Design Procedure Overview

Select section and compute properties (A, Z_x, Z_y, S_x, r_x, r_y, J, C_w).
Compute nominal axial capacity φ_cP_n for both axes using AISC §E3. Governing KL/r controls.
Compute strong-axis moment capacity φ_bM_nx considering LTB zones (§F2).
Compute weak-axis moment capacity φ_bM_ny (§F6, no LTB).
Evaluate interaction ratio using §H1-1. If P_u/φ_cP_n ≥ 0.2 use H1-1a; otherwise H1-1b.

Interaction Equations — AISC 360 §H1-1

H1-1a — High axial (P_u/φP_n ≥ 0.2) P_u / (φ_cP_n) + (8/9)[M_ux/(φ_bM_nx) + M_uy/(φ_bM_ny)] ≤ 1.0

H1-1b — Low axial (P_u/φP_n < 0.2) P_u/(2φ_cP_n) + M_ux/(φ_bM_nx) + M_uy/(φ_bM_ny) ≤ 1.0

φ_c = φ_b = 0.90. The two equations share the same boundary point at P_u/φP_n = 0.2, M/φM_n = 0.9, ensuring a continuous interaction surface. C_b = 1.0 assumed (uniform moment, conservative).

Why two equations? When axial force dominates (≥20% capacity), bending reduces the remaining capacity by 8/9 of its ratio — yielding a steeper envelope. When axial is small, the column behaves more like a beam-column with near-full flexural capacity.

AISC §H1-1 interaction envelope

Axial Compression Capacity — AISC 360 §E3

The critical buckling stress depends on the governing slenderness ratio KL/r (larger of strong and weak axis):

Elastic critical stressF_e = π²E / (KL/r)²

Inelastic KL/r ≤ 4.71√(E/F_y) [typically ≤200 for A992]F_cr = 0.658^(F_y/F_e) × F_y

Elastic KL/r > 4.71√(E/F_y)F_cr = 0.877 × F_e (10% knock-down for imperfections)

Nominal & design strengthP_n = F_cr × A φ_cP_n = 0.90 × P_n

Limit KL/r ≤ 200 is recommended by AISC commentary as a practical upper bound for compression members. Above this, columns are very slender and efficiency drops sharply.

Strong-Axis Flexure & LTB — AISC 360 §F2

Three zones govern depending on the unbraced length L_b (taken as a fraction of column height):

Inelastic LTB (L_p < L_b ≤ L_r) M_nx = C_b[M_p − (M_p − 0.7F_yS_x) × (L_b−L_p)/(L_r−L_p)] ≤ M_p

Elastic LTB (L_b > L_r) F_cr = C_bπ²E / (L_b/r_ts)² × √(1 + 0.078Jc/(S_xh_o) × (L_b/r_ts)²) M_nx = F_crS_x ≤ M_p

Weak-Axis Flexure — AISC 360 §F6

No LTB possible for bending about the weak axis. Capacity is governed by local flange yielding only:

§F6 (compact flanges)M_ny = F_yZ_y ≤ 1.6F_yS_y

The 1.6F_yS_y cap prevents overly optimistic plastic moments for sections with very stocky flanges.

ASCE 7-22 — LRFD Load Combinations

#	Combination
1	1.4D
2	1.2D + 1.6L + 0.5(L_r or S or R)
3	1.2D + 1.6(L_r or S or R) + (L or 0.5W)
4	1.2D + 1.0W + L + 0.5(L_r or S or R)
5	0.9D + 1.0W

ASTM Steel Grades

Grade	F_y (ksi)	F_u (ksi)	Typical use
A992 Gr.50	50	65	W-shapes (standard)
A572 Gr.50	50	65	Plates, HSS
A572 Gr.60	60	75	Higher strength
A36	36	58	Older / misc shapes

Design Procedure Overview

Compute elastic critical axial load N_cr = π²EI/(KL)² for both axes.
Compute non-dimensional slenderness λ̄ = √(Af_y/N_cr) for each axis.
Select buckling curve (a/b/c) from Table 6.2 based on cross-section type and h/b ratio.
Calculate reduction factor χ using the Perry–Robertson formula and get N_b,Rd.
Compute M_cr for LTB, derive χ_LT, get M_b,Rd and M_Rd,z.
Check both interaction equations 6.61 and 6.62 (§6.3.3 Method 2).

Axial Buckling — EN 1993-1-1 §6.3.1

Elastic critical loadN_cr = π²EI / (KL)²

Non-dimensional slendernessλ̄ = √(Af_y / N_cr)

Ayrton–Perry (Robertson) factorφ = 0.5[1 + α(λ̄ − 0.2) + λ̄²]

Reduction factor χχ = 1 / [φ + √(φ² − λ̄²)] ≤ 1.0

Buckling resistanceN_b,Rd = χ × A × f_y / γ_M1 (γ_M1 = 1.0)

The imperfection factor α reflects initial out-of-straightness, residual stresses, and end restraint imperfections. Higher α = more conservative curve.

EN 1993-1-1 Table 6.2 buckling curves

Buckling Curve Selection — EN 1993-1-1 Table 6.2

Section	h/b	t_f	Strong axis (y-y)	Weak axis (z-z)
IPE / HEA / HEB	> 1.2	≤ 40 mm	a (α=0.21)	b (α=0.34)
IPE / HEA / HEB	> 1.2	40–100 mm	b (α=0.34)	c (α=0.49)
HEA / HEB / UC	≤ 1.2	≤ 100 mm	b (α=0.34)	c (α=0.49)
Hot-rolled SHS/RHS	—	—	a (α=0.21)	a (α=0.21)
CHS (hot-rolled)	—	—	a (α=0.21)	a (α=0.21)

Lateral-Torsional Buckling — EN 1993-1-1 §6.3.2.3

Elastic critical momentM_cr = (π/L_b) √(EI_z·GJ) × √(1 + π²EI_w / (GJL_b²))

LTB slendernessλ̄_LT = √(W_y,pl·f_y / M_cr)

LTB factor (§6.3.2.3 rolled, λ̄_LT,0=0.4, β=0.75) If λ̄_LT ≤ 0.4: χ_LT = 1.0 Else: χ_LT = 1 / [φ_LT + √(φ_LT² − βλ̄_LT²)] ≤ min(1.0, 1/λ̄_LT²)

Cross-section	h/b	LTB curve	α_LT
Rolled I/H sections	≤ 2	b	0.34
Rolled I/H sections	> 2	c	0.49
SHS / CHS	—	—	LTB not critical

Interaction Equations — EN 1993-1-1 §6.3.3 Method 2 (Annex B)

Eq. 6.61 (strong-axis check) N_Ed/(χ_yN_Rk/γ_M1) + k_yyM_y,Ed/(χ_LTM_y,Rk/γ_M1) + k_yzM_z,Ed/(M_z,Rk/γ_M0) ≤ 1.0

Eq. 6.62 (weak-axis check) N_Ed/(χ_zN_Rk/γ_M1) + k_zyM_y,Ed/(χ_LTM_y,Rk/γ_M1) + k_zzM_z,Ed/(M_z,Rk/γ_M0) ≤ 1.0

Interaction factors (C_my=C_mz=1.0 for uniform moment, conservative):

Factor	Formula (uniform moment, C_m=1)
k_yy	1 + min(λ̄_y−0.2, 0.8) × n_y
k_zz	1 + min(2λ̄_z−0.6, 1.4) × n_z
k_yz	0.6 × k_zz
k_zy	1 − min(0.133λ̄_z, 0.133) × n_z

n_y = N_Ed/(χ_yN_Rk), n_z = N_Ed/(χ_zN_Rk). γ_M0 = γ_M1 = 1.0 (recommended value).

Both equations must be satisfied. Eq. 6.61 tends to govern when strong-axis moment is large; Eq. 6.62 governs when weak-axis buckling or moment is critical. The calculator reports the maximum of the two ratios.

EN 10025-2 Steel Grades

Grade	f_y (MPa)	f_u (MPa)
S235	235	360
S275	275	430
S355	355	490
S460	460	550

Design Procedure Overview

Compute N_cr = π²EI/(KL)² for both axes. Use E = 200,000 MPa, G = 77,000 MPa.
Compute λ̄ and χ using the same Ayrton–Perry formula as EC3, but with IS 800 Table 7 buckling curves.
Design axial stress f_cd = χ × f_y / γ_m0 (govern by minimum axis). P_d = A × f_cd.
Compute M_cr, λ̄_LT, χ_LT using IS 800 Table 6b LTB curves. M_dx = χ_LT × Z_pl,x × f_y/γ_m0.
Weak-axis: M_dz = Z_pl,z × f_y / γ_m0 (no LTB).
Check linear interaction §9.3.2.2: P/P_d + M_x/M_dx + M_y/M_dz ≤ 1.0.

Axial Compression — IS 800:2007 §7.1

γ_m0 = 1.10 (IS 800 Table 5). The design compressive strength is:

Design axial capacityP_d = A × f_cd where f_cd = χ × f_y / γ_m0

Governing axisf_cd = min(f_cd,x, f_cd,y) — both axes checked independently

IS vs EC3 partial factors: IS 800 uses γ_m0 = 1.10 (EC3 uses 1.00). This makes IS 800 approximately 10% more conservative than EC3 for the same section and yield strength.

IS 800:2007 Buckling Curves — Table 7

Section type	Axis	Buckling curve	α
ISMB / ISLB / ISWB (h/b > 1.2)	Strong (y-y)	a	0.21
ISMB / ISLB / ISWB (h/b > 1.2)	Weak (z-z)	b	0.34
ISHB / ISMC (h/b ≤ 1.2)	Strong (y-y)	b	0.34
ISHB / ISMC (h/b ≤ 1.2)	Weak (z-z)	c	0.49
SHS / RHS (hollow)	Both	a	0.21

Lateral-Torsional Buckling — IS 800:2007 §8.2

Elastic critical moment M_cr M_cr = (π/L_b) √(EI_zGJ) × √(1 + π²EI_w/(GJL_b²))

LTB slenderness & reduction λ̄_LT = √(Z_pl,xf_y/M_cr) → χ_LT using IS 800 Table 6b curves

Section	h/b	LTB curve	α_LT
Rolled I/H	≤ 2	a	0.21
Rolled I/H	> 2	b	0.34

Strong-axis capacityM_dx = χ_LT × Z_pl,x × f_y / γ_m0

Weak-axis capacity (no LTB)M_dz = Z_pl,z × f_y / γ_m0

Combined Loading — IS 800:2007 §9.3.2.2

Linear interaction check P / P_d + M_x / M_dx + M_y / M_dz ≤ 1.0

Comparison with EC3: IS 800 uses a single linear interaction equation, while EC3 §6.3.3 uses two equations with amplification factors k_yy, k_zz, k_yz, k_zy. The IS 800 approach is simpler but may be more conservative for stocky columns (low λ̄) and less conservative than EC3 for slender columns with high bending.

IS 875 Load Combinations (ULS)

#	Combination
1	1.5 DL + 1.5 IL
2	1.5 DL + 1.5 WL
3	1.2 DL + 1.2 IL + 1.2 WL
4	0.9 DL + 1.5 WL (uplift)

IS 2062 Steel Grades

Grade	f_y (MPa)	f_u (MPa)
E250 (Fe 410)	250	410
E300 (Fe 440)	300	440
E350 (Fe 490)	350	490
E410 (Fe 540)	410	540

Overview

TSDS 2016 (Türk Çelik Yapı Tasarım Yönetmeliği) is the Turkish steel design standard. It adopts the AISC 360-10/16 LRFD framework directly, with SI units (MPa, kN, mm). All interaction equations, buckling checks, and resistance factors are identical to AISC — only units and some load combinations differ.

TSDS = AISC 360 in SI: The design procedure is exactly the same as the AISC tab above. Refer to those formulas for full derivations. Key difference: E = 200,000 MPa (vs 29,000 ksi in AISC), F_y in MPa.

Axial Compression — TSDS 2016 §5.3 (= AISC §E3)

Elastic critical stressF_e = π²E / (KL/r)² [MPa] E = 200,000 MPa

Inelastic: KL/r ≤ 4.71√(E/F_y)F_cr = 0.658^(F_y/F_e) × F_y

Elastic: KL/r > 4.71√(E/F_y)F_cr = 0.877 × F_e

Design strengthφ_cP_n = 0.90 × F_cr × A [kN]

Flexure & LTB — TSDS 2016 §6 (= AISC §F)

Same three-zone LTB model as AISC §F2. L_p and L_r formulas identical; all section properties in mm.

Plastic zone (L_b ≤ L_p)φ_bM_nx = 0.90 × F_y × Z_x [kN·m] (Z_x in mm³, divide by 10&sup6;)

Weak axis (§F6)φ_bM_ny = 0.90 × min(F_yZ_y, 1.6F_yS_y) [kN·m]

Interaction — TSDS 2016 §7 (= AISC §H1-1)

H1-1a (P_u/φP_n ≥ 0.2) P_u/(φ_cP_n) + (8/9)[M_ux/(φ_bM_nx) + M_uy/(φ_bM_ny)] ≤ 1.0

H1-1b (P_u/φP_n < 0.2) P_u/(2φ_cP_n) + M_ux/(φ_bM_nx) + M_uy/(φ_bM_ny) ≤ 1.0

TBDY 2018 / TS 498 Load Combinations

#	Combination
1	1.4G
2	1.2G + 1.6Q + 0.5S
3	1.2G + 1.6S + Q
4	1.2G + 1.0W + Q + 0.5S
5	0.9G + 1.0W (uplift)

G = Permanent (DL + self-weight) | Q = Live (LL + roof live) | S = Snow | W = Wind

TS EN 10025-2 Steel Grades (Turkey)

Grade	f_y (MPa)	f_u (MPa)
S235	235	360
S275	275	430
S355	355	490
S450	440	550

Sismik tasarım notu: TBDY 2018 §9 kapsamındaki sismik tasarımda φ faktörleri ve bağlantı kapasiteleri ek kısıtlara tabidir. Bu hesap aracı statik yükler için tasarlanmıştır; deprem yükü kombinasyonları ayrıca doğrulanmalıdır.

Effective Length Factor K — Column End Conditions

The effective length KL represents the distance between inflection points of the buckled shape. K depends on the rotational and translational restraint at each end.

K Factor Reference Table

End Condition	K (theoretical)	K (AISC rec.)	Notes
Pin–Pin (both ends free to rotate)	1.00	1.00	Standard braced frame column
Fixed–Pin (one end restrained)	0.699	0.80	Accounts for realistic fixity
Fixed–Fixed (both ends restrained)	0.500	0.65	No sway assumed
Fixed–Free (cantilever)	2.00	2.10	Free top can translate + rotate
Pin–Fixed (sway permitted)	2.00	2.00	Unbraced frame, conservative

Sway vs. no-sway frames: The theoretical K values above assume no sidesway. In an unbraced (sway) frame, K ≥ 1.0 always. The K values 0.5–0.7 are only valid if lateral drift is prevented by bracing, shear walls, or rigid diaphragms.

Strong vs. Weak Axis & L_b

K_x applies to strong-axis (in-plane) buckling; K_y applies to weak-axis (out-of-plane) buckling. The larger of KL/r_x and KL/r_y governs the axial design. The LTB unbraced length L_b equals the full column height multiplied by the unbraced fraction input (0–1), representing the distance between lateral braces on the compression flange.

Alignment Chart Method (Jackson–Moreland)

For columns in frames, K can be determined from the stiffness ratios G_top and G_bot using the AISC alignment charts (Commentary §C-C2):

G factor at each endG = Σ(EI_col/L_col) / Σ(EI_beam/L_beam)

G = 1.0 for a theoretically fixed base (pinned base: G = 10). The alignment chart then gives K for the braced or unbraced frame case. This calculator uses the K values you input directly — determine them from the alignment chart or engineering judgment for frame columns.

Example — AISC 360-22 LRFD Steel Column under Combined Axial + Biaxial Bending

Given: W10×49 (Grade 50, F_y = 345 MPa / 50 ksi). Storey height L = 4.0 m (13.1 ft). Effective length factors K_x = 1.0, K_y = 1.0. Lateral bracing at top and bottom. Unbraced length for LTB: L_b = 4.0 m.
W10×49 properties (SI): A = 9,290 mm², Z_x = 889 cm³, Z_y = 427 cm³, S_x = 800 cm³, r_x = 106 mm, r_y = 64.8 mm, J = 63.3 cm⁴, C_w = 5,260 cm⁶, L_p = 2.75 m, L_r = 4.95 m.
Factored loads: P_u = 890 kN, M_ux = 95 kN·m (strong axis), M_uy = 30 kN·m (weak axis).

Step 1 — Axial Compressive Strength φ_cP_n

Governing KL/r = max(K_xL/r_x, K_yL/r_y) = max(4000/106, 4000/64.8) = max(37.7, 61.7) = 61.7 (weak axis governs)

Euler stress F_e — §E3

F_e = π²×200,000/61.7² = 1,972,000/3,807 = 518 MPa
4.71√(E/F_y) = 4.71√(200,000/345) = 113.4 > 61.7 → inelastic buckling
F_cr = 0.658^Fy/Fe×F_y = 0.658^345/518×345 = 0.658^0.666×345 = 0.757×345 = 261 MPa
φ_cP_n = 0.90×261×9,290/1,000 = 2,182 kN

Step 2 — Strong-Axis Flexural Strength φ_bM_nx

LTB zone check — §F2

L_p = 2.75 m < L_b = 4.0 m < L_r = 4.95 m → Inelastic LTB
M_p = F_y×Z_x = 345×889×10³/10⁶ = 307 kN·m
M_r = 0.7×F_y×S_x = 0.7×345×800×10³/10⁶ = 193 kN·m
M_nx = M_p−(M_p−M_r)×(L_b−L_p)/(L_r−L_p)
= 307−(307−193)×(4.0−2.75)/(4.95−2.75) = 307−114×0.568 = 307−65 = 242 kN·m
φ_bM_nx = 0.90×242 = 218 kN·m

Step 3 — Weak-Axis Flexural Strength φ_bM_ny

M_py = F_y×Z_y = 345×427×10³/10⁶ = 147 kN·m
No LTB for weak-axis bending → M_ny = M_py = 147 kN·m
φ_bM_ny = 0.90×147 = 133 kN·m

Step 4 — Interaction Check — §H1-1

Load ratio

P_r/P_c = P_u/(φ_cP_n) = 890/2,182 = 0.408
Since 0.408 ≥ 0.20 → use Equation H1-1a:
P_r/P_c + (8/9)×[M_rx/M_cx + M_ry/M_cy] ≤ 1.0
= 0.408 + (8/9)×[95/218 + 30/133]
= 0.408 + 0.889×[0.436 + 0.226]
= 0.408 + 0.889×0.662
= 0.408 + 0.588 = 0.996 < 1.0 ✓

Alternative: W10×54 (additional margin)

W10×54: A = 10,200 mm², Z_x = 980 cm³, Z_y = 475 cm³, r_y = 65.0 mm, L_p = 2.76 m, L_r = 5.14 m.

KL/r_y = 4000/65.0 = 61.5 → F_e = π²×200,000/61.5² = 522 MPa
F_cr = 0.658^345/522×345 = 0.758×345 = 261 MPa
φ_cP_n = 0.90×261×10,200/1,000 = 2,396 kN
M_p = 345×980×10³/10⁶ = 338 kN·m
L_p = 2.76 m < L_b = 4.0 m < L_r = 5.14 m → inelastic LTB
M_nx ≈ 338−(338−M_r)×(4.0−2.76)/(5.14−2.76) ≈ 272 kN·m → φ_bM_nx = 245 kN·m
φ_bM_ny = 0.90×345×475×10³/10⁶ = 148 kN·m
Interaction: 890/2,396 + (8/9)×[95/245 + 30/148] = 0.372 + 0.889×[0.388+0.203] = 0.372+0.525 = 0.897 < 1.0 ✓

Result: W10×49 Grade 50 satisfies H1-1a with interaction ratio = 0.996 under P_u=890 kN, M_ux=95 kN·m, M_uy=30 kN·m. L_p = 1.76×r_y×√(E/F_y) = 2.75 m governs the LTB zone. W10×54 reduces the ratio to 0.90 if additional margin is required.

Steel Column Design — Combined Axial + Biaxial Bending

Input Parameters

Column Design Calculation Report

📚 Steel Column Design — Theory & Design Background

Design Procedure Overview

Interaction Equations — AISC 360 §H1-1

Axial Compression Capacity — AISC 360 §E3

Strong-Axis Flexure & LTB — AISC 360 §F2

Weak-Axis Flexure — AISC 360 §F6

ASCE 7-22 — LRFD Load Combinations

ASTM Steel Grades

Design Procedure Overview

Axial Buckling — EN 1993-1-1 §6.3.1

Buckling Curve Selection — EN 1993-1-1 Table 6.2

Lateral-Torsional Buckling — EN 1993-1-1 §6.3.2.3

Interaction Equations — EN 1993-1-1 §6.3.3 Method 2 (Annex B)

EN 10025-2 Steel Grades

Design Procedure Overview

Axial Compression — IS 800:2007 §7.1

IS 800:2007 Buckling Curves — Table 7

Lateral-Torsional Buckling — IS 800:2007 §8.2

Combined Loading — IS 800:2007 §9.3.2.2

IS 875 Load Combinations (ULS)

IS 2062 Steel Grades

Overview

Axial Compression — TSDS 2016 §5.3 (= AISC §E3)

Flexure & LTB — TSDS 2016 §6 (= AISC §F)

Interaction — TSDS 2016 §7 (= AISC §H1-1)

TBDY 2018 / TS 498 Load Combinations

TS EN 10025-2 Steel Grades (Turkey)

Effective Length Factor K — Column End Conditions

K Factor Reference Table

Strong vs. Weak Axis & L_b

Alignment Chart Method (Jackson–Moreland)

Example — AISC 360-22 LRFD Steel Column under Combined Axial + Biaxial Bending

Step 1 — Axial Compressive Strength φ_cP_n

Step 2 — Strong-Axis Flexural Strength φ_bM_nx

Step 3 — Weak-Axis Flexural Strength φ_bM_ny

Step 4 — Interaction Check — §H1-1

Alternative: W10×54 (additional margin)

Steel Column Design — Combined Axial + Biaxial Bending

Input Parameters

Column Design Calculation Report

📚 Steel Column Design — Theory & Design Background

Design Procedure Overview

Interaction Equations — AISC 360 §H1-1

Axial Compression Capacity — AISC 360 §E3

Strong-Axis Flexure & LTB — AISC 360 §F2

Weak-Axis Flexure — AISC 360 §F6

ASCE 7-22 — LRFD Load Combinations

ASTM Steel Grades

Design Procedure Overview

Axial Buckling — EN 1993-1-1 §6.3.1

Buckling Curve Selection — EN 1993-1-1 Table 6.2

Lateral-Torsional Buckling — EN 1993-1-1 §6.3.2.3

Interaction Equations — EN 1993-1-1 §6.3.3 Method 2 (Annex B)

EN 10025-2 Steel Grades

Design Procedure Overview

Axial Compression — IS 800:2007 §7.1

IS 800:2007 Buckling Curves — Table 7

Lateral-Torsional Buckling — IS 800:2007 §8.2

Combined Loading — IS 800:2007 §9.3.2.2

IS 875 Load Combinations (ULS)

IS 2062 Steel Grades

Overview

Axial Compression — TSDS 2016 §5.3 (= AISC §E3)

Flexure & LTB — TSDS 2016 §6 (= AISC §F)

Interaction — TSDS 2016 §7 (= AISC §H1-1)

TBDY 2018 / TS 498 Load Combinations

TS EN 10025-2 Steel Grades (Turkey)

Effective Length Factor K — Column End Conditions

K Factor Reference Table

Strong vs. Weak Axis & Lb

Alignment Chart Method (Jackson–Moreland)

Example — AISC 360-22 LRFD Steel Column under Combined Axial + Biaxial Bending

Step 1 — Axial Compressive Strength φcPn

Step 2 — Strong-Axis Flexural Strength φbMnx

Step 3 — Weak-Axis Flexural Strength φbMny

Step 4 — Interaction Check — §H1-1

Alternative: W10×54 (additional margin)

Related Calculators & Guides

Strong vs. Weak Axis & L_b

Step 1 — Axial Compressive Strength φ_cP_n

Step 2 — Strong-Axis Flexural Strength φ_bM_nx

Step 3 — Weak-Axis Flexural Strength φ_bM_ny