Steel Column Axial Strength Design

AISC 360 LRFD axial compression design — buckling check per Chapter E. ASCE 7-22 load combinations — P_D, P_L, P_S, P_W. Self-weight automatically applied.

Input Parameters

kip · ft · in

Design Code:

Units:

Steel Grade

Section Selection

Section Type

Section

Column Height L [ft]

Loads

Dead P_D (excl. SW) [kip]

kip

Live P_L [kip]

kip

Roof Live P_Lr [kip]

kip

Snow P_S [kip]

kip

Wind P_W [kip]

kip

⚙

Select a section and enter column height to see results.

📚 Column Axial Compression — Design Theory

Flexural Buckling — AISC 360 §E3

Nominal compressive strength P_n = F_cr × A_g. Design strength φ_cP_n with φ_c = 0.90, E = 29,000 ksi / 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

Euler stressF_e = π²E / (KL/r)²

LRFD checkP_u ≤ φ_cP_n = 0.90 × F_cr × A_g

F_y	4.71√(E/F_y)
36 ksi / 250 MPa	133.7
50 ksi / 345 MPa	113.4
355 MPa	111.8

ASCE 7-22 §2.3.1 — LRFD Load Combinations

#	Combination
1	1.4D
2	1.2D + 1.6L + 0.5S
3	1.2D + 1.6S + L
4	1.2D + 1.0W + L + 0.5S
5	0.9D + 1.0W

D = Dead (incl. self-weight) | L = Live + Roof Live | S = Snow | W = Wind

Effective Length Factor K — AISC Commentary Table C-A-7.1

End Condition	K (theoretical)	K (recommended)
Both ends pinned	1.0	1.00
One pinned, one fixed (no sway)	0.7	0.80
Both ends fixed (no sway)	0.5	0.65
One fixed, one free (cantilever)	2.0	2.10
Both pinned, sway permitted	1.0	1.20
Both fixed, sway permitted	1.0	1.20

Flexural Buckling — EN 1993-1-1 §6.3.1.2

The buckling resistance N_b,Rd uses a reduction factor χ from the Perry-Robertson imperfection model. γ_M1 = 1.00, E = 210,000 MPa.

Non-dimensional slendernessλ̄ = (KL/r) / (π√(E/f_y))

Auxiliary valueΦ = 0.5 [1 + α(λ̄ − 0.2) + λ̄²]

Reduction factorχ = 1 / (Φ + √(Φ² − λ̄²)) ≤ 1.0

Buckling resistanceN_b,Rd = χ · A · f_y / γ_M1

Buckling Curves — EN 1993-1-1 Table 6.2

Section type	Axis	Curve	α
IPE / HE (h/b > 1.2), t_f ≤ 40 mm	y‑y (strong)	a	0.21
IPE / HE (h/b > 1.2), t_f ≤ 40 mm	z‑z (weak)	b	0.34
HEA / HEB (h/b ≤ 1.2), t_f ≤ 100 mm	y‑y (strong)	b	0.34
HEA / HEB (h/b ≤ 1.2), t_f ≤ 100 mm	z‑z (weak)	c	0.49
SHS / RHS (hot-formed)	any	a	0.21
CHS (hot-formed)	any	a	0.21
Welded I, t_f ≤ 40 mm	y‑y	b	0.34
Welded I, t_f ≤ 40 mm	z‑z	c	0.49

EN 1990 §6.4.3 — ULS Load Combinations (Exp. 6.10)

#	Combination
1	1.35 G
2	1.35 G + 1.5 Q
3	1.35 G + 1.5 S + 0.75 Q
4	1.35 G + 1.5 Q + 0.5 S
5	1.35 G + 1.5 W + 0.9 Q
6	1.0 G + 1.5 W

G = Permanent | Q = Imposed (live) | S = Snow | W = Wind. ψ₀ values from EN 1990 Table A1.1; National Annexes may modify γ factors.

Flexural Buckling — TSDS 2016 §5.2

TSDS 2016 adopts the same F_cr formulas as AISC 360 Ch. E in SI units. φ_c = 0.90, E = 200,000 MPa. Slenderness limit: KL/r ≤ 200.

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

Euler stress (SI)F_e = π² × 200,000 / (KL/r)² (MPa)

Design strengthφ_cP_n = 0.90 × F_cr × A (kN)

f_y (MPa)	4.71√(E/f_y)
S235 — 235	137.5
S275 — 275	126.9
S355 — 355	111.8

TSDS 2016 — Strength Design Load Combinations

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

G = Dead load | Q = Live load | S = Snow | W = Wind

Flexural Buckling — IS 800:2007 §7.1.2

IS 800 uses the same Perry-Robertson model as EC3 with γ_m0 = 1.10, E = 200,000 MPa. Slenderness limit: KL/r ≤ 180.

Non-dimensional slendernessλ̄ = (KL/r) / (π√(E/f_y))

Auxiliary valueφ = 0.5 [1 + α(λ̄ − 0.2) + λ̄²]

Reduction factorχ = 1 / (φ + √(φ² − λ̄²)) ≤ 1.0

Design strengthP_d = χ · A · f_y / (γ_m0 · 1000) (kN)

Buckling Classes — IS 800:2007 Table 10

Section type	Axis	Class	α
Hot-rolled I (h/b > 1.2), t_f ≤ 40 mm	y‑y (strong)	a	0.21
Hot-rolled I (h/b > 1.2), t_f ≤ 40 mm	z‑z (weak)	b	0.34
Hot-rolled I (h/b ≤ 1.2), t_f ≤ 100 mm	y‑y (strong)	b	0.34
Hot-rolled I (h/b ≤ 1.2), t_f ≤ 100 mm	z‑z (weak)	c	0.49
SHS / RHS (hot-formed)	any	a	0.21
CHS (hot-formed)	any	a	0.21
Welded I, t_f ≤ 40 mm	y‑y	b	0.34
Welded I, t_f ≤ 40 mm	z‑z	c	0.49

IS 800 buckling classes a/b/c/d match EC3 curves a/b/c/d exactly — imperfection factor α values are identical.

IS 800:2007 Table 4 — ULS Load Combinations

#	Combination
1	1.5 (DL + LL)
2	1.5 (DL + SL)
3	1.5 (DL + WL)
4	1.2 (DL + LL + WL)
5	1.2 (DL + LL + SL)
6	0.9 DL + 1.5 WL

DL = Dead Load | LL = Imposed Load | SL = Snow Load | WL = Wind Load. γ_f from IS 875 (Part 5).

Example — AISC 360-22 LRFD Axial Compression Design

Given: Interior column, storey height L = 4.0 m (13.1 ft). Effective length factors K_x = K_y = 1.0 (pinned–pinned, both axes). Design code: AISC 360-22, Grade A992 steel (F_y = 345 MPa, E = 200,000 MPa).
Factored axial load (governing ASCE 7-22 combo): P_u = 1,650 kN. Select and verify a W-shape column.

Step 1 — Required strength and trial section

Target φ_cP_n ≥ 1,650 kN. Try W10×60: A_g = 11,400 mm², r_x = 112 mm, r_y = 65.8 mm, self-weight ≈ 0.89 kN/m (included in P_u).

Step 2 — Slenderness

KL/r_x = 1.0×4,000/112 = 35.7
KL/r_y = 1.0×4,000/65.8 = 60.8 ← governs (weak axis)
Limit: 4.71√(E/F_y) = 4.71√(200,000/345) = 113.4 > 60.8 → Inelastic buckling

Step 3 — Critical stress F_cr — §E3

Euler stress

F_e = π²E/(KL/r)² = π²×200,000/60.8² = 1,973,921/3,697 = 534 MPa

Inelastic buckling

F_cr = 0.658^F_y/F_e × F_y = 0.658^345/534 × 345 = 0.658^0.6461 × 345
= 0.763 × 345 = 263 MPa

Step 4 — Design compressive strength φ_cP_n

φ_cP_n = 0.90 × F_cr × A_g = 0.90 × 263 × 11,400 / 1,000 = 2,699 kN
DCR = P_u / φ_cP_n = 1,650 / 2,699 = 0.611 < 1.0 ✓

Step 5 — Check slenderness limit

KL/r_y = 60.8 < 200 (AISC Chapter E user note maximum) ✓

Optimise — Try W10×49 (lighter section)

W10×49: A_g = 9,290 mm², r_y = 64.8 mm

KL/r_y = 4,000/64.8 = 61.7 → F_e = π²×200,000/61.7² = 518 MPa
F_cr = 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
DCR = 1,650/2,182 = 0.756 < 1.0 ✓

Section	A (mm²)	r_y (mm)	KL/r_y	F_cr (MPa)	φ_cP_n (kN)	DCR
W10×60	11,400	65.8	60.8	263	2,699	0.611
W10×49	9,290	64.8	61.7	261	2,182	0.756
W10×45	8,520	51.1	78.3	220	1,687	0.978

Result: W10×49 Grade A992 selected. φ_cP_n = 2,182 kN > P_u = 1,650 kN ✓. DCR = 0.756. Weak-axis buckling governs (KL/r_y=61.7 vs KL/r_x=35.7). Inelastic buckling regime — F_cr=261 MPa vs F_y=345 MPa — confirming residual-stress sensitivity. W10×45 (r_y=51.1 mm) would give DCR=0.978 (also acceptable if no combined bending).