Structural Concrete - Hassoun

374 Chapter 11 Members in Compression and Bending
where
A = 0.85f ′ c
b2
B = 0.85f ′ c b(e′ − d)
C = A ′ s (f y − 0.85f ′ c )(e′ − d + d ′ )+87A s e ′
D =−87 A s e ′ β 1 d
Once the values of A, B, C, andD are calculated, a can be determined by trial or directly
by a scientific calculator. Also, the solution of the cubic equation can be obtained by using the
well-known Newton–Raphson method. This method is very powerful for finding a root of f(x) = 0.
It involves a simple technique, and the solution converges rapidly by using the following steps:
1. Let f(a) = Aa 3 + Ba 2 + Ca + D, and calculate A, B, C, andD.
2. Calculate the first derivative of f(a):
f ′ (a) =3Aa 2 + 2Ba + C
3. Assume any initial value of a, say,a 0 , and compute the next value:
a 1 = a 0 − f (a 0)
f ′ (a 0 )
4. Use the obtained value a 1 in the same way to get
a 2 = a 1 − f (a 1)
f ′ (a 1 )
5. Repeat the same steps to get the answer up to the desired accuracy. In the case of the analysis of
columns when compression controls, the value a is greater than the balanced a(a b ). Therefore,
start with a 0 = a b and repeat twice to get reasonable results.
Example 11.5
Repeat Example 11.4 using numerical solution.
Solution
1. Calculate A, B, C, andD and determine f(a):
A = 0.85 × 4 × 14
2 = 23.8
B = 0.85 × 4 × 14(18.5 − 19.5) =−47.6
C = 4(60 − 0.85 × 4)(18.5 − 19.5 + 2.5)+87 × 4 × 18.5
= 6777.6
D =−87 × 4 × 18.5 ×(0.85 × 19.5) =−106,710
f (a) =23.8a 3 − 47.6a 2 + 6777.6a − 106,710
2. Calculate the first derivative:
f ′ (a) =71.4a 2 − 95.2a + 6777.6