CELE Steel & Timber Design — Steel Beams: Flexure and ShearRevision Notes

Formulas

Exam Tips

• Memorize: Mp = FyZx and φb = 0.90. These two lines solve most straightforward board problems.
• If the problem gives Sx only, compute Zx = f × Sx using f ≈ 1.12 for W-shapes (unless exact value is given).
• Double-check whether the problem asks for Mn (nominal) or φbMn (design strength) — boards ask both.
• For Fy = 248 MPa (A36 equivalent) and Fy = 345 MPa (A572 Gr.50) — two most common values in Philippine board exams.

Key Points

• The nominal flexural strength of a compact, fully braced steel beam equals the plastic moment: Mn = Mp = Fy × Zx.
• The plastic section modulus Zx is always larger than the elastic section modulus Sx; their ratio is the shape factor f = Zx/Sx ≈ 1.12 for standard W-shapes.
• The LRFD resistance factor for flexure is φb = 0.90; the ASD safety factor is Ωb = 1.67.
• Design flexural strength (LRFD): φbMn = 0.90 FyZx.
• Allowable flexural strength (ASD): Mn/Ωb = FyZx/1.67.
• Full Mp is only achievable when the section is compact AND the compression flange is braced at intervals Lb ≤ Lp.
• A section is compact if both its flange and web width-to-thickness ratios are at or below the compact limit λp.
• For I-shapes: flange compactness requires bf/(2tf) ≤ λp = 0.38√(E/Fy); web compactness requires h/tw ≤ λp = 3.76√(E/Fy).

Definitions

Section Title

1. Plastic Moment Capacity — Compact, Fully Braced Beams

Common Mistakes

• Using Sx (elastic modulus) instead of Zx (plastic modulus) when computing Mp — this under-estimates capacity by the shape factor (~12% for W-sections).
• Applying φb = 0.90 to Mp without first checking compactness and Lb ≤ Lp.
• Forgetting to convert N·mm to kN·m (divide by 10⁶) — unit errors cause wrong answers in board exams.
• Treating all steel beams as compact without verifying flange and web λ ratios.

Formulas

Exam Tips

• Board-exam LTB problems almost always fall in Zone 1 (Lb ≤ Lp) or require computing Lp — master Lp = 1.76ry√(E/Fy) cold.
• When Lb is given but Lp is not, compute Lp first and compare — this sets up the rest of the solution.
• Remember: 0.7FySx is the lower bound of the inelastic LTB zone — it equals 70% of My, not 70% of Mp.
• For quick estimates: at Fy = 248 MPa and typical ry = 35–50 mm, Lp falls in the 1.7–2.5 m range — beams longer than this need checking.
• Cb is always ≥ 1.0; if your computed Cb < 1.0, recheck — it cannot be less than 1.0 by definition.

Key Points

• LTB occurs when an insufficiently braced compression flange moves laterally and twists, preventing full plastic moment development.
• Three LTB zones are defined by comparing the unbraced length Lb to limit lengths Lp and Lr.
• Zone 1 — No LTB (Lb ≤ Lp): Mn = Mp. Full plastic moment is available.
• Zone 2 — Inelastic LTB (Lp < Lb ≤ Lr): Mn reduces linearly from Mp to 0.7FySx.
• Zone 3 — Elastic LTB (Lb > Lr): Mn = FcrSx, governed by elastic buckling; Mn can fall well below Mp.
• The moment gradient factor Cb amplifies Mn for non-uniform moment diagrams; Cb = 1.0 for uniform moment (most conservative). Cb always caps at Mp.
• For doubly-symmetric I-shapes: Lp = 1.76 ry√(E/Fy) and Lr depends on section warping properties.
• Increasing brace frequency (reducing Lb) is the most direct way to restore Mp capacity.

Definitions

Section Title

2. Lateral-Torsional Buckling (LTB) and Unbraced Length Limits

Common Mistakes

• Forgetting to check whether Lb ≤ Lp before jumping to LTB calculations — always locate the LTB zone first.
• Using Cb = 1.0 in problems that explicitly give moment values at quarter points — compute Cb; it can significantly raise Mn.
• Confusing ry (weak-axis radius of gyration) with rx (strong-axis) when computing Lp.
• Not capping Mn at Mp even when Cb > 1.0 raises the calculated value above Mp.
• Mixing up the inelastic and elastic LTB formulas — the linear interpolation formula applies only between Lp and Lr.

Formulas

Exam Tips

• Most PRC board problems involving standard W-sections implicitly assume compact sections — but if the problem states 'non-compact' or gives unusually thin flanges, reduce Mn accordingly.
• Quick check: if Fy = 248 MPa and the section is a standard W-shape from AISC tables, it is almost certainly compact.
• For Fy = 345 MPa, recompute λp = 0.38√(200,000/345) = 9.15 for flanges — some W-shapes fall non-compact.

Key Points

• Local buckling of flange or web plate elements can reduce Mn below Mp before LTB even occurs.
• Three classification tiers: Compact (λ ≤ λp) → full Mp; Non-compact (λp < λ ≤ λr) → Mp reduced; Slender (λ > λr) → Mn governed by elastic local buckling.
• Flange slenderness: λf = bf / (2tf) compared to λp = 0.38√(E/Fy) and λr = 1.0√(E/Fy).
• Web slenderness: λw = h / tw compared to λp = 3.76√(E/Fy) and λr = 5.70√(E/Fy).
• Most standard W-sections (W-shapes) from AISC tables are compact for Fy = 248 MPa — verify for Fy = 345 MPa.
• Non-compact flange: Mn interpolates linearly between Mp (at λp) and 0.7FySx (at λr) — same form as inelastic LTB.
• Slender sections use critical stress formulas similar to plate buckling.

Definitions

Section Title

3. Local Buckling — Compact vs. Non-Compact vs. Slender Sections

Common Mistakes

• Assuming all W-sections are compact — for higher grades (Fy = 345 MPa or above), λp values are lower and some sections become non-compact.
• Using the full depth d instead of the clear web height h in computing web slenderness λw = h/tw.
• Forgetting that both flange AND web must be compact for the section to qualify as compact overall.

Formulas

Exam Tips

• Shear problems are usually straightforward in board exams: φvVn = 1.0 × 0.6 × Fy × d × tw. Three numbers multiplied — do not overthink it.
• The factor 0.6 comes from von Mises yield criterion: shear yield stress τy = 0.577Fy ≈ 0.6Fy.
• When a problem says 'compact web' or 'stocky web', immediately write φv = 1.0 and Cv = 1.0.
• For Fy = 248 MPa: φvVn = 1.0 × 0.6 × 248 × Aw = 148.8 Aw (N) where Aw in mm². Memorize this shortcut.

Key Points

• Steel beam webs resist shear primarily through shear yielding — the web acts like a shear panel.
• Nominal shear strength: Vn = 0.6 Fy Aw Cv.
• Web area: Aw = d × tw (overall depth d × web thickness tw).
• For stocky webs where h/tw ≤ 2.24√(E/Fy): Cv = 1.0 and φv = 1.0 (LRFD).
• For webs with h/tw > 2.24√(E/Fy): Cv < 1.0 (shear buckling governs) and φv = 0.90.
• Most standard rolled W-shapes satisfy h/tw ≤ 2.24√(E/Fy) for Fy = 248 MPa — use Cv = 1.0 and φv = 1.0.
• LRFD check: Vu ≤ φvVn; ASD check: Va ≤ Vn/Ωv where Ωv = 1.50 (or 1.67 for non-stocky webs).
• Do NOT use net web height for Aw in the AISC shear formula — use full depth d.

Definitions

Section Title

4. Shear Strength of Steel Beams

Common Mistakes

• Using φv = 0.90 for all shear calculations — for rolled W-shapes with stocky webs, φv = 1.0 (AISC 360-10 Chapter G).
• Computing Aw using clear web height instead of total depth d — the AISC shear formula explicitly uses d × tw.
• Forgetting to check the stocky web limit h/tw ≤ 2.24√(E/Fy) before assuming Cv = 1.0.
• Applying shear reduction (Cv < 1.0) to standard rolled W-shapes — they are almost always stocky enough for Cv = 1.0.

Formulas

Exam Tips

• In multi-part board problems, part (a) often asks for flexural strength, part (b) for shear strength — solve systematically, do not mix formulas.
• When a W-section is given by designation (e.g., W310×97), look up its properties from AISC tables — boards provide these in the problem appendix.
• If Lb is not mentioned in a board problem and the section is compact, assume full bracing (Lb ≤ Lp) → use Mn = Mp.
• Always state units clearly: N·mm vs kN·m, mm vs m — unit errors are the #1 arithmetic pitfall.

Key Points

• Steel beam design follows a logical sequence: classify section → determine governing limit state → compute Mn → apply φb → check Mu ≤ φbMn.
• Shear is checked separately: compute φvVn → check Vu ≤ φvVn.
• Deflection checks (serviceability) use unfactored loads and elastic section modulus Ix — separate from strength checks.
• In Philippine practice (NSCP 2015 Section 502), AISC 360 provisions are adopted by reference.
• Beam selection from tables uses φbMpx values directly — verify Lb ≤ Lp and then pick the lightest section satisfying φbMpx ≥ Mu.
• Always specify: Fy, E, section properties (Zx, Sx, ry, d, tw, bf, tf), Lb, and loading pattern before starting calculations.

Definitions

Section Title

5. Design Summary and Step-by-Step Procedure

Common Mistakes

• Skipping the compactness classification and LTB check — assuming all beams achieve Mp without verification.
• Checking flexure but forgetting the shear check, or vice versa — both must be satisfied.
• Using factored loads for deflection calculations — deflection uses service (unfactored) loads.
• Selecting a section from tables without verifying Lb ≤ Lp — table φbMp values assume full bracing.