Welded Joint Fatigue: Method

Detail Categories, Design and Evaluation Checks, Miner Sum, Eurocode Form

Scope

The tool estimates the fatigue life of welded (and bolted or riveted) steel details by the stress-life method with detail categories: each detail is assigned to a category, each category has one S-N line, and the life follows from the nominal stress range at the detail and the number of cycles. Five analyses are chosen by the first dropdown: the bridge design check under truck traffic, the remaining life of an existing detail, a Palmgren-Miner sum over an entered stress-range histogram with the AASHTO categories, the constant-amplitude life, and a Miner sum on the Eurocode form of S-N curve. The analyst picks the category from the detail tables of the specification that governs the structure; those tables are copyrighted and are not reproduced. The tool is a calculator of the published mechanics, not a substitute for the specification.

Units and conventions

Stress ranges and the span length go through the site's unit dictionary and convert with the unit switch. Load factors, traffic counts, years, cycles, the Miner sum and the growth rate in percent are plain numbers. The histogram is read in the unit chosen beside it. The AASHTO constants A are in ksi³ as published; with MPa the page also prints A in MPa³, converted exactly (1 ksi = 6.894757 MPa). The Eurocode-form analysis works in MPa, because its detail categories are named by a stress range in N/mm². Stress range means the algebraic difference between the largest and the smallest nominal stress of a cycle on the gross section at the detail (net section for riveted members), without the local stress concentration of the weld, which is already in the category. Results are stated as demand over capacity; a ratio at or below 1 passes.

Detail categories

The AASHTO categories run from A, the most fatigue resistant, to E′. In the words of the Federal Highway Administration's reference manual (FHWA-NHI-16-016, pages 6.8 to 6.14): A is plain base metal with rolled or cleaned surfaces; B covers continuous longitudinal fillet or full-penetration welds of built-up members, full-penetration butt splices ground smooth, and pretensioned bolted joints; B′ covers longitudinal welds with backing bars left in place and longitudinal partial-penetration welds; C covers transversely loaded welds with reinforcement, copes, re-entrant corners and short attachments; C′ is reserved for the base metal at the toe of transverse stiffener and connection plate fillet welds and has the constant of C with a higher threshold; D is the transition to details governed by geometry (open holes, attachments with radiused ends, short longitudinal attachments); E and E′ are abrupt terminations of cover plates, longitudinal stiffeners and long or thick attachments parallel to the stress, E′ for the thicker plates. The fatigue resistance falls as an attachment grows longer in the direction of the stress and thicker. These lines only recall what each letter stands for; the category of a detail is read from the governing table with its sketches and limits.

Each category has a sloping line N = A / Δf³ and a horizontal constant-amplitude fatigue threshold (ΔF)_TH. The constants used are those printed in FHWA-NHI-16-016 (Tables 6.3.3.2-1 and 6.3.4.2-1) and in the FHWA Steel Bridge Design Handbook (Volume 12, Tables 1 and 2): A = 250, 120, 61, 44, 44, 22, 11 and 3.9 × 10⁸ ksi³ and thresholds of 24, 16, 12, 10, 12, 7, 4.5 and 2.6 ksi for A, B, B′, C, C′, D, E and E′. They round the lower-bound constants of NCHRP Report 286 (Table 8). The sloping lines lie two standard deviations below the mean of the full-scale beam tests, about a 97.5 percent probability of survival; the thresholds are bounds drawn under the long-life tests and are not statistical. Life in these tests means a crack through the flange or a similar loss of section, not first visible cracking.

Design check under truck traffic

The stress range Δf is that of one passage of the fatigue design truck with the dynamic allowance and the distribution factor. Two load combinations are checked. Fatigue I, for infinite life: γ_I Δf ≤ (ΔF)_TH, where γ_I Δf stands for the largest stress range expected (exceeded by about one truck in 10,000). Fatigue II, for finite life: γ_II Δf ≤ (ΔF)_n = (A/N)^1/3 with N = 365 × Y × n × (ADTT)_SL, where γ_II Δf is the effective (root-mean-cube) stress range of the truck population, Y the design life in years (75), n the number of stress cycles per truck passage and (ADTT)_SL = p × ADTT the average daily truck traffic in one lane. The detail passes when either check passes. The load factors are inputs: 1.75 and 0.80 since the 2016 revision, 1.5 and 0.75 before it (the ratio of the two is the ratio of the largest to the effective stress range).

The page also prints the years of the entered traffic that bring the detail to its design line, A / (365 n (ADTT)_SL (γ_II Δf)³); the largest unfactored stress range that passes, the greater of (ΔF)_TH/γ_I and (ΔF)_n/γ_II; and the single-lane truck traffic above which the finite-life resistance falls below the factored threshold, A / (365 Y n (γ_II (ΔF)_TH/γ_I)³), beyond which only the infinite-life check can pass. That last figure is derived here from the two checks; the specification tabulates it for its own factors. A table repeats the check for all eight categories and the page names the lowest category that still passes.

Remaining life of an existing detail

The evaluation follows the procedure of the bridge evaluation manual as presented in chapter 8 of FHWA-NHI-16-016. The largest stress range is (Δf)_max = R_p γ_I Δf; if it does not exceed the threshold the life is infinite. Otherwise the effective stress range is (Δf)_eff = R_s R_p γ_II Δf and the total life in years is

Y = log[ R_R A g (1 + g)^{a − 1} / (365 n (ADTT)_SL (Δf)_eff³) + 1 ] / log(1 + g),

with a the present age, (ADTT)_SL the present single-lane truck traffic and g its annual growth, taken constant over the whole life (the traffic of year y is the present traffic times (1 + g)^{y − a}, and the cycles are summed from the first year). With no growth Y = R_R A / (365 n (ADTT)_SL (Δf)_eff³). R_p is the multiple presence factor, 0.988 + 6.87 × 10⁻⁵ L + 4.01 × 10⁻⁶ (ADTT)_PRESENT + 0.0107/n_L for longitudinal members (L in feet, traffic of both directions, n_L striped lanes; developed for 2 to 4 lanes and 30 to 220 ft) and 1.0 for transverse members. R_s is the stress-range estimate factor, from 1.0 for simplified analysis down to 0.85 for measured strains. R_R turns the design line into less conservative lines: the minimum life (R_R = 1, 98 percent probability of lasting), the Evaluation 1 life (84 percent), the Evaluation 2 life (67 percent) and the mean life (50 percent). The factors are those printed in Table 8.3.2-1 of the FHWA manual and can be overridden.

The remaining life is Y − a. The fatigue serviceability index is Q = ((Y − a)/N) G R I with N the greater of Y and 100 years and G, R, I the load path, redundancy and importance factors (entered). The action bands printed with the index were proposed in NCHRP Report 721 and were not adopted by the evaluation manual. The manual's further step for a detail with negative remaining life and no crack (updating the life distribution for the survived age) is not built.

Histogram and Miner sum

Under variable amplitude each group of n_i cycles at the range S_i is taken to use the fraction n_i/N_i of the life, N_i being the constant-amplitude life at S_i, and failure is expected when the sum D = Σ n_i/N_i reaches 1 (Palmgren, Miner). With one slope of 3 the sum equals N S_re³/A, where N is the number of cycles and S_re = (Σ γ_i S_i³)^1/3 the effective, or root-mean-cube, stress range, γ_i being the fraction of the cycles in bin i. The histogram is entered as pairs of stress range and cycles for the period it covers; the life left is (1 − D₀)/D periods, D₀ being any damage already used. A few high ranges carry much of the damage: in the manual's example the three highest bins hold 6 percent of the cycles and a third of the damage.

Cycles below the threshold do extend a crack once larger cycles have started one, so the tool continues the sloping line below the threshold and by default counts every cycle. The option to drop the ranges below a fraction of the threshold follows the evaluation manual, which forms the effective stress range of a measured histogram from the ranges above 45 percent of the threshold. The threshold is reported separately: when the largest range of a representative record stays at or below it, the life is theoretically infinite, and for a short record the largest range of the life is estimated as 2.2 times the effective range. The pass or fail statement for a required life uses the Miner sum alone. The histogram should come from rainflow counting of a measured or computed stress history; counting is not done here.

Constant amplitude

At a constant stress range the life on the design line is N = A/Δf³ when the range exceeds the threshold and is theoretically unlimited otherwise; the allowable range for a required number of cycles is the greater of (A/N)^1/3 and the threshold. The line meets the threshold at A/(ΔF)_TH³ cycles, between 1.8 million (A) and 22 million (E′).

Eurocode form

The European standard names a detail category by the stress range Δσ_C in MPa that the detail endures for 2 million cycles and uses one shape of curve for all categories (JRC report EUR 23252 EN, Figure 3-8): slope m = 3 down to the constant amplitude fatigue limit Δσ_D at 5 million cycles, slope m = 5 down to the cut-off limit Δσ_L at 100 million cycles, and no damage below. From the two slopes Δσ_D = (2/5)^1/3 Δσ_C = 0.737 Δσ_C and Δσ_L = (5/100)^1/5 Δσ_D = 0.549 Δσ_D. The endurance is N = 2 × 10⁶ (Δσ_C/Δσ)³ above Δσ_D and 5 × 10⁶ (Δσ_D/Δσ)⁵ between the two limits. The stress ranges are multiplied by γ_Ff and the category divided by γ_Mf; both factors are inputs taken from the standard, and the category is taken from its detail tables. The tool sums n_i/N_i over the histogram and gives the life left. For a design life it also prints the damage-equivalent stress range at 2 million cycles on the first slope, γ_Ff Δσ_E,2 = (Δσ_C/γ_Mf) D^1/3, and the fatigue safety level μ_fat = (Δσ_C/γ_Mf)/(γ_Ff Δσ_E,2) of the JRC report's equation 2-1, which equals D^−1/3 with that definition of the equivalent range. A single slope of 3 without limits is offered as the conservative simplification. Size, mean-stress and other adjustments of the standard are not applied.

Hot-spot stress and weld improvement

The nominal stress method needs a category for the detail. Where no category fits (complex plated joints, web gaps under out-of-plane distortion), the local structural, or hot-spot, stress at the weld toe is used with one baseline curve. The FHWA manual recommends AASHTO Category C with the hot-spot stress range measured or computed about ¼ inch from the weld toe, and ignoring the threshold in that method (pages 5.11 and 5.12). To use it here, enter the hot-spot stress range as the stress range with Category C. Deriving the hot-spot stress from a finite element model or strain gauges is outside the tool.

The fatigue resistance of a weld toe can be raised by treating the toe: disc or burr grinding, gas tungsten arc remelting, air hammer peening and ultrasonic impact treatment are the methods listed in the FHWA manual (page 9.15), mostly for the repair and retrofit of existing details. A radiused and ground transition at the end of an attachment moves the detail to a better category. The tool gives no credit for toe treatment: the improved category, if the governing specification allows one, is the analyst's choice.

Checks

The library reproduces the design example of FHWA-NHI-16-016 (Section 6.6: 14.09 and 6.44 ksi, 14,782,500 cycles, 6.68 ksi), its evaluation example (Section 8.3.3: R_p 1.0018, 7.99 and 3.65 ksi, lives of 42, 50, 57 and 63 years), its variable-amplitude table (Table 4.2.3-1: 122.08 ksi³ and the damage shares), its effective truck weights (133.2 and 43.4 kips), and the damage accumulation of the JRC report (Tables 6-3 and 6-4: endurances to the printed thousands, sums of 0.868 and 0.01668, life reached in 1988). The constants agree in three printings. Twenty seeded cases of each analysis are compared with an independent calculation in which the life with traffic growth is summed year by year and the Eurocode-form curve is interpolated between its corner points.

Limits

Not covered: the detail tables themselves, distortion-induced fatigue, fracture toughness, crack growth by fracture mechanics, rainflow counting, the cycles per passage of members other than girders, shear on weld throats and stud connectors (which have their own lines), tubular joints, thickness corrections, corrosion fatigue and elevated temperature.