LTB Control

Step 1 — Plastic Moment

Design value: φ_bM_n = 0.90 · M_n    (LRFD, AISC §F1)

Step 2 — Limiting Unbraced Lengths

Step 3 — Nominal Moment M_n

Geometric Properties Used in LTB

Exact C_w values are tabulated in AISC Steel Construction Manual (15th Ed.) for W sections and in Arcelor-Mittal EN 10365 tables for IPE/HEA/HEB. This calculator uses those tabulated values directly.

Section Database Coverage

AISC 360-22 §F1 — Eq. F1-1

C_b accounts for the non-uniform moment distribution within the unbraced segment. Uniform moment (C_b=1.0) is the most conservative case.

Common C_b Values

Note: C_b > 1.0 is beneficial — it increases the computed M_n, but the result is capped at M_p. When loads are applied at the top flange rather than the shear center, an effective C_b reduction should be applied per AISC Commentary to §F1.

Overview: χ_LT Reduction Factor Approach

Unlike AISC (zone-based, M_n directly), EC3 computes a reduction factor χ_LT via European buckling curves. Design resistance M_b,Rd = χ_LT·W_y·f_y/γ_M1.

Step 1 — Elastic Critical Moment M_cr

Step 2 — Non-Dimensional Slenderness

Step 3 — Buckling Curve (Specific Method §6.3.2.3)

The specific method for rolled I-sections (§6.3.2.3) uses λ̅_LT,0=0.4 and β=0.75 (vs 0.2/1.0 for the general method). Buckling curve from Table 6.5:

Step 4 — Reduction Factor χ_LT

Step 5 — Design Buckling Resistance

C₁ Factor — Moment Shape (NCCI SN003)

AISC vs EC3 Comparison

About the Specification

TSDS 2016 (Turkish Steel Design Specification — formally ÇYTHYE) was published in the Official Gazette No. 29614 on 04 February 2016 and entered into force on 01 September 2016, with a corrective revision in 2018. The specification is technically based on AISC 360-16; the lateral torsional buckling provisions of §F2 are formula-identical to AISC.

TSDS 2016 §F2 — Lateral Torsional Buckling

TSDS notation: f_yk = yield strength (MPa), W_px = plastic section modulus (mm³), W_ex = elastic section modulus (mm³) — equivalent to F_y, Z_x, S_x in AISC.

Steel Grades — TS EN 10025

Comparison with AISC 360-22

About IS 800:2007

IS 800:2007 is the Indian Standard code for general construction in steel, published by the Bureau of Indian Standards (BIS). It replaced the earlier working-stress format (IS 800:1984) with a Limit State Design (LSD) approach broadly aligned with Eurocode 3. The LTB provisions appear in §8.2.2 and use a reduction-factor approach similar to EC3 but with Indian buckling curves and partial safety factors.

IS 800:2007 §8.2 — Design Bending Strength

The design bending strength of a laterally unsupported beam is:

For compact sections (Class 1 & 2): β_b = 1.0 and Z_p = plastic section modulus.

Non-Dimensional Slenderness

E = 200 000 MPa, G = 77 000 MPa (IS 800), I_y = minor-axis second moment, I_w = warping constant, I_t = torsional constant.

Reduction Factor χ_LT — IS 800 §8.2.2

When λ̅_LT ≤ 0.2: no LTB reduction (χ_LT = 1.0).

Buckling Curves — IS 800 Table 8.1

Steel Grades — IS 2062

Comparison with EC3 General Method

Scope of AISC §F2

• Compact sections only — flanges and web must satisfy compactness limits per AISC Table B4.1b. Non-compact sections require §F3/F4.
• Doubly-symmetric I-shaped members loaded in the plane of the web. Singly-symmetric or other shapes require different provisions (§F4–F10).
• Strong-axis bending only. Weak-axis bending of doubly-symmetric I-shapes is governed by §F6 and is not subject to LTB.
• Loads at shear center. Top-flange loading destabilises the beam; bottom-flange loading stabilises it. Use effective C_b for other load heights.
• Bracing requirements — braces must meet stiffness and strength requirements per AISC Appendix 6. Point braces must provide both lateral and torsional restraint to be effective against LTB.
• Not checked: Shear (§G2), flange local buckling (§F3/F4), web local buckling, connection strength, or deflection limits.

Unit System Notes

Example — AISC 360-22 §F2 LTB Check (Inelastic LTB Zone)

Given: W18×55, Grade A992 (F_y = 345 MPa / 50 ksi). Unbraced length L_b = 3.6 m (11.8 ft). Moment gradient C_b = 1.30 (moment varies linearly — one end pinned, mid-span load). Factored moment M_u = 310 kN·m.
W18×55 properties (SI): Z_x = 1,090 cm³, S_x = 980 cm³, r_y = 42.2 mm, r_ts = 47.7 mm, J = 6.86 cm⁴, C_w = 27,000 cm⁶, h₀ = 532 mm, c = 1.0.

1. Plastic moment M_p
   M_p = F_y·Z_x = 345×1,090×10³/10⁶ = 376.1 kN·m
2. Limiting unbraced lengths L_p and L_r — §F2
   L_p = 1.76·r_y·√(E/F_y) = 1.76×42.2×√(200,000/345)/1,000 = 1.76×42.2×24.08/1,000 = 1.789 m
   X = J/(S_x·h₀) = 6.86×10⁴/(980×10³×532) = 1.315×10⁻⁴ mm⁻¹
   Y = 6.76×(0.7F_y/E)² = 6.76×(0.7×345/200,000)² = 6.76×(1.2075×10⁻³)² = 9.866×10⁻⁶
   L_r = 1.95·r_ts·(E/(0.7F_y))·√(X + √(X² + Y))
       = 1.95×47.7×(200,000/241.5)×√(1.315×10⁻⁴ + √((1.315×10⁻⁴)² + 9.866×10⁻⁶))/1,000
       ≈ 5.21 m
   Zone check: L_p=1.79 m < L_b=3.6 m < L_r=5.21 m → Inelastic LTB
3. Nominal LTB capacity M_n — §F2-2
   M_r = 0.7·F_y·S_x = 0.7×345×980×10³/10⁶ = 236.3 kN·m
   M_n = C_b·[M_p − (M_p − M_r)·(L_b−L_p)/(L_r−L_p)] ≤ M_p
   = 1.30×[376.1 − (376.1 − 236.3)×(3.6−1.789)/(5.21−1.789)]
   = 1.30×[376.1 − 139.8×0.529]
   = 1.30×[376.1 − 73.9] = 1.30×302.2 = 392.9 kN·m > M_p → use M_n = M_p = 376.1 kN·m
   (C_b boost hit the M_p cap — this is correct AISC behaviour)
4. Design capacity φ_bM_n
   φ_bM_n = 0.90×376.1 = 338.5 kN·m
   DCR = M_u/φ_bM_n = 310/338.5 = 0.916 < 1.0 ✓