Building Design and Construction Handbook - Merritt - Ventech!

STRUCTURAL THEORY 5.81
L
�L � �R � (5.107)
3EI
L
� � (5.108)
6EI
Carry-Over Factors. The preceding equations can be used to determine carryover
factors for any magnitude of end restraint. The carry-over factors toward fixed
ends, however, are of special importance.
The bending-moment diagram for a continuous span LR that is not loaded except
for a moment M applied at end L is shown in Fig. 5.61a. For determination of the
carry-over factor C R toward R, that end is assumed fixed (no rotation can occur
there). The carry-over moment to R then is C RM. The moment diagram in Fig.
5.61a can be resolved into two components: a simple beam with M applied at L
(Fig. 5.61b) and a simple beam with C RM applied at R (Fig. 5.61c). As indicated
in Fig. 5.61d, M causes an angle change at R of ��. As shown in Fig. 5.61e, C R
M induces an angle change at R of C RM� R. Since the net angle change at R is zero
(Fig. 5.61ƒ), C RM� R � M� � 0, from which
�
C � (5.109)
Similarly, the carry-over factor toward support L is given by
R
FIGURE 5.61 Effect of applying an end moment M to any span of a continuous
beam: (a) An end moment C RM is induced at the opposite end. (b) and (c) The
moment diagram in (a) is resolved into moment diagrams for a simple beam. (d)
and (e) Addition of the end rotations corresponding to conditions (b) and (c) yields
(ƒ), the end rotations induced by M in the beam shown in (a)
� R
�
C � (5.110)
L
Since the carry-over factors are positive, the moment carried over has the same
sign as the applied moment. For prismatic beams, � � L/6EI and � � L/3EI.
Hence,
� L