Data Source and Methodology
Authoritative source: ACI Committee 318. “Building Code Requirements for Structural Concrete (ACI 318-19) and Commentary.” American Concrete Institute, 2019. Official reference.
Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.
The Formula Explained
Gross area:
Rectangular: Ag = b · h
Circular: Ag = \(\frac{\pi D^2}{4}\)
Steel area (percentage mode): \(A_s = \rho \cdot A_g\), where \(\rho = \frac{\text{percent}}{100}\)
Concentric nominal axial strength:
\(P_{0n} = 0.85\,f'_c\,(A_g - A_s) + f_y A_s\)
Strength reduction factor (compression-controlled):
\(\phi = 0.65\) (tied), \(\phi = 0.75\) (spiral)
Pure bending nominal strength (rectangular approximation, symmetric layout, half steel in tension):
\(a = \dfrac{A_{s,t} f_y}{0.85 f'_c\, b}\), \(M_n = A_{s,t} f_y \left( d - \tfrac{a}{2} \right)\)
For bending about x: use width \(b_x = b\), effective depth \(d_x = h - d'\).
For bending about y: use width \(b_y = h\), effective depth \(d_y = b - d'\).
For circular sections (approximation): use \(b_{\text{eff}} = 0.85 D\) and \(d = D - d'\).
Biaxial (Bresler-type) interaction:
\(\left(\dfrac{M_{ux}}{\phi M_{nx0}}\right)^r + \left(\dfrac{M_{uy}}{\phi M_{ny0}}\right)^r \le \left(1 - \dfrac{P_u}{\phi P_{0n}}\right)^r\), where \(1 \le r \le 2\) and \(r = 2 - \dfrac{P_u}{\phi P_{0n}}\)
Glossary of Variables
- b, h — Rectangular section width and depth.
- D — Circular section diameter.
- Ag — Gross cross-sectional area.
- As — Total longitudinal steel area; ρ — steel ratio As/Ag.
- f'c — Concrete compressive strength (psi or MPa).
- fy — Steel yield strength (ksi or MPa).
- d' — Distance from extreme compression fiber to bar centroid at the tension face, approximated as cover + db/2.
- Pu — Factored axial load (compression positive).
- Mux, Muy — Factored moments about x and y axes.
- P0n — Nominal concentric axial strength.
- φ — Strength reduction factor.
- Mnx0, Mny0 — Nominal pure-bending strengths about x and y (at zero axial load), approximated.
- r — Bresler exponent transitioning from 2.0 (bending dominated) to 1.0 (axial dominated).
- U — Utilization ratio = demand/capacity; U ≤ 1.0 indicates pass for the simplified check.
How It Works: A Step-by-Step Example
- Select Units = US, Shape = Rectangular, Ties = Tied (φ = 0.65).
- Enter b = 16 in, h = 20 in; f'c = 4000 psi; fy = 60 ksi.
- Steel mode = As %, ρ = 2.0%; cover c = 1.5 in; bar diameter db = 1.0 in.
- Loads: Pu = 400 kip; Mux = 150 kip-ft; Muy = 80 kip-ft.
- The tool computes:
- Ag = 320 in²; As = 6.4 in²; φ = 0.65.
- φP0 = φ[0.85·4000(320 − 6.4) + 60,000·6.4] ≈ reported in Results.
- d' = 1.5 + 0.5 = 2.0 in; d_x = 20 − 2 = 18 in; d_y = 16 − 2 = 14 in.
- Assuming As,t = As/2 = 3.2 in²: a_x = (3.2·60,000)/(0.85·4000·16), and Mn_x0 = As,t·fy(d_x − a_x/2). Similarly for y with b_y = 20 and d_y = 14.
- Compute r = 2 − Pu/(φP0) clamped [1,2].
- Check (Mux/φMnx0)^r + (Muy/φMny0)^r ≤ (1 − Pu/φP0)^r; report U and pass/fail.
Frequently Asked Questions (FAQ)
Does this calculator include slenderness and second-order effects?
No. Slenderness and P-Δ/P-δ effects per ACI 318 (including moment magnifiers) are not included. Use this tool for preliminary checks and apply full code procedures for final design.
How accurate is the pure-bending approximation?
It is conservative for symmetric reinforcement because it neglects compression steel contribution and assumes half of As acts in tension. For final design, use full strain compatibility analysis with your exact bar layout.
Can I input different steel on each face?
Not in this version. Total As is assumed symmetric. Future updates may allow per-face bar counts and positions.
What if U is slightly above 1.0?
Consider increasing member size, raising steel ratio, improving material strengths (within code limits), or reducing loads. Always re-check with full code procedures.
Why is φ different for tied vs. spiral columns?
Spiral confinement generally improves ductility and strength, allowing a higher φ per ACI 318.
How do I choose b and h vs. x and y axes?
In this tool, Mux uses b as the compression block width and h as the effective depth; for Muy, they swap. Ensure your axis convention matches the input geometry.