Building Design and Construction Handbook - Merritt - Ventech!

5.90 SECTION FIVE
O, the sum of the end moments in each member at O must be equal to U. Furthermore,
all members must rotate at O through the same angle �, since they are
assumed to be rigidly connected there. Hence, by the definition of fixed-end stiffness,
the proportion of U induced in the end of each member at O is equal to the
ratio of the stiffness of that member to the sum of the stiffnesses of all the members
at the joint (Art. 5.11.3).
FIGURE 5.74 Effect of an unbalanced moment
at a joint in a frame.
Suppose a moment of 100 ft-kips is
applied at O, as indicated in Fig. 5.74b.
The relative stiffness (or I/L) is assumed
as shown in the circle on each member.
The distribution factors for the moment
at O are computed from the stiffnesses
and shown in the boxes. For example,
the distribution factor for OA equals its
stiffness divided by the sum of the stiffnesses
of all the members at the joint:
3/(3 � 2 � 4 � 1) � 0.3. Hence, the
moment induced in OA at O is 0.3 �
100 � 30 ft-kips. Similarly, OB gets 10
ft-kips, OC 40 ft-kips and OD 20 ftkips.
Because the far ends of these members
are fixed, one-half of these moments
are carried over to them (Art.
5.11.2). Thus M AO � 0.5 � 30 � 15;
M BO � 0.5 � 10 � 5; M CO � 0.5 �
40 � 20; and M DO � 0.5 � 20 � 10.
Most structures consist of frames
similar to the one in Fig. 5.74, or even
simpler, joined together. Though the
ends of the members are not fixed, the
technique employed for the frame in
Fig. 5.74b can be applied to find end
moments in such continuous structures.
Before the general method is presented, one short cut is worth noting. Advantage
can be taken when a member has a hinged end to reduce the work of distributing
moments. This is done by using the true stiffness of a member instead of the fixedend
stiffness. (For a prismatic beam with one end hinged, the stiffness is threefourth
the fixed-end stiffness; for a beam with variable I, it is equal to the fixedend
stiffness times 1 � C LC R, where C L and C R are the carry-over factors for the
beam.) Naturally, the carry-over factor toward the hinge is zero.
When a joint is neither fixed nor pinned but is restrained by elastic members
connected there, moments can be distributed by a series of converging approximations.
All joints are locked against rotation. As a result, the loads will create
fixed-end moments at the ends of every member. At each joint, a moment equal to
the algebraic sum of the fixed-end moments there is required to hold it fixed. Then,
one joint is unlocked at a time by applying a moment equal but opposite in sign
to the moment that was needed to prevent rotation. The unlocking moment must
be distributed to the members at the joint in proportion to their fixed-end stiffnesses
and the distributed moments carried over to the far ends.
After all joints have been released at least once, it generally will be necessary
to repeat the process—sometimes several times—before the corrections to the fixed-