Column Design
Rectangular RC column — biaxial bending with axial load. Full P-M interaction diagram using strain-compatibility.
Input Parameters
kN-m · mSection Geometry
nx = bars along x (top & bottom rows, parallel to b) · ny = bars along y (left & right columns, parallel to h, incl. corners)
Input Parameters
kN-m · mSection Geometry
nx = bars along x (top & bottom rows, parallel to b) · ny = bars along y (left & right columns, parallel to h, incl. corners)
Input Parameters
kN-m · mSection Geometry
nx = bars along x (top & bottom rows, parallel to b) · ny = bars along y (left & right, parallel to h, incl. corners)
Input Parameters
kN-m · mSection Geometry
Input Parameters
kN-m · mSection Geometry
Input Parameters
kN-m · mSection Geometry
📚 Design Background & Code References
P-M Interaction: Strain Compatibility
The interaction diagram is generated by sweeping the neutral axis depth c from a very large value (near pure compression) down to zero (pure tension), computing the resulting axial force and moment at each step using full strain compatibility.
For each neutral axis position, the extreme compression fibre strain is fixed at εcu (0.003 for ACI, 0.0035 for EC2/IS 456). Each bar's strain is then proportional to its distance from the neutral axis. The bar stress is taken as the lesser of fy/fyd and Es·εbar. The concrete compression resultant is computed using the equivalent rectangular stress block.
- 300 neutral axis positions ensure smooth, accurate curves
- Concrete displaced by bars in compression zone is deducted
- ACI: φ varies continuously with net tensile strain εt (0.65→0.90)
- EC2 / IS 456: design values fcd, fyd applied directly; no additional φ
Axis Convention
The section has width b (horizontal) and depth h (vertical). Moments are defined as follows:
- Mx — bending about the horizontal x-axis: neutral axis is parallel to b, compression/tension varies along h
- My — bending about the vertical y-axis: neutral axis is parallel to h, compression/tension varies along b
ACI 318-25 §22.4 Tied Columns
For compression-controlled sections, ACI 318-25 limits the maximum axial capacity to account for accidental eccentricity:
φ = 0.65 (tied) · φ = 0.75 (spiral)
ACI 318-25 §21.2.2 Variable φ
0.002 < εt < 0.005: φ = 0.65 + (εt−0.002)·(0.25/0.003)
εt ≥ 0.005: φ = 0.90 (tension-controlled)
ACI 318-25 §10.6 Reinforcement Limits
EN 1992-1-1 §6.1 Combined Bending and Axial
EC2 uses design material strengths throughout the interaction diagram. No additional capacity reduction factor (φ) is applied — safety is embedded in the partial factors γc and γs.
fyd = fyk / γs [γs = 1.15]
η = 1.0 (fck ≤ 50 MPa) → block stress = η·fcd
For fck > 50 MPa: λ = 0.8−(fck−50)/400 · η = 1−(fck−50)/200
EN 1992-1-1 §9.5 Reinforcement Limits
As,max = 0.04·Ac (ρg,max = 4%)
IS 456:2000 Cl 39.3 Axial Load and Biaxial Bending
IS 456:2000 uses design material strengths with partial safety factors embedded in material values. No separate capacity reduction factor (φ) is applied — the safety margins are captured through γc = 1.5 and γs = 1.15.
fyd = fy / γs = fy / 1.15 = 0.87·fy
IS 456:2000 Equivalent Rectangular Stress Block
IS 456 specifies a parabolic-rectangular stress distribution for concrete in compression. For computation, this is converted to an equivalent rectangular block with the same resultant force and centroid:
Centroid at 0.416·xu from compression face
Equivalent rectangular block depth: a = λ·xu where λ = 0.832
Equivalent block stress: fc,eff = 0.4327·fck
IS 456:2000 Cl 39.3 Maximum Axial Capacity
Where Ac = Ag − Asc (net concrete area)
IS 456:2000 Cl 26.5.3.1 Reinforcement Limits
ρmax = 6.0% (Asc,max = 0.06·Ag)
IS 456:2000 Cl 39.6 Biaxial Bending
IS 456 Cl 39.6 provides the Bresler-type interaction formula for combined biaxial bending:
αn = 1.0 when Pu/Puz = 0.2
αn = 2.0 when Pu/Puz = 0.8
Linear interpolation between 0.2 and 0.8
Biaxial Bending Check — Bresler Reciprocal Method
When moments exist about both axes simultaneously, the uniaxial interaction diagrams are insufficient. This calculator uses the Bresler reciprocal load contour method to estimate the combined demand-capacity ratio:
Where:
M0x = Mx-capacity from uniaxial diagram at demand P
M0y = My-capacity from uniaxial diagram at demand P
α = exponent depending on P/P0 ratio (typically 1.0–2.0)
The biaxial DCR shown is the left-hand side of the inequality above. Values ≤ 1.0 indicate the section is adequate for combined biaxial loading.
References
- [1]ACI 318-25 — Building Code Requirements for Structural Concrete. American Concrete Institute, 2025. §10.6 (Columns), §21.2.2 (φ factors), §22.4 (Axial strength).
- [2]EN 1992-1-1:2004 — Eurocode 2. CEN, Brussels. §6.1 (Bending and axial force), §9.5 (Columns).
- [3]MacGregor, J.G. & Wight, J.K. — Reinforced Concrete: Mechanics and Design, 7th Ed. Pearson, 2016. Chapter 11.
- [4]Bresler, B. — Design Criteria for Reinforced Columns under Axial Load and Biaxial Bending. ACI Journal, Vol. 57, 1960.
- [5]Mosley, Bungey & Hulse — Reinforced Concrete Design to Eurocode 2, 7th Ed. Palgrave Macmillan, 2012. Chapter 9.
- [6]IS 456:2000 — Plain and Reinforced Concrete — Code of Practice (4th Revision). Bureau of Indian Standards, New Delhi. Cl 26.5.3.1 (Steel limits), Cl 39.3 (Axial capacity), Cl 39.6 (Biaxial bending).
- [7]Pillai, S.U. & Menon, D. — Reinforced Concrete Design, 3rd Ed. Tata McGraw-Hill, 2009. Chapter 13 (Columns under combined loading).
- [8]SP:16(S&T)-1980 — Design Aids for Reinforced Concrete to IS 456:1978. Bureau of Indian Standards, New Delhi. (Interaction diagrams and design charts).