Slender Column & Shear Wall Check — Moment Magnification and Biaxial P–M

Design checks

Calculation

1. 1Longitudinal steelACI318 10.9.1
   A_{st} = (2n_b + 2n_h - 4)\,\tfrac{\pi}{4}d_b^2, \quad \rho_g = A_{st}/A_g
   A_{st} = 88 \times 0.761 = 66.96, \quad \rho_g = 66.96 / 1400 = 4.780\%
   \rho_g=4.78 %
2. 2Concrete modulus and stress-block factorACI318 10.2
   E_c = 57{,}000\sqrt{f'_c}, \quad \beta_1 = 0.85 - 0.05\,\frac{f'_c - 4000}{1000}
   E_c = 4415\text{ ksi}, \quad \beta_1 = 0.750
   E_c=4415 ksi
3. 3Weak axis (M2): gross and steel moments of inertia
   I_g = \frac{w D^3}{12}, \quad I_{se} = \sum A_{s,i}\,(d_i - D/2)^2
   I_g = \frac{100.0 \times 14.0^3}{12} = 22867, \quad I_{se} = 1301\text{ in}^4 \;(4\text{ layers})
   I_g=22867 in⁴
4. 4Weak axis (M2), largest δns — "142 1.2D-1.6WX+0.5L-6": slenderness limitACI318 Eq. (10-7)
   \frac{k l_u}{r} > 34 - 12\,\frac{M_1}{M_2} \;(\le 40), \quad r = 0.3D
   \frac{1 \times 24 \times 12}{0.3 \times 14} = 68.600 > 34 - 12\,\frac{-8.770}{-14.120} = 26.500
   k l_u / r=68.6

   Slenderness must be considered for every compression combination on this axis.

5. 5Weak axis (M2): equivalent moment factorACI318 10.10.6.4
   C_m = 0.6 + 0.4\,\frac{M_1}{M_2}
   C_m = 0.6 + 0.4\,\frac{-8.770}{-14.120} = 0.848
   C_{m2}=0.848

   From the analysed end moments, M1/M2 positive in single curvature; taken as 0 when both are zero. The minimum moment does not enter Cm.

6. 6Weak axis (M2): flexural stiffness, Eq. 10-14ACI318 10.10.6.1
   EI = \frac{0.2E_cI_g + E_sI_{se}}{1 + \beta_{dns}}
   \beta_{dns} = 0.917, \quad EI = 30215441
   EI=30215441 kip·in²
7. 7Weak axis (M2): critical load and magnifierACI318 10.10.6
   P_c = \frac{\pi^2 EI}{(k l_u)^2}, \quad \delta_{ns} = \frac{C_m}{1 - P_u / 0.75P_c} \ge 1.0
   P_c = \frac{\pi^2 \times 30215441}{(1 \times 24 \times 12)^2} = 3595, \quad \delta_{ns} = \frac{0.848}{1 - 1763/(0.75 \times 3595)} = 2.451
   \delta_{ns2}=2.451
8. 8Weak axis (M2): design moment for strengthACI318 10.10.6.5
   M_{2,min} = P_u(0.6 + 0.03h), \quad M_c = \delta_{ns} \max(|M_2|, M_{2,min})
   M_{2,min} = \frac{1763(0.6 + 0.03 \times 14)}{12} = 149.9, \quad M_c = 2.451 \times 149.900 = 367.3
   M_{c2}=367.3 kip·ft
9. 9Strong axis (M3): gross and steel moments of inertia
   I_g = \frac{w D^3}{12}, \quad I_{se} = \sum A_{s,i}\,(d_i - D/2)^2
   I_g = \frac{14.0 \times 100.0^3}{12} = 1166667, \quad I_{se} = 57279\text{ in}^4 \;(42\text{ layers})
   I_g=1166667 in⁴
10. 10Strong axis (M3): slenderness may be neglected for every combination (tightest: "111 1.4D-1")ACI318 Eq. (10-7)
    \frac{k l_u}{r} \le 34 - 12\,\frac{M_1}{M_2} \le 40, \quad r = 0.3D
    \frac{1 \times 24 \times 12}{0.3 \times 100} = 9.600 \le 26.100
    k l_u / r=9.6

    First-order moments are used as they are: no magnifier and no minimum moment.

11. 11Maximum design axial strengthACI318 10.3.6
    \phi P_{n,max} = 0.80\phi\left[0.85f'_c(A_g - A_{st}) + f_yA_{st}\right]
    \phi P_{n,max} = 0.80 \times 0.65 \times 11653 = 6059
    \phi P_{n,max}=6059 kip
12. 12Biaxial strength, governing combination — "142 1.2D-1.6WX+0.5L-6", minimum moment about the weak axisACI318 10.3
    DCR = \frac{|M_{c2}|}{\phi M_{n2}} + \frac{|M_{c3}|}{\phi M_{n3}}
    DCR = \frac{367.300}{1456.300} + \frac{1908.400}{9941.900} = 0.444
    DCR=0.444

    The minimum moment is applied about one axis at a time: each combination is checked with M2,min on the weak axis and the strong axis on its analysed moment, then the reverse, and the larger sum governs.

13. 13Bar spacing along the long face hACI318 7.6.1
    s_{clear} = s - w_{bar} \ge s_{min}
    s = 2.32, \quad s_1.33 = 1.33, \quad s_1.00 = 1.00
    s_{clear}=1.33 in

    Governing minimum: 1 in (§7.6.1).

14. 14Bar spacing along the short face bACI318 7.6.1
    s_{clear} = s - w_{bar} \ge s_{min}
    s = 3.00, \quad s_2.02 = 2.02, \quad s_1.00 = 1.00
    s_{clear}=2.02 in

    Governing minimum: 1 in (§7.6.1).

Assumptions

• Nonsway (braced) member: first-order end moments from the analysis are magnified by δns only; sway magnification δs is not applied.
• Slenderness is neglected, per axis and combination, where klu/r ≤ 34 − 12(M1/M2) ≤ 40 with r = 0.3h (Eq. 10-7); M1/M2 is taken as 0 when both end moments are zero. Such a combination is checked on its first-order moment, without the minimum moment — as PCA Notes Example 11.1 does.
• Cm is taken from the analysed end moments with no lower bound; the minimum moment M2,min only raises the moment that is magnified and checked for strength.
• Rectangular section with bars on the perimeter only, equally spaced along each face; the corner bars are counted once.
• Axial load per combination is taken at the end with the larger compression (the bottom, for gravity-dominated combinations).
• βdns is taken as the combination's dead-load coefficient (read from its name, with ±Ev for the λD combinations) times Pu,1.4D / 1.4, over its Pu, capped at 1.0 — the owner's method.
• Biaxial strength by the linear interaction DCR2 + DCR3 ≤ 1.0 (ACI SP-17) — conservative relative to a full P–M–M surface. The minimum moment is applied about one axis at a time (the other axis on its analysed moment, magnified); the worse of the two cases governs.
• Tied member: φ = 0.65 in compression rising to 0.90 at εt = 0.005; φPn capped at 0.80φPo.