBERS*

Strain energy is generated in structural members when they are acted on by
forces, moments, or deformations. Formulas for strain energy, U, for shear,
torsion and bending in beams, columns, and other structural members are:

Strain Energy in Shear.
For a member subjected to pure shear, strain energy is given by
2
V L
U (2.38)
2AG
AG 2
U (2.39)
2L
where V shear load
shear deformation
L length over which the deformation takes place
A shear area
G shear modulus of elasticity

*Brockenbrough and Merritt—Structural Steel Designer’s Handbook, McGraw-Hill.