Page 233 - Handbook of Civil Engineering Calculations, Second Edition

2.18 REINFORCED AND PRESTRESSED CONCRETE ENGINEERING AND DESIGN

the supports rotate as planes. Refer to Fig. 9b and c: 1 /6; 2 2 1 /3 0.333 ;
3 /b; 4 /c; 5 (1/b 1/c) [ /(AE)](6/a a/6).
3. Select a trial value of a, and evaluate the distances and angles
2
2 0.5
Using a 4.5 ft (1.37 m) as the trial value, we find AE (a 6 ) 7.5 ft (2.28 m);
b 5.63 ft (1.716 m); c 10 ft (3.0 m); 5 ( /7.5)(6/4.5 4.5/6) 0.278 .
4. Develop an equation for the external work W E performed by the
uniform load on a surface that rotates about a horizontal axis
In Fig. 10, consider that the surface ABC rotates about axis AB through an angle while
carrying a uniform load of w lb/sq.ft. (kPa). For the elemental area dA s , the deflection, to-
tal load, and external work are x ; dW wdA; dW E dW x wdA. The total
work for the surface is W E w dA, or
W E w Q (20)

where Q static moment of total area, with respect to the axis of rotation.
5. Evaluate the external and
internal work for the slab
Using the assumed value, we see a 4.5 ft
(1.37 m), EF 16 9 7 ft (2.1 m). The
external work for the two triangles is
2w u ( /4.5)( /6)(12)(4.5) 2 18w u . The
1
external work for the two trapezoids is
2
1
2w u ( /6)( /6)(16 2 7)(6) 60w u .
Then W E w u (18 60) 78w u ; W I
m u (7 2 4 7.5 5 ) 10.67m u .
6. Find the value of m u
corresponding to the assumed
value of a
Equate the external and internal work to
find this value of m u . Thus, 10.67m u
78w u ; m u 7.31w u .
7. Determine the highest value
of m u
Assign other trial values to a, and find the
corresponding values of m u . Continue this
procedure until the highest value of m u is
obtained. This is the true value of the ulti-
mate unit moment.
FIGURE 10

Design of Flexural Members by the
Working-Stress Method

As demonstrated earlier, the analysis or design of a composite beam by the working-stress
method is most readily performed by transforming the given beam to an equivalent homo-
geneous beam. In the case of a reinforced-concrete member, the transformation is made
by replacing the reinforcing steel with a strip of concrete having an area nA s and located
at the same distance from the neutral axis as the steel. This substitute concrete is assumed
capable of sustaining tensile stresses.