Building Design and Construction Handbook - Merritt - Ventech!

5.160 SECTION FIVE
resistance to deformation, and other factors. In general, it is advisable to represent
the actual structure and loading by idealized systems that permit a solution in closed
form (see Arts. 5.18.1 to 5.18.5).
Whenever possible, represent the actual structure by a one-degree system consisting
of an equivalent mass with massless spring. For structures with distributed
mass. simplify the analysis in the elastic range by computing the response only for
one or a few of the normal modes. In the plastic range, treat each stage—elastic,
and plastic—as completely independent; for example, a fixed-end beam may be
treated, when in the elastic-plastic stage, as a simply supported beam.
Choose the parameters of the equivalent system to make the deflection at a
critical point, such as the location of the concentrated mass, the same as it would
be in the actual structure. Stresses in the actual structure should be computed from
the deflections in the equivalent system.
Compute an assumed shape factor � for the system from the shape taken by the
actual structure under static application of the loads. For example, for a simple
beam in the elastic range with concentrated load at midspan, � may be chosen, for
x � L/2, as (Cx/L 3 )(3L 2 � 4x 2 ), the shape under static loading, and C may be set
equal to 1 to make � equal to 1 when x � L/2. For plastic conditions (hinge at
midspan), � may be taken as Cx/L, and C set equal to 2, to make � � 1 when
x � L/2.
For a structure with concentrated forces, let W r be the weight of the rth mass,
� r the value of � for a specific mode at the location of that mass, and F r the
dynamic force acting on W r. Then, the equivalent weight of the idealized system
is
j
2
e � r r
r�1
where j is the number of masses. The equivalent force is
W � W � (5.300)
j
e � r r
r�1
F � F � (5.301)
For a structure with continuous mass, the equivalent weight is
2
We � � w� dx (5.302)
where w is the weight in lb/lin ft. The equivalent force is
F � �q� dx (5.303)
e
for a distributed load q, lb/lin ft.
The resistance of a member or structure is the internal force tending to restore
it to its unloaded static position. For most structures, a bilinear resistance function,
with slope k up to the elastic limit and zero slope in the plastic range (Fig. 5.112a),
may be assumed. For a given distribution of dynamic load, maximum resistance of
the idealized system may be taken as the total load with that distribution that the
structure can support statically. Similarly, stiffness is numerically equal to the total
load with the given distribution that would cause a unit deflection at the point where
the deflections in the actual structure and idealized system are equal. Hence, the