1. Concrete Column Biaxial Strength Calculations

06

Figure 4 - Strains, Forces, and Moment Arms Diagram

1.1. Location of Neutral Axis and Concrete Compression Force

The trial-and-error process for calculating the neutral axis depth and angle α is not required in this example since these values are given (c = 10.05 in. and α = 50o). Where c is the distance from the fiber of maximum compressive strain to the neutral axis and α is the angle of the neutral axis.     ACI 318-19 (22.2.2.4.2)



(for spiral)    ACI 318-19 (Table 21.2.2)

    ACI 318-19 (22.2.2.4.1)

    ACI 318-19 (22.2.2.1)

Where:

a = Depth of equivalent rectangular stress block

    ACI 318-19 (Table 22.2.2.4.3)

   (Compression) ACI 318-19 (22.2.2.4.1)

Where (see the following figure):







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Figure 5 - Cracked Concrete Column Section Centroid Calculations

1.2. Strains and Forces Determination in Reinforcement Layers

The following shows the calculations of forces in the reinforcement layers with the extreme tension (at bar 3) and extreme compression (at bar 6) strains. The calculations for the rest of layers are shown the table at the end of this section.

For extreme tension reinforcement layer (at bar 3):



(Tension)

For extreme tension reinforcement layer (at bar 6):



The area of the reinforcement in this layer is included in the area used to compute Cc (a = 8.04 in. > d6 = 2.83 in.). As a result, it is necessary to subtract 0.85fc’ from fs6 before computing Fs6:

(Compression)

The same procedure shown above can be repeated to calculate the forces in the remaining reinforcement locations, results are summarized in the following table:

Table 1 - Strains, internal force resultants and Moments

Location

d, in

ε, in./in.

fs, psi

Fs, kip

Cc, kip

Moment arm (x), in.

My, kip-ft

Moment arm (y), in.

Mx, kip-ft

Concrete

---

0.00300

---

---

430.03

2.58

92.60

2.17

77.70

Bar 1

4.625

0.00162

46964

18.79*

---

5.25

8.22

0.00

0.00

Bar 2

9.796

0.00008

2211

0.97*

---

2.63

0.21

-4.55

-0.37

Bar 3

13.170

-0.00093

-26998

-11.88

---

-2.63

2.60

-4.55

4.50

Bar 4

11.375

-0.00040

-11456

-5.04

---

-5.25

2.21

0.00

0.00

Bar 5

6.204

0.00115

33296

12.78*

---

-2.63

-2.80

4.55

4.84

Bar 6

2.830

0.00216

60000

24.53*

---

2.63

5.37

4.55

9.29

Axial Force and Biaxial Bending Moments Capacities

Pn, kip

470.19

Mny, kip-ft

110.87

Mnx, kip-ft

93.51

ϕPn, kip

352.64

ϕMny, kip-ft

83.15

ϕMnx, kip-ft

70.13

* The area of the reinforcement in this layer has been included in the area used to compute Cc. As a result, 0.85fc’ is subtracted from fs in the computation of Fs.


1.3. Calculation of Pn, Mnx and Mny


(+) = Compression

(-) = Tension





(+) = Counter Clockwise

(-) = Clockwise





(+) = Counter Clockwise

(-) = Clockwise