Fatigue with mean stress and variable loading

The rotating-beam test applies fully reversed stress, with a mean of zero. Real parts rarely see that. A spring is preloaded, so its stress never falls to zero. A bolt is tightened before any external load arrives. The TP-80 shaft sees fully reversed bending from the impeller load, but its torque is nearly steady, which adds a mean shear stress. And no machine runs at one load forever: the TP-80 runs at its best efficiency point most of the time, but customers also throttle it far back on its curve, start and stop it, and occasionally let it run dry. This lesson extends the fatigue model to mean stresses and to loads that change from cycle to cycle.

Describing a fluctuating stress

A stress that cycles between a maximum and a minimum can be described by its alternating component and its mean component:

Other common terms are the stress range \(\Delta\sigma = 2\sigma_a\), the stress ratio \(R = \sigma_{\min}/\sigma_{\max}\) and the amplitude ratio \(A = \sigma_a/\sigma_m\). Fully reversed stress has R = −1; a stress that rises from zero and returns, such as a bolt in a pressurized flange, has R = 0 and is called repeated or pulsating. Stress concentration factors apply to both components; the usual practice for ductile materials is to apply \(K_f\) to both, provided the peak stress stays below yield.

Experiments show that a tensile mean stress reduces the alternating stress a part can survive, while a compressive mean stress has little effect or even helps. That is why compressive residual stresses from shot peening improve fatigue life: they lower the effective mean stress at the surface where cracks start.

The Haigh diagram and its failure lines

Mean stress effects are shown on a plot of alternating stress against mean stress, often called a Haigh diagram or a modified Goodman diagram. The point \((0, S_e)\) on the vertical axis is fully reversed loading at the endurance limit; the point \((S_{ut}, 0)\) on the horizontal axis is static failure. Test data for steels fall between a straight line and a parabola joining these points. Several criteria are in use:

Goodman is the most widely used: it is conservative, linear and simple. Soderberg is more conservative and is rarely used now except where yielding must be strictly avoided. Gerber and ASME-elliptic fit test data for ductile steels more closely and are used when a less conservative estimate is justified, for example in shaft design to the ASME approach. With any of the fatigue criteria, a separate check for first-cycle yielding, the Langer line, is needed when mean stresses are large, because the Gerber and ASME curves extend above the yield line near the mean stress axis.

Combined stresses: equivalent alternating and mean stresses

When a part sees fluctuating bending and fluctuating torsion together, the distortion energy theory is applied separately to the alternating and mean components to form equivalent stresses:

These are then used in the chosen criterion with the corrected endurance limit \(S_e\) for bending. For the TP-80 shaft, rotating bending gives a purely alternating normal stress and the steady motor torque gives a purely mean shear stress, so \(\sigma_a' = K_f\sigma_a\) and \(\sigma_m' = \sqrt3 K_{fs}\tau_m\).

# Fluctuating stress at the TP-80 shoulder: Goodman, Gerber, ASME-elliptic, Soderberg and Miner's rule
import math
Sut, Sy, Se = 630.0, 530.0, 178.0          # MPa (Se from Lesson 3)
d, M, T = 0.035, 216.0, 59.7
Kf, Kfs = 1.57, 1.35                       # fatigue concentration factors (bending, torsion)
sa = Kf * 32 * M / (math.pi * d**3) / 1e6        # rotating bending -> fully reversed
tm = Kfs * 16 * T / (math.pi * d**3) / 1e6       # steady torque -> mean shear
sa_vm, sm_vm = sa, math.sqrt(3) * tm
print(f"alternating sigma_a' = {sa_vm:.1f} MPa, mean sigma_m' = {sm_vm:.1f} MPa")
n_good = 1 / (sa_vm / Se + sm_vm / Sut)
n_sod = 1 / (sa_vm / Se + sm_vm / Sy)
n_ger = 0.5 * (Sut / sm_vm)**2 * (sa_vm / Se) * (-1 + math.sqrt(1 + (2 * sm_vm * Se / (Sut * sa_vm))**2))
n_asme = 1 / math.sqrt((sa_vm / Se)**2 + (sm_vm / Sy)**2)
n_y = Sy / (sa_vm + sm_vm)
for name, n in (("Goodman", n_good), ("Soderberg", n_sod), ("Gerber", n_ger), ("ASME-elliptic", n_asme), ("first-cycle yield (Langer)", n_y)):
    print(f"  {name:27s} n = {n:.2f}")

# Variable amplitude: a year of duty at the plant, with Miner's rule
a, b = 1653.0, -0.1614                     # S-N line from Lesson 3 (MPa, exponent)
duty = [  # (description, alternating stress MPa, cycles per year)
    ("normal flow",              80,  1760 * 60 * 6000),
    ("flow far left of BEP",    150,  1760 * 60 * 400),
    ("starts with hammer",      320,  2 * 365 * 3),
    ("dry-run trips",           240,  12 * 2000),
]
D = 0.0
for name, s, n in duty:
    N = (s / a) ** (1 / b) if s > Se else float('inf')
    frac = n / N if N != float('inf') else 0.0
    D += frac
    print(f"{name:21s} {s:4d} MPa  n = {n:>12,}  N = {('infinite' if N == float('inf') else f'{N:,.0f}'):>12}  n/N = {frac:.4f}")
print(f"Miner damage per year D = {D:.4f}  -> predicted life about {1/D:.1f} years")

alternating sigma_a' = 80.6 MPa, mean sigma_m' = 16.6 MPa
  Goodman                     n = 2.09
  Soderberg                   n = 2.07
  Gerber                      n = 2.20
  ASME-elliptic               n = 2.20
  first-cycle yield (Langer)  n = 5.46
normal flow             80 MPa  n =  633,600,000  N =     infinite  n/N = 0.0000
flow far left of BEP   150 MPa  n =   42,240,000  N =     infinite  n/N = 0.0000
starts with hammer     320 MPa  n =        2,190  N =       26,203  n/N = 0.0836
dry-run trips          240 MPa  n =       24,000  N =      155,758  n/N = 0.1541
Miner damage per year D = 0.2377  -> predicted life about 4.2 years

Variable amplitude loading and Miner's rule

When the stress amplitude changes from cycle to cycle, the simplest model of damage is the linear damage rule proposed by Palmgren and Miner. Each cycle at stress amplitude \(\sigma_i\) uses up a fraction \(1/N_i\) of the life, where \(N_i\) is the life at that stress from the S-N curve. Failure is predicted when the fractions add to one:

For the TP-80, Farah estimates a year of service at a typical water utility: 6,000 hours at the normal duty with an alternating stress of 80 MPa; 400 hours throttled far left of the best efficiency point, where hydraulic radial loads rise and the stress reaches about 150 MPa; two starts a day with a pressure surge raising the stress briefly to about 320 MPa for three cycles; and a monthly test trip of the dry-run protection, during which rubbing at the wear rings raises the stress to about 240 MPa for about 2,000 cycles. Stresses at or below the endurance limit contribute no damage in the basic rule. The surges and dry-run events do.

Miner's rule is easy to use but has known weaknesses. The order of loading matters in reality: high stresses first, followed by low stresses, are more damaging than the reverse, because a crack started by the high stresses can then grow under stresses that would never have started one. Tests show that failure can occur with D anywhere from about 0.3 to 3. In addition, stresses below the original endurance limit can cause damage once a crack exists. Modified approaches, such as extending the S-N line below the knee with a shallower slope (the Haibach approach), are used in standards for welded structures and some machine elements. A cautious designer uses a damage limit below 1, such as 0.3 to 0.5 for critical parts, or tests the part under a representative load spectrum.

Counting cycles

When the load history is measured rather than estimated, as with strain gauges on a shaft in service, the cycles must be counted before Miner's rule can be applied. The standard method is rainflow counting, which pairs peaks and valleys in a way that matches the closed stress-strain loops seen in the material. Each counted cycle has a range and a mean; mean stress effects are included by converting each cycle to an equivalent fully reversed amplitude with the Goodman or another criterion. Software for rainflow counting is widely available, and ASTM E1049 describes the procedure.

In US units the TP-80 shoulder stresses are 11.7 ksi alternating and 2.4 ksi mean, against \(S_e = 25.8\) ksi and \(S_{ut} = 91.4\) ksi; the Goodman factor is the same 2.09. Variable-speed drives are increasingly common in US water systems, partly because of utility incentive programs and partly because they soften starts and stops, which, as the Miner's rule example shows, can matter more for shaft life than the steady running load.