Stress-Life (S-N) Approach

S-N Curve

The S-N curve, first developed by Wöhler, defines a relationship between stress and number of cycles to failure.

Typically, the S-N curve (and other fatigue properties) of a material is obtained from experiment; through fully reversed rotating bending tests. Due to the large amount of scatter that usually accompanies test results, statistical characterization of the data should also be provided (certainty of survival is used to modify the S-N curve according to the standard error of the curve and a higher reliability level requires a larger certainty of survival).


Figure 1. S-N Data from Testing
When S-N testing data is presented in a log-log plot of alternating nominal stress amplitude S a or range S R versus cycles to failure N , the relationship between S and N can be described by straight line segments. Normally, a one or two segment idealization is used.


Figure 2. One Segment S-N Curves in Log-Log Scale
(1) S = S 1 ( N f ) b 1

for segment 1

Where, S is the nominal stress range, N f are the fatigue cycles to failure, b l is the first fatigue strength exponent, and S I is the fatigue strength coefficient.

The S-N approach is based on elastic cyclic loading, inferring that the S-N curve should be confined, on the life axis, to numbers greater than 1000 cycles. This ensures that no significant plasticity is occurring. This is commonly referred to as high-cycle fatigue.

S-N curve data is provided for a given material using the Materials module.

Multiple SN Curves

HyperLife supports the following Multiple SN curve types:
  • Multi-mean S-N curves: group of S-N curves defined at different mean stress.
  • Multi-ratio S-N curves: group of S-N curves defined at different stress ratio R.
  • Multi-Haigh Diagram: group of Haigh curves defined at different Number of Cycles.


Figure 3. Multi-mean S-N Curve


Figure 4. Multi-ratio S-N Curves


Figure 5. Multi-Haigh Diagrams
Note: Refer Mean Stress = Interpolate, to understand how life is determined when Multiple SN curves are assigned.

Rainflow Cycle Counting

Cycle counting is used to extract discrete simple "equivalent" constant amplitude cycles from a random loading sequence.

One way to understand "cycle counting" is as a changing stress-strain versus time signal. Cycle counting will count the number of stress-strain hysteresis loops and keep track of their range/mean or maximum/minimum values.

Rainflow cycle counting is the most widely used cycle counting method. It requires that the stress time history be rearranged so that it contains only the peaks and valleys and it starts either with the highest peak or the lowest valley (whichever is greater in absolute magnitude). Then, three consecutive stress points (1, 2, and 3) will define two consecutive ranges as Δ S 12 = | S 1 S 2 | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam 4uamaaBaaaleaacaaIXaGaaGOmaaqabaGccqGH9aqpdaabdaqaaiaa dofadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWGtbWaaSbaaSqaai aaikdaaeqaaaGccaGLhWUaayjcSdaaaa@428A@ and Δ S 23 = | S 2 S 3 | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam 4uamaaBaaaleaacaaIYaGaaG4maaqabaGccqGH9aqpdaabdaqaaiaa dofadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWGtbWaaSbaaSqaai aaiodaaeqaaaGccaGLhWUaayjcSdaaaa@428F@ . A cycle from 1 to 2 is only extracted if Δ S 12 Δ S 23 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam 4uamaaBaaaleaacaaIXaGaaGOmaaqabaGccqGHKjYOcqGHuoarcaWG tbWaaSbaaSqaaiaaikdacaaIZaaabeaaaaa@3F7C@ . Once a cycle is extracted, the two points forming the cycle are discarded and the remaining points are connected to each other. This procedure is repeated until the remaining data points are exhausted.

Simple Load History



Figure 6. Continuous Load History
Since this load history is continuous, it is converted into a load history consisting of peaks and valleys only.


Figure 7. Peaks and Valleys for Rainflow Counting. 1, 2, 3, and 4 are the four peaks and valleys
It is clear that point 4 is the peak stress in the load history, and it will be moved to the front during rearrangement (Figure 8). After rearrangement, the peaks and valleys are renumbered for convenience.


Figure 8. Load History after Rearrangement and Renumbering

Next, pick the first three stress values (1, 2, and 3) and determine if a cycle is present.

If S i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGPbaabeaaaaa@37E9@ represents the stress value, point i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaai aadMgaaeqaaaaa@3711@ then:(2) Δ S 12 = | S 1 S 2 | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam 4uamaaBaaaleaacaaIXaGaaGOmaaqabaGccqGH9aqpdaabdaqaaiaa dofadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWGtbWaaSbaaSqaai aaikdaaeqaaaGccaGLhWUaayjcSdaaaa@428A@ (3) Δ S 23 = | S 2 S 3 | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam 4uamaaBaaaleaacaaIYaGaaG4maaqabaGccqGH9aqpdaabdaqaaiaa dofadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWGtbWaaSbaaSqaai aaiodaaeqaaaGccaGLhWUaayjcSdaaaa@428F@
As you can see from Figure 8 , Δ S 12 Δ S 23 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam 4uamaaBaaaleaacaaIXaGaaGOmaaqabaGccqGHLjYScqGHuoarcaWG tbWaaSbaaSqaaiaaikdacaaIZaaabeaaaaa@3F8D@ ; therefore, no cycle is extracted from point 1 to 2. Now consider the next three points (2, 3, and 4).(4) Δ S 23 = | S 2 S 3 | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam 4uamaaBaaaleaacaaIYaGaaG4maaqabaGccqGH9aqpdaabdaqaaiaa dofadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWGtbWaaSbaaSqaai aaiodaaeqaaaGccaGLhWUaayjcSdaaaa@428F@ (5) Δ S 34 = | S 3 S 4 | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam 4uamaaBaaaleaacaaIYaGaaG4maaqabaGccqGH9aqpdaabdaqaaiaa dofadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWGtbWaaSbaaSqaai aaiodaaeqaaaGccaGLhWUaayjcSdaaaa@428F@
Δ S 23 Δ S 34 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam 4uamaaBaaaleaacaaIYaGaaG4maaqabaGccqGHKjYOcqGHuoarcaWG tbWaaSbaaSqaaiaaiodacaaI0aaabeaaaaa@3F80@ , hence a cycle is extracted from point 2 to 3. Now that a cycle has been extracted, the two points are deleted from the graph.


Figure 9. Delete and Reconnect Remaining Points
The same process is applied to the remaining points:(6) Δ S 14 = | S 1 S 4 | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam 4uamaaBaaaleaacaaIYaGaaG4maaqabaGccqGH9aqpdaabdaqaaiaa dofadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWGtbWaaSbaaSqaai aaiodaaeqaaaGccaGLhWUaayjcSdaaaa@428F@ (7) Δ S 45 = | S 4 S 5 | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam 4uamaaBaaaleaacaaIYaGaaG4maaqabaGccqGH9aqpdaabdaqaaiaa dofadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWGtbWaaSbaaSqaai aaiodaaeqaaaGccaGLhWUaayjcSdaaaa@428F@

In this case, Δ S 14 = Δ S 45 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam 4uamaaBaaaleaacaaIXaGaaGinaaqabaGccqGH9aqpcqGHuoarcaWG tbWaaSbaaSqaaiaaisdacaaI1aaabeaaaaa@3ED3@ , so another cycle is extracted from point 1 to 4. After these two points are also discarded, only point 5 remains; therefore, the rainflow counting process is completed.

Two cycles (2→3 and 1→4) have been extracted from this load history. One of the main reasons for choosing the highest peak/valley and rearranging the load history is to guarantee that the largest cycle is always extracted (in this case, it is 1→4). If you observe the load history prior to rearrangement, and conduct the same rainflow counting process on it, then clearly, the 1→4 cycle is not extracted.

Complex Load History

The rainflow counting process is the same regardless of the number of load history points. However, depending on the location of the highest peak/valley used for rearrangement, it may not be obvious how the rearrangement process is conducted.Figure 10 shows just the rearrangement process for a more complex load history. The subsequent rainflow counting is just an extrapolation of the process mentioned in the simple example above, and is not repeated here.


Figure 10. Continuous Load History
Since this load history is continuous, it is converted into a load history consisting of peaks and valleys only:


Figure 11. Peaks and Valleys for Rainflow Counting
Clearly, load point 11 is the highest valued load and therefore, the load history is now rearranged and renumbered.


Figure 12. Load History After Rearrangement and Renumbering

The load history is rearranged such that all points including and after the highest load are moved to the beginning of the load history and are removed from the end of the load history.

  • Complex Load History

Equivalent Nominal Stress

Since S-N theory deals with uniaxial stress, the stress components need to be resolved into one combined value for each calculation point, at each time step, and then used as equivalent nominal stress applied on the S-N curve.

Various stress combination types are available with the default being "Absolute maximum principle stress". "Absolute maximum principle stress" is recommended for brittle materials, while "Signed von Mises stress" is recommended for ductile material. The sign on the signed parameters is taken from the sign of the Maximum Absolute Principal value.

"Critical plane stress" is also available as a stress combination for uniaxial calculations (stress life and strain life).

Normal Stress resolved at each plane θ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37A9@ is calculated by:(8) σ = σ x ( cos 2 θ ) + σ y ( sin 2 θ ) + 2 σ x y ( cos θ sin θ ) θ = 0 , 10 , 20 , 30......170 degrees , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqaHdp WCcqGH9aqpcqaHdpWCdaWgaaWcbaGaamiEaaqabaGcdaqadaqaaiGa cogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaakiabeI7aXbGaay jkaiaawMcaaiabgUcaRiabeo8aZnaaBaaaleaacaWG5baabeaakmaa bmaabaGaci4CaiaacMgacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaeq iUdehacaGLOaGaayzkaaGaey4kaSIaaGOmaiabeo8aZnaaBaaaleaa caWG4bGaamyEaaqabaGcdaqadaqaaiGacogacaGGVbGaai4CaiabeI 7aXjGacohacaGGPbGaaiOBaiabeI7aXbGaayjkaiaawMcaaaqaaiab eI7aXjabg2da9iaaicdacaGGSaGaaGymaiaaicdacaGGSaGaaGOmai aaicdacaGGSaGaaG4maiaaicdacaGGUaGaaiOlaiaac6cacaGGUaGa aiOlaiaac6cacaaIXaGaaG4naiaaicdacaaMe8Uaaeizaiaabwgaca qGNbGaaeOCaiaabwgacaqGLbGaae4CaiaacYcaaaaa@76D1@
HyperLife expects a number of planes (n) as input, which are converted to equivalent θ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37A9@ using the following formula.(9) θ = 180 n 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdeNaey ypa0ZaaSaaaeaacaaIXaGaaGioaiaaicdaaeaacaWGUbGaeyOeI0Ia aGOmaaaaaaa@3D95@

For example, if number of planes requested is 20, then stress is calculated every 10 degrees.

By default, HyperLife also calculates at θ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37A9@ = 45 and 135-degree planes in addition to the requested number of planes. This is to include the worst possible damage if occurring on these planes.

Mean Stress Correction

Generally, S-N curves are obtained from standard experiments with fully reversed cyclic loading. However, the real fatigue loading could not be fully-reversed, and the normal mean stresses have significant effect on fatigue performance of components. Tensile normal mean stresses are detrimental and compressive normal mean stresses are beneficial, in terms of fatigue strength. Mean stress correction is used to take into account the effect of non-zero mean stresses.

The Gerber parabola and the Goodman line in Haigh's coordinates are widely used when considering mean stress influence, and can be expressed as:

Gerber

When SN curve is of the Stress Ration R = -1(10) S e = S r ( 1 ( S m S u ) 2 )
When SN curve is of the Stress Ratio R != -1(11) S a R = 1 + 4 S e 2 1 + R 2 1 R 2 S u 2 1 1 R 2 S u 2 2 S e 1 + R 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ua8aadaWgaaWcbaWdbiaadggacqGHsislcaWGsbaapaqabaGc peGaeyypa0ZaaeWaa8aabaWdbmaakaaapaqaa8qadaqadaWdaeaape GaaGymaiabgUcaRmaalaaapaqaa8qacaaI0aGaeyyXICTaam4ua8aa daqhaaWcbaWdbiaadwgaa8aabaWdbiaaikdaaaGccqGHflY1daqada WdaeaapeGaaGymaiabgUcaRiaadkfaaiaawIcacaGLPaaapaWaaWba aSqabeaapeGaaGOmaaaaaOWdaeaapeWaaeWaa8aabaWdbiaaigdacq GHsislcaWGsbaacaGLOaGaayzkaaWdamaaCaaaleqabaWdbiaaikda aaGccaWGtbWdamaaBaaaleaapeGaamyDaaWdaeqaaOWaaWbaaSqabe aapeGaaGOmaaaaaaaakiaawIcacaGLPaaaaSqabaGccqGHsislcaaI XaaacaGLOaGaayzkaaGaeyyXIC9aaSaaa8aabaWdbmaabmaapaqaa8 qacaaIXaGaeyOeI0IaamOuaaGaayjkaiaawMcaa8aadaahaaWcbeqa a8qacaaIYaaaaOGaam4ua8aadaWgaaWcbaWdbiaadwhaa8aabeaakm aaCaaaleqabaWdbiaaikdaaaaak8aabaWdbiaaikdacaWGtbWdamaa BaaaleaapeGaamyzaaWdaeqaaOWdbiabgwSixpaabmaapaqaa8qaca aIXaGaey4kaSIaamOuaaGaayjkaiaawMcaa8aadaahaaWcbeqaa8qa caaIYaaaaaaaaaa@6D7B@ (12) S e = S a 1 S m S u 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ua8aadaWgaaWcbaWdbiaadwgaa8aabeaak8qacqGH9aqpdaWc aaWdaeaapeGaam4ua8aadaWgaaWcbaWdbiaadggaa8aabeaaaOqaa8 qacaaIXaGaeyOeI0YaaeWaa8aabaWdbiaadofapaWaaSbaaSqaa8qa caWGTbaapaqabaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUv gaiuaak8qacqWFviYGcaWGtbWdamaaBaaaleaapeGaamyDaaWdaeqa aaGcpeGaayjkaiaawMcaa8aadaahaaWcbeqaa8qacaaIYaaaaaaaaa a@4F4B@

Goodman

When SN curve is of the Stress Ratio R = -1

(13) S e = S r ( 1 S m S u )
When SN curve is of the Stress Ratio R != -1(14) S a R = S a S u S u S m + S a 1 + R 1 R MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ua8aadaWgaaWcbaWdbiaadggacqGHsislcaWGsbaapaqabaGc peGaeyypa0ZaaSaaa8aabaWdbiaadofapaWaaSbaaSqaa8qacaWGHb aapaqabaGcpeGaeyyXICTaam4ua8aadaWgaaWcbaWdbiaadwhaa8aa beaaaOqaa8qacaWGtbWdamaaBaaaleaapeGaamyDaaWdaeqaaOWdbi abgkHiTiaadofapaWaaSbaaSqaa8qacaWGTbaapaqabaGcpeGaey4k aSIaam4ua8aadaWgaaWcbaWdbiaadggaa8aabeaak8qadaqadaWdae aapeWaaSaaa8aabaWdbiaaigdacqGHRaWkcaWGsbaapaqaa8qacaaI XaGaeyOeI0IaamOuaaaaaiaawIcacaGLPaaaaaaaaa@5164@

Gerber2

Improves the Gerber method by ignoring the effect of negative mean stress.

When SN curve is of the Stress Ratio R != -1

If S m > 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ua8aadaWgaaWcbaWdbiaad2gaa8aabeaak8qacqGH+aGpcaaI Waaaaa@3A09@ , same as Gerber

If S m 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ua8aadaWgaaWcbaWdbiaad2gaa8aabeaak8qacqGHKjYOcaaI Waaaaa@3AB6@ , S a R = S a MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ua8aadaWgaaWcbaWdbiaadggacqGHsislcaWGsbaapaqabaGc peGaeyypa0Jaam4ua8aadaWgaaWcbaWdbiaadggaa8aabeaaaaa@3D1D@

Soderberg

Is slightly different from GOODMAN; the mean stress is normalized by yield stress instead of ultimate tensile stress.

When SN curve is of the Stress Ratio R = -1(15) S e = S a ( 1 S m S y ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGLbaabeaakiabg2da9maalaaabaGaam4uamaaBaaaleaa caWGHbaabeaaaOqaamaabmaabaGaaGymaiabgkHiTmaalaaabaGaam 4uamaaBaaaleaacaWGTbaabeaaaOqaaiaadofadaWgaaWcbaGaamyE aaqabaaaaaGccaGLOaGaayzkaaaaaaaa@4246@
Where,
S e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGLbaabeaaaaa@37E5@
Equivalent stress amplitude
S a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGHbaabeaaaaa@37E1@
Stress amplitude
S m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGTbaabeaaaaa@37ED@
Mean stress
S y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWG5baabeaaaaa@37F9@
Yield stress
When SN curve is of the Stress Ratio R != -1(16) S a R = S a S y S y S m + S a 1 + R 1 R MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ua8aadaWgaaWcbaWdbiaadggacqGHsislcaWGsbaapaqabaGc peGaeyypa0ZaaSaaa8aabaWdbiaadofapaWaaSbaaSqaa8qacaWGHb aapaqabaGcpeGaeyyXICTaam4ua8aadaWgaaWcbaWdbiaadMhaa8aa beaaaOqaa8qacaWGtbWdamaaBaaaleaapeGaamyEaaWdaeqaaOWdbi abgkHiTiaadofapaWaaSbaaSqaa8qacaWGTbaapaqabaGcpeGaey4k aSIaam4ua8aadaWgaaWcbaWdbiaadggaa8aabeaak8qadaqadaWdae aapeWaaSaaa8aabaWdbiaaigdacqGHRaWkcaWGsbaapaqaa8qacaaI XaGaeyOeI0IaamOuaaaaaiaawIcacaGLPaaaaaaaaa@516C@

FKM

If only one slope field is specified for mean stress correction, the corresponding Mean Stress Sensitivity value ( M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaaaa@36C9@ ) for Mean Stress Correction is set equal to Slope in Regime 2 (Figure 14). Based on FKM-Guidelines, the Haigh diagram is divided into four regimes based on the Stress ratio ( R = S min / S max MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGsbGaeyypa0Jaam4uamaaBaaaleaaciGGTbGaaiyAaiaac6ga aeqaaOGaai4laiaadofadaWgaaWcbaGaciyBaiaacggacaGG4baabe aaaaa@431F@ ) values. The Corrected value is then used to choose the S-N curve for the damage and life calculation stage.


Figure 14.
Note: The FKM equations below illustrate the calculation of Corrected Stress Amplitude ( S e A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWGLbaabaGaamyqaaaaaaa@38AC@ ). The actual value of stress used in the Damage calculations is the Corrected stress range (which is 2 S e A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiabgw SixlaadofadaqhaaWcbaGaamyzaaqaaiaadgeaaaaaaa@3BB2@ ). These equations apply for SN curves input by the user (by default, any user-defined SN curve is expected to be input for a stress ratio of R=1.0).

There are two available options for FKM correction in HyperLife. They are activated by setting FKM MSS to 1 slope/4 slopes in the Assign Material dialog.

If only one slope is defined and if mean stress correction on an SN module is set to FKM:
Regime 1 (R > 1.0)
S e A = S a ( 1 M ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWGLbaabaGaamyqaaaakiabg2da9iaadofadaWgaaWcbaGa amyyaaqabaGcdaqadaqaaiaaigdacqGHsislcaWGnbaacaGLOaGaay zkaaaaaa@3FB3@
Regime 2 (-∞ ≤ R ≤ 0.0)
S e A = S a + M * S m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWGLbaabaGaamyqaaaakiabg2da9iaadofadaWgaaWcbaGa amyyaaqabaGccqGHRaWkcaWGnbGaaiOkaiaadofadaWgaaWcbaGaam yBaaqabaaaaa@4008@
Regime 3 (0.0 < R < 0.5)
S e A = ( 1 + M ) S a + ( M 3 ) S m 1 + M 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWGLbaabaGaamyqaaaakiabg2da9maabmaabaGaaGymaiab gUcaRiaad2eaaiaawIcacaGLPaaadaWcaaqaaiaadofadaWgaaWcba GaamyyaaqabaGccqGHRaWkdaqadaqaamaaliaabaGaamytaaqaaiaa iodaaaaacaGLOaGaayzkaaGaam4uamaaBaaaleaacaWGTbaabeaaaO qaaiaaigdacqGHRaWkdaWccaqaaiaad2eaaeaacaaIZaaaaaaaaaa@4902@
Regime 4 (R ≥ 0.5)
S e A = 3 S a ( 1 + M ) 2 3 + M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWGLbaabaGaamyqaaaakiabg2da9maalaaabaGaaG4maiaa dofadaWgaaWcbaGaamyyaaqabaGcdaqadaqaaiaaigdacqGHRaWkca WGnbaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaGcbaGaaG4m aiabgUcaRiaad2eaaaaaaa@43D9@
Where,
S e A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWGLbaabaGaamyqaaaaaaa@38AC@
Stress amplitude after mean stress correction (Endurance stress)
S m
Mean stress
S a
Stress amplitude
M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaaaa@36C8@
Slope entered for region 2

If all four slopes are specified for mean stress correction, the corresponding Mean Stress Sensitivity values are slopes for controlling all four regimes. Based on FKM-Guidelines, the Haigh diagram is divided into four regimes based on the Stress ratio ( R = S min / S max MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGsbGaeyypa0Jaam4uamaaBaaaleaaciGGTbGaaiyAaiaac6ga aeqaaOGaai4laiaadofadaWgaaWcbaGaciyBaiaacggacaGG4baabe aaaaa@431F@ ) values. The corrected value is then used to choose the S-N curve for the damage and life calculation stage.

If four slopes are defined and mean stress correction is set to FKM:
Regime 1 (R > 1.0)
S e = ( S a + M 1 S m ) ( ( 1 M 2 ) / ( 1 M 1 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWGLbaabaaaaOGaeyypa0ZaaeWaaeaacaWGtbWaaSbaaSqa aiaadggaaeqaaOGaey4kaSIaamytamaaBaaaleaacaaIXaaabeaaki aadofadaWgaaWcbaGaamyBaaqabaaakiaawIcacaGLPaaadaqadaqa amaalyaabaWaaeWaaeaacaaIXaGaeyOeI0IaamytamaaBaaaleaaca aIYaaabeaaaOGaayjkaiaawMcaaaqaamaabmaabaGaaGymaiabgkHi Tiaad2eadaWgaaWcbaGaaGymaaqabaaakiaawIcacaGLPaaaaaaaca GLOaGaayzkaaaaaa@4CA0@
Regime 2 (-∞ ≤ R ≤ 0.0)
S e = S a + M 2 S m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWGLbaabaaaaOGaeyypa0Jaam4uamaaBaaaleaacaWGHbaa beaakiabgUcaRiaad2eadaWgaaWcbaGaaGOmaaqabaGccaWGtbWaaS baaSqaaiaad2gaaeqaaaaa@3F86@
Regime 3 (0.0 < R < 0.5)
S e = ( 1 + M 2 ) S a + M 3 S m 1 + M 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWGLbaabaaaaOGaeyypa0ZaaeWaaeaacaaIXaGaey4kaSIa amytamaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaamaalaaaba Gaam4uamaaBaaaleaacaWGHbaabeaakiabgUcaRiaad2eadaWgaaWc baGaaG4maaqabaGccaWGtbWaaSbaaSqaaiaad2gaaeqaaaGcbaGaaG ymaiabgUcaRiaad2eadaWgaaWcbaGaaG4maaqabaaaaaaa@47E3@
Regime 4 (R ≥ 0.5)
S e = ( ( 1 + 3 M 3 ) S a M 4 ( 1 + 3 M 3 ) S m ) ( ( 1 + M 2 ) / ( ( 1 3 M 4 ) ( 1 + M 3 ) ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGLbaabeaakiabg2da9maabmaabaWaaeWaaeaacaaIXaGa ey4kaSIaaG4maiaad2eadaWgaaWcbaGaaG4maaqabaaakiaawIcaca GLPaaacaWGtbWaaSbaaSqaaiaadggaaeqaaOGaeyOeI0Iaamytamaa BaaaleaacaaI0aaabeaakmaabmaabaGaaGymaiabgUcaRiaaiodaca WGnbWaaSbaaSqaaiaaiodaaeqaaaGccaGLOaGaayzkaaGaam4uamaa BaaaleaacaWGTbaabeaaaOGaayjkaiaawMcaamaabmaabaWaaeWaae aacaaIXaGaey4kaSIaamytamaaBaaaleaacaaIYaaabeaaaOGaayjk aiaawMcaaiaac+cadaqadaqaamaabmaabaGaaGymaiabgkHiTiaaio dacaWGnbWaaSbaaSqaaiaaisdaaeqaaaGccaGLOaGaayzkaaWaaeWa aeaacaaIXaGaey4kaSIaamytamaaBaaaleaacaaIZaaabeaaaOGaay jkaiaawMcaaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@5FC2@
Where,
S e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGLbaabeaaaaa@37E4@
Fully reversed fatigue strength (Endurance stress)
S m
Mean stress
S a
Stress amplitude
M i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGPbaabeaaaaa@37E3@
Slopes at each region

Interpolate

Multi-Mean SN Curves
Life is usually determined by interpolation of two SN curves with respect to mean stress. A log function mentioned below is a 10 base log function.


Figure 15.
Case A
If a cycle has a mean stress of 150MPa at point A, HyperLife locates point 1 and point 2 in Figure 15. Then HyperLife linearly interpolates logN1 and logN2 with respect to mean stress in order to determine logN_A at mean stress 150MPa. Once logN_A is determined, life (N_A) and corresponding damage can be determined.
Case B
If the cycle has a mean stress greater than the maximum mean stress of the curve set (180MPa in this case), HyperLife offers two options to choose its behavior.
Option 1 , Curve Extrapolation = False
Use an SN curve of the maximum mean stress (the SN curve of mean stress 180 MPa in this case). In the example in HyperLife, N1 is the life HyperLife will report.
Option 2 , Curve Extrapolation = True
Extrapolate log(N) of the two SN curves with the highest mean stress values. In the example in Figure 15, log(N) will be extrapolated from log(N1) and log(N2) with respect to mean stress.
Case C
If the cycle has a mean stress less than the minimum mean stress of the curve set (90MPa in this case), HyperLife will use the SN curve of the minimum mean stress to determine life. In the example in Figure 15, life will be N2.
Multi-Stress Ratio SN Curve
Life is usually determined by interpolation of 2 SN curves with respect to mean stress. When multi-stress ratio SN curves are used, HyperLife assumes that you will not define SN curves with stress ratio greater than or equal to 1, which are SN curves with compressive stress or zero stress amplitude. A log function mentioned below is a 10 base log function. R denotes a stress ratio.


Figure 16.
Case A
If a cycle has R = -0.2 at point A, HyperLife locates point 1 and point 2 in Figure 16. Then HyperLife linearly interpolates logN1 and logN2 with respect to mean stress in order to determine logN_A at R = -0.2. Once R value and stress amplitude of the cycle are given, we can always calculate mean stress of the cycle. Once logN_A is determined, life (N_A) and corresponding damage can be determined. It is worthwhile to mention that HyperLife does not use stress ratio for interpolation because R can be an infinite value when maximum stress is zero.
Case B
If the cycle has R greater than the maximum R of the curve set (R=0 in this case), HyperLife offers two options to choose its behavior.
Option 1, Curve Extrapolation = False
Use an SN curve of the maximum R (the SN curve of R= 0 in this case). In the example in Figure 16, N1 is the life HyperLife will report.
Option 2, Curve Extrapolation = True
Extrapolate log(N) of the two SN curves with the highest R values. In the example in Figure 16, log(N) will be extrapolated from log(N1) and log(N2) with respect to mean stress.
Case C
If the cycle has R less than the minimum R of the curve set (R= -1 in this case), HyperLife will use the SN curve of the minimum R to determine life. In the example in Figure 16, life will be N2.
Constant Life Haigh Diagram
Life is usually determined by interpolation of two Haigh diagrams with respect to stress amplitude. A log function mentioned below is a 10 base log function.


Figure 17.
Interpolation on a Constant Mean Stress Line
If you choose constant mean stress line for linear interpolation of Haigh diagram, HyperLife interpolates two Haigh diagrams on a constant mean stress line as described in the following.
Case A
If a cycle has a mean stress and stress amplitude at point A, HyperLife locates point 1 and point 2 in Figure 17. Life of point A should be between 1000 and 100000. HyperLife linearly interpolates log(1000) and log(100000) with respect to stress amplitude along Sm_A constant mean stress line in order to determine logN_A at point A. Once logN_A is determined, life (N_A) and corresponding damage can be determined.
Case B
If a point (mean stress, stress amplitude) is located above or below all the Haigh diagrams, life of the point is calculated by extrapolation of the two highest or two lowest curves. In the example in Figure 17, log(1000) and log(100000) will be extrapolated with respect to stress amplitude along Sm_B constant mean stress line.
Case C
In this case, stress amplitude at point 5 and point 6 may be calculated from extrapolation. Once stress amplitudes become available at the 2 points, a procedure described in case A is applied.
Interpolation on a Constant Stress Ratio Line


Figure 18.
If you choose constant stress ration line for linear interpolation of Haigh diagram, HyperLife interpolates two Haigh diagrams on a constant stress ratio line as described in the following.
Case A
If a cycle has a mean stress and stress amplitude at point A, HyperLife locates point 1 and point 2 in Figure 18. Life of point A should be between 1000 and 100000. HyperLife linearly interpolates log(1000) and log(100000) with respect to stress amplitude along RA constant stress ratio line in order to determine logN_A at point A. Once logN_A is determined, life (N_A) and corresponding damage can be determined.
Case B
If a point (mean stress, stress amplitude) is located above or below all the Haigh diagrams, life of the point is calculated by extrapolation of the two highest or two lowest curves. In the example in Figure 18, log(1000) and log(100000) will be extrapolated with respect to stress amplitude along R=RB constant stress ratio line.
Case C
In this case, stress amplitude at point 5 and point 6 may be calculated from extrapolation. Once stress amplitudes become available at the 2 points, a procedure described in case A is applied on constant stress ratio line R=RC.

Damage Accumulation Model

Palmgren-Miner's linear damage summation rule is used. Failure is predicted when:(17) D i = n i N i f 1.0
Where,
N i f
Materials fatigue life (number of cycles to failure) from its S-N curve at a combination of stress amplitude and means stress level i .
n i
Number of stress cycles at load level i .
D i
Cumulative damage under n i load cycle.

The linear damage summation rule does not take into account the effect of the load sequence on the accumulation of damage, due to cyclic fatigue loading. However, it has been proved to work well for many applications.

The fatigue life or damage obtained for the event specified can be scaled in HyperLife as shown below. Scaled life or scaled damage will be available as additional output from the fatigue evaluation. (18) S c a l e d   L i f e   =   E q u i v a l e n t   l i f e   u n i t s D a m a g e MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaado gacaWGHbGaamiBaiaadwgacaWGKbaeaaaaaaaaa8qacaGGGcWdaiaa dYeacaWGPbGaamOzaiaadwgapeGaaiiOa8aacqGH9aqppeGaaiiOa8 aadaWcaaqaaiaadweacaWGXbGaamyDaiaadMgacaWG2bGaamyyaiaa dYgacaWGLbGaamOBaiaadshapeGaaiiOa8aacaWGSbGaamyAaiaadA gacaWGLbWdbiaacckapaGaamyDaiaad6gacaWGPbGaamiDaiaadoha aeaacaWGebGaamyyaiaad2gacaWGHbGaam4zaiaadwgaaaaaaa@5D93@ Life (which is 1/Damage) is scaled in equivalent units.(19) S c a l e d   D a m a g e   =   D a m a g e A l l o w a b l e   M i n e r   S u m MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaado gacaWGHbGaamiBaiaadwgacaWGKbaeaaaaaaaaa8qacaGGGcGaamir aiaadggacaWGTbGaamyyaiaadEgacaWGLbGaaiiOa8aacqGH9aqppe GaaiiOa8aadaWcaaqaaiaadseacaWGHbGaamyBaiaadggacaWGNbGa amyzaaqaaiaadgeacaWGSbGaamiBaiaad+gacaWG3bGaamyyaiaadk gacaWGSbGaamyza8qacaGGGcWdaiaad2eacaWGPbGaamOBaiaadwga caWGYbWdbiaacckapaGaam4uaiaadwhacaWGTbaaaaaa@5D05@ Linearly accumulated damage can be modified by applying the Allowable Miner sum. Scaled life and scaled damage are supported for SN, EN, Transient Fatigue, Weld Fatigue, and Vibrational Fatigue.

Safety Factor

Safety factor is calculated based on the endurance limit or target stress (at target life) against the stress amplitude from the working stress history.

HyperLife calculates this ratio via two criteria:
  • Mean Stress = Constant
  • Stress Ratio = Constant

The safety factor (SF) based on the mean stress correction applied is given by the following equations.

Mean Stress = Constant
  1. Goodman or Soderberg

    When SN curve is of the Stress Ratio R = -1

    (20) S F = s σ a = s e σ a 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaadA eacqGH9aqpdaWcaaqaaiaadohaaeaacqaHdpWCdaWgaaWcbaGaamyy aaqabaaaaOGaeyypa0ZaaSaaaeaacaWGZbWaaSbaaSqaaiaadwgaae qaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadggadaWgaaadbaGaaGimaaqa baaaleqaaaaaaaa@437B@

    s e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGLbaabeaaaaa@3804@ = Target stress amplitude against the target life from the modified SN curve

    σ a 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadggadaWgaaadbaGaaGimaaqabaaaleqaaaaa@39BD@ = Stress amplitude after mean stress correction



    Figure 19.
    When SN curve is of the Stress Ratio R != -1


    Figure 20.

    σ a MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHdpWCpaWaaSbaaSqaa8qacaWGHbaapaqabaaaaa@3916@ = Stress Amplitude

    σ m MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHdpWCpaWaaSbaaSqaa8qacaWGTbaapaqabaaaaa@3922@ = Mean Stress

    S e R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGtbWdamaaBaaaleaapeGaamyzaiabgkHiTiaadkfaa8aabeaa aaa@39F3@ = Endurance limit obtained from SN curve with R ratio

    S e m MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGtbWdamaaBaaaleaapeGaamyzaiabgkHiTiaad2gaa8aabeaa aaa@3A0E@ = Mean Stress corresponding to S e R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGtbWdamaaBaaaleaapeGaamyzaiabgkHiTiaadkfaa8aabeaa aaa@39F3@

    If R   >   1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaa baaaaaaaaapeGaamOuaiaabccacqGH+aGpcaqGGaGaeyOeI0IaaGym aaWdaeqaaaaa@3B1B@ , s e = S e R 1 s m R U T S MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Ca8aadaWgaaWcbaWdbiaadwgaa8aabeaak8qacqGH9aqpdaWc aaWdaeaapeGaam4ua8aadaWgaaWcbaWdbiaadwgacqGHsislcaWGsb aapaqabaaakeaapeGaaGymaiabgkHiTmaalaaapaqaa8qacaWGZbWd amaaBaaaleaapeGaamyBaiabgkHiTiaadkfaa8aabeaaaOqaa8qaca WGvbGaamivaiaadofaaaaaaaaa@4612@

    (21) s m R = S e R .   1 + R 1 R MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Ca8aadaWgaaWcbaWdbiaad2gacqGHsislcaWGsbaapaqabaGc peGaeyypa0Jaam4ua8aadaWgaaWcbaWdbiaadwgacqGHsislcaWGsb aapaqabaGcpeGaaiOlaiaacckadaWcaaWdaeaapeGaaGymaiabgUca Riaadkfaa8aabaWdbiaaigdacqGHsislcaWGsbaaaaaa@4642@

    If R   <   1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaa baaaaaaaaapeGaamOuaiaabccacqGH8aapcaqGGaGaeyOeI0IaaGym aaWdaeqaaaaa@3B17@ , S e = S e R MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ua8aadaWgaaWcbaWdbiaadwgaa8aabeaak8qacqGH9aqpcaWG tbWdamaaBaaaleaapeGaamyzaiabgkHiTiaadkfaa8aabeaaaaa@3D25@

    If σ m >   0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHdpWCpaWaaSbaaSqaa8qacaWGTbaapaqabaGcdaWgaaWcbaWd biabg6da+iaabccacaaIWaaapaqabaaaaa@3BDC@ , s a = σ a 1 σ m U T S MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Ca8aadaWgaaWcbaWdbiaadggaa8aabeaak8qacqGH9aqpdaWc aaWdaeaapeGaeq4Wdm3damaaBaaaleaapeGaamyyaaWdaeqaaaGcba WdbiaaigdacqGHsisldaWcaaWdaeaapeGaeq4Wdm3damaaBaaaleaa peGaamyBaaWdaeqaaaGcbaWdbiaadwfacaWGubGaam4uaaaaaaaaaa@4438@

    If σ m 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHdpWCpaWaaSbaaSqaa8qacaWGTbaapaqabaGcdaWgaaWcbaWd biabgsMiJkaaicdaa8aabeaaaaa@3BE6@ , s a = σ a MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Ca8aadaWgaaWcbaWdbiaadggaa8aabeaak8qacqGH9aqpcqaH dpWCpaWaaSbaaSqaa8qacaWGHbaapaqabaaaaa@3C64@ (22) SF=  S e S a MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4uaiaadAeacqGH9aqpcaGGGcWaaSaaa8aabaWdbiaadofapaWa aSbaaSqaa8qacaWGLbaapaqabaaakeaapeGaam4ua8aadaWgaaWcba Wdbiaadggaa8aabeaaaaaaaa@3E53@
  2. Gerber
    (23) S F = s σ a = s e σ a 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaadA eacqGH9aqpdaWcaaqaaiaadohaaeaacqaHdpWCdaWgaaWcbaGaamyy aaqabaaaaOGaeyypa0ZaaSaaaeaacaWGZbWaaSbaaSqaaiaadwgaae qaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadggadaWgaaadbaGaaGimaaqa baaaleqaaaaaaaa@437B@


    Figure 21.
    When SN curve is of the Stress Ratio R != -1


    Figure 22.
    (24) S a = σ a 1 σ m U T S 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadabaaaaaaa aapeGaa83ua8aadaWgaaWcbaWdbiaa=fgaa8aabeaak8qacqGH9aqp cqaHdpWCpaWaaSbaaSqaa8qacaWGHbaapaqabaGcpeGaeyyXIC9aae Waa8aabaWdbiaaigdacqGHsisldaqadaWdaeaapeWaaSaaa8aabaWd biaa=n8apaWaaSbaaSqaa8qacaWFTbaapaqabaaakeaapeGaa8xvai aa=rfacaWFtbaaaaGaayjkaiaawMcaa8aadaahaaWcbeqaa8qacaaI YaaaaaGccaGLOaGaayzkaaaaaa@4A0D@ (25) S e = S e R 1 s m R U T S 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadabaaaaaaa aapeGaa83ua8aadaWgaaWcbaWdbiaa=vgaa8aabeaak8qacqGH9aqp caWGtbWdamaaBaaaleaapeGaamyzaiabgkHiTiaadkfaa8aabeaak8 qacqGHflY1daqadaWdaeaapeGaaGymaiabgkHiTmaabmaapaqaa8qa daWcaaWdaeaapeGaam4Ca8aadaWgaaWcbaWdbiaad2gacqGHsislca WGsbaapaqabaaakeaapeGaamyvaiaadsfacaWGtbaaaaGaayjkaiaa wMcaa8aadaahaaWcbeqaa8qacaaIYaaaaaGccaGLOaGaayzkaaaaaa@4C73@ (26) s m R = S e R .   1 + R 1 R MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Ca8aadaWgaaWcbaWdbiaad2gacqGHsislcaWGsbaapaqabaGc peGaeyypa0Jaam4ua8aadaWgaaWcbaWdbiaadwgacqGHsislcaWGsb aapaqabaGcpeGaaiOlaiaacckadaWcaaWdaeaapeGaaGymaiabgUca Riaadkfaa8aabaWdbiaaigdacqGHsislcaWGsbaaaaaa@4642@ (27) SF=  S e S a MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4uaiaadAeacqGH9aqpcaGGGcWaaSaaa8aabaWdbiaadofapaWa aSbaaSqaa8qacaWGLbaapaqabaaakeaapeGaam4ua8aadaWgaaWcba Wdbiaadggaa8aabeaaaaaaaa@3E53@
  3. Gerber2
    1. (28) σ m > 0 : S F = s σ a = s e σ a 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqaHdp WCdaWgaaWcbaGaamyBaaqabaGccqGH+aGpcaaIWaGaaiOoaaqaaiaa ykW7caaMc8Uaam4uaiaadAeacqGH9aqpdaWcaaqaaiaadohaaeaacq aHdpWCdaWgaaWcbaGaamyyaaqabaaaaOGaeyypa0ZaaSaaaeaacaWG ZbWaaSbaaSqaaiaadwgaaeqaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadg gadaWgaaadbaGaaGimaaqabaaaleqaaaaaaaaa@4C03@
    2. (29) σ m 0 : S F = s σ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqaHdp WCdaWgaaWcbaGaamyBaaqabaGccqGHKjYOcaaIWaGaaiOoaaqaaiaa ykW7caaMc8Uaam4uaiaadAeacqGH9aqpdaWcaaqaaiaadohaaeaacq aHdpWCdaWgaaWcbaGaamyyaaqabaaaaaaaaa@45B1@

    When SN curve is of the Stress Ratio R != -1

    If R   >   1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWHsbGaaeiiaiabg6da+iaabccacqGHsislcaWHXaaaaa@3AE3@ (30) S e = S e R 1 s m R U T S 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadabaaaaaaa aapeGaa83ua8aadaWgaaWcbaWdbiaa=vgaa8aabeaak8qacqGH9aqp caWGtbWdamaaBaaaleaapeGaamyzaiabgkHiTiaadkfaa8aabeaak8 qacqGHflY1daqadaWdaeaapeGaaGymaiabgkHiTmaabmaapaqaa8qa daWcaaWdaeaapeGaam4Ca8aadaWgaaWcbaWdbiaad2gacqGHsislca WGsbaapaqabaaakeaapeGaamyvaiaadsfacaWGtbaaaaGaayjkaiaa wMcaa8aadaahaaWcbeqaa8qacaaIYaaaaaGccaGLOaGaayzkaaaaaa@4C73@ (31) s m R = S e R .   1 + R 1 R MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Ca8aadaWgaaWcbaWdbiaad2gacqGHsislcaWGsbaapaqabaGc peGaeyypa0Jaam4ua8aadaWgaaWcbaWdbiaadwgacqGHsislcaWGsb aapaqabaGcpeGaaiOlaiaacckadaWcaaWdaeaapeGaaGymaiabgUca Riaadkfaa8aabaWdbiaaigdacqGHsislcaWGsbaaaaaa@4642@

    If R   <   1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWHsbGaaeiiaiabgYda8iaabccacqGHsislcaWHXaaaaa@3ADF@ , S e = S e R MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadabaaaaaaa aapeGaa83ua8aadaWgaaWcbaWdbiaa=vgaa8aabeaak8qacqGH9aqp caWGtbWdamaaBaaaleaapeGaamyzaiabgkHiTiaadkfaa8aabeaaaa a@3D29@

    If σ m >   0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHdpWCpaWaaSbaaSqaa8qacaWGTbaapaqabaGcdaWgaaWcbaWd biabg6da+iaabccacaaIWaaapaqabaaaaa@3BDC@ , S a = σ a 1 σ m U T S 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadabaaaaaaa aapeGaa83ua8aadaWgaaWcbaWdbiaa=fgaa8aabeaak8qacqGH9aqp cqaHdpWCpaWaaSbaaSqaa8qacaWGHbaapaqabaGcpeWaaeWaa8aaba WdbiaaigdacqGHsisldaqadaWdaeaapeWaaSaaa8aabaWdbiabeo8a Z9aadaWgaaWcbaWdbiaad2gaa8aabeaaaOqaa8qacaWFvbGaa8hvai aa=nfaaaaacaGLOaGaayzkaaWdamaaCaaaleqabaWdbiaaikdaaaaa kiaawIcacaGLPaaaaaa@4843@

    If σ m 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHdpWCpaWaaSbaaSqaa8qacaWGTbaapaqabaGcdaWgaaWcbaWd biabgsMiJkaaicdaa8aabeaaaaa@3BE6@ , s a = σ a MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Ca8aadaWgaaWcbaWdbiaadggaa8aabeaak8qacqGH9aqpcqaH dpWCpaWaaSbaaSqaa8qacaWGHbaapaqabaaaaa@3C64@ (32) SF=  S e S a MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4uaiaadAeacqGH9aqpcaGGGcWaaSaaa8aabaWdbiaadofapaWa aSbaaSqaa8qacaWGLbaapaqabaaakeaapeGaam4ua8aadaWgaaWcba Wdbiaadggaa8aabeaaaaaaaa@3E53@
  4. FKM
    (33) S F = s ' e σ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaadA eacqGH9aqpdaWcaaqaaiaadohacaGGNaWaaSbaaSqaaiaadwgaaeqa aaGcbaGaeq4Wdm3aaSbaaSqaaiaadggaaeqaaaaaaaa@3E47@
    1. (34) σ m < s e 1 m 2 s ' e = m , ( σ m + s e 1 m 2 ) + s e 1 m 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqaHdp WCdaWgaaWcbaGaamyBaaqabaGccqGH8aapdaWcaaqaaiabgkHiTiaa dohadaWgaaWcbaGaamyzaaqabaaakeaacaaIXaGaeyOeI0IaamyBam aaBaaaleaacaaIYaaabeaaaaaakeaacaaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGZbGaai4jamaaBa aaleaacaWGLbaabeaakiabg2da9iabgkHiTiaad2gacaGGSaGaaGPa VpaabmaabaGaeq4Wdm3aaSbaaSqaaiaad2gaaeqaaOGaey4kaSYaaS aaaeaacaWGZbWaaSbaaSqaaiaadwgaaeqaaaGcbaGaaGymaiabgkHi Tiaad2gadaWgaaWcbaGaaGOmaaqabaaaaaGccaGLOaGaayzkaaGaey 4kaSYaaSaaaeaacaWGZbWaaSbaaSqaaiaadwgaaeqaaaGcbaGaaGym aiabgkHiTiaad2gadaWgaaWcbaGaaGOmaaqabaaaaaaaaa@67E9@
    2. (35) s e 1 m 2 σ m < s e 1 + m 2 s ' e = m 2 σ m + s e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWcaa qaaiabgkHiTiaadohadaWgaaWcbaGaamyzaaqabaaakeaacaaIXaGa eyOeI0IaamyBamaaBaaaleaacaaIYaaabeaaaaGccqGHKjYOcqaHdp WCdaWgaaWcbaGaamyBaaqabaGccqGH8aapdaWcaaqaaiaadohadaWg aaWcbaGaamyzaaqabaaakeaacaaIXaGaey4kaSIaamyBamaaBaaale aacaaIYaaabeaaaaaakeaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGZbGaai4jamaaBaaa leaacaWGLbaabeaakiabg2da9iabgkHiTiaad2gadaWgaaWcbaGaaG OmaaqabaGccqaHdpWCdaWgaaWcbaGaamyBaaqabaGccqGHRaWkcaWG ZbWaaSbaaSqaaiaadwgaaeqaaaaaaa@7026@
    3. (36) s e 1 + m 2 σ m < 3 ( 1 + m 3 ) 1 + 3 m 3 · s e 1 + m 2 s ' e = m 3 ( σ m s e 1 + m 2 ) + s e 1 + m 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWcaa qaaiaadohadaWgaaWcbaGaamyzaaqabaaakeaacaaIXaGaey4kaSIa amyBamaaBaaaleaacaaIYaaabeaaaaGccqGHKjYOcqaHdpWCdaWgaa WcbaGaamyBaaqabaGccqGH8aapdaWcaaqaaiaaiodacaGGOaGaaGym aiabgUcaRiaad2gadaWgaaWcbaGaaG4maaqabaGccaGGPaaabaGaaG ymaiabgUcaRiaaiodacaWGTbWaaSbaaSqaaiaaiodaaeqaaaaakiaa ykW7cqWIpM+zcaaMc8+aaSaaaeaacaWGZbWaaSbaaSqaaiaadwgaae qaaaGcbaGaaGymaiabgUcaRiaad2gadaWgaaWcbaGaaGOmaaqabaaa aaGcbaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8Uaam4CaiaacEcadaWgaaWcbaGaamyzaaqaba GccqGH9aqpcqGHsislcaWGTbWaaSbaaSqaaiaaiodaaeqaaOWaaeWa aeaacqaHdpWCdaWgaaWcbaGaamyBaaqabaGccqGHsisldaWcaaqaai aadohadaWgaaWcbaGaamyzaaqabaaakeaacaaIXaGaey4kaSIaamyB amaaBaaaleaacaaIYaaabeaaaaaakiaawIcacaGLPaaacqGHRaWkda WcaaqaaiaadohadaWgaaWcbaGaamyzaaqabaaakeaacaaIXaGaey4k aSIaamyBamaaBaaaleaacaaIYaaabeaaaaaaaaa@8A4C@
    4. (37) 3 ( 1 + m 3 ) 1 + 3 m 3 · s e 1 + m 2 σ m s ' e = m 4 ( σ m 3 ( 1 + m 3 ) 1 + 3 m 3 · s e 1 + m 2 ) + 1 3 ( 3 ( 1 + m 2 ) 1 + 3 m 3 · s e 1 + m 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWcaa qaaiaaiodacaGGOaGaaGymaiabgUcaRiaad2gadaWgaaWcbaGaaG4m aaqabaGccaGGPaaabaGaaGymaiabgUcaRiaaiodacaWGTbWaaSbaaS qaaiaaiodaaeqaaaaakiaaykW7cqWIpM+zcaaMc8+aaSaaaeaacaWG ZbWaaSbaaSqaaiaadwgaaeqaaaGcbaGaaGymaiabgUcaRiaad2gada WgaaWcbaGaaGOmaaqabaaaaOGaeyizImQaeq4Wdm3aaSbaaSqaaiaa d2gaaeqaaaGcbaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8Uaam4CaiaacEcadaWgaaWcbaGaam yzaaqabaGccqGH9aqpcqGHsislcaWGTbWaaSbaaSqaaiaaisdaaeqa aOWaaeWaaeaacqaHdpWCdaWgaaWcbaGaamyBaaqabaGccqGHsislda WcaaqaaiaaiodacaGGOaGaaGymaiabgUcaRiaad2gadaWgaaWcbaGa aG4maaqabaGccaGGPaaabaGaaGymaiabgUcaRiaaiodacaWGTbWaaS baaSqaaiaaiodaaeqaaaaakiaaykW7cqWIpM+zcaaMc8+aaSaaaeaa caWGZbWaaSbaaSqaaiaadwgaaeqaaaGcbaGaaGymaiabgUcaRiaad2 gadaWgaaWcbaGaaGOmaaqabaaaaaGccaGLOaGaayzkaaGaey4kaSYa aSaaaeaacaaIXaaabaGaaG4maaaadaqadaqaamaalaaabaGaaG4mai aacIcacaaIXaGaey4kaSIaamyBamaaBaaaleaacaaIYaaabeaakiaa cMcaaeaacaaIXaGaey4kaSIaaG4maiaad2gadaWgaaWcbaGaaG4maa qabaaaaOGaaGPaVlabl+y6NjaaykW7daWcaaqaaiaadohadaWgaaWc baGaamyzaaqabaaakeaacaaIXaGaey4kaSIaamyBamaaBaaaleaaca aIYaaabeaaaaaakiaawIcacaGLPaaaaaaa@A594@


    Figure 23.
  5. No Mean Stress Correction
    (38) S F = s e σ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaadA eacqGH9aqpdaWcaaqaaiaadohadaWgaaWcbaGaamyzaaqabaaakeaa cqaHdpWCdaWgaaWcbaGaamyyaaqabaaaaaaa@3D9C@
Stress Ratio = Constant
  1. Goodman

    When SN curve is of the Stress Ratio R = -1

    (39) S F = O B O A = 1 ( σ a s e + σ m U T S ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaadA eacqGH9aqpdaWcaaqaaiaad+eacaWGcbaabaGaam4taiaadgeaaaGa eyypa0ZaaSaaaeaacaaIXaaabaWaaeWaaeaadaWcaaqaaiabeo8aZn aaBaaaleaacaWGHbaabeaaaOqaaiaadohadaWgaaWcbaGaamyzaaqa baaaaOGaey4kaSYaaSaaaeaacqaHdpWCdaWgaaWcbaGaamyBaaqaba aakeaacaWGvbGaamivaiaadofaaaaacaGLOaGaayzkaaaaaaaa@4AAD@


    Figure 24.

    When SN curve is of the Stress Ratio R != -1

    If R   >   1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGsbGaaeiiaiabg6da+iaabccacqGHsislcaaIXaaaaa@3AE0@ , s e = S e R 1 s m R U T S MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Ca8aadaWgaaWcbaWdbiaadwgaa8aabeaak8qacqGH9aqpdaWc aaWdaeaapeGaam4ua8aadaWgaaWcbaWdbiaadwgacqGHsislcaWGsb aapaqabaaakeaapeGaaGymaiabgkHiTmaalaaapaqaa8qacaWGZbWd amaaBaaaleaapeGaamyBaiabgkHiTiaadkfaa8aabeaaaOqaa8qaca WGvbGaamivaiaadofaaaaaaaaa@4612@ (40) s m R = S e R .   1 + R 1 R MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Ca8aadaWgaaWcbaWdbiaad2gacqGHsislcaWGsbaapaqabaGc peGaeyypa0Jaam4ua8aadaWgaaWcbaWdbiaadwgacqGHsislcaWGsb aapaqabaGcpeGaaiOlaiaacckadaWcaaWdaeaapeGaaGymaiabgUca Riaadkfaa8aabaWdbiaaigdacqGHsislcaWGsbaaaaaa@4642@

    If R   <   1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOuaiaabccacqGH8aapcaqGGaGaeyOeI0IaaGymaaaa@3AD1@ , s e = S e R MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Ca8aadaWgaaWcbaWdbiaadwgaa8aabeaak8qacqGH9aqpcaWG tbWdamaaBaaaleaapeGaamyzaiabgkHiTiaadkfaa8aabeaaaaa@3D45@

    If σ m >   0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaBaaaleaapeGaamyBaaWdaeqaaOWaaSbaaSqaa8qa cqGH+aGpcaqGGaGaaGimaaWdaeqaaaaa@3BD1@ , S F = 1 σ a S e + σ m U T S MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4uaiaadAeacqGH9aqpdaWcaaWdaeaapeGaaGymaaWdaeaapeWa aSaaa8aabaWdbiabeo8aZ9aadaWgaaWcbaWdbiaadggaa8aabeaaaO qaa8qacaWGtbWdamaaBaaaleaapeGaamyzaaWdaeqaaaaak8qacqGH RaWkdaWcaaWdaeaapeGaeq4Wdm3damaaBaaaleaapeGaamyBaaWdae qaaaGcbaWdbiaadwfacaWGubGaam4uaaaaaaaaaa@4602@

    If σ m 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaBaaaleaapeGaamyBaaWdaeqaaOWaaSbaaSqaa8qa cqGHKjYOcaaIWaaapaqabaaaaa@3BDB@ , S F =   S e σ a MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4uaiaadAeacqGH9aqpcaGGGcWaaSaaa8aabaWdbiaadofapaWa aSbaaSqaa8qacaWGLbaapaqabaaakeaapeGaeq4Wdm3damaaBaaale aapeGaamyyaaWdaeqaaaaaaaa@3F3E@

  2. Gerber

    When SN curve is of the Stress Ratio R = -1

    1. (41) If σ m = 0 : S F = s e σ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaqGjb GaaeOzaiaaykW7cqaHdpWCdaWgaaWcbaGaamyBaaqabaGccqGH9aqp caaIWaGaaiOoaaqaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caWGtbGaamOraiabg2da9maalaaabaGaam4Cam aaBaaaleaacaWGLbaabeaaaOqaaiabeo8aZnaaBaaaleaacaWGHbaa beaaaaaaaaa@5EFC@
    2. (42) If σ m 0 : S F = 1 2 ( U T S σ m ) 2 · σ a s e [ 1 + 1 + ( 2 s e σ m U T S σ a ) 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaqGjb GaaeOzaiaaykW7cqaHdpWCdaWgaaWcbaGaamyBaaqabaGccqGHGjsU caaIWaGaaiOoaaqaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caWGtbGaamOraiabg2da9maalaaabaGaaGymaa qaaiaaikdaaaWaaeWaaeaadaWcaaqaaiaadwfacaWGubGaam4uaaqa aiabeo8aZnaaBaaaleaacaWGTbaabeaaaaaakiaawIcacaGLPaaada ahaaWcbeqaaiaaikdaaaGccaaMc8UaeS4JPFMaaGPaVpaalaaabaGa eq4Wdm3aaSbaaSqaaiaadggaaeqaaaGcbaGaam4CamaaBaaaleaaca WGLbaabeaaaaGcdaWadaqaaiabgkHiTiaaigdacqGHRaWkdaGcaaqa aiaaigdacqGHRaWkdaqadaqaamaalaaabaGaaGOmaiaadohadaWgaa WcbaGaamyzaaqabaGccqaHdpWCdaWgaaWcbaGaamyBaaqabaaakeaa caWGvbGaamivaiaadofacaaMc8UaeyyXICTaaGPaVlabeo8aZnaaBa aaleaacaWGHbaabeaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaa ikdaaaaabeaaaOGaay5waiaaw2faaaaaaa@8814@
    When SN curve is of the Stress Ratio R != -1(43) S e = S e R 1 S m R U T S 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadabaaaaaaa aapeGaa83ua8aadaWgaaWcbaWdbiaa=vgaa8aabeaak8qacqGH9aqp caWFtbWdamaaBaaaleaapeGaa8xzaiabgkHiTiaa=jfaa8aabeaak8 qadaqadaWdaeaapeGaaGymaiabgkHiTmaabmaapaqaa8qadaWcaaWd aeaapeGaa83ua8aadaWgaaWcbaWdbiaa=1gacqGHsislcaWFsbaapa qabaaakeaapeGaa8xvaiaa=rfacaWFtbaaaaGaayjkaiaawMcaa8aa daahaaWcbeqaa8qacaaIYaaaaaGccaGLOaGaayzkaaaaaa@49E5@ (44) s m R = S e R .   1 + R 1 R MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Ca8aadaWgaaWcbaWdbiaad2gacqGHsislcaWGsbaapaqabaGc peGaeyypa0Jaam4ua8aadaWgaaWcbaWdbiaadwgacqGHsislcaWGsb aapaqabaGcpeGaaiOlaiaacckadaWcaaWdaeaapeGaaGymaiabgUca Riaadkfaa8aabaWdbiaaigdacqGHsislcaWGsbaaaaaa@4642@

    If σ m 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaBaaaleaapeGaamyBaaWdaeqaaOWaaSbaaSqaa8qa cqGHGjsUcaaIWaaapaqabaaaaa@3BED@ , S F = 1 2 U T S σ m 2 σ e S e 1 + 1 + 2 σ m S e U T S σ a 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadabaaaaaaa aapeGaa83uaiaa=zeacqGH9aqpdaWcaaWdaeaapeGaaGymaaWdaeaa peGaaGOmaaaadaqadaWdaeaapeWaaSaaa8aabaWdbiaa=vfacaWFub Gaa83uaaWdaeaapeGaa83Wd8aadaWgaaWcbaWdbiaa=1gaa8aabeaa aaaak8qacaGLOaGaayzkaaWdamaaCaaaleqabaWdbiaaikdaaaGccq GHflY1daWcaaWdaeaapeGaa83Wd8aadaWgaaWcbaWdbiaa=vgaa8aa beaaaOqaa8qacaWFtbWdamaaBaaaleaapeGaa8xzaaWdaeqaaaaak8 qacqGHflY1daqadaWdaeaapeGaeyOeI0IaaGymaiabgUcaRmaakaaa paqaa8qacaaIXaGaey4kaSYaaeWaa8aabaWdbmaalaaapaqaa8qaca aIYaGaa83Wd8aadaWgaaWcbaWdbiaa=1gaa8aabeaak8qacaWFtbWd amaaBaaaleaapeGaa8xzaaWdaeqaaaGcbaWdbiaa=vfacaWFubGaa8 3uaiaa=n8apaWaaSbaaSqaa8qacaWFHbaapaqabaaaaaGcpeGaayjk aiaawMcaa8aadaahaaWcbeqaa8qacaaIYaaaaaqabaaakiaawIcaca GLPaaaaaa@5FD3@

    If σ m = 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaBaaaleaapeGaamyBaaWdaeqaaOWaaSbaaSqaa8qa cqGH9aqpcaaIWaaapaqabaaaaa@3B2C@ , S F =   S e σ a MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4uaiaadAeacqGH9aqpcaGGGcWaaSaaa8aabaWdbiaadofapaWa aSbaaSqaa8qacaWGLbaapaqabaaakeaapeGaeq4Wdm3damaaBaaale aapeGaamyyaaWdaeqaaaaaaaa@3F3E@

  3. Gerber2
    1. (45) If  σ m 0 : S F = s e σ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaqGjb GaaeOzaiaabccacqaHdpWCdaWgaaWcbaGaamyBaaqabaGccqGHKjYO caaIWaGaaiOoaaqaaiaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaG jbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8Uaam4uaiaadAeacqGH 9aqpdaWcaaqaaiaadohadaWgaaWcbaGaamyzaaqabaaakeaacqaHdp WCdaWgaaWcbaGaamyyaaqabaaaaaaaaa@5722@
    2. (46) If σ m 0 : S F = 1 2 ( U T S σ m ) 2 · σ a s e [ 1 + 1 + ( 2 s e σ m U T S σ a ) 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaqGjb GaaeOzaiaaykW7cqaHdpWCdaWgaaWcbaGaamyBaaqabaGccqGHLjYS caaIWaGaaiOoaaqaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caWGtbGaamOraiabg2da9maalaaabaGaaGymaa qaaiaaikdaaaWaaeWaaeaadaWcaaqaaiaadwfacaWGubGaam4uaaqa aiabeo8aZnaaBaaaleaacaWGTbaabeaaaaaakiaawIcacaGLPaaada ahaaWcbeqaaiaaikdaaaGccaaMc8UaeS4JPFMaaGPaVpaalaaabaGa eq4Wdm3aaSbaaSqaaiaadggaaeqaaaGcbaGaam4CamaaBaaaleaaca WGLbaabeaaaaGcdaWadaqaaiabgkHiTiaaigdacqGHRaWkdaGcaaqa aiaaigdacqGHRaWkdaqadaqaamaalaaabaGaaGOmaiaadohadaWgaa WcbaGaamyzaaqabaGccqaHdpWCdaWgaaWcbaGaamyBaaqabaaakeaa caWGvbGaamivaiaadofacaaMc8UaeyyXICTaaGPaVlabeo8aZnaaBa aaleaacaWGHbaabeaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaa ikdaaaaabeaaaOGaay5waiaaw2faaaaaaa@8813@

    When SN curve is of the Stress Ratio R != -1

    If R   >   1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaCOuaiaabccacqGH+aGpcaqGGaGaeyOeI0IaaCymaaaa@3AD8@ (47) S e = S e R 1 s m R U T S 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadabaaaaaaa aapeGaa83ua8aadaWgaaWcbaWdbiaa=vgaa8aabeaak8qacqGH9aqp caWGtbWdamaaBaaaleaapeGaamyzaiabgkHiTiaadkfaa8aabeaak8 qacqGHflY1daqadaWdaeaapeGaaGymaiabgkHiTmaabmaapaqaa8qa daWcaaWdaeaapeGaam4Ca8aadaWgaaWcbaWdbiaad2gacqGHsislca WGsbaapaqabaaakeaapeGaamyvaiaadsfacaWGtbaaaaGaayjkaiaa wMcaa8aadaahaaWcbeqaa8qacaaIYaaaaaGccaGLOaGaayzkaaaaaa@4C73@ (48) s m R = S e R .   1 + R 1 R MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Ca8aadaWgaaWcbaWdbiaad2gacqGHsislcaWGsbaapaqabaGc peGaeyypa0Jaam4ua8aadaWgaaWcbaWdbiaadwgacqGHsislcaWGsb aapaqabaGcpeGaaiOlaiaacckadaWcaaWdaeaapeGaaGymaiabgUca Riaadkfaa8aabaWdbiaaigdacqGHsislcaWGsbaaaaaa@4642@

    If R   <   1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaCOuaiaabccacqGH8aapcaqGGaGaeyOeI0IaaCymaaaa@3AD4@ , S e = S e R MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadabaaaaaaa aapeGaa83ua8aadaWgaaWcbaWdbiaa=vgaa8aabeaak8qacqGH9aqp caWGtbWdamaaBaaaleaapeGaamyzaiabgkHiTiaadkfaa8aabeaaaa a@3D29@

    If σ m >   0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaBaaaleaapeGaamyBaaWdaeqaaOWaaSbaaSqaa8qa cqGH+aGpcaqGGaGaaGimaaWdaeqaaaaa@3BD1@ , S F = 1 2 U T S σ m 2 σ a S e 1 + 1 + 2 σ m S e U T S σ a 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadabaaaaaaa aapeGaa83uaiaa=zeacqGH9aqpdaWcaaWdaeaapeGaaGymaaWdaeaa peGaaGOmaaaadaqadaWdaeaapeWaaSaaa8aabaWdbiaa=vfacaWFub Gaa83uaaWdaeaapeGaa83Wd8aadaWgaaWcbaWdbiaa=1gaa8aabeaa aaaak8qacaGLOaGaayzkaaWdamaaCaaaleqabaWdbiaaikdaaaGccq GHflY1daWcaaWdaeaapeGaa83Wd8aadaWgaaWcbaWdbiaa=fgaa8aa beaaaOqaa8qacaWFtbWdamaaBaaaleaapeGaa8xzaaWdaeqaaaaak8 qacqGHflY1daqadaWdaeaapeGaeyOeI0IaaGymaiabgUcaRmaakaaa paqaa8qacaaIXaGaey4kaSYaaeWaa8aabaWdbmaalaaapaqaa8qaca aIYaGaa83Wd8aadaWgaaWcbaWdbiaa=1gaa8aabeaak8qacaWFtbWd amaaBaaaleaapeGaa8xzaaWdaeqaaaGcbaWdbiaa=vfacaWFubGaa8 3uaiaa=n8apaWaaSbaaSqaa8qacaWFHbaapaqabaaaaaGcpeGaayjk aiaawMcaa8aadaahaaWcbeqaa8qacaaIYaaaaaqabaaakiaawIcaca GLPaaaaaa@5FCF@

    If σ m 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaBaaaleaapeGaamyBaaWdaeqaaOWaaSbaaSqaa8qa cqGHKjYOcaaIWaaapaqabaaaaa@3BDB@ , S F =   S e σ a MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4uaiaadAeacqGH9aqpcaGGGcWaaSaaa8aabaWdbiaadofapaWa aSbaaSqaa8qacaWGLbaapaqabaaakeaapeGaeq4Wdm3damaaBaaale aapeGaamyyaaWdaeqaaaaaaaa@3F3E@

  4. FKM
    (49) S F = s e σ a 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaadA eacqGH9aqpdaWcaaqaaiaadohadaWgaaWcbaGaamyzaaqabaaakeaa cqaHdpWCdaWgaaWcbaGaamyyamaaBaaameaacaaIWaaabeaaaSqaba aaaaaa@3E8E@

    σ a 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadggadaWgaaadbaGaaGimaaqabaaaleqaaaaa@39BD@ = Corrected Stress Amplitude in Constant R mean stress correction

  5. No Mean Stress Correction
    (50) S F = s e s a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaadA eacqGH9aqpdaWcaaqaaiaadohadaWgaaWcbaGaamyzaaqabaaakeaa caWGZbWaaSbaaSqaaiaadggaaeqaaaaaaaa@3CD1@
  6. Interpolate
    Safety Factor with Multi-Mean
    To calculate safety factor, HyperLife creates an internal Haigh diagram for the target life using multi-mean SN curve by finding stress amplitude-mean stress pairs at the target life. Using the internally created Haigh diagram, HyperLife calculates safety factor as described in section Safety Factor in Chapter Haigh diagram. The number of data points of the Haigh diagram is the number of curves. Thus the more number of curves, the better result. When Haigh diagram is not available in mean stress ranges, HyperLife extrapolates the Haigh diagram.


    Figure 25. Conversion of Multi-Mean Curve to Haigh Diagram
    Safety Factor with Multi-Ratio
    To calculate safety factor, HyperLife create an internal Haigh diagram for the target life using multi-mean SN curve by finding stress amplitude-mean stress pairs at the target life. The number of data points of the Haigh diagram is the number of curves. Thus, the more number of curves, the better result. When Haigh diagram is not available in mean stress ranges, HyperLife extrapolates the Haigh diagram.


    Figure 26. Conversion of Multi-Mean Curve to Haigh Diagram
    Safety Factor with Haigh
    Safety factor (SF) is calculated in the following manner in Figure 27.


    Figure 27.
    When target life is 100000:
    • Constant R : SF = OB/OA
    • Constant mean : SF = OD/OC
    If Haigh diagram for a target life is not defined, HyperLife creates Haigh diagram for the target life. In Figure 27, if target life is 10000, and Haigh diagram for N=10000 is not defined, HyperLife will created dashed curve to calculate Safety factor.

Safety Factor with Multi-Mean

To calculate safety factor, HyperLife creates an internal Haigh diagram for the target life using multi-mean SN curve by finding stress amplitude-mean stress pairs at the target life. Using the internally created Haigh diagram, HyperLife calculates safety factor as described in section Safety Factor in Chapter Haigh diagram. The number of data points of the Haigh diagram is the number of curves. Thus the more number of curves, the better result. When Haigh diagram is not available in mean stress ranges, HyperLife extrapolates the Haigh diagram.


Figure 28. Conversion of Multi-Mean Curve to Haigh Diagram

Safety Factor with Multi-Ratio

To calculate safety factor, HyperLife create an internal Haigh diagram for the target life using multi-mean SN curve by finding stress amplitude-mean stress pairs at the target life. The number of data points of the Haigh diagram is the number of curves. Thus, the more number of curves, the better result. When Haigh diagram is not available in mean stress ranges, HyperLife extrapolates the Haigh diagram.


Figure 29. Conversion of Multi-Mean Curve to Haigh Diagram

Safety Factor with Haigh

Safety factor (SF) is calculated in the following manner in Figure 30.


Figure 30.
When target life is 100000:
  • Constant R : SF = OB/OA
  • Constant mean : SF = OD/OC

If Haigh diagram for a target life is not defined by user, HyperLife creates Haigh diagram for the target life. In Figure 30, if target life is 10000, and Haigh diagram for N=10000 is not defined, HyperLife will created dashed curve to calculate Safety factor.