S16-1 PAT318, Chapter 16, March 2005 SECTION 16 STRESS-LIFE (S-N) THEORY

S16-2 PAT318, Chapter 16, March 2005

S16-3 PAT318, Chapter 16, March 2005 Stress-Life (S-N) Theory n The S-N approach estimates total life without distinguishing crack initiation from crack propagation n It usually requires that the test data relate to the geometry of the structure under assessment (structure S-N curves) n Material S-N curves can also be generated from smooth specimen test data; they are subsequently modified to reflect the effects of notches, surface conditions, etc. of the real structure

S16-4 PAT318, Chapter 16, March 2005 Assumptions and Definitions n Deformation can be separated between an elastic (fully recoverable) and plastic (permanent) component n The Stress Range is the algebraic difference between maximum and minimum stress in a cycle n The Stress Amplitude is half of the Range

S16-5 PAT318, Chapter 16, March 2005 n Input is cycles of STRESS n Also known as High Cycle Fatigue or Nominal Stress Approach n Nominal stress cycles must be elastic (hence high cycle) though local stresses at the critical location will be plastic n In MSC.Fatigue SN analysis, elastic FE results are used directly (no plasticity correction) Measured nominal stresses Actual Stress at Critical location S-N Analysis

S16-6 PAT318, Chapter 16, March 2005 S-N Curve

S16-7 PAT318, Chapter 16, March 2005 Wohlers Railway Component Test Rig (1852 to 1870)

S16-8 PAT318, Chapter 16, March 2005 StressAmplitude NotchedShaft UnnotchedShaft Log (fatigue life) Some of Wohlers data for rotating bending tests

S16-9 PAT318, Chapter 16, March 2005 S-N Approach n The S-N approach uses the (assumed elastic) nominal stress range (S) as a measure of the severity of fatigue loading n Life to failure (two pieces) is recorded in experiments n Tests at several levels of stress range characterise the S-N curve n Such a curve can be derived for smooth specimens, for individual components, for sub-assemblies, or for complete structures

S16-10 PAT318, Chapter 16, March 2005 Use of S-N Approach n The uses of the S-N approach include: u Establishing a well defined fatigue curve for the purposes of design u Determination of a fatigue strength at a specified life u Demonstration of improved fatigue resistance from a material or surface treatment u Acceptance of material for manufacturing purposes u Answering questions posed by a service failure

S16-11 PAT318, Chapter 16, March 2005 Material S-N Curves n Steels tested with constant amplitude loading normally exhibit a fatigue limit - a stress below which no fatigue damage appears to occur. n The fatigue limit is associated with the difficulty a crack has in getting past the first grain boundary, or dominant microstructural barrier. It can be reduced or eliminated after e.g. a few large loads, or in corrosive environment, etc. n Aluminum alloys do not seem to exhibit no such limit

S16-12 PAT318, Chapter 16, March 2005 Material S-N Curves Log(Stress) Log(Life) Steel or Ti Al alloy or steel in seawater

S16-13 PAT318, Chapter 16, March 2005 Scatter in S-N curves n Due to the statistical nature of the test, any given S-N curve is associated with a certain probability of failure n More insight on the S-N relationship can be obtained representing scatter bands (e.g +/- 3 Standard errors) together with the mean curve (50% Certainty of Survival)

S16-14 PAT318, Chapter 16, March 2005 The analytical S-N curve Static Limitations 1 m2m2 S t r e s s r a n g e ( ) l o g s c a l e Endurance N (cycles) - log scale 10 7 ( ) *10 8 n The S-N curve can be expressed by the power law: N*S m =const m1m1 m 2 =0 identifies Se as the fatigue limit If m 2 >0, a conventional fatigue limit could be set at N=5*E8

S16-15 PAT318, Chapter 16, March 2005 The analytical S-N curve in MSC.Fatigue* *MSC.Fatigue also gives an option to provide a family of curves by points with the multi mean fatigue curve option n For the S-N curve, MSC.Fatigue uses a variation on the conventional power law ( N*S m =const) : S=SRI1*N b (Note: m=-1/b) n Two straight line can be defined through the b1 and b2 fatigue strength exponents (see next image)

S16-16 PAT318, Chapter 16, March 2005 The analytical S-N curve in MSC.Fatigue cont m1m1 1 b1b1 1 S=SRI1*N b (with m=-1/b)

S16-17 PAT318, Chapter 16, March 2005 Component S-N curves n For some components or features, especially structural joints such as welds, there are so many things modifying the behaviour of the base material that there is little point in applying corrections to a material S-N curve n In cases like this it is best to use a nominal stress- life curve which applies particularly to that component or feature, hence the definition of Component S-N Curve

S16-18 PAT318, Chapter 16, March 2005 Component S-N Curves: use of remote nominal stress P P Nominal Stress P _ A A CLASS F WELD DETAIL (BS7608)

S16-19 PAT318, Chapter 16, March 2005 Example of component curve: BS7608 Welds

S16-20 PAT318, Chapter 16, March 2005 S-N Method - Similitude The life of this is the same as the life of this..... if both are subject to the same nominal stress nom nom

S16-21 PAT318, Chapter 16, March 2005 n The S-N method assumes that the life of a component or structure is the same as that of a laboratory test specimen if both are subject to the same nominal stresses. n If the conditions in the test are different to those in the structure, similitude breaks down, and we need to make corrections for factors such as mean stress, environment, surface finish, etc. The fine art of the Fatigue expert is the struggle to justify and force a similitude assumption between experimental data and real life scenarios S-N Method - Similitude

S16-22 PAT318, Chapter 16, March 2005 Exercise Perform Quick Start Guide MSC.Fatigue 2005, Chapter 2 Exercise, A Simple S-N Analysis. Be sure to ask for help if theres anything you dont understand

S16-23 PAT318, Chapter 16, March 2005 Fully Reversed Loading P = 10,000 N S-N Analysis of a Keyhole Specimen

S16-24 PAT318, Chapter 16, March 2005 FEA Model of the half Keyhole Symmetric BC. Symmetric Half Model

S16-25 PAT318, Chapter 16, March 2005 Loading Info Setup

S16-26 PAT318, Chapter 16, March 2005

S16-27 PAT318, Chapter 16, March 2005 Plot Simple Loading

S16-28 PAT318, Chapter 16, March 2005 Select the Created Loading

S16-29 PAT318, Chapter 16, March 2005 Material Info Setup

S16-30 PAT318, Chapter 16, March 2005 Display Life Contours

S16-31 PAT318, Chapter 16, March 2005 Variable Amplitude Loads - Miners Rule and Rainflow Counting

S16-32 PAT318, Chapter 16, March 2005 Miners Rule - Block Loading Miners rule assigns a damage of 1/Nf to each cycle where Nf is the number of cycles to failure at that load level (determined from an S-N curve) Failure is predicted to occur when the total damage reaches a value of 1. If total damage D < 1 life is predicted to be 1/D repeats

S16-33 PAT318, Chapter 16, March 2005 Range Mean Material Life Curve LifedamagedAccumulate% Cycles 100 MPa i f i N N Damage Damage Counting with Miner

S16-34 PAT318, Chapter 16, March 2005 Palmgren–Miner Damage Summation Law

S16-35 PAT318, Chapter 16, March 2005 n 1 N 1 N 1 S 1 Original S-N Curve S-N Curve after Application of Stress for Cycles S 1 n 1 Cycles to Failure (log scale) S t r e s s A m p l i t u d e ( l o g s c a l e ) Effect of Miners rule on S-N curve

S16-36 PAT318, Chapter 16, March 2005 Advantages & Disadvantages of the Linear (Miner) Damage Theory Advantages: 1. Simple 2. Generally falls within the ball park of tests e.g. varies between 0.61 to Mean is 1.0 Disadvantage: Assumes that the level of stress has no effect on the damage ratio, for example: tests do indicate that high stress cycles followed by low stress cycles cause more damage than the other way around.

S16-37 PAT318, Chapter 16, March 2005 Non-Linear Damage Theory Advantages: - D = n f /N f i ) p takes into account both sequence & load level effects. -if p is known well experiments evidence suggests we get somewhat better results. Disadvantages: - p has to be determined experimentally from a family of stress curves of a given material and so is very difficult to obtain. -for most situations load histories are pseudo- random, i.e. we dont know load history. -finding p is difficult-need many tests at different stress levels. Conclusion: Nonlinear theory does not buy us much and is difficult to use. Consequently it is not used in practice, and therefore is not in MSC.Fatigue.

S16-38 PAT318, Chapter 16, March 2005 Variable Amplitude Loads - Estimating Lifetime -

S16-39 PAT318, Chapter 16, March 2005 Stress or Strain Cycles: Time HistoryPeak Valley Extraction Rainflow Cycle Counting Require Cycle Range & Mean What Drives the Fatigue Crack?

S16-40 PAT318, Chapter 16, March 2005 Rainflow Cycle Counting

S16-41 PAT318, Chapter 16, March 2005 Rainflow Cycle Counting n The story goes Matsuishi and Endo got the idea for the method while watching rain water cascading down a pagoda roof. n Basic rules: rain flows down from each turning point and continues until either: u it is interrupted by flow from above, or u it reaches a turning point which is larger that the one it started from and in the same sense n Good way of representing cycles is Rainflow Cycle Count Matrix

S16-42 PAT318, Chapter 16, March 2005 Cycle Count Matrix

S16-43 PAT318, Chapter 16, March 2005 Rainflow Counting and Stress/Strain Space

S16-44 PAT318, Chapter 16, March 2005 Practical Rainflow Counting: Reservoir method n A Reservoir is created using the peek stress as starting point (outer histeresys loop) n The water is drained from the absolute minimum and the the depth of each residual water pocket is recorded

S16-45 PAT318, Chapter 16, March 2005 n Materials under cyclic loading exhibit material memory effect (they remember the largest previously reached stress-strain state) n What is stress-strain curve in monotonic loading is hysteresis loop in cyclic loading n Rainflow counting identifies closed hysteresis loops as cycles u Some cycles stand within the largest hysteresis loop and some hang; this depends on cycle sequence Rainflow Counting and Stress/Strain Space

S16-46 PAT318, Chapter 16, March 2005 Range Mean Material Life Curve LifedamagedAccumulate% Cycles 100 MPa i f i N N Damage Damage Counting with Miner

S16-47 PAT318, Chapter 16, March 2005 Time HistoryPeak Valley Extraction Rainflow Cycle Counting Damage CountingDamage Histogram LIFE Analysis Route - An Overview Loose Frequency Information Loose Sequence Information Life

S16-48 PAT318, Chapter 16, March 2005 Influences on Fatigue Life

S16-49 PAT318, Chapter 16, March 2005 Factors Influencing Fatigue Life Mean stress

S16-50 PAT318, Chapter 16, March 2005 Mean Stresses Stress amplitude N = constant for all points

S16-51 PAT318, Chapter 16, March 2005 Stress ratio: R = min / max n Most fatigue tests are conducted at R = -1 (fully reversed loading). n If we have cycles with other R values we should make corrections to the stress range in order to be able to compare the cycles to the S-N curve determined at R=-1. n Note: compressive mean stresses do not influence fatigue life. Mean Stresses

S16-52 PAT318, Chapter 16, March 2005 Mean Stress Corrections

S16-53 PAT318, Chapter 16, March 2005 Mean Stress Corrections Haigh Diagram Un-Safe Unsafe Safe

S16-54 PAT318, Chapter 16, March 2005 n Most popular mean stress corrections are Goodman and Gerber methods. n Real test data tend to lie between the two, with the Goodman method being more conservative (i.e. safer). Mean Stress Corrections

S16-55 PAT318, Chapter 16, March 2005 Correcting for the Effect of Mean Stress n Goodman method n Gerber method a e m u SS 1 2 a e m u SS 1 a m u em stressamplitude meanstress Sultimatetensilestress Sequivalentstressfor 0

S16-56 PAT318, Chapter 16, March 2005 Mean Stress Corrections Haigh Diagram

S16-57 PAT318, Chapter 16, March 2005 Mean Stresses Stress amplitude N = constant for all points Goodman Gerber

S16-58 PAT318, Chapter 16, March 2005 Factors Influencing Fatigue Life Mean stress Component size

S16-59 PAT318, Chapter 16, March 2005 Component Size Small laboratory specimens and large engineering structures Influence of Specimen Size on Endurance Limit:

S16-60 PAT318, Chapter 16, March 2005 Component Size The endurance limit used for design (S e ) can be calculated from the experimental endurance limit (S e ) from any size specimen: S e =S e C size

S16-61 PAT318, Chapter 16, March 2005 Factors Influencing Fatigue Life Mean stress Component size Type of loading

S16-62 PAT318, Chapter 16, March 2005 Type of Loading Problem: Data from rotating bend tests Structure sees tension or torsion n A conservative estimate of the ratio of endurance limits for axial to bending is.7 Se (axial) ~.70 Se(bending) n A similar ratio can be given for torsion: Se (torsion) ~.577 Se(bending)

S16-63 PAT318, Chapter 16, March 2005 Factors Influencing Fatigue Life Mean stress Component size Type of loading Notches and discontinuities

S16-64 PAT318, Chapter 16, March 2005 Notches

S16-65 PAT318, Chapter 16, March 2005 n Another factor that will reduce the life of a component is a notch or stress concentration. n Usually, unless the metal is of very high strength, the fatigue limit of the component is not reduced by as much as you might expect from the K t factor. n The difference between K t and K f is due to the notch sensitivity of the material, which is greatest for high strength metals. Notches

S16-66 PAT318, Chapter 16, March 2005 Dealing with Stress Concentrations Measured nominal stress = S Actual Stress at Critical location = S. K t n It is seldom possible to stick the strain gauges at the critical location. n In practice put the strain gauges close to the critical location and use a stress correction factor K t to scale them up to the critical value.

S16-67 PAT318, Chapter 16, March 2005 Effect of Stress Concentration in Fatigue In fatigue, the effect of a stress concentrating notch is to reduce the fatigue stress at a given life. This is defined as the Fatigue Strength Reduction Factor and is given the symbol K f. Strictly, K f can only be obtained from long life fatigue tests and is a ratio: Un-notched fatigue strength K f = Fatigue strength for the notch It is dependant on material as well as local geometry and is generally less than K t.

S16-68 PAT318, Chapter 16, March 2005 Relationship between K f and K t K t depends on geometry only and is relatively easy to obtain but K f depends on material as well, and in theory, should be measured for all possible combinations of both. Can we derive K f from K t ? First, we define the parameter, q, the notch sensitivity factor as: q = (K f - 1) / (K t - 1) For notch insensitive materials, K f =1 and q=0. For perfectly notch sensitive materials K f = K t and q=1.

S16-69 PAT318, Chapter 16, March 2005 K f = 1 + ( K t - 1 ) / ( 1 + a / r ) Relationship between K f and K t EMPIRICALLY, it has been found that: q = 1 / ( 1 + a / r ) where r is notch root radius and a is a function of material UTS: a = ( 2079 / UTS ) 1.8 MPa & mm units Combining gives an EMPIRICAL rule for Kf from Kt:

S16-70 PAT318, Chapter 16, March 2005 Two Ways of Using K t (SN Analysis) n Calculate new time history by multiplying the original by Kt. n This appears the easiest but could take a long time to compute with large time history files. n Reduce the fatigue life curve. n This uses a value called the fatigue reduction factor Kf. n Kf is a function of Kt and a materials susceptibility to notches. n Conservatively take Kf = Kt Modify the time historyModify the Fatigue Life Curve

S16-71 PAT318, Chapter 16, March E31E41E51E61E71E SMOOTH Kt=3, Kf=2.67 Cross Plot of Data : KFEFFECT NOTCHEDUNNOTCHED Life(Cycles) Amplitude(MPa) The Effect of K t and K f on Fatigue Life

S16-72 PAT318, Chapter 16, March 2005 n The notch does not have such a large effect at short lives as it does at long. n This is often dealt with by having a separate Kf factor at 1000 cycles. The Effect of K t and K f on Fatigue Life

S16-73 PAT318, Chapter 16, March 2005 Effect of Notch Factor 1000 cycles Transition Life unnotched notched Stress Life Kf

S16-74 PAT318, Chapter 16, March 2005 Curve for Estimation of Kf (From Juvinall)

S16-75 PAT318, Chapter 16, March 2005 Factors Influencing Fatigue Life Mean stress Component size Type of loading Notches and discontinuities Surface treatment & finish

S16-76 PAT318, Chapter 16, March 2005 Surface Treatment & Finish n Fatigue cracks usually start at the surface, therefore the condition of the surface can have a large impact on the life of a component. n The smoother the surface, the longer it takes to initiate a fatigue crack. n Residual stresses in the surface can also affect the rate of initiation. Residual compression will delay the crack initiation in high cycle load cases. Surface treatments are used to induce residual surface stresses.

S16-77 PAT318, Chapter 16, March 2005 Dealing with Surface Effects

S16-78 PAT318, Chapter 16, March 2005 Surface Finish Note: the curves are for steels only.

S16-79 PAT318, Chapter 16, March 2005 n The effect of surface finish is typically obtained from curves such as on the previous slide. The strength reduction factor is related to the surface finish factor and the strength of the steel. Sometimes the curves are for qualitative finishes such as good machined. n The effect of surface roughness is typically accounted for by applying a reduction factor to the stress at the endurance or fatigue limit. On a log-log plot, the slope of the stress life curve is adjusted, with the stress at 1000 cycles being unaffected. Surface Finish

S16-80 PAT318, Chapter 16, March 2005 Correction for Surface Finish 1000 cycles Transition Life polished rough Stress Life

S16-81 PAT318, Chapter 16, March 2005 The Effect of Residual Compression TensionCompressionTensionCompression TensionCompression Surface Compression Stress Oscillating bending Stress Resulting surface stress never goes into tension therefore surface crack doesnt initiate += This effect only works for high cycle cases where the applied surface stress is insufficient to overcome the residual pre-compression.

S16-82 PAT318, Chapter 16, March 2005 How Can We Get Pre-Compression? n Shot Peening u Fire ball bearings at the surface to induce pre- compression n Cold Rolling u Roll the component surface to induce pre- compression in the surface n Nitriding u Heat up component in an ammonia environment. The component expands and Nitrates from the gas react with the metal. The component contracts on cooling and is compressed.

S16-83 PAT318, Chapter 16, March 2005 Goodman Based Factor of Safety (f) = Endurance Limit ; = Ultimate stress The factor by which we can increase our alternating stress (for a given mean stress), without causing any fatigue failure. f S e a m u – =

S16-84 PAT318, Chapter 16, March 2005 Goodman Based Factor of Safety (f) Calculation Goodman Based: Factor of Safety = Gerber Based: Factor of Safety =

S16-85 PAT318, Chapter 16, March 2005 Estimates the total fatigue life to catastrophic failure. Makes no distinction between crack initiation and crack growth. Uses local or nominal stress as the control parameter Fatigue life computed from the log stress vs. log cycles (S-N) curve. Fatigue life estimates are associated with a probability of failure due to the large amount of scatter in the S-N curve. Reduces complex random waveforms to a list of cycles with a given range and mean using Rainflow cycle counting Mean stress effects are taken into account by Goodman or Gerber algorithms. Summary Total Life Method

S16-86 PAT318, Chapter 16, March 2005 Summary Total Life Method (Contd) S-N method is appropriate for assessing damage in: Long life fatigue problems where there is little plasticity since the S-N method is based on nominal elastic stress Components where the crack initiation and growth models are not appropriate, e.g. composites and welds. Situations where a large amount of preexistent S-N data is available Components which are required by a control body to be designed for fatigue using standard data such as the MIL handbook data

S16-87 PAT318, Chapter 16, March 2005 Stress-Life in MSC.Fatigue n Features u Elastic Stresses u Rainflow Cycle Counting u Mean Stress Correction u Welded Structures u Statistical Confidence Parameters u Palmgren-Miner Linear Damage u User Defined Life u Material and Component S-N u Surface Conditions u Factor of Safety Analysis u Biaxiality Indicators

S16-88 PAT318, Chapter 16, March 2005 Exercise Perform Quick Start Guide MSC.Fatigue 2005, Chapter 3 Exercise, Rainflow Cycle Counting. Perform Quickstart Guide Chapter 4 Exercise, Component S-N Analysis Be sure to ask for help if theres anything you dont understand