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Cheatsheet: Steel Design
1. Design Philosophy and Methods
1.1 LRFD vs ASD
| Method | Design Equation |
|---|---|
| LRFD (Load and Resistance Factor Design) | φRn ≥ Σγi Qi (φ < 1.0,="" γi=""> 1.0) |
| ASD (Allowable Stress Design) | Rn/Ω ≥ Σ Qi (Ω > 1.0, safety factor) |
1.2 Load Combinations
1.2.1 LRFD Load Combinations (ASCE 7)
- 1.4D
- 1.2D + 1.6L + 0.5(Lr or S or R)
- 1.2D + 1.6(Lr or S or R) + (L or 0.5W)
- 1.2D + 1.0W + L + 0.5(Lr or S or R)
- 1.2D + 1.0E + L + 0.2S
- 0.9D + 1.0W
- 0.9D + 1.0E
1.2.2 ASD Load Combinations (ASCE 7)
- D
- D + L
- D + (Lr or S or R)
- D + 0.75L + 0.75(Lr or S or R)
- D + (0.6W or 0.7E)
- D + 0.75(0.6W) + 0.75L + 0.75(Lr or S or R)
- 0.6D + 0.6W
- 0.6D + 0.7E
2. Material Properties
2.1 Steel Grades and Properties
| Steel Grade | Fy (ksi) / Fu (ksi) |
|---|---|
| A36 | 36 / 58 |
| A572 Grade 50 | 50 / 65 |
| A992 | 50 / 65 |
| A500 Grade B (rectangular HSS) | 46 / 58 |
| A500 Grade C (round HSS) | 46 / 62 |
| A53 Grade B (pipe) | 35 / 60 |
2.2 Modulus and Constants
- Modulus of Elasticity: E = 29,000 ksi
- Shear Modulus: G = 11,200 ksi
- Poisson's Ratio: ν = 0.3
- Coefficient of Thermal Expansion: α = 6.5 × 10⁻⁶ /°F
3. Tension Members
3.1 Tensile Strength
| Limit State | Nominal Strength |
|---|---|
| Yielding of gross section | Pn = Fy Ag; φ = 0.90 (LRFD), Ω = 1.67 (ASD) |
| Rupture of net section | Pn = Fu Ae; φ = 0.75 (LRFD), Ω = 2.00 (ASD) |
| Block shear rupture | Rn = 0.6Fu Anv + Ubs Fu Ant ≤ 0.6Fy Agv + Ubs Fu Ant; φ = 0.75, Ω = 2.00 |
3.2 Effective Net Area
- Ae = An U, where U = shear lag factor
- An = Ag - Σ(hole diameter × thickness)
- Bolt hole diameter = db + 1/16" (standard holes)
- Chain of holes: s²/(4g) method for staggered holes
3.3 Shear Lag Factor U
| Connection Type | U Value |
|---|---|
| All elements connected | U = 1.0 |
| Bolts/rivets, x̄ ≤ bf/2 | U = 1 - x̄/L |
| Welds, L ≥ w | U = 1.0 |
| Welds, w ≥ 2L | U = 0.75 |
| Single angle, 4+ bolts | U = 0.80 |
| Single angle, 3 bolts | U = 0.60 |
x̄ = distance from centroid to plane of connection; L = connection length; w = plate width; bf = flange width
3.4 Block Shear
- Ubs = 1.0 when tension stress uniform
- Ubs = 0.5 when tension stress nonuniform
- Agv = gross area in shear; Anv = net area in shear
- Ant = net area in tension; Agt = gross area in tension
4. Compression Members
4.1 Column Buckling
| Parameter | Equation |
|---|---|
| Flexural buckling stress (Fe > 0.44Fy) | Fcr = [0.658^(Fy/Fe)] Fy |
| Flexural buckling stress (Fe ≤ 0.44Fy) | Fcr = 0.877Fe |
| Elastic buckling stress | Fe = π²E / (KL/r)² |
| Nominal strength | Pn = Fcr Ag; φ = 0.90 (LRFD), Ω = 1.67 (ASD) |
4.2 Effective Length Factor K
| End Conditions | K (Theoretical) |
|---|---|
| Pinned-Pinned | 1.0 |
| Fixed-Fixed | 0.5 |
| Fixed-Pinned | 0.7 |
| Fixed-Free | 2.0 |
| Sidesway inhibited frame | 0.5 ≤ K ≤ 1.0 |
| Sidesway uninhibited frame | K ≥ 1.0 |
4.3 Local Buckling - Width-Thickness Ratios
| Element Type | λ |
|---|---|
| Flanges: unstiffened, Fy = 50 ksi | λp = 0.56√(E/Fy) = 13.5; λr = 1.03√(E/Fy) = 24.8 |
| Webs: stiffened, Fy = 50 ksi | λp = 1.49√(E/Fy) = 35.9 |
| Round HSS: D/t | λp = 0.07E/Fy = 40.6 (Fy = 50 ksi) |
| Rectangular HSS: b/t | λp = 1.12√(E/Fy) = 27.0 (Fy = 50 ksi) |
4.4 Built-Up Members
- Modified slenderness: (KL/r)m = √[(KL/r)² + 0.82(a/ri)²(h/ri)²]
- a = spacing between connectors; ri = minimum radius of gyration of individual component
- h = distance between centroids of individual components
- Snug-tight connectors: a ≤ 0.40 times least radius of gyration of built-up member
5. Flexural Members
5.1 Bending Strength
| Limit State | Condition |
|---|---|
| Yielding | Mn = Mp = Fy Zx ≤ 1.6Fy Sx; φ = 0.90, Ω = 1.67 |
| Lateral-Torsional Buckling (Lb ≤ Lp) | Mn = Mp; φ = 0.90, Ω = 1.67 |
| LTB (Lp < lb="" ≤=""> | Mn = Cb[Mp - (Mp - 0.7Fy Sx)(Lb - Lp)/(Lr - Lp)] ≤ Mp |
| LTB (Lb > Lr) | Mn = Fcr Sx ≤ Mp; Fcr = Cb π²E / (Lb/rts)² √[1 + 0.078(Jc/Sxho)(Lb/rts)²] |
5.2 Unbraced Lengths
| Parameter | Equation |
|---|---|
| Lp | 1.76ry √(E/Fy) |
| Lr | 1.95rts (E/0.7Fy) √[(Jc/Sxho)√(1 + √(1 + 6.76(0.7Fy Sxho/E Jc)²))] |
- For doubly symmetric I-shapes: rts² ≈ √(Iy Cw) / Sx
- For compact doubly symmetric I-shapes: Lr ≈ πry √(E/0.7Fy)
5.3 Moment Gradient Factor Cb
- Cb = 12.5Mmax / (2.5Mmax + 3MA + 4MB + 3MC)
- Mmax = absolute maximum moment; MA, MB, MC = moments at 1/4, 1/2, 3/4 points
- Uniform moment: Cb = 1.0
- End moments only: Cb ≤ 1.0 (use formula)
- Conservative: Cb = 1.0
5.4 Shear Strength
| Web Slenderness | Nominal Shear Strength |
|---|---|
| h/tw ≤ 2.24√(E/Fy) | Vn = 0.6Fy Aw Cv; Cv = 1.0 |
| 2.24√(E/Fy) < h/tw="" ≤=""> | Vn = 0.6Fy Aw Cv; Cv = 2.24√(E/Fy) / (h/tw) |
| h/tw > 2.45√(E/Fy) | Vn = 0.6Fy Aw Cv; Cv = 2.45E / [Fy(h/tw)²] |
- Aw = d tw for I-shapes; φ = 0.90 (LRFD), Ω = 1.67 (ASD)
- h = clear distance between flanges for rolled shapes
5.5 Deflection Limits
| Type | Limit |
|---|---|
| Roof beams (live load) | L/240 |
| Floor beams (live load) | L/360 |
| Supporting plaster (total load) | L/360 |
5.6 Compact Section Criteria
Flange: bf/2tf ≤ λpf = 0.38√(E/Fy); Web: h/tw ≤ λpw = 3.76√(E/Fy)
6. Combined Loading
6.1 Interaction Equations
6.1.1 Axial Compression and Flexure
| Condition | Equation |
|---|---|
| Pr/Pc ≥ 0.2 | Pr/Pc + 8/9(Mrx/Mcx + Mry/Mcy) ≤ 1.0 |
| Pr/Pc <> | Pr/(2Pc) + (Mrx/Mcx + Mry/Mcy) ≤ 1.0 |
- Pr = required axial strength; Pc = design axial strength (φPn or Pn/Ω)
- Mr = required flexural strength; Mc = design flexural strength (φMn or Mn/Ω)
6.1.2 Axial Tension and Flexure
- Pr/Pt + (Mrx/Mcx + Mry/Mcy) ≤ 1.0
- Pt = design tensile strength
6.2 Second-Order Effects
| Effect | Amplification Factor |
|---|---|
| P-δ (member) | B1 = Cm / (1 - α Pr/Pe1) ≥ 1.0 |
| P-Δ (story) | B2 = 1 / (1 - α ΣPr/ΣPe2) ≥ 1.0 |
- α = 1.0 (LRFD), 1.6 (ASD)
- Pe1 = π²EI / (K1L)²; Pe2 = RM ΣHL / Δ
- Cm = 0.6 - 0.4(M1/M2) for end moments; Cm = 1.0 for transverse loads
- RM = 1 - 0.15(ΣPmf/ΣPstory)
7. Connections
7.1 Bolt Strength
| Bolt Type | Fnv (ksi) |
|---|---|
| A325 (shear, threads included) | 54 |
| A325 (shear, threads excluded) | 68 |
| A490 (shear, threads included) | 68 |
| A490 (shear, threads excluded) | 84 |
| Nominal Strength | Equation |
|---|---|
| Shear | rn = Fnv Ab; φ = 0.75, Ω = 2.00 |
| Bearing (deformation consideration) | rn = 1.2Lc t Fu ≤ 2.4d t Fu; φ = 0.75, Ω = 2.00 |
| Bearing (no deformation limit) | rn = 1.5Lc t Fu ≤ 3.0d t Fu; φ = 0.75, Ω = 2.00 |
| Tension | rn = Fnt Ab; φ = 0.75, Ω = 2.00 |
- Ab = nominal bolt area (gross area); Lc = clear distance to edge or next bolt
- Fnt (A325) = 90 ksi; Fnt (A490) = 113 ksi
- Combined shear and tension: (frv/φrn)² + (frt/φrn)² ≤ 1.0
7.2 Bolt Spacing and Edge Distance
| Parameter | Minimum |
|---|---|
| Spacing | 2.67d (preferred 3d) |
| Edge distance (sheared edge) | 1.25d to 1.75d (varies by bolt size) |
| Edge distance (rolled edge) | 1.0d to 1.25d (varies by bolt size) |
7.3 Weld Strength
| Electrode | FEXX (ksi) |
|---|---|
| E60 | 60 |
| E70 | 70 |
| E80 | 80 |
| Weld Type | Nominal Strength |
|---|---|
| Fillet weld (shear) | Fnw = 0.60FEXX(1.0 + 0.50sin¹·⁵θ); φ = 0.75, Ω = 2.00 |
| Base metal shear | Fnw = 0.60Fu; φ = 0.75, Ω = 2.00 |
| Fillet weld strength | Rn = Fnw Awe; Awe = effective throat area = 0.707w × L |
- w = fillet weld leg size; L = weld length
- θ = angle of loading measured from weld axis
- Minimum fillet weld size depends on base metal thickness
- Maximum fillet weld size = t - 1/16" (for t ≥ 1/4")
7.4 Weld Size Requirements
| Base Metal Thickness (in) | Minimum Fillet Weld Size (in) |
|---|---|
| t ≤ 1/4 | 1/8 |
| 1/4 < t="" ≤=""> | 3/16 |
| 1/2 < t="" ≤=""> | 1/4 |
| t > 3/4 | 5/16 |
7.5 Eccentrically Loaded Connections
- Bolts: use elastic method or instantaneous center of rotation method
- Welds: treat as line elements; resultant force = √[(ΣFv)² + (ΣFh)² + (ΣFtorsion)²]
- Instantaneous center location C determined from table coefficients
8. Plate Girders
8.1 Web Design
| Parameter | Value/Equation |
|---|---|
| Maximum h/tw (without stiffeners) | 260 |
| Transverse stiffener spacing | a ≤ 3h (h/tw > 2.46√(E/Fy)) |
| Tension field action | Available when a/h ≤ 3.0 and h/tw > 2.46√(E/Fy) |
8.2 Flexural Strength
- Same provisions as rolled shapes with slender web considerations
- Compression flange must be checked for local buckling
- Reduction factor Rpg applied when h/tw > λpw
8.3 Stiffener Requirements
| Stiffener Type | Requirement |
|---|---|
| Transverse (area) | Ast ≥ [0.15Dhtw(1 - Cv)√(Fy/Fyst)] - 18tw² |
| Bearing (thickness) | tbearing ≥ (bf/3)√(Fy/E) |
| Moment of inertia | Ist ≥ (h/50)⁴ (for single stiffener) |
9. Connection Design Details
9.1 Simple Shear Connections
| Connection Type | Typical Use |
|---|---|
| Single-plate (shear tab) | Beam-to-column or beam-to-girder, assumes simple support |
| Double-angle | Beam-to-column or beam-to-girder, flexible connection |
| Seated connection | For beams with coped top flanges |
| Single-angle | Light loads, simple framing |
9.2 Moment Connections
- Fully restrained (FR): resists full moment capacity
- Partially restrained (PR): develops partial moment resistance
- Typical FR connections: welded flange plates, bolted end plates, welded unreinforced flanges
9.3 Beam Copes and Blocks
| Limit State | Check Required |
|---|---|
| Local buckling at cope | c/ho ≤ 1.0; ho/2tf ≤ λp |
| Flexural strength (cope) | Mn = Fy Snet; reduced section modulus at cope |
| Shear yielding | Rn = 0.60Fy Agv |
| Block shear | Check per tension member provisions |
- c = cope length; ho = depth of remaining section
10. Composite Construction
10.1 Composite Beam Strength
| Parameter | Equation |
|---|---|
| Effective slab width | Lesser of: L/4, beam spacing, overhang + 8t |
| Compressive force (concrete) | C = 0.85f'c a b ≤ As Fy |
| Depth of compression block | a = As Fy / (0.85f'c b) |
| Nominal moment (PNA in slab) | Mn = As Fy (d/2 + t - a/2) |
10.2 Shear Studs
| Parameter | Equation/Value |
|---|---|
| Nominal strength | Qn = 0.5Asc √(f'c Ec) ≤ Rg Rp Asc Fu |
| Number required | N = V'/Qn (V' = horizontal shear force) |
| Resistance factors | φ = 0.75 (LRFD), Ω = 2.00 (ASD) |
| Group effect Rg | Rg = 1.0 (one stud), 0.85 (two studs), 0.7 (three studs) |
| Position effect Rp | Rp = 0.75 (deck ribs perpendicular), 0.6 (parallel weak) |
- Asc = cross-sectional area of stud; Ec = modulus of concrete = 57√f'c (ksi)
- Minimum stud diameter: 1/2 inch; maximum: 2.5tf
- Stud height: ≥ 1.5d above deck after welding
10.3 Deflection Considerations
- Use unshored construction: steel section alone carries wet concrete + construction loads
- Composite section carries superimposed dead load + live load
- Lower bound moment of inertia: Ieff = ILB (per AISC specifications)
11. Base Plates
11.1 Axial Compression
| Parameter | Equation |
|---|---|
| Bearing strength (concrete) | Pp = 0.85f'c A1 √(A2/A1) ≤ 1.7f'c A1; φ = 0.65, Ω = 2.31 |
| Required plate thickness | tp ≥ l √(2Pu / 0.9Fy BN) |
| Critical dimension | l = max of [m, n, λn'] |
- A1 = base plate area; A2 = supporting concrete area (max 2:1 slope)
- m = (N - 0.95d)/2; n = (B - 0.8bf)/2
- λ = (2√X)/(1 + √(1-X)); X = (4dbf)/(d + bf)²
- n' = √(db f)/4
11.2 Anchor Bolts
| Failure Mode | Strength |
|---|---|
| Steel tension | Nsa = Ase,N futa; φ = 0.75, Ω = 2.0 |
| Concrete breakout (tension) | Ncb = (Anc/Anco) ψec,N ψed,N ψc,N ψcp,N Nb; φ = 0.70, Ω = 2.14 |
| Pullout | Npn = 8Abrg fc'; φ = 0.70, Ω = 2.14 |
| Steel shear | Vsa = Ase,V futa; φ = 0.65 (grout), Ω = 2.31 |
- Ase,N = effective tensile area; Ase,V = effective shear area
- Nb = basic concrete breakout strength = 24√f'c hef^1.5
- hef = effective embedment depth
12. Special Topics
12.1 Fatigue
| Detail Category | Constant Amplitude Threshold (Fth, ksi) |
|---|---|
| A | 24 |
| B | 16 |
| C | 10 |
| D | 7 |
| E | 4.5 |
| E' | 2.6 |
- Stress range: Δf = fmax - fmin
- Check: Δf ≤ (Δf)r (design stress range)
- Loading cycles: N = (365)(years)(n)(ADTT/100)
12.2 Ponding
- Occurs when roof deflection creates depression that collects water
- Primary member flexibility: Cp = 32Ls⁴ / (10⁷ Ip)
- Secondary member flexibility: Cs = 32Ss⁴ / (10⁷ Is)
- Check: 0.25 + 0.9Cp Cs ≤ 1.0
- U = CpCsγw(S/1000); U < 1.0="" prevents="">
12.3 Serviceability
| Consideration | Typical Limit |
|---|---|
| Drift (lateral displacement) | H/400 to H/600 (wind); H/200 (seismic) |
| Vibration frequency (floors) | ≥ 3 Hz to 5 Hz |
| Camber | 80% to 100% of dead load deflection |
12.4 Bracing Requirements
| Bracing Type | Stiffness Requirement |
|---|---|
| Nodal (beams) | βbr = 1.0Pf / φLb (Cb = 1.0); Pf = flange force |
| Relative (beams) | βbr = 4.0Pf Cd / (φh₀) (Cb = 1.0) |
| Columns | βbr = 2ΣPr / φ |
- φ = 0.75 for brace design; Cd = 1.0 (single curvature)
- Required strength of brace: Pbr = 0.004Pf (beams), 0.01Pr (columns)
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