Building Design and Construction Handbook - Merritt - Ventech!

5.148 SECTION FIVE
FIGURE 5.109 Coefficients for computing natural circular frequencies � and natural periods of
vibration T, s, of prismatic beams. w � weight of beam, lb/lin ft; L � span, ft; E � modulus of
elasticity of the beam material, psi; I � moment of inertia of the beam cross section, in 4 .
of natural circular frequency � and natural period T for the first four modes of
cantilever simply supported, fixed-end, and fixed-hinged beams. To obtain �, select
4
the appropriate constant from Fig. 5.109 and multiply it by �EI/wL . where L �
span of beam, ft. To get T, divide the appropriate constant by . 4 �EI/wL
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 quantities, such as
deflection and bending moment, at the center of each element. Start with an assumed
characteristic shape and apply Eq. (5.255).
Methods are available for dynamic analysis of continuous beams. (R. Clough
and J. Penzien, ‘‘Dynamics of Structures,’’ McGraw-Hill Book Company, New
York; D. G. Fertis and E. C. Zobel, ‘‘Transverse Vibration Theory,’’ The Ronald
Press Company, New York.) But even for beams with constant cross section, these
procedures are very lengthy. Generally, approximate solutions are preferable.
(J. M. Biggs, ‘‘Introduction to Structural Dynamics,’’ McGraw-Hill Book Company,
New York; N. M. Newmark and E. Rosenblueth, ‘‘Fundamentals of Earthquake
Engineering,’’ Prentice-Hall, Englewood Cliffs, N.J.)
5.18.3 Impact and Sudden Loads
Under impact, there is an abrupt exchange or absorption of energy and drastic
change in velocity. Stresses caused in the colliding members may be several times
larger than stresses produced by the same weights applied statically.