vestigation. Calculate fbp and ftp by expressing these
stresses as functions of the kern distances of the sec-
tion.
Calculation Procedure:
- Consider the prestressing force to be
applied at each kern point, and evaluate
the kern distances
Let Qb and Qt denote the points at which a compres-
sive force must be applied to induce a zero stress in the
top and bottom fiber, respectively. These are referred to
as the kern points of the section, and the distances kb
and kt, from the centroidal axis to these points are
called the kern distances.
Consider the prestressing force to be applied at each
FIGURE 36. Kern points. kem point in ^m Set me stresses/^, and/6/? equal to
zero to evaluate the kern distances kb and kt , respec-
tively. Thus 4 = FJA - FMS, = O, Eq. a;fbp = F 1 IA -
Ffit/Sh = O, Eq. b. Then
St Sb
kb=- and kt=^- (55)
a A
And, kb = 14,400/850 = 16.9 in (429.26 mm); kt = 11,400/850 = 13.4 in (340.36 mm).
- Express the stresses fbp and ftp associated with the actual
eccentricity as functions of the kern distances
By combining the stress equations with Eqs. a and b, the following equations are ob-
tained:
Ffa + e) Ft(kb + e)
fbP= c and ftp= c (^56 )
^b ^t
Substituting numerical values gives fbp = 630,000(13.4 + 24)711,400 = +2067 lb/in^2
(+14,252.0 kPa);^ = 630,000(16.9 - 24)14,400 = -311 lb/in^2 (-2144.3 kPa).
- Alternatively, derive Eq. 56 by considering the increase
in prestress caused by an increase in eccentricity
Thus, A/Jp = F^e/Sb; therefore,/^ = Ffa + e)/Sb.
MAGNEL DIAGRAM CONSTRUCTION
The data pertaining to a girder having curved tendons are A = 500 in
2
(3226.0 cm
2
); Sb =
5000 in
3
(81,950 cm
3
); St = 5340 in
3
(87,522.6 cm
3
); Mw = 3600 in-kips (406.7 kN-m);
M 5 = 9500 in-kips (1073.3 kN-m). The allowable stresses are: initial, + 2400 and -190
lb/in
2
(+ 16,548 and-1310.1 kPa); final, + 2250 and-425 lb/in
2
(+15,513.8 and-2930.4
kPa). (a) Construct the Magnel diagram for this member, (b) Determine the minimum
prestressing force and its eccentricity by referring to the diagram, (c) Determine the pre-
stressing force if the eccentricity is restricted to 18 in (457.2 mm).
