Cycle Counting in Fatigue Analysis

Cycle Counting in Fatigue Analysis — Benchmark

In this benchmark model for the Rainflow counting algorithm, values computed by the Fatigue Module in COMSOL are compared with the ASTM standard E1049-85 (Ref. 1). An extension to the benchmark compares the Palmgren-Miner cumulative damage model to analytical expressions.

Model Definition

A flat test specimen is subjected to a repeated load cycle. The material has elastic properties defined by Young’s modulus E = 69 GPa and Poisson’s ratio ν = 0.34. The thickness of the specimen is 6.25 mm. The remaining dimensions are given in Figure 1.

Figure 1: Flat test specimen. Dimensions are given in mm.

The Wöhler diagram (S-N curve) is given by the expression

where σa is the stress amplitude, R is the R-value and N is the number of cycles to failure for a constant stress cycle defined by σa and R. The relation is valid for

, while the parameter

is seen as infinite life and it should not be taken into account in damage calculations.

An ASTM cycle (Ref. 1) is evaluated in the example. The load history of the cycle is presented in Table 1.

Table 1: Fatigue cycle.
Step	Load units
1	-2
2	1
3	-3
4	5
5	-1
6	3
7	-4
8	4
9	2

The unit load can be chosen arbitrarily and is here selected so that one unit load corresponds to 10 MPa stress in the central cross section.

The ASTM example is further extended to examine the fatigue damage caused by 100000 blocks of the cycle.

In the test specimen, the stress state in the central part away from the fillets does not vary with the position. Therefore, any point in the central thin cross section can be chosen for evaluation.

Results and Discussion

Because of the symmetry, only a quarter of the test specimen is modeled. The 2D Solid Mechanics interface is used with a plane stress assumption. Figure 2 shows the axial stress caused by a unit load. Although a quarter of the specimen was modeled, the results for the whole specimen can be examined using datasets of the type .

Figure 2: Axial stress in the specimen caused by a unit load.

The feature generates results of the cycle counting algorithm as well as a damage calculation. A point in the thin central cross section is evaluated. The applied load cycle, see Table 1, is quantified with the Rainflow Counting algorithm and shown in Figure 3. The ASTM results (Ref. 1) are shown in Table 2

Table 2: ASTM Rainflow counting results.
σa (MPa)	σm (MPa)	n
20	-10	0.5
15	-5	0.5
40	0	0.5
45	5	0.5
40	10	0.5
30	10	0.5
1	10	1.0

here, σm is the cycle mean stress amplitude and n is number of cycles.

Figure 3: Load cycle quantified with the Rainflow Counting option.

The Rainflow Counting results by ASTM and COMSOL are in perfect agreement.

The evaluation of the cumulative damage is now compared against analytical expressions. In the Palmgren-Miner model, the damage is calculated for each stress bin. The number of cycles to failure for a constant cycle is taken from the S-N curve which is evaluated at the center of the bin.

With the chosen bin discretization the damage is evaluated at following bin stress centers:

= 17.1 MPa, 21.4 MPa, 25.7 MPa, 30.0 MPa, 34.3 MPa, 38.6 MPa, 42.9 MPa

and

= -8.0 MPa, -4.0 MPa, 0.0 MPa, 4.0 MPa, 8.0 MPa

The key values for the damaging bins are presented in Table 3. The superscript ‘b’ denotes that the variable is evaluated at the bin center.

Table 3: Damaging bin DATA.
σa (MPa)	σm (MPa)	σab (MPa)	σmb (MPa)	Rb	nb	Nb
20.0	-10.0	21.4	-8.0	-2.19	0.500	Inf
15.0	-5.0	17.1	-4.0	-1.61	0.500	Inf
40.0	0.0	38.6	0.0	-1.00	0.500	3.44e7
45.0	5.0	42.9	4.0	-0.829	0.500	2.31e6
40.0	10.0	38.6	8.0	-0.656	0.500	5.84e5
30.0	10.0	30.0	8.0	-0.579	0.500	1.45e6
20.0	10.0	21.4	8.0	-0.456	1.00	2.47e6

The fatigue usage factor fus following the Palmgren-Miner linear damage rule is calculated using the expression

(1)

where m is the number or repeatable block and p is the number of bins.

Following Equation 1 the fatigue usage factor of the ASTM cycles is

. The COMSOL result is fus= 0.182. The small discrepancy between the results is attributed to the evaluation of the S-N curve.

The relative damage contribution from each bin is shown in Figure 4.

Figure 4: Contribution of each stress bin to the fatigue usage.

The Cumulative Damage model is based on the Palmgren-Miner linear damage rule. It is a discrete model in the sense that the calculations are based on stress bins which hold all stress cycles within a certain stress amplitude and mean stress range.

In this example, the stress amplitude range in a bin is 20 MPa/5=4 MPa and the mean stress range in a bin is 30 MPa/7 MPa = 4.3 MPa. By changing the bin discretization, the graph showing counted stress cycles in Figure 3, appears with a different resolution.

As for the results of the relative damage in Figure 4, a change in discretization can affect the results significantly. The damage is evaluated based on the bin stresses and not true cycle stresses. Consider the cycle defined by σa = 45.0 MPa and σm = 5.0 MPa. It is evaluated in a bin defined by

= 42.9 MPa and

= 4.0 MPa. Since bin stresses are lower than true stresses in that cycle, they predict less damage and thus give a nonconservative contribution to the fatigue usage factor.

Notes About the COMSOL Implementation

In the Cumulative Damaged feature, you can use three different types of S-N curves — , , or . In this model, the S-N curve is defined using the R-value definition. The arguments to this S-N curve are specified by first giving the R-value followed by the number of cycles, as it is done in the Analytic function in the Arguments field.

Reference

1. ASTM International, Standard Practices for Cycle Counting in Fatigue Analysis, Designation: E1049-85 (Reapproved 2011).

Fatigue_Module/Verification_Examples/cycle_counting_benchmark

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

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In the tree, select .

Click .

Click

In the tree, select .

Click

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Rectangle 1 (r1)

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In the window for , locate the section.

In the text field, type 10.

In the text field, type 100.

Rectangle 2 (r2)

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In the window for , locate the section.

In the text field, type 10-6.25.

In the text field, type 50-sqrt(12.5^2-8.75^2).

Locate the section. In the text field, type 6.25.

Circle 1 (c1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 6.25+12.5.

In the text field, type 50-sqrt(12.5^2-8.75^2).

Locate the section. In the text field, type 12.5.

Difference 1 (dif1)

In the toolbar, click

and choose .

From the bigger rectangle subtract the smaller rectangle and the circle.

Global Definitions

Specify the load cycle.

Interpolation 1 (int1)

In the toolbar, click

and choose .

In the window for , locate the section.

In the table, enter the following settings:


t	f(t)
1	-2
2	1
3	-3
4	5
5	-1
6	3
7	-4
8	4
9	-2

Solid Mechanics (solid)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Locate the section. In the d text field, type 0.00625.

Symmetry 1

In the toolbar, click

and choose .

Select Boundaries 1 and 2 only.

Boundary Load 1

In the toolbar, click

and choose .

Select Boundary 3 only.

In the window for , locate the section.

From the list, choose .

Specify the Ftot vector as


0	x
F*int1(case)	y

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
F	106.2512.5/2	390.63	Load unit
case	1	1	Load case

Materials

Material 1 (mat1)

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	E	69e9	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	0.34	1	Young’s modulus and Poisson’s ratio
Density	rho	2700	kg/m³	Basic