Steel Beams: Flexure and Shear - Cheat Sheet
Your last-minute revision companion for AISC 360 steel beam design. Focus on plastic moment, lateral-torsional buckling limits, and shear capacity — exam staples.
Plastic Moment Capacity (Compact, Fully Braced)
Formulas
M_p = F_y × Z_x
Meaning
M_p = plastic moment (N·mm); F_y = yield strength (MPa); Z_x = plastic section modulus (mm³)
Watch out
Use PLASTIC Z_x, NOT elastic S_x. Z_x ≈ 1.12 × S_x for I-shapes. Forgetting this is the #1 error.
When to use
When beam is compact AND compression flange continuously braced or L_b ≤ L_p
φ_b × M_n = 0.90 × M_p
Meaning
φ_b = 0.90 (LRFD resistance factor); M_n = nominal flexural strength
Watch out
Do NOT use φ_b for ASD. ASD allowable = M_p / Ω_b = M_p / 1.67
When to use
LRFD design method (RA 544 / NSCP 2015 default); ASD uses Ω_b = 1.67
Common values
Value
≈ 1.10–1.15
Symbol
Z_x / S_x
Quantity
Plastic shape factor (I-shape)
Value
0.90
Symbol
φ_b
Quantity
LRFD resistance factor (flexure)
Value
1.67
Symbol
Ω_b
Quantity
ASD safety factor (flexure)
Important facts
- Plastic moment M_p is the MAXIMUM flexural strength for a steel section (ignoring buckling).
- Full M_p available ONLY if: (1) section is compact, AND (2) compression flange is braced sufficiently (L_b ≤ L_p).
- Brace point = support location preventing lateral displacement of compression flange.
- Z_x tables in section properties include plastic modulus; do NOT calculate from S_x unless given.
- Once L_b > L_p, lateral-torsional buckling reduces M_n below M_p — see LTB section.
Key definitions
Compact Section
Example
Most rolled I-beams (e.g., IPE, HEB in Philippine stock) are compact under typical F_y.
Section where flange and web width-to-thickness ratios ≤ λ_p; can develop full plastic moment without local buckling.
Plastic Section Modulus (Z_x)
Example
For IPE 300: Z_x ≈ 557 cm³ vs. S_x ≈ 557 cm³ (shape factor ~1.12)
Second moment about neutral axis divided by distance to extreme fiber; represents bending capacity at full plasticity.
Diagrams to know
- Moment-curvature diagram: linear elastic → nonlinear plastic → local buckling drop.
- Stress distribution at section: elastic (linear), then plastic (rectangular ±F_y).
Lateral-Torsional Buckling (LTB) & Unbraced Length Limits
Formulas
L_p = 1.76 × r_y × √(E / F_y)
Meaning
L_p = limiting unbraced length for full plastic moment (mm); r_y = radius of gyration about weak axis (mm); E = modulus of elasticity (MPa); F_y = yield strength (MPa)
Watch out
This is r_y (WEAK axis), not r_x. Common confusion: r_y < r_x for I-shapes.
When to use
Calculate maximum brace spacing to develop M_p without LTB. If L_b ≤ L_p, no LTB reduction.
L_r = 1.95 × r_y × √(E / (0.7 × F_y)) × √(J × c / (S_x × h_o))
Meaning
L_r = limiting unbraced length separating inelastic from elastic LTB; J = torsional constant; c, h_o = section constants
Watch out
L_r is COMPLEX; AISC design aids provide pre-calculated L_r. Do NOT derive from first principles in exam.
When to use
For L_p < L_b ≤ L_r: inelastic LTB (M_n varies); L_b > L_r: elastic LTB formula applies.
M_n = M_p − (M_p − 0.7 × F_y × S_x) × (L_b − L_p) / (L_r − L_p) [for L_p < L_b ≤ L_r]
Meaning
Linear interpolation between plastic (L_p) and inelastic LTB limits.
Watch out
Denominator is (L_r − L_p), not just L_b. Verify L_b is between L_p and L_r before using this formula.
When to use
Inelastic LTB range; moment decreases linearly from M_p to 0.7F_y S_x.
M_n = F_cr × S_x [for L_b > L_r]
Meaning
F_cr = elastic LTB critical stress (MPa); S_x = elastic section modulus (mm³)
Watch out
F_cr requires complex calculation (AISC Equation F4-4 or aids). For exams, use provided L_r and inelastic formula if L_b in that range.
When to use
Elastic LTB range (long unbraced spans); M_n → 0 as L_b increases.
C_b × (M_p − M_n) ≤ M_p [moment-gradient modification]
Meaning
C_b = moment-gradient factor (1.0 ≤ C_b ≤ 2.3 typically); increases M_n for non-uniform loading.
Watch out
C_b CAPS M_n at M_p (cannot exceed plastic). For uniform load, C_b ≈ 1.0; for end moments, C_b > 1.0.
When to use
When bending moment varies along span (e.g., cantilever, partial loading). C_b = 1.0 for conservative estimate.
Common values
Value
200,000 MPa
Symbol
E
Quantity
E (steel modulus of elasticity)
Value
248 MPa
Symbol
F_y
Quantity
F_y (Philippine Grade 250 steel)
Value
345 MPa
Symbol
F_y
Quantity
F_y (Philippine Grade 350 steel)
Value
≈ 2.0–2.5 m
Symbol
L_p
Quantity
Typical L_p (IPE 300, F_y=248)
Important facts
- Three LTB regimes: L_b ≤ L_p (no LTB), L_p < L_b ≤ L_r (inelastic), L_b > L_r (elastic).
- Inelastic LTB occurs when residual stresses and partial plasticity influence buckling.
- Elastic LTB follows ELASTIC theory; assume section remains elastic (σ < F_y on average).
- L_p depends on E, F_y, and r_y only; independent of loading or span.
- For most Philippine rolled I-beams (F_y = 248 MPa), L_p ≈ 1.5–2.5 m typically.
- Bracing at L_b ≤ L_p guarantees M_n = M_p (no reduction calculations needed).
- C_b typically found using AISC design tables or calculated per AISC Appendix 1.
- Lateral bracing includes: floor slabs, roof diaphragms, cross-braces, moment connections.
Key definitions
Lateral-Torsional Buckling (LTB)
Example
Long, unbraced beam under pure bending: top flange buckles sideways before material yields.
Instability mode where compression flange moves laterally and section twists, reducing bending capacity below M_p.
Unbraced Length (L_b)
Example
Roof beam spanning 6 m with purlins at 2 m intervals: L_b = 2 m (not 6 m).
Distance between consecutive points of lateral support (e.g., between floor diaphragms, braces, columns).
Moment-Gradient Factor (C_b)
Example
Cantilever beam (moment larger near support): C_b > 1.0, so M_n is higher than uniform load case.
Factor that increases LTB strength when bending moment is non-uniform along span.
Diagrams to know
- M_n vs. L_b diagram: flat at M_p for L_b ≤ L_p, linear drop to 0.7F_y S_x over [L_p, L_r], then hyperbolic descent for L_b > L_r.
- Cross-section showing compression flange buckling mode (lateral + torsional).
Compactness & Local Buckling Limits
Formulas
Compact if: λ_f ≤ λ_pf AND λ_w ≤ λ_pw
Meaning
λ_f = b_f / (2t_f) flange width-to-thickness; λ_w = h / t_w web height-to-thickness; λ_p values in AISC Table B4.1
Watch out
Use b_f / (2t_f), not b_f / t_f. Web uses h (clear height between flanges), not full depth d.
When to use
Check if section can develop M_p (compact) or is non-compact / slender.
λ_pf = 0.38 × √(E / F_y) [flange compact limit]
Meaning
Threshold for flange compactness; if actual λ_f exceeds this, flange can buckle locally.
Watch out
This is the PLASTIC limit. Non-compact and slender limits are higher (different coefficients).
When to use
Verify flange does not buckle before full plasticity.
λ_pw = 3.76 × √(E / F_y) [web compact limit]
Meaning
Threshold for web compactness; prevents web buckling within plastic hinge region.
Watch out
Web compact limit is much higher than flange; webs rarely buckle before flanges for rolled I-shapes.
When to use
Ensure web can reach full plasticity without local buckling.
Common values
Value
0.38
Symbol
C_pf
Quantity
Flange compact limit coefficient
Value
3.76
Symbol
C_pw
Quantity
Web compact limit coefficient
Important facts
- Most Philippine rolled I-beams (IPE, HEB, HEM) are COMPACT at F_y = 248 or 345 MPa.
- Compactness is a SECTION property, independent of loading or unbraced length.
- Non-compact sections cannot reach M_p; maximum strength = M_y = F_y × S_x or lower (if local buckling limit is tighter).
- Slender sections require plate-buckling analysis; not common in design exams.
- AISC tables classify sections as compact/non-compact; do NOT assume without checking.
Key definitions
Compact Section
Example
IPE 300 with F_y = 248: typically compact (pre-verified in section tables).
All plate elements have width-to-thickness ratios ≤ compact limits; section can develop M_p.
Non-Compact Section
Example
Built-up beam with thin flange or deep thin web.
At least one element (flange or web) exceeds compact limit but remains below slender limit; local buckling limits M_n.
Slender Section
Example
Thin-walled custom shapes; rare in standard rolled stock.
Elements exceed slender limits; very low bending strength; treated as stiffened or unstiffened plates.
Diagrams to know
- Width-to-thickness ratio chart: compact ↔ non-compact ↔ slender boundaries.
Shear Strength (Web Shear)
Formulas
V_n = 0.6 × F_y × A_w × C_v
Meaning
V_n = nominal shear strength (N); F_y = yield strength (MPa); A_w = web area (mm²); C_v = web shear buckling coefficient
Watch out
A_w = d × t_w (FULL depth × web thickness), NOT h (clear web height). This is exam mistake #2.
When to use
Always; primary shear failure mode for rolled I-beams.
A_w = d × t_w
Meaning
d = overall beam depth (mm); t_w = web thickness (mm)
Watch out
Do NOT use h (clear web height). Use full d to match AISC 360 definition.
When to use
Calculate web area for shear formula.
C_v = 1.0 [if h/t_w ≤ 2.24 × √(E/F_y)]
Meaning
Stocky web (no shear buckling); full shear strength available.
Watch out
Check h/t_w ratio; if it exceeds limit, C_v < 1.0 and V_n reduces. For exams, problem usually specifies C_v or state 'stocky'.
When to use
Most rolled I-beams fall here; C_v = 1.0 simplifies V_n calculation.
φ_v × V_n = 1.0 × V_n [rolled I-beams, compact web]
Meaning
φ_v = 1.0 (LRFD resistance factor for shear, typical); ASD uses Ω_v = 1.5.
Watch out
Some non-compact or slender webs have φ_v = 0.90; check section properties table.
When to use
LRFD design; convert to ASD as V_n / 1.5 if needed.
Common values
Value
1.0
Symbol
C_v
Quantity
Shear coefficient (stocky web)
Value
1.0
Symbol
φ_v
Quantity
LRFD resistance factor (shear)
Value
1.5
Symbol
Ω_v
Quantity
ASD safety factor (shear)
Value
≈ 40–60
Symbol
h/t_w
Quantity
Typical h/t_w for rolled I-beam
Important facts
- Shear strength rarely controls I-beam design; V_n typically >> applied shear (V).
- V_n = 0.6 F_y A_w C_v is the web shear yield / buckling criterion.
- For rolled I-beams, C_v = 1.0 (stocky web) is standard; assume unless told otherwise.
- φ_v = 1.0 for typical rolled sections; check design table if φ_v ≠ 1.0.
- Shear rarely governs; flexure (LTB, local buckling) is the critical limit state.
- Interaction (flexure + shear): check if V > 0.5 V_n; if yes, slight capacity reduction in flexure.
Key definitions
Shear Buckling
Example
Very tall, thin web (h/t_w > 2.24√(E/F_y)) buckles diagonally before yielding.
Web instability under high shear stress; occurs when h/t_w is too large and C_v < 1.0.
Web Area (A_w)
Example
IPE 300: d = 300 mm, t_w ≈ 7.1 mm → A_w ≈ 2130 mm².
Cross-sectional area of web = d × t_w; used in shear formula per AISC 360-16.
Diagrams to know
- Diagonal shear buckling pattern on web (45° tension field).
- V_n vs. h/t_w graph: flat at 0.6 F_y A_w for stocky, drops as web slenderness increases.
Combined Flexure + Shear Interaction
Formulas
If V ≤ 0.5 V_n: No shear interaction; use M_n as calculated.
Meaning
Low shear does not reduce flexural capacity.
Watch out
If V > 0.5 V_n, flexural strength REDUCES per AISC 360 Sec. H1.
When to use
Check shear first; if V ≤ 0.5 V_n, ignore shear-flexure coupling.
If V > 0.5 V_n: M_n, web reduced = M_n × [1 − ((2×V / V_n) − 1)²]
Meaning
Shear forces cause vertical web distortion, reducing moment capacity.
Watch out
This is AISC H1.2; rarely tested in exams unless problem explicitly combines high V + high M.
When to use
High shear (V > 0.5 V_n) reduces flexural strength; rare in typical building design.
Important facts
- In most building frames, V ≤ 0.5 V_n; interaction effect is negligible.
- Interaction only matters for short spans with very high loads.
- For typical exam problems, assume V ≤ 0.5 V_n unless explicitly stated otherwise.
- Check both M and V separately first; verify shear does not exceed 0.5 V_n before worrying about interaction.
Key definitions
Shear-Flexure Interaction
Example
Short, heavily loaded cantilever: both M and V large → flexural strength must be reduced.
Reduction in flexural capacity when shear force is very large (V > 0.5 V_n).
Design Procedure Summary
Important facts
- STEP 1: Check SECTION COMPACTNESS. If not compact, reduce M_n for local buckling.
- STEP 2: Calculate L_p. If L_b ≤ L_p → M_n = M_p (use plastic Z_x).
- STEP 3: If L_b > L_p, find L_r and determine LTB regime (inelastic or elastic).
- STEP 4: Apply moment-gradient factor C_b if moment is non-uniform.
- STEP 5: Calculate design strength φ_b M_n (LRFD) or M_n / Ω_b (ASD).
- STEP 6: Check SHEAR. If V > applied load, shear is OK. If V ≤ 0.5 V_n, no interaction.
- STEP 7: Compare (M_u / φ_b M_n) and (V_u / φ_v V_n). Both ratios ≤ 1.0 for adequacy.
Worked Exam Examples
Important facts
- EXAMPLE 1 — Plastic Moment: Z_x = 1.2 × 10⁶ mm³, F_y = 248 MPa → M_p = 248 × 1.2 × 10⁶ = 297.6 kN·m; φ_b M_n = 0.90 × 297.6 = 267.8 kN·m.
- EXAMPLE 2 — L_p Calculation: r_y = 40 mm, F_y = 248, E = 200,000 → L_p = 1.76 × 40 × √(200,000/248) = 1.76 × 40 × 28.4 = 2000 mm = 2.0 m.
- EXAMPLE 3 — Shear Strength: d = 450 mm, t_w = 10 mm, F_y = 248, C_v = 1.0 → A_w = 4500 mm²; V_n = 0.6 × 248 × 4500 × 1.0 = 669.6 kN; φ_v V_n = 669.6 kN.
- EXAMPLE 4 — LTB Inelastic: L_b = 3.5 m, L_p = 2.0 m, L_r = 5.5 m → Use linear formula: M_n = M_p − (M_p − 0.7F_y S_x) × (3.5−2.0)/(5.5−2.0).
- EXAMPLE 5 — Design Check: M_u = 150 kN·m, φ_b M_n = 267.8 kN·m → ratio = 150/267.8 = 0.56 < 1.0 ✓. V_u = 200 kN, φ_v V_n = 669.6 kN → ratio = 200/669.6 = 0.30 < 1.0 ✓.
Must remember
- 1. Use PLASTIC modulus Z_x for M_p, NOT elastic S_x. Forgetting this is the most common exam error.
- 2. L_p = 1.76 r_y √(E/F_y) is the MAXIMUM unbraced length for full plastic moment. If L_b > L_p, lateral-torsional buckling reduces M_n.
- 3. A_w = d × t_w (FULL depth × web thickness) for shear formula. Using h (clear height) is a major mistake.
- 4. Compact sections can reach M_p IF (1) compactness λ ≤ λ_p AND (2) unbraced length L_b ≤ L_p.
- 5. Three LTB regimes: L_b ≤ L_p (M_n = M_p), L_p < L_b ≤ L_r (linear drop), L_b > L_r (elastic formula).
- 6. φ_b = 0.90 (LRFD flexure), φ_v = 1.0 (LRFD shear typical); ASD uses Ω_b = 1.67, Ω_v = 1.5.
- 7. Moment-gradient factor C_b > 1.0 for non-uniform loading (cantilever, partial loads); increases M_n but caps at M_p.
- 8. Shear buckling: if h/t_w > 2.24√(E/F_y), then C_v < 1.0 and V_n reduces. Most rolled I-beams: C_v = 1.0.
- 9. Shear-flexure interaction: if V ≤ 0.5 V_n, no reduction in M_n. If V > 0.5 V_n, flexural capacity reduces (rare).
- 10. Design procedure: (1) Check compactness, (2) Calculate L_p and identify LTB regime, (3) Apply C_b, (4) Find M_n, (5) Calculate φ_b M_n, (6) Verify ratio (M_u / φ_b M_n) ≤ 1.0 and (V_u / φ_v V_n) ≤ 1.0.
Last minute tips
- TIP 1 — Section Tables Are Your Friend: AISC and local Philippine section tables pre-calculate Z_x, r_y, L_p, L_r. Do NOT recalculate unless problem forces it; use the table values directly.
- TIP 2 — Assume 'Stocky' Unless Told Otherwise: Most exam problems on rolled I-beams assume C_v = 1.0 and compact section. If the problem doesn't mention slender or non-compact, treat as compact and use M_p.
- TIP 3 — Shear Is Almost Never Critical: For typical spans (< 15 m), V is small compared to V_n. Calculate it for completeness, but focus your effort on M_n and LTB checks — that's where exams get tricky.
- TIP 4 — Check L_b Early: Before calculating M_n, determine L_b and L_p. If L_b ≤ L_p, skip LTB math entirely and use M_n = M_p. This saves 5 minutes on exam.
- TIP 5 — Double-Check Units: Steel formulas mix mm and MPa. A common error: entering L_b in meters instead of mm in the linear LTB interpolation. Always convert to consistent units (PREFERRED: mm, MPa, N) before plugging into formulas.
Comparison tables
LTB Regimes vs. Design Approach
| Condition | L_b Range | M_n Formula | C_b Applies? | Common for Rolled I-Beams? |
|---|---|---|---|---|
| No LTB | L_b ≤ L_p | M_n = M_p = F_y Z_x | No (already at M_p cap) | YES — typical braced beam |
| Inelastic LTB | L_p < L_b ≤ L_r | Linear interpolation | YES — C_b modifies the moment-gradient term | POSSIBLE — moderate spans |
| Elastic LTB | L_b > L_r | M_n = F_cr × S_x | YES — reduces F_cr | RARE — very long unbraced beams |
Compactness Classification & Bending Capacity
| Classification | λ Limits | Max Moment Limit | Local Buckling? | Example / Note |
|---|---|---|---|---|
| Compact | λ_f ≤ λ_pf AND λ_w ≤ λ_pw | M_n = M_p (or reduced by LTB if L_b > L_p) | No local buckling; reaches full plasticity | Most rolled I-beams (IPE, HEB) |
| Non-Compact | λ_pf < λ_f ≤ λ_r (or similar for web) | M_n < M_p; limited by local buckling | Flanges or web buckle before full plasticity | Custom built-up beams, thin-walled tubes |
| Slender | λ > λ_r | M_n << M_p; very low capacity | Severe local buckling governs | Rare in practice; requires stiffening or plate design |
LRFD vs. ASD Design Factors
| Parameter | LRFD (φ factor) | ASD (Ω factor) | Formula | Use Case |
|---|---|---|---|---|
| Flexure | φ_b = 0.90 | Ω_b = 1.67 | M_d = φ_b M_n (LRFD); M_a = M_n / Ω_b (ASD) | Primary limit state |
| Shear | φ_v = 1.0 (typical) | Ω_v = 1.5 | V_d = φ_v V_n; V_a = V_n / Ω_v | Secondary; rarely governs |
| NSCP 2015 Default | LRFD (preferred) | ASD (conservative alternative) | Use LRFD for new designs | RA 544 aligns with AISC LRFD |
Critical Formulas at a Glance
| Concept | Formula | When to Use | Key Variable |
|---|---|---|---|
| Plastic moment | M_p = F_y Z_x | Compact, braced (L_b ≤ L_p) | Z_x (plastic modulus) |
| Unbraced limit | L_p = 1.76 r_y √(E/F_y) | Determine if LTB applies | r_y (weak-axis radius) |
| Inelastic LTB | M_n = M_p − (M_p − 0.7 F_y S_x) (L_b−L_p)/(L_r−L_p) | When L_p < L_b ≤ L_r | L_b, L_p, L_r |
| Shear strength | V_n = 0.6 F_y A_w C_v | Always check; rarely controls | A_w = d t_w (web area) |
Diagrams
Steel Beam Design Flowchart — Flexure Strength Determination
Decision tree for determining nominal flexural strength M_n. Starts with compactness check, then evaluates unbraced length L_b against L_p and L_r to identify which LTB regime applies (none, inelastic, or elastic). Shows where moment-gradient factor C_b enters and final design strength calculation.
Steel Beam Flexure & Shear — Complete Concept Map
Hierarchical overview of all major topics in steel beam flexure and shear design. Shows the primary branches (Flexure M_n, Shear V_n, Design Checks) and sub-topics under each. Useful for rapid mental organization before exam.
Shear Strength Calculation & Interaction Check
Step-by-step procedure for calculating nominal shear strength V_n and checking for shear-flexure interaction. Identifies web slenderness classification and applies appropriate C_v. Includes final interaction check at 0.5 V_n threshold.