About this calculator
This tool traces the axial force–moment (P-M) interaction envelope for a tied rectangular reinforced-concrete column using strain-compatibility analysis, then checks a supplied load point (Pu, Mu) against that envelope. It is meant for engineers designing or verifying uniaxial column capacity per ACI 318 / NSCP 2015.
Method & formulas
Plane sections remain plane: the concrete strain is capped at εcu=0.003 at the extreme compression fiber, and steel strain at each bar layer varies linearly with distance from the neutral axis at depth c:
εs = 0.003·(c − di)/c fs = Es·εs, capped at ±fy (Es=200,000 MPa)
Concrete compression is idealized with the Whitney equivalent stress block (a=β1c), and axial/moment capacity are summed layer-by-layer for each trial c, sweeping from pure compression to pure tension:
Pn = 0.85f'c·a·b + ΣFsi Mn = 0.85f'c·a·b·(h/2 − a/2) + ΣFsi(h/2 − di)
Axial capacity is capped to represent the minimum-eccentricity provision of ACI 318 §22.4.2:
P0 = 0.85f'c(Ag−Ast) + fyAst Pn,max = 0.80P0 (tied) or 0.85P0 (spiral)
The strength-reduction factor φ transitions with the net tensile strain εt in the extreme layer of tension steel: φ=0.90 for εt≥0.005 (tension-controlled), φ=0.65 tied/0.75 spiral for εt≤εy (compression-controlled), and linear interpolation in between. The balanced point is where εcu=0.003 coincides with εt=εy in the extreme tension bar.
Worked example
Using the tool's defaults — 400×400 mm section, cover to bar centroid=60 mm, f'c=28 MPa, fy=420 MPa, 8-Ø20 bars (tied): Ag=160,000 mm², Ast=8×314.2=2513 mm² (ρg≈1.57%, within the 1–8% code range). P0 = 0.85(28)(160,000−2513) + 420(2513) ≈ 3,748,190 + 1,055,586 ≈ 4804 kN. With φmin=0.65 and the 0.80 tied cap, φPn,max = 0.65×0.80×4804 ≈ 2498 kN. The default load point Pu=1500 kN sits comfortably below φPn,max, so the tool then interpolates the swept envelope at that axial level to check whether Mu=200 kN·m falls inside it.
Assumptions & limitations
- Uniaxial bending about one axis only — biaxial demand (e.g. corner columns) needs a separate Bresler reciprocal-load or load-contour check.
- Cross-sectional capacity only: slenderness, second-order (P-Δ) effects, and moment magnification are not evaluated.
- Reinforcement is lumped into exactly two layers (top and bottom, equal area); distributed 4-face bar cages will shift the true balanced point and moment capacity.
- Tie/spiral confinement, spacing, and seismic detailing requirements are not checked.
- Verify results against the governing code and a licensed engineer's judgment.
FAQ
- What is the "balanced point" on the diagram?
- It is the (Pn, Mn) pair where the concrete reaches its crushing strain exactly as the extreme tension bar reaches yield — the transition between compression-controlled and tension-controlled behavior, and typically the point of maximum moment capacity.
- Why is there a flat cap near pure compression?
- ACI 318 limits usable axial strength to 80% (tied) or 85% (spiral) of the theoretical pure-compression capacity P0, since every real column carries some unavoidable minimum eccentricity.
- My load point plots outside the envelope — what are my options?
- Increase the section size, raise f'c, add more longitudinal steel, or switch to a spiral column (higher φ and axial cap), then re-check the point against the new envelope.