26 Analysis of Rigid

When analyzing the frame by moment distribution, begin by lockingthe joints b and c. Fig. 12 illustrates the static conditions from which thefixed end moment at b is determined. It can be shown that

F. E. M. at b =

(3 X 106 X 122) X 0.02

E XDX Sb X £ X (l

-

rj^

j

17.88X 13.3 X

(1/12) X 2.503

X \ 13.3 X 6.0/17.88354,000 ft.lb.

According to Problem 2, the formula for both corner and crown moment —

when deck is straight —may be written as

Moment = 354,000 X [(1.0000 - 0.8590 - 0.0094) - 0.0905] =

354,000 X 0.0411 = 14,500 ft.lb.

Final moments —allowing for curvature of deck —equal(

17 88 \217 88 + 1 5

oj = 12,300 fUb.

/ 17 88 \Crown: 14,500 X ( 17 88 + 1 5

o) =

13>400 ft lb-+ 1.50,

The foregoing procedure is simple and general; it is

applicable to frames with relatively high and slender legs.

In most rigid frame bridges, however, the legs are generallyso short and stubby that it is preferable to determinemoments due to change in deck length by an independentmethod. The reason may be illustrated by an example.Consider the case in which the coefficients in the same frameas above have been determined with less degree of accuracy,and substitute in the original formula the slightly differentvalues for the frame coefficients as follows :

Moment = 354,000 X [ (1.00 - 0.85 - 0.01) - 0.09] = 354,000 X0.05 = 17,700 ft.lb.

By comparing the original with the assumed case, it is seen that theslight changes in the frame coefficients create a variation in the momentwhich exceeds 20 per cent.

A considerable amount of analytical work was made in order to substitute for moment distribution a simpler procedure for change in decklength. The following empirical formula was developed for single-span,hinged frames with proportions as indicated in Chart I :

Corner moment = 4.35 X 106 X ^2

X (Crown thickness)2{tl-\-K)

D is the relative displacement of each hinge (see Fig. 12) ; and the constant is based upon a value of modulus of elasticity equal to 3 X 106

X 122 lb. per sq. ft. The numerical values are to be inserted in feet, and themoment is in ft.lb. per linear foot of width of the frame. The formula

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