Page 91 - Civil Engineering Formulas

46 CHAPTER TWO

reciprocal deflections, we obtain the end moments of the deflected beam in
Fig. 2.9 as
d
F
F
F
M L K L (1 C R ) (2.1)
L
d
F
F
F
M R K R (1 C L ) (2.2)
L
In a similar manner the fixed-end moment for a beam with one end hinged and
the supports at different levels can be found from
d
F
M K (2.3)
L
where K is the actual stiffness for the end of the beam that is fixed; for beams of
F
F
variable moment of inertia K equals the fixed-end stiffness times (1 C L C R ) .
ULTIMATE STRENGTH OF CONTINUOUS BEAMS
Methods for computing the ultimate strength of continuous beams and frames
may be based on two theorems that fix upper and lower limits for load-carrying
capacity:
1. Upper-bound theorem. A load computed on the basis of an assumed link
mechanism is always greater than, or at best equal to, the ultimate load.
2. Lower-bound theorem. The load corresponding to an equilibrium condition
with arbitrarily assumed values for the redundants is smaller than, or at best
equal to, the ultimate loading—provided that everywhere moments do not
exceed M P . The equilibrium method, based on the lower bound theorem, is
usually easier for simple cases.
For the continuous beam in Fig. 2.10, the ratio of the plastic moment for the
end spans is k times that for the center span (k 1).
Figure 2.10(b) shows the moment diagram for the beam made determinate
by ignoring the moments at B and C and the moment diagram for end moments
M B and M C applied to the determinate beam. Then, by using Fig. 2.10(c), equi-
librium is maintained when
wL 2 1 1
M P M B M C
4 2 2
wL 2

4 kM P
wL 2
(2.4)
4(1 k)