Building Design and Construction Handbook - Merritt - Ventech!

STRUCTURAL THEORY 5.89
� �
4EI 1 6EI d
F
ML � �L � �R � M L � (5.135)
L 2 L L
4EI 1 6EI d
F
MR � ��R � �L� � M R � (5.136)
L 2 L L
where E � modulus of elasticity
I � moment of inertia of the cross section
Note that if end L moves downward with respect to R, the sign for d in the
preceding equations is changed.
If the end moments M L and M R are known and the end rotations are to be
determined, Eqs. (5.131) to (5.134) can be solved for � L and � R or derived by
superposition of simple-beam components, as is done in Art. 5.11.4. For beams
with moment of inertia varying along the span:
d
F F
�L � (ML � M L) �L � (MR � M R)
� � (5.137)
L
d
F F
�R � (MR � M R) �R � (ML � M L)
� � (5.138)
L
where � is given by Eq. (5.105) and � by Eq. (5.106). For beams with constant
moment of inertia:
L L d
F F
�L � (ML � M L) � (MR � M R)
�
3EI 6EI L
(5.139)
L L d
F F
�R � (MR � M R) � (ML � M L)
�
3EI 6EI L
(5.140)
The slope-deflection equations can be used to determine end moments and rotations
of the spans of continuous beams by writing compatibility and equilibrium
equations for the conditions at each support. For example, the sum of the moments
at each support must be zero. Also, because of continuity, the member must rotate
through the same angle on both sides of every support. Hence, M L for one span,
given by Eq. (5.133) or (5.135), must be equal to �M R for the adjoining span,
given by Eq. (5.134) or (5.136), and the end rotation � at that support must be the
same on both sides of the equation. One such equation with the end rotations at
the supports as the unknowns can be written for each support. With the end rotations
determined by simultaneous solution of the equations, the end moments can be
computed from the slope-deflection equations and the continuous beam can now
be treated as statically determinate.
See also Arts. 5.11.9 and 5.13.2.
(C. H. Norris et al., ‘‘Elementary Structural Analysis,’’ 4th ed., McGraw-Hill
Book Company, New York.)
5.11.6 Moment Distribution
The frame in Fig. 5.74 consists of four prismatic members rigidly connected together
at O at fixed at ends A, B, C, and D. If an external moment U is applied at