CIVIL ENGINEERING FORMULAS

46 CHAPTER TWO

reciprocal deflections, we obtain the end moments of the deflected beam in

Fig. 2.9 as

(2.1)

(2.2)

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

(2.3)

whereKis the actual stiffness for the end of the beam that is fixed; for beams of

variable moment of inertia Kequals the fixed-end stiffness times.

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 MP. 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 ktimes 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 BandCand the moment diagram for end moments

MBandMCapplied to the determinate beam. Then, by using Fig. 2.10(c), equi-

librium is maintained when

 (2.4)

wL^2

4(1k)



wL^2

4 kMP

MP

wL^2

4



1

2

MB

1

2

MC

(1CLFCRF)

MFK

d

L

MRFKRF (1CLF)

d

L

MFLKFL(1CFR)

d

L