Page 291 - Handbook of Civil Engineering Calculations, Second Edition

2.76 REINFORCED AND PRESTRESSED CONCRETE ENGINEERING AND DESIGN

FIGURE 47. Parabolic arc.

2. Present the results
The equations are

2 x
x
y 2(y 1 2y 2 y 3 ) (3y 1 4y 2 y 3 ) y 1 (66)
L
L
dy x 3y 1 4y 2 y 3

m 4(y 1 2y 2 y 3 ) (67)
dx L 2 L
2
dm d y 4
(y 1 2y 2 y 3 ) (68)
dx dx 2 L 2
(3y 1 4y 2 y 3 )
m 1 (69a)
L

y 1 4y 2 3y 3
m 3 (69b)
L
(L/4)(3y 1 4y 2 y 3 )
x s (70a)
y 1 2y 2 y 3
(1/8)(3y 1 4y 2 y 3 ) 2
y s y 1 (70b)
y 1 2y 2 y 3

ALTERNATIVE METHODS OF ANALYZING A
BEAM WITH PARABOLIC TRAJECTORY

The beam in Fig. 48 is subjected to an initial prestressing force of 860 kips (3825.3
kN) on a parabolic trajectory. The eccentricities at the left end, midspan, and right end,