Structural Concrete - Hassoun

376 Chapter 11 Members in Compression and Bending
c. Then P n calculated by the Whitney equation is not a conservative value in this example, and
the value of P n = 643 K is greater than the more accurate value of 612.9 K calculated by statics
in Example 11.4.
2. Solution by Hsu equation (Eq. 11.21):
a. For a balanced condition, P b = 453.4 K and M b = 6810.8 K ⋅ in. (Example 11.2).
b. P 0 = 0.85f c ′(A g − A st )+A stf = 0.85(4)(14 × 22 − 8)+8(60) =1500 K
( )
P
c. n − 453.4
1.5
1500 − 453.4 + 10Pn
= 1
6810.8
Multiply by 1000 and solve for P n .
0.9555P n + 0.05626P 1.5
n = 1433.2K
By trial, P n = 611 K, which is very close to 612.9 K, as calculated by statics.
11.9 INTERACTION DIAGRAM EXAMPLE
In Example 11.2, the balanced loads P b , M b ,ande b were calculated for the section shown in
Fig. 11.6 (e b = 15 in.). Also, in Examples 11.3 and 11.4, the load capacity of the same section
was calculated for the case when e = 20 in. (tension failure) and when e = 10 in. (compression
failure). These values are shown in Table 11.1.
To plot the load–moment interaction diagram, different values of φP n and φM n were calculated
for various e values that varied between e = 0ande = maximum for the case of pure moment
when P n = 0. These values are shown in Table 11.1. The interaction diagram is shown in Fig. 11.11.
The load φP n0 = 975 K represents the theoretical axial load when e = 0, whereas 0.8φP n0 = 780 K
represents the maximum axial load allowed by the ACI Code based on minimum eccentricity. Note
that for compression failure, e < e b and P n > P b , and for tension failure, e > e b and P n < P b .Thelast
Table 11.1
Summary of Load Strength of Column Section in Previous Examples
e (in.) a (in.) φ P n
(K) φP n
(K) φM n
(K ⋅ ft) Notes
0 — 0.65 1500 975 0.0 φP n0
2.25 19.39 0.65 1200 780 146.3 0.8φP n0
4 16.82 0.65 1018 661.7 220.6 Compression
6 14.19 0.65 843.3 548.1 274.0 Compression
10 a 11.43 0.65 612.9 398.4 332.0 Compression
12 10.63 0.65 538.0 349.7 349.7 Compression
15 a 9.81 0.65 453.4 294.7 368.9 Balanced
20 a 7.10 0.81 324.4 263.4 439.0 Transition
30 5.06 0.90 189.4 170.5 426.2 Tension
50 4.01 0.90 100.6 90.5 377.2 Tension
80 3.59 0.90 58.8 52.9 352.0 Tension
PM b 3.08 0.90 0.0 0.0 352.0 Tension
PM 3.08 0.65 0.0 0.0 254.2 PM (X) c
a Values calculated in Examples 11.2, 11.3 and 11.4.
b PM is pure moment.
c X indicates not applicable, for comparison only.