Serviceability & Fatigue

Why Structural Steel?

Deflection (L/360), drift (H/400), floor vibration (DG-11), and fatigue design per AISC 360-22 Appendix 3 — stress ranges, detail categories A–E', and Miner's rule.

⏱ 120 min★ Core✎ 7 objectives

§01Section 01

Engineering story

Chapter 18 · Serviceability & Fatigue

Deflection (L/360), drift (H/400), floor vibration (DG-11), and fatigue design per AISC 360-22 Appendix 3 — stress ranges, detail categories A–E', and Miner's rule.

A real project narrative for this chapter will be authored as this chapter migrates to the v3.0 structured schema.

§02Section 02

Learning objectives

After this chapter you will be able to

Apply deflection limits (L/360 floors, L/240 total, L/480 cladding)
Check story drift (H/400 wind, Δa seismic per ASCE 7 Tbl 12.12-1)
Check floor vibration per AISC Design Guide 11 (fn, ap/g)
Determine when AISC 360-22 Appendix 3 fatigue check applies
Assign detail categories A–E' from Table A-3.1
Apply AISC Eq. A-3-1: F_SR = (Cf·10⁸/N)^(1/3) ≥ F_TH
Combine variable-amplitude spectra with Miner's rule

§03Section 03

Engineering motivation

§04Section 04

Failure mechanisms

Failure mechanisms & lessons learned

Photographs and lessons-learned case studies for this topic will be added during chapter migration.

AISC Reference Box

AISC 360-22 Ch. L — Serviceability limit states — deflection, drift, vibration, durability
AISC 360-22 App. 3 — Design for Fatigue — stress-range approach, Table A-3.1 detail categories
AISC Design Guide 3 — Serviceability Design Considerations for Steel Buildings
AISC Design Guide 11 — Vibrations of Steel-Framed Structural Systems Due to Human Activity
ASCE 7-22 §12.12 — Drift and deformation — seismic story-drift limits Δa
IBC 2021 §1604.3 — Serviceability — deflection Table 1604.3, drift enforcement

Why This Chapter Matters

Two non-collapse limit states that still ruin a structure: serviceability (a strong floor bounces, brittle cladding cracks, a moment frame drifts and jams elevators) and fatigue (repeated cyclic loading cracks a member at stresses far below Fy, with little warning). Both are checked with unfactored service loads and both hinge on detail geometry, not just member size — which is why they cause the majority of steel retrofits worldwide.

Learning Objectives

Apply live-load deflection limits (L/360 floors, L/240 roofs, L/480 cladding) and total-load limits.
Compute wind drift limit H/400 to H/500 per AISC Design Guide 3, and seismic drift Δa per ASCE 7 Table 12.12-1.
Check floor vibration per AISC Design Guide 11 (fn, W, ap/g ≤ 0.005 for offices).
Compute camber (≈ 0.80 · ΔD) for long-span composite beams.
Determine when AISC 360-22 Appendix 3 fatigue check applies: net tensile stress AND N > 20,000 AND SR > FTH.
Assign fatigue detail category (A → B → B' → C → C' → D → E → E') per Table A-3.1 based on geometry and inspection.
Apply the AISC A-3-1 fatigue equation: FSR = (Cf·10⁸/N)^(1/3) ≥ FTH, then check SR ≤ FSR.
Combine variable-amplitude cycles using Miner's rule: SRe = [Σ αi SRi³]^(1/3).
Recommend retrofits: weld-toe grinding, stop-hole drilling, UIT (+2 category), bolted splice.

Where This Chapter Is Used

Every floor beam and every high-rise building for serviceability. Fatigue governs the design of crane runways, highway and railway bridges, communication towers, and support structures for reciprocating machinery. Historical drivers: Silver Bridge collapse (WV, 1967 — eyebar fatigue, 46 dead), Yellow Mill Pond bridge cracks (I-95, 1970s — cover-plate ends), Hoan Bridge web fracture (Milwaukee, 2000). Feeds the capstone drift/vibration checks and any bridge-design elective.

Lecture Notes

Chapter 18 — Serviceability & Fatigue (AISC 360-22 Ch. L & Appendix 3)

Chapter focus. Two limit states that don't cause collapse but still ruin a building: (1) serviceability — the floor bounces, the cladding cracks, the elevator jams — and (2) fatigue — under repeated cyclic loading, a member fails at a stress far below F_y. Both are checked with unfactored (service) loads, both are governed by detail geometry, and both are the number-one reason existing steel structures need retrofits.

Highway bridge plate girder with welded stiffeners and bolted splices

Part A. Serviceability (AISC 360-22 Ch. L)

1. What Is a Serviceability Limit State?

A serviceability limit state is any condition that impairs the intended use of a structure without threatening collapse. AISC 360-22 Ch. L identifies four: (i) deflection, (ii) drift, (iii) vibration, and (iv) durability (corrosion, connection slip). Because occupant comfort — not safety — is at stake, checks use unfactored service loads: D, L, W, or S with load factors of 1.0.

Serviceability ≠ strength. A W-shape that passes every strength limit state in Ch. D–H can still be unusable. On long-span composite floors, live-load deflection controls the member size more often than φM_n.

2. Deflection Limits (Commentary Ch. L, Table CL-1)

AISC does not mandate specific deflection limits — instead the Commentary tabulates industry practice, and IBC §1604.3 makes them enforceable:

Member	Live-load Δ_L	Total Δ_D+L
Floor beam supporting plaster / drywall	L/360	L/240
Floor beam, flexible finish (tile, wood)	L/240	L/180
Roof beam, no ceiling	L/240	L/180
Cladding-support beam (brittle facade)	L/480	L/240
Cantilever, floor	L/180	L/120

For a simply-supported uniform beam the closed-form deflection is:

$Δ_{L} = \frac{5 w _{L} L ^{4}}{384 E I _{x}} (Eq. 18.1)$

Units check: w in kip/in, L in inches, E = 29,000 ksi, I in in⁴ → Δ in inches. The most common student error is mixing kip/ft with L in feet.

3. Story Drift (ASCE 7-22 §12.12 / IBC §1604.3)

Wind: Δ_x/h_sx ≤ H/400 for occupant comfort (10-yr wind); H/500 for masonry cladding.
Seismic: Δ_x/h_sx ≤ Δ_a from ASCE 7 Table 12.12-1, typically 0.020·h_sx for Risk Category II buildings.
Seismic drift is computed on amplified displacements C_d·δ_xe/I_e, not elastic.

4. Floor Vibration — AISC Design Guide 11

Long-span (≥ 30 ft) composite floors and steel-joist floors can pass every deflection check but still bounce disturbingly under normal walking. DG 11 checks two quantities: the fundamental frequency f_n and the peak acceleration ratio a_p/g.

$f_{n} = 0.18 \frac{g}{Δ _{total}} (Eq. 18.2)$

$\frac{a _{p}}{g} = \frac{P _{o} e ^{- 0.35 f_{n}}}{β W} \leq \frac{a _{o}}{g} (Eq. 18.3)$

where g = 386 in/s², P_o = 65 lb (walking excitation), β = 0.03 (open office), W = effective panel weight, and a_o/g = 0.005 (offices/residences), 0.015 (shopping malls), 0.05 (rhythmic activities).

Fig. 18.1 — Serviceability envelope. Deflection (L/360), drift (H/400), and vibration (DG-11) are three independent checks, all under unfactored service loads.

5. Camber

Fabricators can pre-cambering a beam upward to offset dead-load deflection. AISC Manual Part 3 recommends camber = 80 % of Δ_D for spans ≥ 24 ft. Never camber less than ¾ in — thermal drift and roll-tolerance eat it. Composite beams almost always call for camber because wet-concrete Δ_D is large and shouldn't be seen by the occupants.

Part B. Fatigue (AISC 360-22 Appendix 3)

6. What Is Fatigue?

Fatigue is progressive, localized damage accumulated in a metal under repeated cyclic loading, culminating in a crack that propagates until the remaining net section can no longer carry the peak load — at which point the member fails brittlely with little or no warning. Steel members can and do fail in fatigue at nominal stresses far below F_y: a Category E' detail in A992 (F_y = 50 ksi) has a constant-amplitude fatigue threshold of only 2.6 ksi.

Fatigue-fractured welded steel connection showing beach marks radiating from a corner initiation site

7. When Does Appendix 3 Apply?

AISC 360-22 §3.1 requires a fatigue check whenever all three conditions are met:

Cyclic live load produces a net tensile stress at the detail (fatigue is a tension-driven crack phenomenon; pure compression cycles are exempt).
The number of stress cycles N exceeds 20,000 over the service life.
The stress range S_R exceeds the detail's constant-amplitude fatigue limit F_TH (also called the CAFL).

Wind-loaded members on ordinary buildings almost never satisfy (1) & (2) & (3) simultaneously; crane runways, highway bridges, railway bridges, and support structures for reciprocating machinery almost always do.

Stress range, not peak stress. Fatigue is governed by the range S_R = σ_max − σ_min, calculated from the unfactored (service) cyclic loads. Mean stress and dead-load stress have no effect on Appendix 3 (residual welding stress is assumed tensile and yields the mean).

Fig. 18.3 — Constant-amplitude stress cycle. Only the stress range S_R enters the fatigue equation.

8. Stress-Life (S-N) Curves and Detail Categories

AISC Appendix 3 organizes every welded / bolted / mechanically fastened detail into eight Fatigue Detail Categories: A (best) → B → B' → C → C' → D → E → E' (worst). Category is a property of geometry, not material — the same A992 W-shape can house Category B, C, D, and E details, all under the same service load.

AISC 360-22 Appendix 3 S-N design curves for detail categories A, B, C, D, E

The allowable stress range for finite life is:

$F_{S R} = (\frac{C _{f} \times 1 0 ^{8}}{N})^{1/3} \geq F_{T H} (AISC Eq. A-3-1, Eq. 18.4)$

where C_f is the constant-life constant (Table A-3.1, ksi³·cycles/10⁸), N is the projected total number of stress cycles, and F_TH is the constant-amplitude threshold below which the detail has infinite fatigue life. The design requirement is:

$S_{R} \leq F_{S R} (Eq. 18.5)$

9. Detail Category Table — Key Values

Category	C_f (×10⁸)	F_TH (ksi)	Representative detail
A	250	24	Rolled base metal, smooth-machined edges
B	120	16	Full-penetration groove weld, ground flush & NDT-inspected
B'	61	12	Full-penetration groove weld as-welded (reinforcement in place)
C	44	10	Transverse stiffener fillet welds on beam flange
C'	44	12	Reinforcement of groove welds, ends of PJP welds
D	22	7	Ends of welded cover plates thinner than the flange
E	11	4.5	Longitudinal fillet welds > 4 in. long, cover-plate ends on thick flange
E'	3.9	2.6	Cover-plate end welds on flange plate > 0.8 in. thick, no transverse weld

Values shown are the constants used in AISC 360-22 Table A-3.1. Always look up the exact detail description in the Specification — a “fillet weld” alone is not a category; the geometry, load direction, and inspection level together define the class.

Four AISC fatigue detail categories: B (full-pen groove ground), C (transverse stiffener), D (cover plate end), E (longitudinal fillet weld)

10. Crack Initiation & Propagation

Every fatigue failure follows the same four-stage sequence:

Stress concentration — a weld toe, a bolt hole, a re-entrant corner, an undercut. Local peak stress can be 3–6× nominal.
Micro-crack nucleation — cyclic plastic slip forms an intrusion, typically at a discontinuity or slag inclusion within the first few percent of life.
Stable crack growth — the crack advances a few grain diameters per cycle (Paris law da/dN = C·ΔK^m). Beach marks record the progression.
Final overload fracture — the remaining net section reaches its ultimate load; failure is sudden and brittle.

Four-stage diagram of fatigue crack initiation and propagation from weld toe stress concentration to brittle overload fracture

11. Variable-Amplitude Loading (Miner's Rule)

Real bridges do not see one stress range — an interstate girder sees a full spectrum of truck weights. Palmgren-Miner linear damage summation converts a mixed spectrum to an equivalent constant-amplitude stress range S_Re:

$S_{R e} = [i \sum α_{i} (S_{R i})^{3}]^{1/3} (Eq. 18.6)$

where α_i is the fraction of cycles at stress range S_Ri (Σα_i = 1). Design then applies Eq. 18.4 with N = total cycles and S_R = S_Re. The cube exponent means the largest stress ranges dominate — reducing peaks pays outsized dividends.

Miner's Rule in words. Each cycle at stress range S_Ri uses up 1/N_i of the fatigue life, where N_i is the constant-amplitude life at that stress range. Damage sums linearly; when Σ(n_i/N_i) = 1.0, the detail has been "used up."

12. Bolted & Pin-Connected Details (§3.4)

Bolts loaded in tension use a special S-N curve because thread stress-concentration dominates. Appendix 3 §3.4 gives:

$F_{S R}^{bolt} = (\frac{3.9 \times 1 0 ^{8}}{N})^{1/3} \geq 7 ksi (based on tensile stress area) (Eq. 18.7)$

Pretensioned bolts (slip-critical connections) are largely shielded from cyclic tension because the flange separation must exceed the pretension before the bolt sees any additional stress — a strong argument for pretensioned bolts wherever fatigue governs.

13. Design Workflow

Confirm Appendix 3 applies: net tensile stress, N > 20,000, and S_R > F_TH. If any fails, no check required.
Identify every detail in the tensile fiber of the member — the base metal and each weld/bolt/hole around it. Assign a Category (A–E') from Table A-3.1.
Compute S_R at each detail from the unfactored service live-load stress range. For variable-amplitude loading, apply Eq. 18.6.
Compute F_SR from Eq. 18.4 using the projected total N (per-day truck count × 365 × service life is typical).
Check S_R ≤ F_SR. If not, improve the detail (grind to Category B, remove cover plates, add stop-hole retrofit) or resize.
Design for infinite life when possible: set S_R ≤ F_TH and no cycle count matters.

14. Retrofit & Repair Strategies

Peening / grinding the weld toe (moves detail up 1 category, e.g. C → B).
Stop-hole drilled at the crack tip — a 1-in. hole arrests the crack and restarts the initiation clock.
Cover-plate removal and replacement with a bolted splice (D → B).
Post-weld ultrasonic impact treatment (UIT) — modern high-value bridge retrofit; documented +2 category improvement.
External post-tensioning to reduce live-load stress range.

⚠ Common Mistakes (fatigue + serviceability)

Using factored loads for a deflection or fatigue check. Both use SERVICE loads.
Comparing seismic drift to the wind drift limit (H/400) — seismic uses Δ_a from ASCE 7 Table 12.12-1.
Assigning a fatigue category from “fillet weld” alone — orientation, size, inspection, and load direction all matter.
Forgetting that fatigue is driven by stress range — dead-load stress does not enter.
Skipping the vibration check on long-span (≥ 30 ft) composite floors — DG-11 catches many designs that pass L/360.
Detailing a Category E' cover-plate end (0.8 in flange, no transverse weld) — infinite-life S_R is only 2.6 ksi.

15. Chapter Summary

Serviceability uses unfactored loads. Strength uses factored loads.
Deflection: L/360 live / L/240 total (plaster). Drift: H/400 (wind), Δ_a ≈ 0.020h (seismic).
Vibration (DG-11): f_n = 0.18·√(g/Δ), a_p/g ≤ 0.005 for offices.
Fatigue applies when net tension, N > 20,000, and S_R > F_TH.
S_R ≤ F_SR = (C_f·10⁸ / N)^1/3 ≥ F_TH. (AISC Eq. A-3-1)
Fatigue category depends on geometry, not on F_y.
Variable amplitude: Miner's rule with cube exponent → largest stress ranges dominate.

Project case study — Cardinal Square — 4-story braced-frame office

Every chapter's worked example is one step in the design of the same building: Plan: 4 bays N–S × 3 bays E–W, each 30 ft × 30 ft. Stories: 4 @ 13 ft (52 ft roof). Composite floor: 4.5 in NW concrete on 3 VLI20 deck. Roof: 1.5 in B-deck + insulation + membrane. Materials: Wide-flange members A992 (Fy = 50 ksi, Fu = 65 ksi). Plates A572 Gr. 50. HSS bracing A500 Gr. C. Bolts A325-N 7/8 in dia. Welds E70XX. Concrete f'c = 4 ksi. Anchor rods F1554 Gr. 36.

Chapter 18 — Serviceability (deflection)

Same composite filler beam under live load only

Demand carried forward

wL = 0.50 klf. ΔL = 5 wL L⁴ / (384 E Ix-comp).

This chapter contributes

Checks ΔL ≤ L/360 = 30·12/360 = 1.0 in. If not met, re-pick a stiffer section and loop back to Chapter 6.

Serviceability limits: LL Δ ≤ L/360, total Δ ≤ L/240, wind drift Δ ≤ H/400 (AISC Ch. L + DG-11).

Formula Sheet

Name	Equation	AISC Ref
Simply-supported live-load deflection	\Delta_L = \frac{5\, w_L\, L^4}{384\, E\, I_x}	Beam theory / IBC Table 1604.3
Wind drift limit	\Delta_x / h_{sx} \le 1/400	AISC Ch. L Commentary / ASCE 7
Seismic drift limit	\Delta_x / h_{sx} \le \Delta_a	ASCE 7-22 Table 12.12-1
DG-11 fundamental frequency	f_n = 0.18\,\sqrt{g / \Delta_{total}}	AISC DG-11
DG-11 peak acceleration	a_p/g = P_o\, e^{-0.35 f_n} / (\beta\, W) \le a_o/g	AISC DG-11
Allowable fatigue stress range (AISC Eq. A-3-1)	F_{SR} = (C_f \times 10^{8} / N)^{1/3} \ge F_{TH}	AISC 360-22 App. 3.2
Fatigue design check	S_R \le F_{SR}	AISC App. 3.2
Miner's rule — variable amplitude	S_{Re} = [\sum \alpha_i\, S_{Ri}^3]^{1/3}	AISC App. 3.3
Bolt tension fatigue	F_{SR}^{bolt} = (3.9 \times 10^{8} / N)^{1/3} \ge 7\;\text{ksi}	AISC App. 3.4

Worked Example

Chapter 18 — Worked Examples (Serviceability & Fatigue)

How to use. All examples use unfactored (service) stress ranges. Every derivation is written as an aligned KaTeX chain so you can follow it one line at a time. The set covers four contexts:

A. Serviceability — Ex. 18.1 (composite floor deflection + DG-11 vibration).
B. Buildings & Cranes — Ex. 18.2 (crane runway) and Ex. 18.7 (rooftop AHU frame, wind-induced fatigue).
C. Bridges — Ex. 18.3 (cover-plate end, Cat. E) and Ex. 18.4 (variable-amplitude Miner's rule).
D. Machinery & Industrial — Ex. 18.5 (reciprocating compressor bracket) and Ex. 18.6 (vibrating-screen support beam, bolted vs welded).

Section A — Serviceability (Buildings)

Worked Example 18.1 — Composite Floor Beam Deflection & DG-11 Vibration

Given. Composite W18×35 filler beam, L = 30 ft, tributary width s = 10 ft, I_eff = I_LB = 1230 in⁴ (AISC Manual Table 3-20). Service loads w_D = 0.75 klf, w_L = 0.50 klf. Ceiling finish: gypsum board → limit Δ_L ≤ L/360.

Fig. 18.1a — Live-load deflection Δ_L at midspan.

Step 1 — Live-Load Deflection (Eq. 18.1)

Convert to consistent units: w_L = 0.500/12 = 0.04167 k/in, L = 360 in, L⁴ = 1.68 × 10¹⁰ in⁴, E = 29 000 ksi.

X X X Δ_{L} = \frac{5 w _{L} L ^{4}}{384 E I _{eff}}

X X X = \frac{5 ( 0.04167 ) ( 1.68 \times 1 0 ^{10} )}{384 ( 29000 ) ( 1230 )}

X X X = \frac{3.50 \times 1 0 ^{9}}{1.37 \times 1 0 ^{10}}

X X X = 0.259 in

Δ_L = 0.259 in < L/360 = 1.00 in ✓ (26 % utilization).

Step 2 — Total Deflection (Eq. 18.1 scaled)

X X X Δ_{D} = \frac{w _{D}}{w _{L}} Δ_{L}

X X X = \frac{0.75}{0.50} (0.259)

X X X = 0.389 in

X X X Δ_{D + L} = Δ_{D} + Δ_{L}

X X X = 0.389 + 0.259

X X X = 0.648 in < L /240 = 1.50 in

Step 3 — Fundamental Frequency (Eq. 18.2)

X X X Δ_{tot} = Δ_{D} + 0.25 Δ_{L}

X X X = 0.389 + 0.25 (0.259)

X X X = 0.454 in

f_{n} = 0.18 \frac{g}{Δ _{tot}}

X X X = 0.18 \frac{386}{0.454}

X X X = 0.18 (29.15)

X X X = 5.25 Hz

f_n < 9 Hz → DG-11 walking-excitation criterion applies.

Step 4 — Peak Acceleration Ratio (Eq. 18.3)

W = (w_{D} + 0.11 w_{L}) L s

X X X = (0.75 + 0.055) (30) (10)

X X X = 241, 500 lb

X X X \frac{a _{p}}{g} = \frac{P _{o} e ^{- 0.35 f_{n}}}{β W}

X X X = \frac{65 e ^{- 0.35 (5.25)}}{0.03 ( 241500 )}

X X X = \frac{65 ( 0.160 )}{7245}

X X X = 0.00143

a_p/g ≈ 0.14 % < a_o/g = 0.5 % ✓ (office criterion).

Decision. W18×35 satisfies both static (L/360, L/240) and DG-11 vibration criteria for a typical office floor. No camber required for this span; if camber is specified, use 0.80·Δ_D ≈ 5⁄16 in.

Section B — Buildings & Cranes (Fatigue)

Worked Example 18.2 — Crane Runway Girder, Fatigue Check

Given. A992 W24×84 monorail crane runway supports 10 trolley crossings per hour, 12 h/day, 300 days/yr for a 40-year service life. Each trolley crossing produces one stress cycle with S_R = 12 ksi in the bottom flange (unfactored, computed from service crane load). Two details are relevant: (a) bottom-flange base metal (Category B for rolled shapes), (b) transverse stiffener fillet weld to bottom flange (Category C). Check both for infinite life and finite life.

Fig. 18.2a — Crane runway girder with transverse stiffener welded to the tension flange.

Step 1 — Compute N

N = n_{cy c / h r} \cdot h / d a y \cdot d a y / y r \cdot y r

X X X = 10 \cdot 12 \cdot 300 \cdot 40

X X X = 1.44 \times 1 0^{6} cycles

N > 20 000 → Appendix 3 applies.

Step 2 — Category B (bottom-flange rolled base metal)

Table A-3.1: C_f = 120 × 10⁸, F_TH = 16 ksi.

F_{S R, B} = (\frac{C _{f}}{N})^{1/3}

X X X = (\frac{120 \times 1 0 ^{8}}{1.44 \times 1 0 ^{6}})^{1/3}

X X X = (8333)^{1/3}

X X X = 20.3 ksi

S_R = 12 ksi < F_TH,B = 16 ksi ⇒ infinite life at the base metal.

Step 3 — Category C (transverse-stiffener weld toe)

Table A-3.1: C_f = 44 × 10⁸, F_TH = 10 ksi.

F_{S R, C} = (\frac{44 \times 1 0 ^{8}}{1.44 \times 1 0 ^{6}})^{1/3}

X X X = (3056)^{1/3}

X X X = 14.5 ksi

S_R = 12 ksi < 14.5 ksi ✓ but > F_TH,C = 10 ⇒ finite life. Solve for cycles-to-failure:

N_{f} = \frac{C _{f}}{S _{R}^{3}}

X X X = \frac{44 \times 1 0 ^{8}}{1 2 ^{3}}

X X X = 2.55 \times 1 0^{6} cycles

X X X \approx 71 yr

Decision. The base metal has infinite fatigue life; the stiffener-to-flange fillet weld has ≈ 71 years of finite life at N = 1.44×10⁶ over 40 years — marginal. Recommend either (a) grinding the weld toe to move to Category B (infinite life), (b) locating stiffeners on the compression (top) flange, or (c) reducing S_R below F_TH,C = 10 ksi by upsizing to W24×94.

Section C — Bridges (Fatigue)

Worked Example 18.3 — Highway Bridge Cover-Plate End (Category E)

Given. Simple-span composite plate girder, 90 ft span. Bottom flange plate 20 × ½ in with a partial-length cover plate 8 × ½ in welded to the outside face. The cover-plate ends 25 ft from each support; the cover-plate end weld is Category E. Fatigue truck (HL-93K) produces a stress range at the cover-plate end of S_R = 4.8 ksi in the bottom flange. ADTT (single-lane average daily truck traffic) = 500 trucks/day. Design life = 75 yr. Check the detail for infinite life; if not, compute finite life.

Fig. 18.3a — Cover-plate end weld — Category E fatigue detail on tension flange.

Step 1 — Cycles per 75 yr

N = ADTT \cdot 365 \cdot yr

X X X = 500 \cdot 365 \cdot 75

X X X = 1.37 \times 1 0^{7} cycles

Step 2 — Category E

Table A-3.1: C_f = 11 × 10⁸, F_TH = 4.5 ksi.

F_{S R, E} = (\frac{11 \times 1 0 ^{8}}{1.37 \times 1 0 ^{7}})^{1/3}

X X X = (80.3)^{1/3}

X X X = 4.32 ksi

S_R = 4.80 ksi > 4.32 ksi ⇒ NG. Solve for finite life:

N_{f} = \frac{11 \times 1 0 ^{8}}{S _{R}^{3}}

X X X = \frac{11 \times 1 0 ^{8}}{4. 8 ^{3}}

X X X = 9.94 \times 1 0^{6} cycles

X X X \approx 54 yr

Step 3 — Options to Fix

Remove the cover plate and upsize the flange to 20 × ⅝. Bottom-flange base metal is Category B (F_TH = 16 ksi) → infinite life.
Replace weld with a bolted splice: pretensioned bolts → Category B behavior.
Post-weld ultrasonic impact treatment (UIT): documented +2 category shift → Category C, F_TH = 10 ksi. Instantly passes.
Reduce S_R below F_TH,E = 4.5 ksi by increasing the flange plate.

Decision. Redesign — either eliminate the Category E cover-plate end weld or specify UIT retrofit. This is the exact detail that motivated AASHTO's rule against unshaped cover-plate ends on new bridges (post-1990 practice).

Worked Example 18.4 — Variable-Amplitude Truck Spectrum (Miner's Rule)

Given. Same cover-plate-end Category E detail. Instead of a single design truck, WIM (weigh-in-motion) data gives a variable-amplitude spectrum:

Fig. 18.4a — Measured stress-range spectrum from WIM data at a rural interstate.

Stress range S_Ri (ksi)	Fraction α_i	α_i·S_Ri³
4.0	0.55	0.55·64 = 35.2
6.0	0.30	0.30·216 = 64.8
9.0	0.12	0.12·729 = 87.5
14.0	0.03	0.03·2744 = 82.3
Sum	1.00	269.8

Step 1 — Equivalent Constant-Amplitude Stress Range (Eq. 18.6)

S_{R e} = [i \sum α_{i} S_{R i}^{3}]^{1/3}

X X X = (269.8)^{1/3}

X X X = 6.46 ksi

Step 2 — Compare to F_SR,E for N = 1.37 × 10⁷

From Ex. 18.3, F_SR,E = 4.32 ksi. S_Re = 6.46 ksi > 4.32 ⇒ NG.

D = i \sum \frac{n _{i}}{N _{i}} = \frac{N \sum α _{i} S _{R i}^{3}}{C _{f}}

X X X = \frac{( 1.37 \times 1 0 ^{7} ) ( 269.8 )}{11 \times 1 0 ^{8}}

X X X = 3.36 ≫ 1.0

Step 3 — Observe the cube-law leverage

The 3 % of trucks at 14 ksi contribute 82.3 / 269.8 = 31 % of the damage. Legal-load screening that removes the top few percent of overloads is the most effective fatigue mitigation for existing bridges.

Decision. Retrofit required. UIT of the cover-plate end welds combined with an enforcement program that eliminates the 14-ksi overload tail typically buys another 40+ years of life.

Section D — Machinery & Industrial (Fatigue)

Worked Example 18.5 — Reciprocating Compressor Support Bracket

Given. A cantilever steel bracket supports a natural-gas reciprocating compressor rotating at 720 rpm. The dynamic unbalanced force produces a bending stress range S_R = 7 ksi at the Category C weld toe between the bracket web and the pedestal cap plate. Machine runs continuously (24 / 7). Owner requires a 25-year fatigue life plus a check of the constant-amplitude fatigue limit (CAFL) because the load is truly constant-amplitude and effectively infinite-cycle.

Fig. 18.5a — Reciprocating compressor pedestal bracket with cyclic bending at the weld toe.

Step 1 — Cycles per 25 years

N = n_{r p m} \cdot 60 \cdot 24 \cdot 365 \cdot yr

X X X = 720 \cdot 60 \cdot 24 \cdot 365 \cdot 25

X X X = 9.46 \times 1 0^{9} cycles

N is well beyond the transition at ≈ 5×10⁶ cycles ⇒ the only criterion that matters is the CAFL (F_TH).

Step 2 — Category C threshold (Eq. A-3-1)

Table A-3.1 for Cat. C: F_TH = 10 ksi. Because N is essentially infinite, AISC forces S_R ≤ F_TH:

S_{R} = 7 ksi

X X X \leq F_{T H, C} = 10 ksi ✓

The finite-life curve would give F_SR = (44×10⁸ / 9.46×10⁹)^1/3 ≈ 0.76 ksi — meaningless. The CAFL governs.

Step 3 — What if a resonance amplifies S_R to 12 ksi?

S_{R} = 12 ksi

X X X > F_{T H, C} = 10 ksi \Rightarrow finite life only

N_{f} = \frac{C _{f}}{S _{R}^{3}}

X X X = \frac{44 \times 1 0 ^{8}}{1 2 ^{3}}

X X X = 2.55 \times 1 0^{6}

X X X \approx 2.5 days

This is the classic rotating-equipment lesson: any stress excursion above the CAFL destroys the infinite-life assumption in days, not years. Structural fatigue design for machine supports is really a vibration problem — keep S_R below F_TH under the worst credible operating condition (start-up, resonance sweep, load rejection).

Decision. S_R = 7 ksi is well under F_TH,C = 10 ksi → infinite life. Require the vibration monitor to alarm at S_R = 9 ksi (0.9 F_TH) and trip at 10 ksi. Best practice: upgrade the weld toe to Category B (grind + UIT) so F_TH = 16 ksi and gives a real safety margin against resonance.

Worked Example 18.6 — Vibrating-Screen Support Beam (Bolted vs Welded)

Given. A W16×36 beam supports a mining vibrating screen operating at 16 Hz. Peak dynamic reaction at the beam midspan produces a bending stress range S_R = 8 ksi. Duty cycle: 20 h/day, 350 day/yr, 25-yr life. Compare two attachment details of the screen frame to the beam top flange: (a) pretensioned high-strength bolts (Category B) and (b) 3⁄8-in fillet weld to a longitudinal stiffener (Category E).

Fig. 18.6a — Same beam, two attachment strategies, dramatically different fatigue life.

Step 1 — Cycles

N = f \cdot 3600 \cdot h / d a y \cdot d a y / y r \cdot y r

X X X = 16 \cdot 3600 \cdot 20 \cdot 350 \cdot 25

X X X = 1.01 \times 1 0^{10} cycles

Again essentially infinite — CAFL governs.

Step 2 — Option (a): Pretensioned bolted attachment (Cat. B)

S_{R} = 8 ksi

X X X \leq F_{T H, B} = 16 ksi ✓ \Rightarrow infinite life

Step 3 — Option (b): Welded longitudinal stiffener (Cat. E)

S_{R} = 8 ksi

X X X > F_{T H, E} = 4.5 ksi \Rightarrow finite life only

N_{f} = \frac{C _{f, E}}{S _{R}^{3}}

X X X = \frac{11 \times 1 0 ^{8}}{8 ^{3}}

X X X = 2.15 \times 1 0^{6} cycles

t_{f} = \frac{N _{f}}{f \cdot 3600 \cdot h / d a y}

X X X = \frac{2.15 \times 1 0 ^{6}}{16 \cdot 3600 \cdot 20}

X X X \approx 1.9 days of operation

Decision. Welded attachment fails in under 2 operating days; bolted attachment gives infinite life. This is why every vibrating-screen manufacturer catalog specifies bolted or clamped connections between the screen and its supporting steel — never welded. Same beam, same stress, five orders of magnitude difference in life because of the detail category.

Worked Example 18.7 — Rooftop AHU Steel Frame, Wind-Induced Fatigue

Given. A rooftop air-handling unit (AHU) is supported on four HSS 6×6×3/8 posts welded (CJP) to base plates bolted to the roof structure. Wind buffeting produces measured across-wind oscillations at f = 2.4 Hz with an equivalent stress range S_R = 5 ksi at the CJP weld toe (Cat. C' — CJP with backing bar left in place). Building operates in windy conditions ≈ 40 % of the time; design life = 50 yr. Check fatigue.

Fig. 18.7a — Wind-induced across-wind buffeting excites the HSS post at its base weld.

Step 1 — Effective cycles

N = f \cdot 3600 \cdot 24 \cdot 365 \cdot yr \cdot p_{w in d}

X X X = 2.4 \cdot 3600 \cdot 24 \cdot 365 \cdot 50 \cdot 0.40

X X X = 1.51 \times 1 0^{9} cycles

Step 2 — Category C' — CAFL check (essentially infinite N)

Table A-3.1 for CJP with backing bar (Cat. C'): F_TH = 7 ksi.

S_{R} = 5 ksi

X X X \leq F_{T H, C^{'}} = 7 ksi ✓

Step 3 — Sensitivity — what if buffeting doubles under a 50-yr wind?

S_{R} = 10 ksi

X X X > F_{T H, C^{'}} = 7 ksi \Rightarrow finite life

N_{f} = \frac{44 \times 1 0 ^{8}}{1 0 ^{3}}

X X X = 4.4 \times 1 0^{6} cycles

t_{f} = \frac{N _{f}}{f \cdot 3600 \cdot 24 \cdot p _{w in d}}

X X X = \frac{4.4 \times 1 0 ^{6}}{2.4 \cdot 3600 \cdot 24 \cdot 0.40}

X X X \approx 53 days of storm exposure

Rare storms move the AHU frame from "infinite" to "days" — so infinite-life design at the mean wind is not conservative for wind-buffeted equipment. Best practice is to check the fatigue-limit stress under the 50-yr wind gust, not just the mean.

Decision. At the operating condition (S_R = 5 ksi) the frame satisfies fatigue by CAFL. Recommend: (1) remove the backing bar and back-gouge to upgrade to Category B (F_TH = 16 ksi), (2) add vibration isolators between AHU and frame to cut S_R by 60–80 %.

Section E — PE-Style Practice Problems (AISC 360 Appendix 3)

Worked Example 18.8 — Finite Fatigue Life: Highway Bridge Cover-Plate End (Cat. E′)

Given. A welded cover-plate termination on the tension flange of a highway bridge beam. Traffic averages 3 truck cycles/day over a 50-year design life. Stresses at the detail: dead-load σ_D = 12.0 ksi (steady), live-load σ_max = +14.5 ksi, σ_min = −2.0 ksi. Determine whether the beam is adequate for fatigue per AISC 360 Appendix 3.

Fig. 18.8a — Welded cover-plate termination on wide-flange tension flange (Category E′).

Step 1 — Design Cycles

N = n_{cy c / d a y} \cdot 365 \cdot yr

X X X = 3 \cdot 365 \cdot 50

X X X = 54, 750 cycles

Because 54,750 > 20,000 cycles, a formal fatigue evaluation is required per AISC 360 Appendix 3.

Step 2 — Fatigue Parameters (Table A-3.1, Section 5)

Welded cover-plate termination on a rolled-shape flange:

Stress Category: E′
Detail constant C_f = 3.9 × 10⁸
Threshold F_TH = 2.6 ksi

Step 3 — Allowable Stress Range F_SR

F_{S R} = (\frac{C _{f}}{N})^{0.333}

X X X = (\frac{3.9 \times 1 0 ^{8}}{54 , 750})^{0.333}

X X X = (7123.3)^{0.333}

X X X \approx 19.24 ksi

F_SR = 19.24 ksi > F_TH = 2.6 ksi, so the finite-life value governs.

Step 4 — Applied Live-Load Stress Range Δf

Dead-load stress (12.0 ksi) is steady and omitted from the fatigue range.

X X X Δ f = σ_{ma x} - σ_{min}

X X X = 14.5 - (- 2.0)

X X X = 16.5 ksi

Step 5 — Adequacy Check

X X X Δ f = 16.5 ksi

X X X \leq F_{S R} = 19.24 ksi ✓

✅ Final Answer. F_SR = 19.24 ksi and Δf = 16.5 ksi. The beam is Adequate for finite-life fatigue at the Category E′ cover-plate end.

Worked Example 18.9 — Infinite Fatigue Life: Crane Runway Slip-Critical Bolt Hole (Cat. B)

Given. A crane runway girder with a slip-critical pretensioned bolted splice. The base metal at the bolt hole is Category B. Duty: 15 cycles/hr · 16 h/day · 250 day/yr · 20-yr life. Live-load moment range M_max = 320 k-ft, M_min = 0. Section modulus S_x = 244 in³. Check fatigue adequacy.

Fig. 18.9a — Crane runway girder with slip-critical bolted splice (Category B base metal at bolt hole).

Step 1 — Design Cycles

N = n_{cy c / h r} \cdot h / d a y \cdot d a y / y r \cdot yr

X X X = 15 \cdot 16 \cdot 250 \cdot 20

X X X = 1, 200, 000 cycles

Step 2 — Fatigue Parameters (Table A-3.1, Section 2)

Stress Category: B
C_f = 12 × 10⁸
F_TH = 16.0 ksi

Step 3 — Allowable Stress Range F_SR

F_{S R, f ini t e} = (\frac{C _{f}}{N})^{0.333}

X X X = (\frac{12 \times 1 0 ^{8}}{1 , 200 , 000})^{0.333}

X X X = (1000)^{0.333}

X X X \approx 10.0 ksi

Per AISC Appendix 3, F_SR cannot be less than the constant-amplitude threshold F_TH:

F_{S R} = max (F_{S R, f ini t e}, F_{T H})

X X X = max (10.0, 16.0)

X X X = 16.0 ksi

Step 4 — Applied Live-Load Stress Range Δf

X X X Δ M = M_{ma x} - M_{min}

X X X = 320 - 0

X X X = 320 k-ft = 3, 840 k-in

X X X Δ f = \frac{Δ M}{S _{x}}

X X X = \frac{3 , 840}{244}

X X X \approx 15.74 ksi

Step 5 — Adequacy Check

X X X Δ f = 15.74 ksi

X X X \leq F_{S R} = 16.0 ksi ✓

✅ Final Answer. The threshold F_TH = 16.0 ksi governs and Δf = 15.74 ksi. The girder is Adequate with a razor-thin margin — a small increase in wheel load would push the detail into finite life. Recommend upsizing S_x or specifying a higher-category detail.

Worked Example 18.10 — PE-Style Challenge: W8×31 Hanger with Slip-Critical Bolt Splice

Given. A structural hanger uses a W8×31 in pure axial tension, spliced at its base by 4 × 7/8-in A325 bolts (two per flange, longitudinal pitch 3.0 in), pretensioned in slip-critical configuration. Service axial loads: P_D = 45.0 k (steady); cyclic live load P_L,max = 60.0 k, P_L,min = 10.0 k. Duty: 20 cycles/day, 25-yr life. Determine whether the base metal at the bolt-hole cross-section is fatigue-adequate per AISC 360 Appendix 3.

Fig. 18.10a — W8×31 tension hanger with 4 pretensioned high-strength bolts in a slip-critical splice.

Table references you must synthesize.

Manual Part 1, Table 1-1 — W-shape geometric properties (A_g, t_f).
Specification Table J3.3 — nominal hole dimensions.
Specification §B4.3b — extra 1/16 in on hole for net-area computation.
Appendix 3, Table A-3.1 — fatigue detail category, C_f, F_TH.

Step 1 — W8×31 Geometry (Table 1-1)

A_g = 9.13 in²
t_f = 0.435 in

Step 2 — Effective Hole Width (Table J3.3 + §B4.3b)

Standard hole for a 7/8-in bolt is 15/16 in; add 1/16 in for damage per §B4.3b.

W_{h} = d_{h o l e} + 1/16

X X X = 15/16 + 1/16

X X X = 1.00 in

Step 3 — Net Section Area A_n

4 hole penetrations cross the critical failure plane (2 per flange).

A_{n} = A_{g} - n_{h o l es} \cdot W_{h} \cdot t_{f}

X X X = 9.13 - 4 \cdot 1.00 \cdot 0.435

X X X = 9.13 - 1.74

X X X = 7.39 in^{2}

Step 4 — Design Cycles N

N = 20 \cdot 365 \cdot 25

X X X = 182, 500 cycles

N > 20,000 ⇒ fatigue check required.

Step 5 — Fatigue Parameters (Table A-3.1, Section 2)

Base metal at slip-critical pretensioned bolted joint:

Stress Category: B
C_f = 12.0 × 10⁸
F_TH = 16.0 ksi

Step 6 — Allowable Fatigue Stress Range F_SR

F_{S R, f ini t e} = (\frac{C _{f}}{N})^{0.333}

X X X = (\frac{12.0 \times 1 0 ^{8}}{182 , 500})^{0.333}

X X X = (6575.34)^{0.333}

X X X \approx 18.72 ksi

F_{S R} = max (F_{S R, f ini t e}, F_{T H})

X X X = max (18.72, 16.0)

X X X = 18.72 ksi

Step 7 — Applied Live-Load Stress Range Δf

Dead load is steady and omitted; fatigue uses net area for base-metal-at-hole checks.

X X X Δ P_{L} = P_{L, ma x} - P_{L, min}

X X X = 60.0 - 10.0

X X X = 50.0 k

X X X Δ f = \frac{Δ P _{L}}{A _{n}}

X X X = \frac{50.0}{7.39}

X X X \approx 6.77 ksi

Step 8 — Adequacy Check

X X X Δ f = 6.77 ksi

X X X \leq F_{S R} = 18.72 ksi ✓

✅ Final Answer. Δf = 6.77 ksi << F_SR = 18.72 ksi. The W8×31 base metal at the slip-critical bolt-hole cross-section is Adequate for fatigue over the 25-year design life.

Worked Example 18.11 — PE-Style Challenge: WT7×34 Tension Chord with Longitudinal Fillet Welds Only (Cat. E)

Given. A WT7×34 (A992) tension chord is connected to a gusset by longitudinal fillet welds along the flange heels only (no transverse weld, stem unattached). Weld length l = 9.5 in. Service axial loads: P_D = 35.0 k (steady); cyclic P_L,max = +90.0 k, P_L,min = −10.0 k (reversal). N = 2.2 × 10⁶ cycles over 20-yr life. Check fatigue adequacy of the base metal at the weld termination.

Fig. 18.11a — WT7×34 with longitudinal fillet welds on flange heels only; stem unattached ⇒ shear lag governs.

Table references you must synthesize.

Manual Part 1, Table 1-8 — WT-shape geometric properties (A_g, b_f, ȳ).
Specification Table D3.1 — shear lag factor U (Case 2 and Case 4 alternatives).
Appendix 3, Table A-3.1, Section 3 — fatigue category for termination of longitudinal fillet welds.

Step 1 — WT7×34 Geometry (Table 1-8)

A_g = 10.0 in²
b_f = 10.03 in
ȳ = 1.29 in (connection eccentricity x̄)

Step 2 — Shear Lag Factor U (Table D3.1)

Case 2 (general): U = 1 − x̄/l.

U_{Case 2} = 1 - \frac{x ˉ}{l}

X X X = 1 - \frac{1.29}{9.5}

X X X = 1 - 0.1358

X X X = 0.864

Case 4 (single tees, longitudinal welds only) requires l ≥ b_f. Check length thresholds:

l ≥ 2b_f ? 9.5 ≥ 20.06 → No
2b_f > l ≥ 1.5b_f ? 20.06 > 9.5 ≥ 15.05 → No
1.5b_f > l ≥ b_f ? 15.05 > 9.5 ≥ 10.03 → No

Case 4 length thresholds all fail ⇒ Case 2 governs: U = 0.864.

Step 3 — Effective Net Area A_e

Welded connection ⇒ A_n = A_g (no hole reductions).

A_{e} = A_{n} \cdot U

X X X = 10.0 \cdot 0.864

X X X = 8.64 in^{2}

Step 4 — Fatigue Parameters (Table A-3.1, Section 3)

Base metal at the termination of longitudinal fillet welds where b_f > l (weld shorter than flange width): Category E.

Stress Category: E
C_f = 1.1 × 10⁸
F_TH = 4.5 ksi

Step 5 — Allowable Fatigue Stress Range F_SR

F_{S R, f ini t e} = (\frac{C _{f}}{N})^{0.333}

X X X = (\frac{1.1 \times 1 0 ^{8}}{2 , 200 , 000})^{0.333}

X X X = (50.0)^{0.333}

X X X \approx 3.68 ksi

F_{S R} = max (F_{S R, f ini t e}, F_{T H})

X X X = max (3.68, 4.5)

X X X = 4.5 ksi

Step 6 — Applied Live-Load Stress Range Δf

Dead load is steady and omitted; use A_e so shear-lag stress concentration is reflected.

X X X Δ P_{L} = P_{L, ma x} - P_{L, min}

X X X = 90.0 - (- 10.0)

X X X = 100.0 k

X X X Δ f = \frac{Δ P _{L}}{A _{e}}

X X X = \frac{100.0}{8.64}

X X X \approx 11.57 ksi

Step 7 — Adequacy Check

X X X Δ f = 11.57 ksi

X X X > F_{S R} = 4.5 ksi \times

❌ Final Answer. Δf = 11.57 ksi >> F_SR = 4.5 ksi. The WT7×34 base metal is Inadequate and will fail in fatigue well before the 20-year design life. Remedies: (1) extend weld length to l ≥ 2b_f to raise the category, (2) add transverse weld across the flange tip to move out of Cat. E, (3) upsize to a WT with higher A_g and connect the stem to eliminate shear lag, or (4) apply post-weld UIT to shift +2 categories.

Section F — Additional Deflection (Serviceability) Examples

Worked Example 18.12 — W24×55 Floor Beam Live-Load Deflection Check

Given. A W24×55 (A992) filler beam spans L = 21 ft, simply supported, and carries a service live UDL w_L = 3.0 k/ft. I_x = 1350 in⁴, E = 29 000 ksi. Ceiling finish attached ⇒ limit Δ_L ≤ L/360.

Fig. 18.12 — Live-load deflection at midspan of a W24×55.

Step 1 — Convert Units

w_{L}

X X X = \frac{3.0 k/ft}{12 in/ft}

X X X = 0.250 k/in

L

X X X = 21 ft \cdot 12

X X X = 252 in

Step 2 — Midspan Deflection (uniform load)

X X X Δ_{L} = \frac{5 w _{L} L ^{4}}{384 E I _{x}}

X X X = \frac{5 ( 0.250 ) ( 252 ) ^{4}}{384 ( 29000 ) ( 1350 )}

X X X = \frac{5.037 \times 1 0 ^{9}}{1.504 \times 1 0 ^{10}}

X X X \approx 0.335 in

Step 3 — Compare to Serviceability Limit

X X X \frac{L}{360}

X X X = \frac{252}{360}

X X X = 0.700 in

X X X Δ_{L} = 0.335 in

X X X < 0.700 in ✓

✅ OK — 48 % utilization. The W24×55 satisfies the L/360 live-load deflection limit with generous margin.

Worked Example 18.13 — W18×35 Roof Beam vs L/360

Given. A W18×35 (A992) roof filler spans L = 30 ft, simply supported. Service live UDL w_L = 0.55 k/ft. I_x = 510 in⁴.

Fig. 18.13 — Roof beam under uniform live load.

Step 1 — Compute Deflection

w_{L}

X X X = 0.55/12

X X X = 0.04583 k/in

X X X Δ_{L} = \frac{5 w _{L} L ^{4}}{384 E I _{x}}

X X X = \frac{5 ( 0.04583 ) ( 360 ) ^{4}}{384 ( 29000 ) ( 510 )}

X X X = \frac{3.85 \times 1 0 ^{9}}{5.68 \times 1 0 ^{9}}

X X X \approx 0.678 in

Step 2 — Check L/360

X X X \frac{L}{360}

X X X = \frac{360}{360}

X X X = 1.00 in

0.678

X X X < 1.00 ✓

✅ OK — 68 % utilization. W18×35 passes L/360. If a plaster ceiling is added, re-check against the stricter L/480 limit (0.75 in) — still marginally OK.

Worked Example 18.14 — Maximum P_L on a W14×68 by Deflection

Given. A W14×68 (A992) beam spans L = 24 ft simply supported and carries a single concentrated live load P_L at midspan (dead load steady, not part of live deflection). I_x = 722 in⁴. Ceiling attached ⇒ Δ_L ≤ L/360. Determine the largest P_L the beam can carry without violating serviceability.

Fig. 18.14 — Midspan point-load deflection.

Step 1 — Deflection Formula (concentrated load at midspan)

X X X Δ_{L}

X X X = \frac{P _{L} L ^{3}}{48 E I _{x}}

Step 2 — Set Δ_L = L/360, solve for P_L

X X X \frac{P _{L} L ^{3}}{48 E I _{x}}

X X X = \frac{L}{360}

P_{L, m a x}

X X X = \frac{48 E I _{x}}{360 L ^{2}}

X X X = \frac{E I _{x}}{7.5 L ^{2}}

Step 3 — Substitute Numbers

P_{L, m a x}

X X X = \frac{( 29000 ) ( 722 )}{7.5 ( 288 ) ^{2}}

X X X = \frac{2.094 \times 1 0 ^{7}}{6.220 \times 1 0 ^{5}}

X X X \approx 33.7 k

Step 4 — Verification

X X X Δ_{L}

X X X = \frac{33.7 ( 288 ) ^{3}}{48 ( 29000 ) ( 722 )}

X X X = \frac{8.05 \times 1 0 ^{8}}{1.005 \times 1 0 ^{9}}

X X X = 0.800 in = L /360 ✓

✅ Final Answer. P_L,max ≈ 33.7 kips at midspan governs the W14×68 by serviceability. Strength (φ_bM_p) must still be checked — for A992, Z_x = 115 in³, φ_bM_p = 431 k-ft ≫ P_uL/4 with typical load factors, so deflection governs.

Independent Practice

A pedestrian steel bridge girder has a partial-length longitudinal fillet weld (Cat. E) attached to the tension flange. Estimated N = 2×10⁶ cycles over 50 yr, S_R = 3.2 ksi. Does the detail have infinite life? If not, what N_f does it have?
A composite W24×55 floor beam spans 32 ft with w_L = 0.60 klf service. I_eff = 1650 in⁴. Check Δ_L against L/360.
A moment-frame column has a story height h_sx = 13 ft. Under service wind, δ_x = 0.48 in. Compare to the H/400 wind-drift limit.
A support bracket for a reciprocating compressor sees S_R = 15 ksi at a Category C weld toe, N = 10 million cycles/yr, 30-yr life. Check infinite and finite life.
A wind-turbine tower base weld (Cat. C', F_TH = 7 ksi) sees S_R = 6 ksi at 0.6 Hz nacelle motion, operating 8000 h/yr, 20-yr life. Infinite life?
A vibrating-conveyor support beam (f = 12 Hz) uses a bolted (Cat. B) clip and a welded (Cat. E) gusset carrying the same S_R = 9 ksi. Compare fatigue life of each detail.

FE-Style Worked Examples(6)

Each example mirrors the NCEES FE Civil Reference Handbook style: brief givens, a labeled figure, AISC section reference, step-by-step numeric solution, and a single boxed answer.

Given

W18×35: Ix=510 in⁴, L=30 ft, wL=1.2 k/ft (service live).

AISC Reference

AISC §L3

Step-by-step solution

ΔL
5wL⁴/(384EI) = 5(1.2/12)(360)⁴/(384(29000)(510)) = 1.51 in
Limit
L/360 = 360/360 = 1.0 in → NG (1.51 > 1.0)

Answer Increase Ix — try W18×46 (I=712 in⁴) → Δ=1.08 still slightly NG; try W21×44 (I=843)→ Δ=0.91 ✓.

Course Materials — Lecture & Worked Examples

Lecture and examples below are extracted from the instructor's 'Deflection of Beams' notes. Service-load deflections (not factored!) are compared to code limits L/360, L/240, L/180.

Lecture highlights (from instructor notes)

Why limit deflection: protect finishes (plaster), preserve appearance, prevent psychological discomfort, and avoid ponding/load-sharing problems.
Use SERVICE loads (no load factors) in every deflection calc. Convert: w in k/in, L in inches.
Simply-supported UDL: δ = 5wL⁴/(384 EI). Concentrated load at midspan: δ = PL³/(48 EI). Combine cases by superposition.
Make sure I corresponds to the bending axis — I_xx for major-axis bending, I_yy for minor-axis bending.
AISC modified equation (simple span I-shapes & channels): δ = ML²/(C₁·I), where M is in k-ft, L in ft, I in in⁴, and C₁ depends on the loading pattern (Manual Fig. 3-2). For UDL on a simple span C₁ = 161; for point load at midspan C₁ = 201.
Dead-load deflection is typically removed by cambering the beam, so only live-load deflection matters in service.

Given

Roof beams supporting plaster ceiling: δLL ≤ L/360. Floor beams: δLL ≤ L/360. Roof not supporting ceiling: δLL ≤ L/180.

AISC Reference

IBC 2018 Table 1604.3; AISC §L3

Step-by-step solution

Use the table
Read the column matching the load (LL only vs D+L vs S/W) and the row for the member type.

Answer Limits range from L/180 to L/360 depending on member type and load type.

Allowable δ_LL, δ_(D+L), and δ_(snow/wind) limits by member type.

Interactive Calculator

Beam Deflection (Simply Supported, UDL)

AISC Design Guide / serviceability

w (service)kip/ftL (span)ftEksiIxin⁴Limit L/

Δ = 5wL⁴/(384 E I)1.232 in

Allowable = L/3601.000 inNG

§11Section 11

Chapter summary

§13Section 13

FE exam preparation

Concept review

Concept summary coming soon.

Calculator tips

Calculator tips coming soon.

Common exam traps

Traps coming soon.

Time management

Aim for ~3 minutes per FE problem; skip and return to any item that takes longer than 5 minutes.

Serviceability & Fatigue

Why Structural Steel?

Why This Chapter Matters

Learning Objectives

Where This Chapter Is Used

Lecture Notes

Chapter 18 — Serviceability & Fatigue (AISC 360-22 Ch. L & Appendix 3)

Part A. Serviceability (AISC 360-22 Ch. L)

1. What Is a Serviceability Limit State?

2. Deflection Limits (Commentary Ch. L, Table CL-1)

3. Story Drift (ASCE 7-22 §12.12 / IBC §1604.3)

4. Floor Vibration — AISC Design Guide 11

5. Camber

Part B. Fatigue (AISC 360-22 Appendix 3)

6. What Is Fatigue?

7. When Does Appendix 3 Apply?

8. Stress-Life (S-N) Curves and Detail Categories

9. Detail Category Table — Key Values

10. Crack Initiation & Propagation

11. Variable-Amplitude Loading (Miner's Rule)

12. Bolted & Pin-Connected Details (§3.4)

13. Design Workflow

14. Retrofit & Repair Strategies

⚠ Common Mistakes (fatigue + serviceability)

15. Chapter Summary

Formula Sheet

Worked Example

Chapter 18 — Worked Examples (Serviceability & Fatigue)

Worked Example 18.1 — Composite Floor Beam Deflection & DG-11 Vibration

Step 1 — Live-Load Deflection (Eq. 18.1)

Step 2 — Total Deflection (Eq. 18.1 scaled)

Step 3 — Fundamental Frequency (Eq. 18.2)

Step 4 — Peak Acceleration Ratio (Eq. 18.3)

Worked Example 18.2 — Crane Runway Girder, Fatigue Check

Step 1 — Compute N

Step 2 — Category B (bottom-flange rolled base metal)

Step 3 — Category C (transverse-stiffener weld toe)

Worked Example 18.3 — Highway Bridge Cover-Plate End (Category E)

Step 1 — Cycles per 75 yr

Step 2 — Category E

Step 3 — Options to Fix

Worked Example 18.4 — Variable-Amplitude Truck Spectrum (Miner's Rule)

Step 1 — Equivalent Constant-Amplitude Stress Range (Eq. 18.6)

Step 2 — Compare to FSR,E for N = 1.37 × 10⁷

Step 3 — Observe the cube-law leverage

Worked Example 18.5 — Reciprocating Compressor Support Bracket

Step 1 — Cycles per 25 years

Step 2 — Category C threshold (Eq. A-3-1)

Step 3 — What if a resonance amplifies SR to 12 ksi?

Worked Example 18.6 — Vibrating-Screen Support Beam (Bolted vs Welded)

Step 1 — Cycles

Step 2 — Option (a): Pretensioned bolted attachment (Cat. B)

Step 3 — Option (b): Welded longitudinal stiffener (Cat. E)

Worked Example 18.7 — Rooftop AHU Steel Frame, Wind-Induced Fatigue

Step 1 — Effective cycles

Step 2 — Category C' — CAFL check (essentially infinite N)

Step 3 — Sensitivity — what if buffeting doubles under a 50-yr wind?

Worked Example 18.8 — Finite Fatigue Life: Highway Bridge Cover-Plate End (Cat. E′)

Step 1 — Design Cycles

Step 2 — Fatigue Parameters (Table A-3.1, Section 5)

Step 3 — Allowable Stress Range FSR

Step 4 — Applied Live-Load Stress Range Δf

Step 5 — Adequacy Check

Worked Example 18.9 — Infinite Fatigue Life: Crane Runway Slip-Critical Bolt Hole (Cat. B)

Step 1 — Design Cycles

Step 2 — Fatigue Parameters (Table A-3.1, Section 2)

Step 3 — Allowable Stress Range FSR

Step 4 — Applied Live-Load Stress Range Δf

Step 5 — Adequacy Check

Worked Example 18.10 — PE-Style Challenge: W8×31 Hanger with Slip-Critical Bolt Splice

Step 1 — W8×31 Geometry (Table 1-1)

Step 2 — Effective Hole Width (Table J3.3 + §B4.3b)

Step 3 — Net Section Area An

Step 4 — Design Cycles N

Step 5 — Fatigue Parameters (Table A-3.1, Section 2)

Step 6 — Allowable Fatigue Stress Range FSR

Step 7 — Applied Live-Load Stress Range Δf

Step 8 — Adequacy Check

Worked Example 18.11 — PE-Style Challenge: WT7×34 Tension Chord with Longitudinal Fillet Welds Only (Cat. E)

Step 1 — WT7×34 Geometry (Table 1-8)

Step 2 — Compare to F_SR,E for N = 1.37 × 10⁷

Step 3 — What if a resonance amplifies S_R to 12 ksi?

Step 3 — Allowable Stress Range F_SR

Step 3 — Allowable Stress Range F_SR

Step 3 — Net Section Area A_n

Step 6 — Allowable Fatigue Stress Range F_SR

Step 3 — Effective Net Area A_e

Step 5 — Allowable Fatigue Stress Range F_SR

Worked Example 18.14 — Maximum P_L on a W14×68 by Deflection

Step 2 — Set Δ_L = L/360, solve for P_L