Fatigue

Fatigue Life Estimation using S-N Curves

The definition fatigue according to Metal fatigue in Engineering by Ali Fatemi is 

• The process of localized progressive permanent structural change occurring in a  material subjected to conditions which produce fluctuating stresses and strains at point or points and which may culminate in cracks or complete fracture after a sufficient number of fluctuations”.

 Fatigue Failures occur at the load levels lower than the original ultimate loads.

 Fatigue failure can also be global as in the case of Widespread Fatigue Damage (WFD)

The cyclic loads are defined by the parameters shown in the figure.

Parameters Defining the Cyclic Loading

S-N Curves

A typical S-N Curve exhibits linear behaviour in log-log scale (S-N Curve obey power law). The vertical axis (Ordinate) represents alternating stress (Not maximum stress. Most of the beginners do mistake here). The horizontal axis (Abscissa) is represents the fatigue life. S-N Curves can be represented by polynomial equation or using Power law.  Power law equation for representing the median SN behaviour is given below. The dotted line is the extrapolation of the curve. At the dotted region, the experimental data can not be obtained. Stain based approach is more appropriate in this region.

The statistical reliability and confidence levels are always included in the S-N curves due to scattering of the experimental (test) data. Most of the curves are either 95/95  (95% Confidence, 95% Reliability) or 50/95 (50% Confidence or 95% Reliability).

For the Aircraft design an appropriate safety factor needs to be accounted (Ref. FAR/CS 25.571)

Typical S-N Curves under log-log Scale

Mean Stress Correction

The fatigue life depends on the stress amplitude and also the mean stresses. The draw back the S-N curves is that it gives the relation between the Stress amplitude and the stress life. The effect of the mean stress needs to be accounted separately. The tensile mean stresses are detrimental and the compressive mean stresses are beneficial. In Aerospace Industry, the effect of mean stress is accounted using Goodman relation. The Goodman relation gives the linear relation between the alternating stress and the mean stress. Knowing the alternating stress (Stress amplitude) and the mean stress, an equivalent alternating stress can be calculated as

S_eq.alt =S_alt(1-S_mean/S_ult)

Where,

• S_eq.alt -> Equivalent alternating stress
• S_mean -> Mean stress of the stress segment
• S_ult-> Ultimate strength of the material

Knowing the equivalent alternating stress, the fatigue life can be estimated using the S-N Curves.

Factors Affecting Fatigue

Many factors affect the fatigue behavior of the material. Some of the Factors affecting the fatigue behavior are listed below

• Surface Finish : Smooth surfaces exhibit better fatigue behavior than the rough surfaces.
• Stress Gradient or Stress concentrations : Higher stress concentration factor decrease the fatigue life of a component. The design should aim at reducing the stress concentration. This topic is described more in detail below.
• Size : The probability of the fatigue crack increases with the increase in size of the component. The smaller components exhibit better fatigue behavior than the larger components.
• Manufacturing process : Manufacturing processes such as casting introduce porosity in the component which reduce the fatigue behavior. The processed like forging through reduce the porosity but they can not eliminate the porosity completely. Rolled components introduce unsymmetrical fatigue behaviour with the rolling direction showing the better fatigue behavior than the other directions.
• Surface Treatments : The surface treatments like anodising, plating reduce the fatigue behavior of the components due to the introduction of a brittle layer on the surface. Shot peening improve the fatigue behaviour by adding residual stresses on the surface of the component. It may be noticed that most of cracks appear from the surface of the components.  (Question: Where does the fatigue crack initiate in the coil springs?)
• Temperature : The temperatures affect the fatigue behavior in many aspects. In the hybrid design (bi-metallic or Metallic-Composite interface), the change in temperature with respect to the assembly line temperature introduce thermal loading. Again, the temperature below the Ductile Brittle Transition Temperature (DBTT) can be very catastrophic. (Question: What caused Titanic to fail?)
• Environment : The environments conductive to corrosion and hydrogen embrittlement reduce the fatigue behavior of the components. The aluminium is more susceptible to corrosion when exposed to moisture.
• Loading :  Uniaxial, Multiaxial, Bending, Shear etc.

Geometrical Stress Concentration Factor

Stress Concentration is measured as a ratio of Stress level at the discontinuity to the far field stress. Stress Concentration factor (K_t) is purely a function of geometry and the type of loading (Tension/Compression, Bending, Shear, bearing etc.).

The stress concentration factor to be considered for the analysis should be consistent with the test coupon. The S-N curves have built-in K_t depending on the test coupon. It is necessary to know the Kt value in the Coupon and the way the S-N curve is normalised.

K_t calculated based on the net stresses is called K_t.net. The Kt calculated based on the Gross stresses is called the Gross Kt.

For the S-N curves generated based on gross stresses, then the analysis must use Gross k_t. For the S-N curves generated based on net stresses, the K_t.net must be applied.

For the estimation of the fatigue life, it is important that the K_t built-in the S-N Curves to be removed before applying the K_t of the structure.

The stress concentration factor is dependent on the load transfer in the Joint. The picture shows influence of load transfer on the stress concentration factor.

Stress Concentration Factor

The stress concentration factor at the 0 degree position in the open hole in a plate of infinite width is 3.0. Where as for the same configuration with medium load transfer the stress concentration factor is approx. 6.0.

Closing Note: The stress concentration factor due to the open hole in a plate of infinite width subjected to pure shear is 4. Why?

Joint Load Transfer

As described above, the load transfer in the filled hole influence the stress concentration factor. Therefore, it is necessary to establish the load transfer in the joint in order to estimate the stress concentration factor in the joint.

The load transfer can be calculated analytically or using FEM.

The load transfer depends on the following

• Flexibility of the Fastener
• Thickness of the joint
• Fastener Pitch
• The stiffness of the plates

The flexibility of the Fastener itself depends on the following

• Diameter of the fastener
• Thickness of the plates

There are many equations to calculate the fastener flexibility in the literature.

• Huth Criteria (Ref [1]),
• Swift (Douglas)
• Tate & Rosenfeld
• Delft University
• ESDU 98012
• The companies BOEING, Vought, Grumman have their own empirical formulas to calculate faster flexibilities based on their trade practices.

It is important to note that every approach leads to different fastener flexibility values.

Joint Load transfer Calculation

The bearing and bypass loads can be calculated by establishing the global equilibrium in the joint as well as by establishing the compatibility between each fastener pitch.

Equilibrium equation for the above joint is established by equating the sum of the the shear load transfer in the fasteners with the applied load.

P=P₁+P₂+P₃

Compatibility between upper and lower plates between the fastener 1 and 2 is given by

δ_fast1+δ_{lwr.plate(1-2)}=δ_fast2+δ_{upr.plate(1-2)}

f₁×P₁+(P-P₁)/(A₁×E₁/l₁)=f₂× P₂+P₁/(A₂×E₂/l₁)

(A_i×E_i/l_i)  – is the stiffness of the plate at segment i

f_i – are the fastener flexibilities

Compatibility between upper and lower plates between the fastener 2 and 3 is given by

δ_fast2+δ_{lwr.plate(2-3)}=δ_fast3+δ_{upr.plate(2-3)}

f₂×P₂+(P-P₁-P₂)/(A₁×E₁/l₂)=f₃× P₃+(P₁+P₂)/(A₂×E₂/l₂)

The bending contribution due to the eccentricity can be ignored for the joints made of thin plates or for the stable joints. The bending stresses can be superimposed after calculating the load transfer in each segment.

Once the load transfer in the joint is calculated, the stress concentration factor due to load transfer effect can be calculated. The joints with load transfer more than 50% are generally called high load transfer joints and the joints with load transfer less than 5% are called low load transfer joints.

Joints Classification

For the practical purpose, the joints are classified as High, Medium or Low load transfer joints.

High load transfer (HLT) joints are the ones where the load transfer in the fastener is around 50% . For practical purpose, the joints with 2 or 3 fastener rows are considered as high load transfer joints. (E.g. Longitudinal lap joint of fuselage).

Joint with less than 5% load transfer are classified as low load transfer (LLT) joints. The joints load transfer anything between the HLT and LLT are called medium transfer joints.

Variable Amplitude Loading

The S-N Curves are generated from the test considering the constant amplitude loading

The loads experienced by the structure in general is not constant amplitude. Typical Ground-Air-Ground (GAG) spectrum of the lower skin of an aircraft is shown above

The GAG cycle shown above has a lot of minor cycles which cause significant damage to the structure. Therefore, the damaging effect of both the major of minor cycles needs to be accounted while estimating the fatigue life.

The cycle counting techniques help in identifying the major and minor cycles.

There are many cycle counting techniques defined in ASTM E1049-85 (Ref [2]). However, Rainflow counting is more popular in the Aerospace Industry.

Rainflow Counting

The rainflow cycle counting (Ref [2]) is the most popular method for counting the cycles. The ASTM algorithm which is based 3 points comparison method. This is suffering from the fact that the spectrum needs to rearranged to start with lowest valley or highest peak. Most popular method of rainflow counting is the 4 point method.

References

1. Influence of Fastener Flexibility on the Prediction of Load Transfer and Fatigue Life for Multiple-Row Joints, ASTM STP927, 1986.
2. Standard Practices for Cycle Counting in Fatigue Analysis,  ASTM E1049-85 (Reapproved 1997)