Simplified Design for Columns - Simplified Design of Reinforced Concrete Buildings - page 159

⎡ ⎤

⎛
⎜

⎞
⎟

3
∑

C 2 d i
d 1

φ P n = 0.65

C 1 hd 1 + 87 A si 1 −

⎢ ⎥

⎣ ⎦

i = 1

= 0.65{(2.89 × 18 × 15.56) + 87[(3 × 0) + (2 × 0.42) + (3 × 0.69)]}

= 0.65 (809.4 + 253.2)

= 690 kips

⎡ ⎤

⎛
⎜

⎞
⎟

) + 87 A si 1 − C 2 d i
d 1

3
∑

h
2 −

⎛
⎜

⎞
⎟

(

φ M n = 0.65

0.5C 1 hd 1 C 3 d 1 d i

⎢ ⎥

⎣ ⎦

i = 1

= 0.65 {(0.5 × 2.89 × 18 × 15.56)[18 - (0.85 × 15.56)]

+ 87[(3.0 × 0(9 - 15.56)) + (2.0 × 0.42(9 - 9)) + (3.0 × 0.69(9 - 2.44))]}/12

= 0.65 (1932.1 + 1181.4) /12

= 169 ft-kips

(3) Point 3 (f s1 = 0.5 f y ):

In this case, C 1 = 2.14, C 2 = 1.35, and C 3 = 0.63 (Table 5-1) (Table 5-1)

C 2 d 1

Layer 1:

1 −

d 1 =

1 − 1.35 =− 0.35

C 2 d 2

9.00
15.56

⎛
⎜

⎞

Layer 2:

1 −

d 1 =

1 − 1.35

⎟ =−

0.22

C 2 d 2

⎛
⎜

2.44
15.56

⎞

Layer 3:

1 −

d 1 =

1 − 1.35

⎟ =

0.79 > 0.69 Use 0.69

φ P n = 0.65 {(2.14 × 18 × 15.56) + 87[(3 - (-0.35)) + (2 × 0.22) + (3 × 0.69)]}

= 0.65 (599.4 + 127.0) = 472 kips

φ M n = 0.65 {(0.5 × 2.14 × 18 × 15.56) [18 - (0.63 × 15.56)]

+ 87 [(3.0(-0.35) × (9 - 15.56)) + 0 + (3.0 × 0.69 (9 - 2.44))]}/12

= 0.65 (2456.6 + 1780.7) /12 = 229 ft-kips

(4) Point 4 (f s1 = f y ):

In this case, C 1 = 1.70, C 2 = 1.69, and C 3 = 0.50

Similar calculations yield the following:

φ P n = 312 kips

φ M n = 260 ft-kips