CIVIL ENGINEERING FORMULAS

76 CHAPTER TWO

multiply it by. To get T, divide the appropriate constant by.

In these equations,

 natural frequency, rad/s

Wbeam weight, lb per linear ft (kg per linear m)

Lbeam length, ft (m)

Emodulus of elasticity, lb/in^2 (MPa)

Imoment of inertia of beam cross section, in^4 (mm^4 )

Tnatural 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

(2.38)

(2.39)

whereVshear load

shear deformation

Llength over which the deformation takes place

Ashear area

Gshear modulus of elasticity

U

AG 2

2 L

U

V^2 L

2 AG

EI/wL^4 EI/wL^4

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