Steel Beam Flexural Design

AISC LRFD flexural design for compact sections with full lateral bracing. Full ASCE 7-22 load combinations — D, L, S, L_r, W. Self-weight automatically applied.

Input Parameters

kip · ft · in

Design Code:

Units:

Steel Grade

γ_M0 — partial factor (EC3 §6.1 / IS 800 §5)

—

Section Selection

Section Type

Profile

Span L [ft]

End Conditions

Loads — ASCE 7-22

Cover Loads D [kip/ft]

kip/ft

Live Load L [kip/ft]

kip/ft

Snow Load S [kip/ft]

kip/ft

Roof Live L_r [kip/ft]

kip/ft

Wind Load W [kip/ft]

kip/ft

Beam Orientation

Inclined beam — biaxial bending from gravity loads

Roof slope θ [° from horizontal]

📋

Select a section, enter loads, and click Calculate.

📚 AISC LRFD — Theory & Load Combinations

Design Method↑ Top

AISC LRFD — φM_n (AISC §F2)

For a compact section with full lateral bracing (L_b ≤ L_p), the design moment capacity is:

Plastic moment — American units φM_n = 0.9 × F_y × Z_x / 12 [kip·ft]

Plastic moment — SI units φM_n = 0.9 × F_y × W_pl,y / 1000 [kN·m]

Maximum Moment by Boundary Condition

UDL Design Moments by Boundary Condition Pin – Pin: M_u = w_uL²/8 (midspan)
Pin – Fixed: M_u = 9w_uL²/128 (max positive at 3L/8) — fixed-end negative = w_uL²/8 (check separately)
Fixed – Fixed: M_u = w_uL²/12 (at supports)
Cantilever (Fixed – Free): M_u = w_uL²/2 (at fixed support)

Deflection

Deflection (service loads) Pin–Pin: δ = 5wL⁴/(384EI) | Pin–Fixed: δ ≈ wL⁴/(185EI) | Fixed–Fixed: δ = wL⁴/(384EI) | Cantilever (free end): δ = wL⁴/(8EI)

Limits (Pin–Pin / Pin–Fixed / Fixed–Fixed): live δ ≤ L/360, total δ ≤ L/240 (D+L+S+L_r).
Limits (Cantilever): live δ ≤ L/180, total δ ≤ L/120 — equivalent to 2L/360 and 2L/240 per AISC DG3.

Inclined Beam — Biaxial Bending (AISC §H1-1)

A beam inclined at angle θ from horizontal receives gravity loads in both principal directions. The UDL is resolved into strong- and weak-axis components:

Load decomposition w_y = w_u · cosθ (strong axis) w_z = w_u · sinθ (weak axis)

AISC §H1-1 biaxial interaction (P_r = 0) M_u,y / φM_n,y + M_u,z / φM_n,z ≤ 1.0

For pure bending (no axial force), §H1-1a and §H1-1b both reduce to a simple linear sum of the two demand-to-capacity ratios. This is the standard AISC biaxial bending check for beams.

ASCE 7-22 Combos↑ Top

ASCE 7-22 §2.3.1 — LRFD Strength Combinations

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

D = Cover loads + self-weight | W = Wind UDL on beam | R = Rain (not included)

Steel Grades

American (ASTM) A992 Gr50: F_y=50 ksi | A572 Gr50: 50 ksi | A572 Gr60: 60 ksi | A36: 36 ksi

European (EN 10025-2) S235: 235 MPa | S275: 275 MPa | S355: 355 MPa | S460: 460 MPa

EN 1993-1-1 (EC3) — Design Method↑ Top

Section Classification — EN 1993-1-1 Table 5.2

Classification uses ε = √(235/f_y) to normalise slenderness limits:

Section type	Ratio	Class 1	Class 2	Class 3
I/H flange (outstand)	c_f/t_f	≤ 9ε	≤ 10ε	≤ 14ε
I/H web (pure bending)	d/t_w	≤ 72ε	≤ 83ε	≤ 124ε
SHS/RHS flange (compression)	c/t	≤ 33ε	≤ 38ε	≤ 42ε
SHS/RHS web (bending)	c/t	≤ 72ε	≤ 83ε	≤ 124ε
CHS	D/t	≤ 50ε²	≤ 70ε²	≤ 90ε²

c_f = (b − t_w − 2r)/2 for I-sections | d = h − 2t_f − 2r | c = a − 3t for hot-formed SHS (r ≈ 1.5t)

Flexural Resistance — EN 1993-1-1 §6.2.5

Class 1 & 2 (plastic) M_c,Rd = W_pl,y × f_y / γ_M0 [kN·m]

Class 3 (elastic) M_c,Rd = W_el,y × f_y / γ_M0 [kN·m]

γ_M0 = 1.00 (recommended, may be modified by national annex)

EN 1990 Eq. 6.10 — ULS Load Combinations

#	Combination
1	1.35G + 1.5Q_L
2	1.35G + 1.5Q_L + 0.75Q_S (Q_L dominant + snow, ψ_0,S=0.5)
3	1.35G + 1.5Q_L + 0.9Q_W (Q_L dominant + wind, ψ_0,W=0.6)
4	1.35G + 1.05Q_L + 1.5Q_S (Q_S dominant, ψ_0,L=0.7)
5	1.35G + 1.05Q_L + 1.5Q_W (Q_W dominant, ψ_0,L=0.7)
6	0.9G + 1.5Q_W (uplift)

G = permanent (dead + self-weight) | Q_L = live + roof live | Q_S = snow | Q_W = wind

Inclined Beam — Biaxial Bending (EN 1993-1-1 §6.2.9)

For an inclined beam under gravity load w_Ed, the load is resolved into strong- and weak-axis components (w_y = w_Ed·cosθ, w_z = w_Ed·sinθ), producing moments about both principal axes simultaneously.

EN 1993-1-1 §6.2.9(6) gives the interaction formula with section-dependent exponents α and β (for N_Ed = 0):

General form [M_y,Ed / M_cy,Rd]^α + [M_z,Ed / M_cz,Rd]^β ≤ 1.0

Section type	α	β	Interaction shape
I and H sections (IPE, HEA, HEB …)	2	1	Parabolic in M_y, linear in M_z
Circular hollow (CHS)	2	2	Circular
Rectangular hollow (RHS, SHS)	1.66	1.66	Rounded convex
Channels and other sections	1	1	Linear (conservative)

Why not linear for I-sections? EC3 recognises that in an I-section the flanges carry both strong- and weak-axis bending independently. The true biaxial plastic interaction surface is convex, not flat — so a linear formula is overly conservative. AISC §H1-1 uses a linear sum for all section types regardless of shape, which is simpler but can be 15–25% more conservative than EC3 for I-sections under high strong-axis utilisation.

Example — I-section with M_y/M_cy = 0.7, M_z/M_cz = 0.3:
Linear (AISC / conservative): 0.7 + 0.3 = 1.00 ← exactly at limit
EC3 §6.2.9(6) I-section (α=2, β=1): 0.7² + 0.3 = 0.49 + 0.30 = 0.79 ← 21% reserve

Serviceability Deflection Limits — EN 1990 Annex A1.4

Simply supported & propped (recommended) Variable load δ_Q ≤ L/300 | Total δ_max ≤ L/250

Cantilever Variable load δ_Q ≤ L/150 | Total δ_max ≤ L/125

E = 210 000 N/mm² (EN 1993-1-1 §3.2.6)

IS 800:2000 (BIS) — Design Method↑ Top

Section Classification — IS 800:2000 Table 2

Classification uses ε = √(250/f_y):

Section type	Ratio	Class 1 — Plastic	Class 2 — Compact	Class 3 — Semi-compact
I/H flange outstand b = (B−t_w)/2	b/t_f	≤ 9.4ε	≤ 10.5ε	≤ 15.7ε
I/H web (pure bending)	d/t_w	≤ 84ε	≤ 105ε	≤ 126ε
Channel flange outstand b = B−t_w	b/t_f	≤ 9.4ε	≤ 10.5ε	≤ 15.7ε

d = h − 2t_f (clear web, fillet excluded for simplicity) | Class 4 (Slender) not covered here.

Flexural Design Strength — IS 800:2000 Cl. 8.2.1

Class 1 & 2 (Plastic / Compact) M_d = β_b × Z_px × f_yd [kN·m] (β_b = 1.0)
Cap (simply supported): M_d ≤ 1.2 × Z_ex × f_yd
Cap (cantilever): M_d ≤ 1.5 × Z_ex × f_yd

Class 3 (Semi-compact) M_d = Z_ex × f_yd [kN·m]

f_yd = f_y / γ_m0 | γ_m0 = 1.10 (IS 800:2000 Table 5) | Z_px, Z_ex from IS 808:1989

IS 800:2000 Table 4 — ULS Load Combinations (IS 875)

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

DL = dead + self-weight | IL = imposed (live + roof live) | SL = snow | WL = wind

Inclined Beam — Biaxial Bending (IS 800:2000 Cl. 9.3.1)

Gravity load is resolved into strong- and weak-axis components. IS 800 uses a linear interaction for pure bending:

Biaxial check M_u,y / M_d,y + M_u,z / M_d,z ≤ 1.0

Indian Steel Grades — IS 2062

IS 2062 grades E250A: f_y = 250 MPa | E300: f_y = 300 MPa | E350: f_y = 350 MPa | E410: f_y = 410 MPa

Serviceability Deflection Limits — IS 800:2000 Table 6

Simply supported / propped Imposed load δ_IL ≤ L/300 | Total δ_max ≤ L/250

Cantilever Imposed load δ_IL ≤ L/150 | Total δ_max ≤ L/125

E = 200 000 N/mm² (IS 800:2000 §2.2.4.1)

TSDS 2016 — Design Method↑ Top

TSDS 2016 (Turkish Steel Design Code) is the Turkish steel design regulation for steel structures. It uses the LRFD (Load and Resistance Factor Design) method.

Section Classification — TSDS 2016 Table 5.1B

Slenderness limits normalised by ε = √(235/f_y):

Section type	Ratio	Class 1	Class 2	Class 3
I/H flange (outstand)	c_f/t_f	≤ 9ε	≤ 10ε	≤ 14ε
I/H web (pure bending)	d/t_w	≤ 72ε	≤ 83ε	≤ 124ε
SHS/RHS flange (compression)	c/t	≤ 33ε	≤ 38ε	≤ 42ε
SHS/RHS web (bending)	c/t	≤ 72ε	≤ 83ε	≤ 124ε
CHS	D/t	≤ 50ε²	≤ 70ε²	≤ 90ε²

c_f = (b − t_w − 2r)/2 | d = h − 2t_f − 2r | c = a − 3t (hot-rolled)

Flexural Capacity — TSDS 2016 (AISC 360 §F2)

Class 1 & 2 (Plastic / Compact) φM_n = φ × W_pl,y × f_y [kN·m]

Class 3 (Semi-compact) φM_n = φ × W_el,y × f_y [kN·m]

φ = 0.90 (resistance factor for bending, AISC 360 §F1) | Full lateral bracing assumed; LTB not checked.

TSDS 2016 LRFD — ULS Load Combinations

#	Combination
1	1.4G
2	1.2G + 1.6Q + 0.5S
3	1.2G + 1.6S + Q
4	1.2G + 1.6S + 0.5W
5	1.2G + 1.0W + Q + 0.5S
6	0.9G + 1.0W (uplift)

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

Biaxial Bending — AISC 360 H1-1 (linear)

For inclined beams, gravity load generates moments about both principal axes. TSDS follows AISC H1-1 linear interaction for pure bending (P_r = 0):

Biaxial check M_u,y / φM_n,y + M_u,z / φM_n,z ≤ 1.0

Vertical Deflection Limits — TSDS §15.1

Simply supported / Pin–Fixed / Fixed–Fixed Total deflection under G+Q: δ ≤ L/300

Cantilever Total deflection under G+Q: δ ≤ L/150

E = 200 000 N/mm² | Single service check under G+Q; no separate variable-load deflection limit.

Steel Grades — TS EN 10025-2

European grades (used in Turkey) S235: f_y = 235 MPa | S275: 275 MPa | S355: 355 MPa | S460: 460 MPa

Limitations↑ Top

Assumptions & Limitations

Full lateral bracing: LTB not checked. Ensure L_b ≤ L_p.
Compact section: W-shapes compact for F_y ≤ 50 ksi. Verify for other types or higher grades.
UDL only: Point loads, partial UDL, and cantilevers not handled.
Z_x estimate: S, M, C, MC shapes use Z_x ≈ 1.12 × S_x. HSS and CHS types use tabulated Z_x.
Wind as UDL: W is treated as a vertical UDL adding to gravity (e.g., roof wind pressure). For lateral-only wind on columns, set W = 0.
Self-weight: Taken from the selected section’s tabulated weight and added to D automatically.

Steel Beam Flexural Design

Input Parameters

Calculation Report

📚 AISC LRFD — Theory & Load Combinations

AISC LRFD — φMn (AISC §F2)

Maximum Moment by Boundary Condition

Deflection

Inclined Beam — Biaxial Bending (AISC §H1-1)

ASCE 7-22 §2.3.1 — LRFD Strength Combinations

Steel Grades

Section Classification — EN 1993-1-1 Table 5.2

Flexural Resistance — EN 1993-1-1 §6.2.5

EN 1990 Eq. 6.10 — ULS Load Combinations

Inclined Beam — Biaxial Bending (EN 1993-1-1 §6.2.9)

Serviceability Deflection Limits — EN 1990 Annex A1.4

Section Classification — IS 800:2000 Table 2

Flexural Design Strength — IS 800:2000 Cl. 8.2.1

IS 800:2000 Table 4 — ULS Load Combinations (IS 875)

Inclined Beam — Biaxial Bending (IS 800:2000 Cl. 9.3.1)

Indian Steel Grades — IS 2062

Serviceability Deflection Limits — IS 800:2000 Table 6

Section Classification — TSDS 2016 Table 5.1B

Flexural Capacity — TSDS 2016 (AISC 360 §F2)

TSDS 2016 LRFD — ULS Load Combinations

Biaxial Bending — AISC 360 H1-1 (linear)

Vertical Deflection Limits — TSDS §15.1

Steel Grades — TS EN 10025-2

Assumptions & Limitations

Related Calculators

AISC LRFD — φM_n (AISC §F2)