Column Design

Rectangular RC column — biaxial bending with axial load. Full P-M interaction diagram using strain-compatibility.

Input Parameters

kN-m · m

Units:

Column:

Section Geometry

Width b [m]

Depth h [m]

Clear Cover to main bar [m]

Material Properties

f'c [MPa]

MPa

f_y [MPa]

MPa

E_s [MPa]

MPa

φ (tied column)

—

φ = 0.65 (compression-controlled) to 0.90 (tension-controlled). ACI 318-25 §21.2.2 — diagram applies transition automatically.

Factored Loads

Axial P_u [kN] (compression +)

M_u,x [kN·m]

kN·m

Bending about x-axis → NA ∥ b, depth along h

M_u,y [kN·m]

kN·m

Bending about y-axis → NA ∥ h, depth along b

Reinforcement Layout

Bar Dia. [mm]

n_x = bars along x (top & bottom rows, parallel to b) · n_y = bars along y (left & right columns, parallel to h, incl. corners)

n_x (along x)

n_y (along y)

—

📊

Configure inputs and click Calculate.

Input Parameters

kN-m · m

Units:

Column:

Section Geometry

Width b [m]

Depth h [m]

Clear Cover to main bar [m]

Material Properties

f_ck [MPa]

MPa

f_yk [MPa]

MPa

E_s [MPa]

MPa

γ_c

Recommended: 1.50

γ_s

Recommended: 1.15

Design Loads (ULS)

Axial N_Ed [kN] (compression +)

M_Ed,x [kN·m]

kN·m

Bending about x-axis → depth along h

M_Ed,y [kN·m]

kN·m

Bending about y-axis → depth along b

Reinforcement Layout

Bar Dia. [mm]

n_x = bars along x (top & bottom rows, parallel to b) · n_y = bars along y (left & right columns, parallel to h, incl. corners)

n_x (along x)

n_y (along y)

—

📊

Configure inputs and click Calculate.

Input Parameters

kN-m · m

Units:

Column:

Section Geometry

Width b [m]

Depth h [m]

Clear Cover to main bar [m]

Material Properties

f_ck [MPa]

MPa

f_y [MPa]

MPa

E_s [MPa]

MPa

γ_c = 1.5 · γ_s = 1.15 (IS 456 Table 1) · f_cd = 0.447 f_ck · f_yd = 0.87 f_y

Typical grades: M15/M20/M25/M30 · Fe250 / Fe415 / Fe500

Factored Loads (Limit State)

Axial P_u [kN] (compression +)

M_ux [kN·m]

kN·m

Bending about x-axis → depth along h

M_uy [kN·m]

kN·m

Bending about y-axis → depth along b

Reinforcement Layout

Bar Dia. [mm]

n_x = bars along x (top & bottom rows, parallel to b) · n_y = bars along y (left & right, parallel to h, incl. corners)

n_x (along x)

n_y (along y)

—

📊

Configure inputs and click Calculate.

Input Parameters

kN-m · m

Units:

Column:

Section Geometry

Width b_w [m]

Total depth h [m]

Clear cover c_v [mm]

d = h − c_v − d_t − d_bl/2 (computed internally)

Material Properties

f'c [MPa]

MPa

f_yt ties [MPa]

MPa

λ (1.0=NW concrete)

—

ρ_l longit. [%]

Factored Loads

V_u,x (→h) [kN]

V_u,y (→b) [kN]

N_u [kN] comp+

Transverse Reinforcement (Ties)

Tie dia. d_t [mm]

Main bar d_bl [mm]

Legs n_x (across b_w)

—

Legs n_y (across h)

—

Spacing s [mm]

🔩

Configure inputs and click Calculate.

Input Parameters

kN-m · m

Units:

Column:

Section Geometry

Width b_w [m]

Total depth h [m]

Clear cover c_v [mm]

d = h − c_v − d_t − d_bl/2 (computed internally)

Material Properties

f_ck [MPa]

MPa

f_yk ties [MPa]

MPa

ρ_l longit. [%]

cot θ ?

—

Factored Loads

V_Ed,x (→h) [kN]

V_Ed,y (→b) [kN]

N_Ed [kN] comp+

Seismic Design — Eurocode 8 (EC8)

Transverse Reinforcement (Links)

Link dia. d_t [mm]

Main bar d_bl [mm]

Legs n_x (across b_w)

—

Legs n_y (across h)

—

Spacing s [mm]

🔩

Configure inputs and click Calculate.

Input Parameters

kN-m · m

Units:

Column:

Section Geometry

Width b [m]

Total depth D [m]

Clear cover c_v [mm]

d = D − c_v − d_t − d_bl/2 (computed internally)

Material Properties

f_ck [MPa]

MPa

f_y ties [MPa]

MPa

ρ_l = 100·A_st/(b·d) [%]

Used for τ_c lookup from IS 456 Table 19. Range 0.15–3.0%.

Factored Loads

V_u,x (→h) [kN]

V_u,y (→b) [kN]

P_u [kN] comp+

Transverse Reinforcement (Ties)

Tie dia. d_t [mm]

Main bar d_bl [mm]

Legs n_x (across b)

—

Legs n_y (across D)

—

Spacing s_v [mm]

🔩

Configure inputs and click Calculate.

📚 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 ε_cu (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 f_y/f_yd and E_s·ε_bar. 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 ε_t (0.65→0.90)
EC2 / IS 456: design values f_cd, f_yd applied directly; no additional φ

Key Points on the Interaction Diagram

The interaction diagram has several characteristic points that define the full range of section behaviour:

Point	Description	Governing Strain State
Pure compression P₀	Maximum axial capacity, M = 0	ε = ε_cu uniform across section
Maximum P_n,max	ACI: 0.80·P₀ (tied); IS 456: Cl 39.3	Accounts for accidental eccentricity
Balanced point P_b, M_b	ε_cu and ε_y reached simultaneously	c = ε_cu/(ε_cu+ε_y) · d
Pure bending P = 0	Beam-type flexure only	ε_s ≫ ε_y
Pure tension P_t0	Maximum tension capacity, M = 0	ε = ε_y uniform, concrete ignored

The transition from compression-controlled to tension-controlled behaviour occurs at the balanced point. Sections below and to the right of the balanced point are tension-controlled (steel yields first); sections above and to the left are compression-controlled (concrete crushes first).

Slenderness Effects and Second-Order Moments

This calculator generates the short-column interaction diagram based on section geometry only. For slender columns, the lateral deflection under eccentric load amplifies the applied moments — the so-called P-δ effect. If the column is part of a laterally unbraced frame, additional global P-Δ effects must also be considered.

Slenderness Check — ACI 318-25 §6.2.5

Short column condition: kl_u/r ≤ 22 (braced frame, M1/M2 ≥ 0)
Short column condition: kl_u/r ≤ 34 + 12(M₁/M₂) ≤ 40 (braced, non-zero ratio)
r = radius of gyration = 0.30h (rectangular) or 0.25D (circular)
k = effective length factor (1.0 braced, ≥ 1.0 unbraced)

When kl_u/r exceeds these limits, the design moment M_c must be amplified by the moment magnifier δ_s or δ_ns per §6.6.4. Slenderness effects are ignored in this calculator — verify slenderness separately before applying results.

Minimum Eccentricity — ACI §6.6.4.5.4

e_min = 0.6 + 0.03·h [mm] (ACI minimum eccentricity for braced frame columns)
IS 456 Cl 25.4: e_min = max(l/500 + D/30, 20 mm) (l = unsupported length, D = depth)

Minimum eccentricity accounts for unintended construction tolerances and load eccentricities not explicitly modelled. A column designed for pure axial load P_u must be checked for P_u combined with M_min = P_u·e_min.

Axis Convention

The section has width b (horizontal) and depth h (vertical). Moments are defined as follows:

M_x — bending about the horizontal x-axis: neutral axis is parallel to b, compression/tension varies along h
M_y — bending about the vertical y-axis: neutral axis is parallel to h, compression/tension varies along b

For uniaxial bending, only one moment acts — enter the other as zero. The calculator draws the full 3D surface by combining the uniaxial diagrams for both axes; the biaxial demand point (P_u, M_x, M_y) is checked against this surface using the Bresler reciprocal method.

ACI 318-25 §22.4 Tied Columns

For compression-controlled sections, ACI 318-25 limits the maximum axial capacity to account for accidental eccentricity:

Maximum Axial Capacity (Tied)

φP_n,max = φ · 0.80 · [0.85·f'c·(A_g−A_st) + f_y·A_st]
φ = 0.65 (tied) · φ = 0.75 (spiral)

ACI 318-25 §21.2.2 Variable φ

Strength Reduction Factor

ε_t ≤ 0.002: φ = 0.65 (compression-controlled)
0.002 < ε_t < 0.005: φ = 0.65 + (ε_t−0.002)·(0.25/0.003)
ε_t ≥ 0.005: φ = 0.90 (tension-controlled)

ε_t = net tensile strain in outermost tension steel at ULS. This transition is applied automatically in the interaction diagram.

ACI 318-25 §10.6 Reinforcement Limits

Steel Ratio Limits

0.01 ≤ ρ_g = A_st/A_g ≤ 0.08

Minimum ensures ductility; maximum prevents congestion. §10.6.1.1

ACI 318-25 §6.6.4 Moment Magnification (Braced Frames)

For slender columns in braced (non-sway) frames where kl_u/r exceeds the short-column limit, the design moment must be amplified to account for P-δ effects:

Nonsway Magnification — §6.6.4.5

M_c = δ_ns · M₂
δ_ns = C_m / (1 − P_u/0.75·P_c) ≥ 1.0
P_c = π²·EI_eff / (k·l_u)² (Euler critical load)
C_m = 0.6 − 0.4·M₁/M₂ (single curvature: C_m → 1.0)

EI_eff = 0.4·E_c·I_g (simplified per §6.6.4.4.4). The interaction diagram generated by this calculator applies to M_c, not the first-order M₂. For short columns (kl_u/r below the limit), δ_ns = 1.0 and M_c = M₂.

ACI 318-25 §10.7.6.2 Transverse Reinforcement Spacing

Maximum Tie Spacing (Tied Columns)

s ≤ min(16·d_bl, 48·d_tie, least dimension of column)
d_bl = longitudinal bar diameter · d_tie = tie bar diameter

Tie spacing governs longitudinal bar buckling after concrete cover spalls. ACI §18.7.5 imposes tighter confinement requirements for special moment frame columns in seismic regions.

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 γ_c and γ_s.

Design Strengths

f_cd = α_cc · f_ck / γ_c [α_cc = 1.0; γ_c = 1.5]
f_yd = f_yk / γ_s [γ_s = 1.15]

Stress Block Parameters (§3.1.7)

λ = 0.8 (f_ck ≤ 50 MPa) → block depth = λ·x
η = 1.0 (f_ck ≤ 50 MPa) → block stress = η·f_cd
For f_ck > 50 MPa: λ = 0.8−(f_ck−50)/400 · η = 1−(f_ck−50)/200

These factors reduce the effective stress block for high-strength concrete, which is why the moment capacity may decrease at higher f_ck if the reinforcement ratio is fixed.

EN 1992-1-1 §9.5 Reinforcement Limits

Steel Ratio Limits

A_s,min = 0.10·N_Ed/f_yd ≥ 0.002·A_c
A_s,max = 0.04·A_c (ρ_g,max = 4%)

EN 1992-1-1 §5.8 Slenderness and Second-Order Effects

EC2 §5.8.3 defines the slenderness ratio λ = l₀/i where l₀ is the effective length and i is the radius of gyration of the uncracked section (i = h/√12 for a rectangular section). Second-order effects may be ignored if:

Short Column Limit — §5.8.3.1

λ ≤ λ_lim = 20·A·B·C / √n
A = 1/(1+0.2φ_ef) (≈ 0.7 simplified)
B = √(1+2ω) (≈ 1.1 simplified, ω = A_sf_yd/(A_cf_cd))
C = 1.7 − M₀₁/M₀₂ (≈ 0.7 simplified for unknown ratio)
n = N_Ed/(A_c·f_cd) (relative normal force)

Simplified conservative estimate: λ_lim ≈ 20·0.7·1.1·0.7/√n = 10.78/√n. For n = 0.5 (typical), λ_lim ≈ 15. This is the condition to check before applying this interaction diagram directly to slender columns.

EN 1992-1-1 §9.5.3 Transverse Reinforcement

Link Diameter and Spacing

d_link,min ≥ max(6 mm, d_bl/4)
s_cl,max = min(20·d_bl, b or h of section, 400 mm) (general)
Reduce to 0.60·s_cl,max above/below beam-column junctions (distance = max(h_col, l_b/6))

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 γ_c = 1.5 and γ_s = 1.15.

Design Strengths (IS 456 Table 1)

f_cd = 0.67·f_ck / γ_c = 0.67·f_ck / 1.5 = 0.447·f_ck
f_yd = f_y / γ_s = f_y / 1.15 = 0.87·f_y

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:

IS 456 Stress Block Parameters

C_c = 0.36·f_ck·b·x_u (resultant force)
Centroid at 0.416·x_u from compression face

Equivalent rectangular block depth: a = λ·x_u where λ = 0.832
Equivalent block stress: f_c,eff = 0.4327·f_ck

Derived so that: f_c,eff·λ·x_u·b = 0.36·f_ck·b·x_u (same Cc) and centroid at a/2 = 0.416·x_u (same lever arm). Maximum compressive strain ε_cu = 0.0035.

IS 456:2000 Cl 39.3 Maximum Axial Capacity

P_0,max (IS 456 Cl 39.3)

P₀ = 0.4·f_ck·A_c + 0.67·f_y·A_sc
Where A_c = A_g − A_sc (net concrete area)

The 0.4·fck factor (rather than 0.447·fck) accounts for the IS 456 long-term loading allowance and implicit minimum eccentricity of 0.05·h per Cl 39.2. This is IS 456's approach to accidental eccentricity, equivalent to the ACI k-factor.

IS 456:2000 Cl 26.5.3.1 Reinforcement Limits

Steel Ratio Limits

ρ_min = 0.8% (A_sc,min = 0.008·A_g)
ρ_max = 6.0% (A_sc,max = 0.06·A_g)

Minimum steel controls long-term creep and shrinkage effects; maximum prevents concrete placement difficulty. For lapped splices at the same location, the local maximum reduces to 4%.

IS 456:2000 Cl 39.6 Biaxial Bending

IS 456 Cl 39.6 provides the Bresler-type interaction formula for combined biaxial bending:

IS 456 Biaxial Check (Cl 39.6)

(M_ux/M_ux1)^α_n + (M_uy/M_muy1)^α_n ≤ 1.0

α_n = 1.0 when P_u/P_uz = 0.2
α_n = 2.0 when P_u/P_uz = 0.8
Linear interpolation between 0.2 and 0.8

This calculator uses α = 1 + P/P₀, a conservative approximation consistent with the IS 456 spirit. For precise α_n per Cl 39.6, the P_uz reference value should be checked explicitly.

IS 456:2000 Cl 25.1.2 & 25.1.3 Slenderness and Effective Length

IS 456 Cl 25.1.2 defines columns as short when both the ratios l_ex/D and l_ey/b are less than 12, where l_ex and l_ey are the effective lengths in the respective planes and D, b are the corresponding depths. Columns with either ratio ≥ 12 are classified as slender and require additional moment consideration per Cl 39.7.

Additional Moments in Slender Columns — Cl 39.7.1

M_ax = P_u · D/(2000) · (l_ex/D)²
M_ay = P_u · b/(2000) · (l_ey/b)²
Design moments: M_ux,total = M_ux + M_ax · k

k = (P_uz − P_u)/(P_uz − P_bal) ≤ 1.0 is a reduction factor accounting for the beneficial effect of axial force on the additional moment. These additional moments are added to the primary moments before applying the biaxial bending check.

Effective Length — Table 28 (Cl 25.2)

Fixed-fixed (both ends restrained): l_eff = 0.50 · l
Fixed-pinned (one end restrained): l_eff = 0.70 · l
Fixed-free (cantilever): l_eff = 2.00 · l
Pinned-pinned (both ends unrestrained): l_eff = 1.00 · l

📚 Design Background & Code References — Shear

Shear in Reinforced Concrete Columns

Columns subjected to seismic or lateral loads must be designed for shear in addition to axial-flexural demands. The nominal shear strength is the sum of the concrete contribution V_c (or τ_c) and the steel contribution V_s from transverse reinforcement.

General Shear Strength

φV_n = φ(V_c + V_s) ≥ V_u

V_s = A_v · f_yt · d / s (rectangular)
V_s = A_v · f_yt · d / s (circular, b_w = D)

For circular columns, the ACI approach approximates the section as rectangular with b_w = diameter D. The effective depth d = D − c_v − d_t − d_bl/2.

Effect of Axial Load on Shear Capacity

Compressive axial force generally increases concrete shear capacity, while tensile axial force reduces it. Each code handles this differently:

ACI 318-25: V_c depends on N_u/A_g via the detailed table method (§22.5.5.1) or simplified formula (§22.5.5.3)
EC2: σ_cp = N_Ed/A_c directly modifies the V_Rd,c formula
IS 456: δ = 1 + 3P_u/(A_g·f_ck) ≤ 1.5 scales up the basic shear strength τ_c

Column Design

Input Parameters

Calculation Report

📈 P-M Interaction Diagram

Input Parameters

Calculation Report

📈 P-M Interaction Diagram

Input Parameters

Calculation Report

📈 P-M Interaction Diagram

Input Parameters

Shear Design Report

Input Parameters

Shear Design Report

Input Parameters

Shear Design Report

📚 Design Background & Code References

P-M Interaction: Strain Compatibility

Key Points on the Interaction Diagram

Slenderness Effects and Second-Order Moments

Axis Convention

ACI 318-25 §22.4 Tied Columns

ACI 318-25 §21.2.2 Variable φ

ACI 318-25 §10.6 Reinforcement Limits

ACI 318-25 §6.6.4 Moment Magnification (Braced Frames)

ACI 318-25 §10.7.6.2 Transverse Reinforcement Spacing

EN 1992-1-1 §6.1 Combined Bending and Axial

EN 1992-1-1 §9.5 Reinforcement Limits

EN 1992-1-1 §5.8 Slenderness and Second-Order Effects

EN 1992-1-1 §9.5.3 Transverse Reinforcement

IS 456:2000 Cl 39.3 Axial Load and Biaxial Bending

IS 456:2000 Equivalent Rectangular Stress Block

IS 456:2000 Cl 39.3 Maximum Axial Capacity

IS 456:2000 Cl 26.5.3.1 Reinforcement Limits

IS 456:2000 Cl 39.6 Biaxial Bending

IS 456:2000 Cl 25.1.2 & 25.1.3 Slenderness and Effective Length

Biaxial Bending Check — Bresler Reciprocal Method

References

📚 Design Background & Code References — Shear

Shear in Reinforced Concrete Columns

Effect of Axial Load on Shear Capacity

ACI 318-25 §22.5 Shear Strength

ACI 318-25 §26.6.3.2 Minimum Shear Reinforcement

ACI 318-25 §10.7.6.2 Tie Spacing

EN 1992-1-1 §6.2 Shear Design

EN 1992-1-1 §9.2.2 Minimum Links

IS 456:2000 Cl 40 Shear Design

IS 456:2000 Cl 26.5.3.2 Lateral Ties

References — Shear Design

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