CELE Steel & Timber Design — Steel Beams: Flexure and ShearSummary

When a compact steel section is continuously braced against lateral-torsional buckling (compression flange restrained at spacing L_b ≤ L_p), all fibres yield in bending, and the section develops its plastic moment: M_p = F_y Z_x. The plastic section modulus Z_x exceeds the elastic modulus S_x by the shape factor (typically 1.10–1.17 for I-shapes). The design flexural strength is φ_b M_n = 0.90 × M_p per AISC 360 (φ_b = 0.90 for LSD; Ω_b = 1.67 for ASD, giving allowable M_allow = M_p / 1.67). This represents the maximum flexural capacity achievable under ideal conditions—no buckling, no local element instability.

As the unbraced length L_b of the compression flange increases, the beam becomes more vulnerable to sideways (lateral) and twisting (torsional) buckling before the full plastic moment is reached. AISC 360 defines two critical lengths: (1) L_p = 1.76 r_y √(E/F_y) — the unbraced length at which M_n = M_p with no LTB reduction (plastic zone); and (2) L_r — the unbraced length separating inelastic LTB (M_p > M_n > 0.7 F_y S_x) from elastic LTB (M_n ≤ 0.7 F_y S_x). For L_b ≤ L_p, the full M_p is available. For L_p < L_b ≤ L_r, moment capacity decreases linearly. For L_b > L_r, elastic buckling theory dominates and M_n becomes very small. These limits depend on beam geometry (r_y, L_p ∝ r_y), material properties (F_y, E), and bracing strategy.

The moment-gradient factor C_b accounts for non-uniform bending moments along a beam's length. For uniform moment (cantilever with end load, or simply-supported beam with concentrated midspan load), C_b = 1.0. For decreasing moment towards the free end or other gradient conditions, C_b may be 1.0 to ~2.5 (capped at 1.0 when M_n would exceed M_p). The formula is C_b = 12.5 M_max / (2.5 M_max + 3 M_A + 4 M_B + 3 M_C), where M_A, M_B, M_C are quarter-span moments. A larger C_b increases M_n in the inelastic LTB range, reflecting the fact that regions of lower moment resist LTB better. C_b = 1.0 is conservative and often used if the exact moment distribution is unknown.

A cross-section is classified as compact, non-compact, or slender based on the width-to-thickness (λ = b/t) ratios of its flanges and web compared to the limiting ratios λ_p (compact limit) and λ_r (non-compact limit). For a compact section, all elements remain elastic up to the plastic moment; flange and web do not buckle locally before M_p is reached. Non-compact sections develop less than M_p due to local buckling; the nominal moment M_n is reduced from M_p toward the 'flange local buckling' limit (M_f). Slender sections are further compromised. AISC 360 provides detailed tables for λ_p and λ_r as functions of F_y; for example, for rolled I-beams with F_y = 248 MPa (Grade 250), the flange limit λ_p ≈ 10.75. If the beam's actual flange b/t ≤ 10.75, it is compact (no FLB); if 10.75 < b/t < ~17, it is non-compact (FLB reduces M_n); if b/t ≥ 17, it is slender (significant loss).

The nominal shear strength of a steel beam web is V_n = 0.6 F_y A_w C_v, where A_w = d × t_w is the web area (overall depth times web thickness) and C_v is the shear coefficient. For stocky webs (h/t_w ≤ 2.24 √(E/F_y)), C_v = 1.0 (web can yield in shear; no buckling). For slender webs (h/t_w > 2.24 √(E/F_y)), C_v < 1.0 (web-shear buckling reduces strength). Most rolled I-beams have stocky webs, yielding C_v = 1.0 and V_n = 0.6 F_y A_w. The design shear strength is φ_v V_n = 1.0 × V_n for rolled I-shapes (φ_v = 0.90 for built-up sections with slender webs or plates). The shear capacity is typically much higher than required, so shear often does not govern; however, deep, thin-walled beams or short spans with high loads can be shear-critical.

The design flexural strength is the product φ_b × M_n, where M_n is the nominal moment (computed based on compactness, unbraced length, and C_b) and φ_b = 0.90 is the LRFD resistance factor per AISC 360 and NSCP 2015. For ASD, the allowable moment is M_allow = M_n / Ω_b with Ω_b = 1.67. The designer verifies that φ_b M_n ≥ M_u (LRFD; M_u = factored moment) or M_allow ≥ M_service (ASD; M_service = unfactored service moment). The computation of M_n depends on beam class: (1) Compact, braced (L_b ≤ L_p): M_n = M_p. (2) Compact, partially braced (L_p < L_b ≤ L_r): M_n interpolated linearly from M_p to 0.7 F_y S_x. (3) Compact, unbraced (L_b > L_r): M_n = F_cr S_x (elastic LTB). (4) Non-compact or slender: M_n further reduced by local-buckling factors. This decision tree is the crux of flexural design.

The shear-buckling coefficient C_v accounts for the slenderness of the web in shear. For a stocky web, defined as h/t_w ≤ 2.24 √(E/F_y) [where h is the clear web height and t_w is web thickness], C_v = 1.0; the web yields in shear before buckling. At F_y = 248 MPa and E = 200 GPa, this limit is h/t_w ≤ 2.24 × √(200,000/248) ≈ 101. Most rolled I-beams fall in this category. For h/t_w between 2.24 √(E/F_y) and 3.08 √(E/F_y), the web is transitional (inelastic shear buckling), and C_v is computed via a transition formula. For very slender webs (h/t_w > 3.08 √(E/F_y)), elastic shear buckling dominates and C_v is calculated from the elasticity formula. The reduction in C_v for slender webs can significantly lower V_n, which may govern in thin-wall or fabricated beams.

AISC 360 and NSCP 2015 employ limit-state design with resistance factors (LRFD): φ_b = 0.90 for flexure and φ_v = 1.0 for shear (rolled I-shapes) or 0.90 (built-up sections). In ASD (Allowable Stress Design, older approach but still used in some contexts), the design is verified using safety factors: Ω_b = 1.67 for flexure (so M_allow = M_n / 1.67) and Ω_v = 1.50 for shear (so V_allow = V_n / 1.50). The relationship is φ = 1/Ω (approximately). LRFD is the modern, preferred approach in the Philippines (per NSCP 2015 and PRC guidelines). Reviewees must know both systems but prioritize LRFD for current exams. The design inequality is M_u ≤ φ_b M_n (LRFD) or M_service ≤ M_n / Ω_b (ASD).

A section achieves the plastic moment M_p = F_y Z_x only if it is compact (flanges and web satisfy λ ≤ λ_p). Non-compact sections cannot reach M_p; they are limited by local buckling (flange or web instability).

As L_b increases beyond L_p, LTB reduces M_n from M_p toward 0.7 F_y S_x (inelastic zone) and eventually to elastic F_cr S_x (very small). Continuous lateral bracing keeps L_b ≤ L_p and preserves M_p.

If the section is compact AND L_b ≤ L_p (well-braced), then M_n = M_p (full capacity). If compact but L_b > L_p, M_n drops due to LTB. If non-compact, local buckling limits M_n regardless of L_b.

Non-uniform moment distributions (e.g., cantilevers, unequal-span beams) allow higher C_b > 1.0, which increases M_n in the inelastic LTB zone. However, C_b M_n cannot exceed M_p.

Stocky webs (h/t_w ≤ 2.24 √(E/F_y)) have C_v = 1.0 and full shear capacity V_n = 0.6 F_y A_w. Slender webs have reduced C_v, lowering V_n. Most rolled beams are stocky.

φ_b M_n = 0.90 M_n for flexure; φ_v V_n = 1.0 V_n for shear (rolled I-shapes). The resistance factor φ accounts for material variability and model uncertainty. Design must satisfy φ_n S_n ≥ E (LRFD: E = factored demand).

Higher F_y (e.g., 345 vs. 248 MPa) reduces L_p (stronger beams buckle laterally at shorter spans), reduces λ_p (sections become non-compact more easily), but increases V_n (higher shear yield). Design limits are inversely proportional to F_y in many cases.

I-beams have large r_y (relative to r_x) and low torsional stiffness, making them prone to LTB. Box sections have higher torsional rigidity and larger r_y, resulting in larger L_p and better LTB resistance. This is why composite action (concrete deck on steel beams) provides substantial LTB benefit.

A typical industrial warehouse in the Philippines uses hot-rolled IPE or local-equivalent I-beams spanning 8 m, with the concrete roof deck bolted to the top flange every 2 m. Since L_b = 2 m and typical L_p ≈ 2.0–2.5 m for common sections, the beam achieves M_p and develops full plastic-moment capacity. The design checks: (1) Compactness: likely compact for standard grades. (2) L_b ≤ L_p: satisfied by deck bracing. (3) Shear: rarely critical for typical loads. Result: economical beam size with high moment utilization.

A cantilever over a shop entrance has no lateral bracing of the top (compression) flange, extending 3 m. The moment is maximum at the fixed end and zero at the free end (C_b > 1.0). If L_p ≈ 2.0 m, the cantilever length L_b = 3 m > L_r, pushing the beam into elastic LTB, severely reducing M_n. The designer must either: (1) add a lateral brace partway along the cantilever, (2) use a larger, stiffer section (higher r_y), or (3) reduce the design load. Moment-gradient benefit (C_b) partially offsets the penalty but often cannot overcome the LTB reduction.

A fabricated plate-girder (e.g., 1000 mm deep, 10 mm web thickness) for a 20 m span bridge or stadium roof requires detailed checks: (1) Flange compactness: thick flanges may be non-compact, reducing M_n. (2) Web slenderness h/t_w = 980/10 = 98, near or above the stocky limit (~101 at F_y = 248), so C_v may be slightly less than 1.0. (3) Lateral bracing: if only at supports (L_b = 20 m) and L_p ≈ 2–3 m, the girder is severely limited by LTB. The design typically adds intermediate bracing (e.g., every 4 m) to keep L_b manageable or increases section size. This illustrates the combined complexity of compactness, LTB, and shear buckling for long-span beams.

A short canopy beam spanning 3 m supports a concentrated load of 300 kN near the support. Shear at the support is V = 300 kN. If the beam has depth d = 400 mm and web thickness t_w = 8 mm, then A_w = 400 × 8 = 3200 mm². Assuming C_v = 1.0 and F_y = 248 MPa, V_n = 0.6 × 248 × 3200 = 477 kN. The design shear strength is φ_v V_n = 1.0 × 477 = 477 kN > 300 kN (OK). However, if the web were thinner (t_w = 5 mm), A_w = 2000 mm², V_n = 298 kN, barely adequate. This illustrates that shear can govern in short, heavily loaded spans and that even modest reductions in web thickness matter.

A 20 m office-building corridor can be served by two 10 m simply-supported spans or one continuous span. (1) Simply-supported: M_max at midspan ≈ wL²/8. With L_b = 10 m and typical L_p ≈ 2 m, the beam suffers LTB reduction; M_n is significantly less than M_p, requiring a larger section. (2) Continuous: moment at midspan is lower (≈ wL²/10), and the moment diagram is more complex (C_b > 1.0 in regions of lower moment). Although the continuous beam has a lower maximum positive moment, support moments (negative, hogging) are also higher. The trade-off requires detailed analysis, but continuous beams often allow smaller sections due to lower midspan moment and benefit from varied C_b. This is a classic design decision in practice.

A deep I-beam section (d = 900 mm) fabricated with thin flanges (b_f = 200 mm, t_f = 8 mm) for a high-strength application (F_y = 345 MPa) is checked for flange local buckling. The flange λ = b_f/(2t_f) = 200/16 = 12.5. At F_y = 345, λ_p ≈ 9.5 (from AISC tables), so 12.5 > 9.5 and the section is non-compact. The nominal moment M_n is reduced from M_p by a local-buckling formula, e.g., M_n ≈ M_p - (M_p - 0.7F_y S_x) × k(λ). The reduction can be 10–30% depending on slenderness. This is why ultra-deep or high-strength beams require careful section design to balance economy with compactness.

To keep L_b ≤ L_p and preserve M_p, engineers use: (1) Continuous lateral bracing (concrete deck bolted to top flange every meter or so). (2) Partial-height bracing (e.g., purlins bolted to flanges at ~2 m intervals). (3) Sway-frame action (if the beam is part of a moment-resisting frame, lateral displacement of one end is coupled to the other). Each strategy has cost and feasibility trade-offs. For a typical 248 MPa section with r_y ≈ 40–50 mm, L_p ≈ 2.0–2.5 m. If architectural or structural constraints prevent bracing at this interval, the designer must use a stiffer section (higher r_y, larger I_y) or accept reduced moment.