Section 2 Evaluation of test results

2.1 Principles

2.1.1 Prior to fatigue testing, the crankshaft must be tested as required by quality control procedures, e.g. for chemical composition, mechanical properties, surface hardness, hardness depth and extension, fillet surface finish, etc.

2.1.2 The test samples should be prepared so as to represent the ‘lower end’ of the acceptance range, e.g. for induction hardened crankshafts this means the lower range of acceptable hardness depth, the shortest extension through a fillet, etc. Otherwise the mean value test results should be corrected with a confidence interval: a 90 per cent confidence interval may be used both for the sample mean and the standard deviation.

2.1.3 The test results, when applied as required by Pt 5, Ch 2, 3 Crankshaft Design of the Rules and Regulations for the Classification of Ships, July 2022, shall be evaluated to represent the mean fatigue strength, with or without taking into consideration the 90 per cent confidence interval mentioned above. The standard deviation should be considered by taking the 90 per cent confidence into account. Subsequently, the result to be used as the fatigue strength is then the mean fatigue strength minus one standard deviation.

2.1.4 If the evaluation aims to find a relationship between (static) mechanical properties and the fatigue strength, the relation must be based on the real (measured) mechanical properties, not on the specified minimum properties.

2.1.5 The calculation technique presented in Ch 2, 2.4 Calculation of sample mean and standard deviation was developed for the original staircase method. However, since there is no similar method dedicated to the modified staircase method, the same is applied for both.

2.2 Staircase method

2.2.1 In the original staircase method, the first specimen is subjected to a stress corresponding to the expected average fatigue strength. If the specimen survives 10⁷ cycles, it is discarded and the next specimen is subjected to a stress that is one increment above the previous, i.e. a survivor is always followed by the next using a stress one increment above the previous. The increment should be selected to correspond to the expected level of the standard deviation.

2.2.2 When a specimen fails prior to reaching 10⁷ cycles, the obtained number of cycles is noted and the next specimen is subjected to a stress that is one increment below the previous. With this approach, the sum of failures and run-outs is equal to the number of specimens.

2.2.3 This original staircase method is only suitable when a high number of specimens are available. Through simulations it has been found that the use of about 25 specimens in a staircase test leads to a sufficient accuracy in the result.

2.3 Modified staircase method

2.3.1 When a limited number of specimens are available, it is advisable to apply the modified staircase method. With this method, the first specimen is subjected to a stress level that is most likely well below the average fatigue strength. When this specimen has survived 10⁷ cycles, this same specimen is subjected to a stress level one increment above the previous. The increment should be selected to correspond to the expected level of the standard deviation. This is continued with the same specimen until failure. Then the number of cycles is recorded, and the next specimen is subjected to a stress that is at least two increments below the level where the previous specimen failed.

2.3.2 With this approach the number of failures usually equals the number of specimens. The number of run-outs, counted as the highest level where 10⁷ cycles were reached, also equals the number of specimens.

2.3.3 The acquired result of a modified staircase method should be used with care, since some results available indicate that testing a run-out on a higher test level, especially at high mean stresses, tends to increase the fatigue limit. However, this ‘training effect’ is less pronounced for high strength steels (e.g. UTS > 800 MPa).

2.3.4 If the confidence calculation is desired or necessary, the minimum number of test specimens is three.

2.4 Calculation of sample mean and standard deviation

2.4.1 A hypothetical example of tests for five crank throws is presented further in the subsequent text. When using the modified staircase method and the evaluation method of Dixon and Mood, the number of samples will be 10, meaning five run-outs and five failures, i.e.:

Number of samples,

n= 10

Furthermore, the method distinguishes between:

	Less frequent event is failures	C=1
	Less frequent event is run-outs	C=2

2.4.2 The method uses only the less frequent occurrence in the test results, i.e. if there are more failures than run-out, then the number of run-outs is used, and vice versa.

2.4.3 In the modified staircase method, the number of run-outs and failures are usually equal. However, the testing can be unsuccessful, e.g. the number of run-outs can be less than the number of failures if a specimen with two increments below the previous failure level goes directly to failure. On the other hand, if this unexpected premature failure occurs after a rather high number of cycles, it is possible to define the level below this as a run-out.

2.4.4 Dixon and Mood’s approach, derived from the maximum likelihood theory, which may also be applied here, especially on tests with few samples, presents some simple approximate equations for calculating the sample mean and the standard deviation from the outcome of the staircase test. The sample mean can be calculated as follows:

when C=1

when C=2

The standard deviation can be found by:

where:

S_a0 is the lowest stress level for the less frequent occurrence

d is the stress increment

F	=	Σ f_i
A	=	Σ i·f_i
B	=	Σ i²·f_i

i is the stress level numbering

f_i is the number of samples at stress level i

2.4.5 The formula for the standard deviation is an approximation and can be used when:

and

If any of these two conditions are not fulfilled, a new staircase test should be considered, or the standard deviation should be taken quite large in order to be on the safe side.

2.4.6 If increment d is much higher than the standard deviation s, the procedure leads to a lower standard deviation and a slightly higher sample mean, both compared to values calculated when the difference between the increment and the standard deviation is relatively small. Respectively, if increment d is much less than the standard deviation s, the procedure leads to a higher standard deviation and a slightly lower sample mean.

2.5 Confidence interval for mean fatigue limit

2.5.1 If the staircase fatigue test is repeated, the sample mean and the standard deviation will most likely be different from the previous test. Therefore it is necessary to assure with a given confidence that the repeated test values will be above the chosen fatigue limit by using a confidence interval for the sample mean.

2.5.2 The confidence interval for the sample mean value with unknown variance is known to be distributed according to the t-distribution (also called Student’s t-distribution), which is a distribution symmetric around the average.

Figure 2.2.1 Student's t-distribution

2.5.3 If S_a is the empirical mean and s is the empirical standard deviation over a series of n samples, in which the variable values are normally distributed with an unknown sample mean and unknown variance, the (1 - α) 100 per cent confidence interval for the mean is:

The resulting confidence interval is symmetric around the empirical mean of the sample values, and the lower endpoint can be found as:

which is the mean fatigue limit (population value) to be used to obtain the reduced fatigue limit where the limits for the probability of failure are taken into consideration.

2.6 Confidence interval for standard deviation

2.6.1 The confidence interval for the variance of a normal random variable is known to possess a chi-square distribution with n - 1 degrees of freedom.

Figure 2.2.2 Chi-square distribution

2.6.2 An assumed fatigue test value from n samples is a normal random variable with a variance of σ² and has an empirical variance s². Therefore a (1 - α) 100 per cent confidence interval for the variance is:

2.6.3 A (1 – α). 100 per cent confidence interval for the standard deviation is obtained by the square root of the upper limit of the confidence interval for the variance, and can be found by:

2.6.4 This standard deviation (population value) is to be used to obtain the fatigue limit, where the limits for the probability of failure are taken into consideration.