FEXLA

Fatigue: from S-N curves to weld design curves

From the smooth-bar S-N curve to the welded joint: stress range, cycles, damage sums and weld design curves.

Principle 17 of 20 · 6 min read ·

An S-N curve on log scales (illustrative)Stress range against cycles to failure, both on log scales, for an illustrative smooth steel bar: a straight sloping line, Δσ = 800 MPa at a thousand cycles falling with a slope of one in eight, which flattens at a million cycles into an endurance limit of about 340 MPa. A marked point shows about 450 MPa at a hundred thousand cycles.200500100010²10⁴10⁶10⁸about 450 MPa at 10⁵slope 1 in mendurance limitStress range Δσ, MPa (log scale, illustrative)Cycles to failure N (log scale)
Figure 1. An S-N curve on log scales (illustrative).
Contents

Fatigue is cracking under a stress that is repeated many times, even when each cycle stays well below the stress the metal could carry once. An S-N curve gives how many times a stress range can be repeated before failure, and welded joints have their own, lower curves because the toe of a weld already acts as a sharp notch with a high residual stress.

This article goes from the smooth test bar to the welded joint: what the curve shows, how mean stress and notches enter, why welds are rated by detail and stress range, and how mixed cycles are added up.

Photo: mixer shaft and impeller turning inside a stainless steel vessel
Figure 2. Photo: a mixer shaft and impeller turning inside a vessel. Each turn bends the nozzle once, which is why agitators are a fatigue question. (Free-licence photo, see the image credits.)

Repeated stress, not high stress

A part loaded once to a stress below yield is unharmed. Load it and unload it a million times and a crack can start at the surface, grow a little on every cycle, and finally cut through the section. Nothing in a single cycle looks dangerous. The damage is in the repetition.

What drives it is mainly the stress range, the difference between the highest and lowest stress in a cycle, written Δσ. Half the range is the stress amplitude. A steady stress with a small swing on top does far less harm than a stress that goes from zero to the same peak and back, because the swing is what opens and closes the crack.

Fatigue life has two parts: starting a crack, and growing it. In a smooth, polished bar most of the life is spent starting the crack. In a welded joint a crack-like flaw is usually there from the start, so most of the life is crack growth.

The S-N curve

Test many bars, each at its own constant stress range, and count the cycles to failure. Plotted with both scales logarithmic, the results fall about a straight line over a wide range of lives. That line is the S-N curve, as in the drawing at the top of this page.

N = C / Δσ^mN cycles to failure at range Δσ; C and m fitted to tests; m is the slope on log scales

The slope m says how strongly life depends on range. With m = 3, doubling the range cuts the life by 2³ = 8, and a rise of only 26 % in range halves it. For many steels in clean air the curve flattens at high cycle counts: below a certain range, the endurance limit, cracks do not grow in practice. Corrosion, high temperature and variable loading can remove that flat part, so design curves often keep sloping down.

Mean stress and notches

For the same range, a higher mean stress shortens life, because the crack spends more of the cycle held open. A simple way to account for it is the straight-line rule named after Goodman, which trades amplitude against mean stress up to the tensile strength.

σa / Se + σm / Su = 1σa amplitude, σm mean, Se endurance at zero mean, Su tensile strength

A notch raises the local stress above the nominal value by the stress concentration factor Kt. A round hole in a wide plate under tension has Kt = 3, so the edge of the hole sees three times the stress in the plate. The extra stress is confined to a small zone and does not matter for collapse, but it is where the crack starts, so fatigue is judged on the peak stress at the notch.

Primary and secondary stress: Peak: very local, matters for fatigueA plate with an attachment and fillet welds. Lines of stress crowd together where the welds meet the plate, marked as the weld toes.σ = PD / 2tPressureHoop stress in the wall holds itStrainStressYieldLoad held: strain runs onHot part: free, it would reach hereCool partJoined: the hot part is squeezed,the cool part is stretchedStrainStressYieldStrain set by the fitStress stops at yieldLoadPeak stress at the weld toes
Figure 3. Peak stress: a very local rise at a notch or weld toe. It adds nothing to the strength check but sets the fatigue life.
Where each stress category sits round a nozzleHalf a section through a nozzle wall meeting a shell wall, with a fillet weld at the outer corner. Far along the shell the zone is marked general membrane. Next to the joint, on both the shell and the nozzle wall, a shaded zone is marked local membrane and bending. Dots at the two weld toes are marked peak.nozzle axisGeneral membraneaway from the jointLocal membrane (primary)and bending (secondary)Peak at the weld toescounted for fatigue
Figure 4. Round a nozzle the peak stress sits at the weld toes, on top of the membrane and bending stresses of the shell.

Why welds get their own curves

A weld toe is a notch that cannot be avoided: the surface changes direction sharply, there are small undercuts and slag traces, and the metal next to it has cooled from melting. Cooling leaves residual tensile stress close to yield in and around the weld. Two things follow.

  • Life is mostly crack growth from a flaw that is already there, and crack growth rises with about the third power of the stress range. So weld curves have a slope close to m = 3, flatter than a smooth bar's curve.
  • Because the residual stress is already near yield, the real stress at the toe cycles down from yield whatever the applied mean stress is. For as-welded joints the applied mean stress is mostly ignored and the check uses the range alone.
  • Each kind of joint, such as a butt weld ground flush, a fillet weld toe on a plate, or the root of a one-sided weld, gets its own class. The classes form a family of parallel lines.
A smooth bar and three welded details (illustrative)Stress range against cycles on log scales from a hundred thousand to a hundred million cycles. The illustrative smooth-bar curve starts near 450 MPa and flattens at about 340 MPa. Below it three parallel straight lines for welded details A, B and C, each falling with a slope of one in three, pass through 130, 90 and 60 MPa at two million cycles.205010020050010⁵10⁶10⁷10⁸ABCsmooth barWelded details A, B, C:parallel lines, slope m = 3Stress range, MPa (log scale, illustrative)Cycles N (log scale)
Figure 5. A smooth-bar curve beside three welded detail classes. The weld lines are lower, parallel to each other and flatter. Illustrative constants, not any code's curves.

Thicker plates have shorter lives at the same nominal range, because the stress field at a thick toe is steeper and a crack finds more highly stressed metal ahead of it; many rules reduce the allowed range above a reference thickness. Grinding the toe to a smooth radius, or peening it to leave compression at the surface, can raise the life. Whether and how much credit is given is code-specific.

Adding up mixed cycles

Real equipment sees many different ranges: full start-ups, partial load changes, small daily swings. The usual way to combine them is the linear damage rule named after Miner. Each block of n cycles at a range whose life is N uses up the fraction n / N of the life, and the fractions are added.

D = Σ nᵢ / Nᵢ ≤ 1D cumulative damage; design rules may set a lower limit than 1

The rule ignores the order of the cycles and any effect of one block on the next, and test scatter is wide. It is used because it is simple and, with the margins in design curves, has served well.

How design checks use it

A fatigue check needs three things: the stress range at the place where the crack would start, the number of cycles of each range, and a design curve for that material or weld detail. Design curves sit below the mean of the tests, with margins on both stress and cycles. Some rules first screen out parts with few or small cycles, so a full check is only needed where it matters.

How the stress range is found differs between rules: a peak stress with a notch factor for smooth parts, or a structural stress at the weld toe used with a weld curve. The curves, the margins, the thickness and mean-stress corrections and the screening criteria are all code-specific, so check the governing code.

Key points

  • Fatigue cracks grow under a repeated stress range, even when every cycle stays below yield.
  • An S-N curve gives the cycles to failure for each range; on log scales it is close to a straight line of slope m.
  • Mean stress and notches shorten life; fatigue is judged on the peak stress where the crack starts.
  • Welds have lower, flatter curves rated by detail class on stress range, because of the toe notch and residual stress near yield.
  • Mixed cycles are added with a damage sum D = Σ n/N, kept below the code's limit.

Where this matters in practice

  • Nozzle and attachment weld toes on vessels that start up, shut down or pulse in pressure.
  • Agitator and mixer nozzles, where rotating loads give many cycles over the life.
  • Lugs and supports on equipment that vibrates or is moved often.
  • Head-to-shell and cone junctions with thermal transients on every start-up.

FEXLA's local check at a nozzle or clip compares the combined local stress with limits built from the allowable stress S. It does not run a fatigue analysis.

See the nozzle check

Related principles

Next principle: How a structure collapses: mechanisms and the limit load

Educational article. It explains principles and is not design guidance; follow your governing code and edition.