Building Design and Construction Handbook - Merritt - Ventech!

5.80 SECTION FIVE
of each member. Hence, if an end moment is represented by a curved arrow, the
arrow will point in the direction in which the moment is to be plotted.
5.11.2 Carry-Over Moments
When a member of a continuous beam or frame is loaded, bending moments are
induced at the ends of the member as well as between the ends. The magnitude of
the end moments depends on the magnitude and location of the loads, the geometry
of the member, and the amount of restraint offered to end rotation of the member
by other members connected to it. Because of the restraint, end moments are induced
in the connecting members, in addition to end moments that may be induced
by loads on those spans.
If the far end of a connecting member is restrained by support conditions against
rotation, a resisting moment is induced at that end. That moment is called a carryover
moment. The ratio of the carry-over moment to the other end moment is called
carry-over factor. It is constant for the member, independent of the magnitude and
direction of the moments to be carried over. Every beam has two carry-over factors,
one directed toward each end.
As pointed out in Art. 5.10.6, analysis of a continuous span can be simplified
by treating it as a simple beam subjected to applied end moments. Thus, it is
convenient to express the equations for carry-over factors in terms of the end rotations
of simple beams: Convert a continuous member LR to a simple beam with
the same span L. Apply a unit moment to one end (Fig. 5.60). The end rotation at
the support where the moment is applied is �, and at the far end, the rotation is �.
By the dummy-load method (Art. 5.10.4), if x is measured from the � end,
L 2 1 x
� � � dx (5.105)
2 L 0 EIx L 1 x(L � x)
2
0
� � � dx (5.106)
L EIx in which I x � moment of inertia at a section a distance of x from the � end
E � modulus of elasticity
In accordance with the reciprocal theorem (Art. 5.10.5) � has the same value regardless
of the beam end to which the unit moment is applied (Fig. 5.60). For
prismatic beams (I x � constant),
FIGURE 5.60 End rotations of a simple beam LR when a unit moment
is applied (a) at end L and (b) at end R.