Page 280 - Handbook of Civil Engineering Calculations, Second Edition

PRESTRESSED CONCRETE 2.65

Calculation Procedure:

1. Set the initial stress in the bottom fiber at midspan equal to
or less than its allowable value, and solve for the reciprocal of F i
In this situation, the superimposed load is given, and the sole objective is to minimize the
prestressing force. The Magnel diagram is extremely useful for this purpose because it
brings into sharp focus the relationship between F i and e. In this procedure, let f bi and f bf
and so forth represent the allowable stresses.
Thus,

1 k t e
(57a)
F i M w f bi S b
2. Set the final stress in the bottom fiber at midspan equal to
or algebraically greater than its allowable value, and solve for the
reciprocal of F i
Thus,
1 (k t e)
(57b)
F i M w M s f bf S b

3. Repeat the foregoing procedure with respect to the top fiber
Thus,

1 e k b
(57c)
F i M w f ti S
t
and
1 (e k b )
(57d)
b
F i M w M s f tf S
4. Substitute numerical values, expressing F i in thousands of kips
Thus, 1/F i (10 e)/15.60, Eq. a; 1/F i (10 e)/12.91, Eq. b; 1/F i (e 10.68)/4.61,
Eq. c; 1/F i (e 10.68)/1.28, Eq. d.
5. Construct the Magnel diagram
In Fig. 37, consider the foregoing relationships as equalities, and plot the straight lines
that represent them. Each point on these lines represents a set of values of 1/F i and e at
which the designated stress equals its allowable value.
When the section moduli are in excess of those corresponding to balanced design, as
they are in the present instance, line b makes a greater angle with the e axis than does a,
and line d makes a greater angle than does c. From the sense of each inequality, it follows
that 1/F i and e may have any set of values represented by a point within the quadrilateral
CDEF or on its circumference.
6. To minimize F i , determine the coordinates of point E
at the intersection of lines b and c
Thus, 1/F i (10 e)/12.91 (e 10.68)/4.61; so e 22.2 in. (563.88 mm); F i 401
kips (1783.6 kN).
The Magnel diagram confirms the third design guide presented earlier in the section.