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The Multislope model: A new description for the fatigue strength of glass fibre reinforced plastic
International Journalof Fatigue
International Journal of Fatigue 29 (2007) 1571–1576
www.elsevier.com/locate/ijfatigue
Technical note
The Multislope model: A new description for the fatigue strength of glass fibre reinforced plastic G.K. Boerstra
*
AE-Rotortechniek B.V, Goudstraat 15, 7554 NG Hengelo, The Netherlands Received 28 April 2006; received in revised form 19 October 2006; accepted 14 November 2006 Available online 3 January 2007
Abstract A set of equations is proposed to describe the fatigue behaviour of E-glass laminates under loads with various means and amplitudes. The equations enable a simple method for life time predictions of laminate structures subjected to fatigue loads with continuously varying mean stress. The shape parameters describing the new model will be optimised by minimising the least square sum of the shortest distance. An extensive data set of measurements with constant amplitudes is used to demonstrate the accuracy of the model and clearly shows the fundamental difference in damage accumulation between a mean stress on tension and a mean stress on compression. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Composites; Mechanical properties; Life time prediction; Goodman
1. Introduction For the life time calculations of structures of glass fibre reinforced plastics such as rotor blades of wind turbines it is essential to have access to accurate information about the fatigue resistance of the material. These data should comprise the number of cycles to failure under a specific combination of stress amplitude and mean stress. The fatigue data can be presented as a descriptive model which is essentially a three-dimensional figure of which the familiar diagrams such as the S–N lines and the Constant-Life lines (or shortly CL lines) are two-dimensional sections. The S– N lines are sections parallel to the N axis and the CL lines are presented in the Sm–Sa plane and are valid for one number of cycles. Traditionally these figures are kept as simple as possible. S–N lines are preferably straight on log–log scale while the CL lines are triangular: Goodman diagrams [1]. The advantages were obvious in pre-computer times but nowadays more sophisticated models with more parameters may *
Tel.: +31 742552647. E-mail address: [email protected]
0142-1123/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2006.11.007
be preferred if this can improve the accuracy of the fatigue life prediction. This report introduces a new method for combining fatigue measurements together with a new type of threedimensional model and compares it with the traditional model in terms of accuracy that can be achieved. 2. Data processing in the traditional way The first descriptions of the influence of the mean stress in the fatigue behaviour of metals with an empirical, material dependent, parameter were made by Morrow [2], Walker [3] and Schu¨tz [4]. For the fatigue of composites the most common practise until recently was as to collect the test results with the same stress ratio R in one group. Per group the results are plotted in a graph with the achieved number of cycles on the horizontal axis and the stress amplitude on the vertical axis. Through the data points a regression line is fitted that is preferably linear when both axes are scaled logarithmically. The result can be described with a simple equation: S 1 =S 2 ¼ ðN 1 =N 2 Þ
ð1=mÞ
ð1Þ
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G.K. Boerstra / International Journal of Fatigue 29 (2007) 1571–1576
Nomenclature D m0 m n N Ne Np R Sa SA1
skewness parameter for the dependency of m slope of S–N line on log–log scale for Sm = 0 slope of S–N line for mean stress Sm number of data points number of cycles to failure expected number of cycles based on model reference number of cycles for defined shape of CL line stress ratio: Smin/Smax stress amplitude of load cycle apex amplitude for N = 1 and Sm = 0
SAp Sap Sm SDt aC aT DSa Dn Dt m
apex amplitude for Np cycles and Sm = 0 amplitude for Np cycles and mean stress Sm mean stress of load cycle standard deviation of Dt exponent for CL line for Sm < 0 exponent for CL line for Sm > 0 logarithmic deviation of stress amplitudes Sap logarithmic deviation of number of cycles ‘‘shortest distance’’ combination of DSa and Dn coefficient of variation
This result in regression lines with a different slope for each R-value. The stress values of the regression lines at round numbers of cycles are read from these graphs and plotted in CL lines with the mean stress on the horizontal axis and the amplitude on the vertical axis. For each number of cycles the stress points are connected together with the static values on the horizontal axis. The result for a data set with five different R-values looks rather odd (Fig. 1) and seems not suited for any mathematical description. Some researcher are comfortable with this situation e.g. Sutherland [5] or state that: ‘‘..for composites no empirical or analytical relationship between mean values and amplitudes at fixed numbers of cycles could be established.’’ (Brønstedt et al. [6, p. 529]). The only acceptable way to fit the CL lines into a mathematic model was to draw triangles with an apex that decreases with the same power law as the S–N line for R = 1. This may result in the well known Goodman diagram with the apex at Sm = 0 (Fig. 1) or a less familiar triangle with the apex in the middle between UTS and UCS (Fig. 2) [11].
It is easy to see that the original figure has undergone a drastic change in order to find something that can be used for mathematical operations. The result is however far from satisfying. For this reason the model has to be expanded with more parameters. A very promising empirical model was developed by Beheshty et al. [7] describing CL lines with three parameters. In this model the shape of the CL line for each observed number of cycles can be different which is in good correlation with reality. A drawback of the model is however that for each number of cycles a different set of parameters is needed which have no mathematical relation with those of other numbers. This drawback is not present in the model presented here.
Fig. 1. Constant life diagram based on constant R-values.
Fig. 2. Simple constant life diagram scaled from static values.
3. The Multislope model The shape of the CL lines will be based on the Gerber parabola [8] but is modified by replacing the exponent 2 by a variable exponent. It seems most appropriate to take different exponents for the tension and the compression
G.K. Boerstra / International Journal of Fatigue 29 (2007) 1571–1576
side: aT for the tension and aC for the compression side. This yields the general formulas: aT
ð2aÞ
aC
ð2bÞ
For S m > 0 :
S ap ¼ S Ap ð1 ðS m =UTSÞ Þ
For S m < 0 :
S ap ¼ S Ap ð1 ðS m =UCSÞ Þ:
The second expansion of the original triangular model will be made by introducing variable slopes for the S–N lines with various mean stress values. As can be seen from Fig. 1 at the tension side of the diagram the CL lines are spread more widely, suggesting a steep S–N line with small m value and on the compression side the CL lines are more concentrated with a flatter S–N line and higher value for m. A practical solution for the dependency of m can be to apply an exponential relation with the mean stress Sm: m ¼ m0 eðS m =DÞ :
ð3Þ
It should be noted that for the triangular (Goodman) model the slope of the S–N lines for different R-values is already variable. The slope is minimal for R = 1 and rises on either side to 1 when R approaches +1. When both expansions are combined it becomes necessary to choose a reference number of cycles for which the CL line will be defined by the parameters aT and aC. For all other numbers the shape of the CL lines follows from the defined change in S–N slope. If the skewness parameter D equals 1 the shape of CL lines is the same for all numbers of cycles. The complete three-dimensional model with dimension Sa, Sm and ln(N) will look now as follows: (Fig. 3) Clearly can be seen that the shape of the CL lines is different for the three values of N: 1 cycle, Np, and N cycles. The arrowed line between the measuring point (Sa, Sm,N) and the Np plane represents the S–Ns line for this particular value of Sm.
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4. Constructing the model from measurements When constructing the model for a specific material first the static strengths UTS and UCS have to be determined. Subsequently fatigue tests on coupons with various values of stress amplitude Sa and mean stress Sm will be performed. Now a preliminary model can be based on a chosen value for Np and estimates for the values for the parameters m0, D, aT and aC. At first the value for SAp (the apex of the CL line in Np) will be calculated as follows: Calculate for each Sm the expected value for m with (3) and project each measured point along the S–N slope onto the Np plane with (1). Transpose each point Sap in the Np plane along the CL lines onto the Sm = 0 axis using (2a) or (2b). The average of all individual points will be used as the SAp in the model. The parameters Np, m0, D, aT and aC will now be optimised until the best fit is obtained of the model through the data points. The least square method is followed for the shortest distance of each measuring point to the S–N line for its particular mean stress Sm. The shortest distance is composed of the deviation in stress amplitude DSa and the deviation in the number of cycles Dn. The reason to do so is as follows: On the tensile side of the mean values the S–N slope is rather steep and on the compressive side more gentle. This will cause that measuring points on the compressive side will become more dominant than those on the tensile side when the least squares of the number of cycles is used for determination of the best fitting regression line. Likewise can be seen that points on the tensile side will become more dominant than those on the compressive side when the least squares of the stress is used for the regression analysis. To balance the influence of both directions with each other the regression can best be performed by minimising the combination of both deviations. First the deviation on the Stress amplitude DSa is calculated for each point as the difference between the logarithm of the projected amplitude in the Np plane and the logarithm of the amplitude value following from the model for this Sm, using (2a) or (2b). DSa ¼ lnðS ap Þ lnðS ap;mod Þ
ð4Þ
Secondly the deviation on the number of cycles Dn is calculated for each measured point as the difference between the realised number of cycles and the predicted number of cycles Ne Fig. 3. The complete Multislope model.
Dn ¼ lnðN Þ lnðN e Þ
ð5Þ
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G.K. Boerstra / International Journal of Fatigue 29 (2007) 1571–1576 Table 2 Influence of Np on the model parameters
Table 1 Model sophistication and obtained accuracy
Np m0 aT aC D SDt SA1
Traditional Goodman diagram
Constant slope, curved CL lines
Variable slope, straight CLD for Np
All parameters free
n.a. 10.10 1 1 1 0.173 471
1 10.08 1.09 2.90 1 0.126 389
2245 10.47 1.00 1.00 234 0.0945 358
100 10.54 2.06 1.04 244 0.0787 389
Since DSa and Dn were both calculated from logarithms they have become dimensionless and may now be combined to the shortest distance to the S–N line Dt: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð6Þ Dt ¼ signðDSaÞ ð1=ð1=DSa2 þ 1=Dn2 ÞÞ The standard deviation of all values of Dt will now be called the total standard deviation SDt. The best fit will be reached by minimising the SDt by varying the estimated parameters m0; D; aT and aC. 5. Analysis of existing data set To test the efficiency of the expanded model a set of existing data is taken of fatigue tests on glassfibre-polyester coupons measured at ECN and TU Delft. These data are part of the database FACT and already analysed in [9– 11]. It comprised material with 50% UD and 50% ± 45 fibres known as GP 0/45. The static strength of this material was: UTS = 370 MPa and UCS = 286 MPa The S–N line for R = 1 showed a slope of 1:10.
Np
10
100
1000
m0 aT aC D SDt
10.51 3.37 0.94 268 0.0798
10.54 2.06 1.04 244 0.0787
10.57 1.36 1.22 250 0.0797
Starting with the traditional Goodman triangular model with the apex on Sm = 0 apex and the same slope for all values of Sm now the influence of the various parameters in the new model is investigated by optimising to a lowest value for SDt. The most sophisticated expanded model with all parameters free is compared with the more simple models with restricted parameters for aT, aC and D. The result can be summarised as follows (Table 1). As can be seen from Table 1, giving up the triangular shapes already gives a remarkable reduction of the scatter just as introducing a variable slope. Combination of both expansions gives the best results, as was expected. The choice of Np, the reference number for the projection plane, is not very critical for the achieved SDt values (Table 2). For all values between 1 and 104 the SDt was about 0.08 with almost the same values for m0, and D. Only the values for aT and aC adapted themselves to the chosen number Np. From the point of view of the lowest standard deviation it is best to take Np equal to 100. Furthermore with this value the bundle of CL lines showed the best resemblance with Fig. 1 (Fig. 4). The effectiveness of the model can be illustrated when all measuring points are projected along the straight SN slopes onto the Np = 100 plane together with the CL line that follows from the model (Fig. 5).
100
400
1000
Sa 350
1E+4
300
1E+5
250
1E+6 1E+7
200 1E+8 150
Static
100 50 0 -300
-200
-100
0
100
200
Fig. 4. Constant life lines generated by the model.
300
Sm 400
G.K. Boerstra / International Journal of Fatigue 29 (2007) 1571–1576
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350 Sa 300
250
200
150
100
50
0 -400
-300
-200
-100
0
100
200
Sm
300
400
Fig. 5. CL diagram for Np and the projected data points.
The dashed line represents the calculated CL line and the continuous line is the line for 95% probability with 95% confidence drawn as a reference. This so called 95/ 95 line was calculated from the standard deviation on the stress amplitude using the equation according DIN 55303: 1 S 95=95 ¼ S 1 1:645 m 1 þ pffiffiffi n in which n = number of data points, m = coefficient of variation. A second way for illustration of the effectiveness is the transposition of all measuring points along the CL lines to the Sm = 0 plane and to plot them together with the calculated S–N line for R = 1 (Fig. 6): Again the 95/95 limit is drawn as a reference. From Figs. 5 and 6 can be seen that the fit of the model is remarkably well. Even the test results with failure cycles 1000.0 SA
100.0
R2 = 0.96146
10.0 1.E+00
1.E+02
1.E+04
1.E+06
1.E+08
N
1.E+10
Fig. 6. SN line for Sm = 0 and the transposed data points.
below 100 can still be described accurately in spite of the fact that for higher mean stresses the model predicts an amplitude larger than the Goodman line for one cycle: the static limit (see Fig. 4). 6. Conclusions and discussion The proposed Multislope model gives a very good fit with measured data points. The standard deviation of all points is more than half the standard deviation in the traditional model with triangular CL lines and a constant S–N slope. The model is relatively insensitive for ‘‘good’’ or ‘‘bad’’ static values. The aT and aC can simply flex the CL curves to the best fit whatever the static values are. This certainly is an advantage considering the difficulty to perform reliable compressive tests. The model uses 7 parameters for its description. One can argue that this lack of simplicity or elegance is proof that it can never be a true description of reality. The model however does not pretend to give a theory explaining why failure takes place but is merely a practical description for engineering purposes. A drawback of the model could be that it looses accuracy for very low numbers of cycles and ignores the static strength. This could be solved by introducing S–N lines shaped as Sendeckyj lines but it is well justified to state that fatigue has nothing to do with static strength and it is allowed to split the problem in two. The fatigue model can be declared valid for numbers of cycles larger than 100 and will be based on measurements with this minimum lifetime. A static strength verification must be performed separately using the static values. The model fitting is based on reducing the scatter in stress as well as in the number of cycles. Normally for
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G.K. Boerstra / International Journal of Fatigue 29 (2007) 1571–1576
the fit of the S–N line for one R-value a regression calculation of N on S is used since S is the independent variable and N is the dependent. This is however not completely true since only the average stress is known. Internal stress distribution can be different for various test coupons so local stresses can cause scatter as well. Another reason is that also the calculation of the 95/95 limit is performed on stress level using the SD on the stress.
[4]
[5]
[6] [7]
References [1] Goodman J. Mechanics applied engineering. London: Longman, Green and Co.; 1899. [2] Morrow J. Fatigue properties of metals. In: Proceedings of Division 4 of the SAE Iron and Steel Technical Committee. November 4, 1964. [3] Walker K. The Effect of Stress Ratio During Crack Propagation and Fatigue for 2024-T3 and 7075-T6 Aluminum, Effects for Environment and Complex load History on Fatigue Life. ASTM
[8] [9] [10]
[11]
STP 462. West Conshohocken (PA): American Society For Testing and Materials; 1970. Schu¨tz W. Gesichtspunkte zur Werkstoffauswahl fu¨r Schwingbeanschpruchte Konstruktionen. LBF Kolloquium, Darmstadt, 1968. Lab. Fu¨r Betriebsfestigkeit, Bericht Nr. TB-80; 1968. Sutherland HJ, Mandell JF. Updated Goodman diagrams for fibreglass composites materials using the DOE/MSU fatigue database. Global Windpower 2004, Paper #18983 , AWEA/EWEA; 2004. Brønstedt P, Lilholt H, Lystrup A. Composite materials for wind power turbine blades. Annu Rev Mater Res 2005;35:505–38. Beheshty MH et al. An empirical fatigue life model for highperformance fibre composites with and without impact damage. Composites: Part A 1999;30:971–87. Gerber WZ. Bayer Architect Ing 1874;6:101. Bach PW. Fatigue lifetime of glass–polyester laminates for wind turbine blades. ECN-C-94-020, Petten; 1994. Bach PW, De Smet BJ. Database FACT, Fatigue of composites for wind turbines. ECN-C-94-045, Petten; 1994.. Bach PW, De Smet BJ. Life time predictions of glass fibre reinforced polyester with database FACT. ECN-C-94-044, Petten; 1994.
International Journal of Fatigue 29 (2007) 1571–1576
www.elsevier.com/locate/ijfatigue
Technical note
The Multislope model: A new description for the fatigue strength of glass fibre reinforced plastic G.K. Boerstra
*
AE-Rotortechniek B.V, Goudstraat 15, 7554 NG Hengelo, The Netherlands Received 28 April 2006; received in revised form 19 October 2006; accepted 14 November 2006 Available online 3 January 2007
Abstract A set of equations is proposed to describe the fatigue behaviour of E-glass laminates under loads with various means and amplitudes. The equations enable a simple method for life time predictions of laminate structures subjected to fatigue loads with continuously varying mean stress. The shape parameters describing the new model will be optimised by minimising the least square sum of the shortest distance. An extensive data set of measurements with constant amplitudes is used to demonstrate the accuracy of the model and clearly shows the fundamental difference in damage accumulation between a mean stress on tension and a mean stress on compression. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Composites; Mechanical properties; Life time prediction; Goodman
1. Introduction For the life time calculations of structures of glass fibre reinforced plastics such as rotor blades of wind turbines it is essential to have access to accurate information about the fatigue resistance of the material. These data should comprise the number of cycles to failure under a specific combination of stress amplitude and mean stress. The fatigue data can be presented as a descriptive model which is essentially a three-dimensional figure of which the familiar diagrams such as the S–N lines and the Constant-Life lines (or shortly CL lines) are two-dimensional sections. The S– N lines are sections parallel to the N axis and the CL lines are presented in the Sm–Sa plane and are valid for one number of cycles. Traditionally these figures are kept as simple as possible. S–N lines are preferably straight on log–log scale while the CL lines are triangular: Goodman diagrams [1]. The advantages were obvious in pre-computer times but nowadays more sophisticated models with more parameters may *
Tel.: +31 742552647. E-mail address: [email protected]
0142-1123/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2006.11.007
be preferred if this can improve the accuracy of the fatigue life prediction. This report introduces a new method for combining fatigue measurements together with a new type of threedimensional model and compares it with the traditional model in terms of accuracy that can be achieved. 2. Data processing in the traditional way The first descriptions of the influence of the mean stress in the fatigue behaviour of metals with an empirical, material dependent, parameter were made by Morrow [2], Walker [3] and Schu¨tz [4]. For the fatigue of composites the most common practise until recently was as to collect the test results with the same stress ratio R in one group. Per group the results are plotted in a graph with the achieved number of cycles on the horizontal axis and the stress amplitude on the vertical axis. Through the data points a regression line is fitted that is preferably linear when both axes are scaled logarithmically. The result can be described with a simple equation: S 1 =S 2 ¼ ðN 1 =N 2 Þ
ð1=mÞ
ð1Þ
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G.K. Boerstra / International Journal of Fatigue 29 (2007) 1571–1576
Nomenclature D m0 m n N Ne Np R Sa SA1
skewness parameter for the dependency of m slope of S–N line on log–log scale for Sm = 0 slope of S–N line for mean stress Sm number of data points number of cycles to failure expected number of cycles based on model reference number of cycles for defined shape of CL line stress ratio: Smin/Smax stress amplitude of load cycle apex amplitude for N = 1 and Sm = 0
SAp Sap Sm SDt aC aT DSa Dn Dt m
apex amplitude for Np cycles and Sm = 0 amplitude for Np cycles and mean stress Sm mean stress of load cycle standard deviation of Dt exponent for CL line for Sm < 0 exponent for CL line for Sm > 0 logarithmic deviation of stress amplitudes Sap logarithmic deviation of number of cycles ‘‘shortest distance’’ combination of DSa and Dn coefficient of variation
This result in regression lines with a different slope for each R-value. The stress values of the regression lines at round numbers of cycles are read from these graphs and plotted in CL lines with the mean stress on the horizontal axis and the amplitude on the vertical axis. For each number of cycles the stress points are connected together with the static values on the horizontal axis. The result for a data set with five different R-values looks rather odd (Fig. 1) and seems not suited for any mathematical description. Some researcher are comfortable with this situation e.g. Sutherland [5] or state that: ‘‘..for composites no empirical or analytical relationship between mean values and amplitudes at fixed numbers of cycles could be established.’’ (Brønstedt et al. [6, p. 529]). The only acceptable way to fit the CL lines into a mathematic model was to draw triangles with an apex that decreases with the same power law as the S–N line for R = 1. This may result in the well known Goodman diagram with the apex at Sm = 0 (Fig. 1) or a less familiar triangle with the apex in the middle between UTS and UCS (Fig. 2) [11].
It is easy to see that the original figure has undergone a drastic change in order to find something that can be used for mathematical operations. The result is however far from satisfying. For this reason the model has to be expanded with more parameters. A very promising empirical model was developed by Beheshty et al. [7] describing CL lines with three parameters. In this model the shape of the CL line for each observed number of cycles can be different which is in good correlation with reality. A drawback of the model is however that for each number of cycles a different set of parameters is needed which have no mathematical relation with those of other numbers. This drawback is not present in the model presented here.
Fig. 1. Constant life diagram based on constant R-values.
Fig. 2. Simple constant life diagram scaled from static values.
3. The Multislope model The shape of the CL lines will be based on the Gerber parabola [8] but is modified by replacing the exponent 2 by a variable exponent. It seems most appropriate to take different exponents for the tension and the compression
G.K. Boerstra / International Journal of Fatigue 29 (2007) 1571–1576
side: aT for the tension and aC for the compression side. This yields the general formulas: aT
ð2aÞ
aC
ð2bÞ
For S m > 0 :
S ap ¼ S Ap ð1 ðS m =UTSÞ Þ
For S m < 0 :
S ap ¼ S Ap ð1 ðS m =UCSÞ Þ:
The second expansion of the original triangular model will be made by introducing variable slopes for the S–N lines with various mean stress values. As can be seen from Fig. 1 at the tension side of the diagram the CL lines are spread more widely, suggesting a steep S–N line with small m value and on the compression side the CL lines are more concentrated with a flatter S–N line and higher value for m. A practical solution for the dependency of m can be to apply an exponential relation with the mean stress Sm: m ¼ m0 eðS m =DÞ :
ð3Þ
It should be noted that for the triangular (Goodman) model the slope of the S–N lines for different R-values is already variable. The slope is minimal for R = 1 and rises on either side to 1 when R approaches +1. When both expansions are combined it becomes necessary to choose a reference number of cycles for which the CL line will be defined by the parameters aT and aC. For all other numbers the shape of the CL lines follows from the defined change in S–N slope. If the skewness parameter D equals 1 the shape of CL lines is the same for all numbers of cycles. The complete three-dimensional model with dimension Sa, Sm and ln(N) will look now as follows: (Fig. 3) Clearly can be seen that the shape of the CL lines is different for the three values of N: 1 cycle, Np, and N cycles. The arrowed line between the measuring point (Sa, Sm,N) and the Np plane represents the S–Ns line for this particular value of Sm.
1573
4. Constructing the model from measurements When constructing the model for a specific material first the static strengths UTS and UCS have to be determined. Subsequently fatigue tests on coupons with various values of stress amplitude Sa and mean stress Sm will be performed. Now a preliminary model can be based on a chosen value for Np and estimates for the values for the parameters m0, D, aT and aC. At first the value for SAp (the apex of the CL line in Np) will be calculated as follows: Calculate for each Sm the expected value for m with (3) and project each measured point along the S–N slope onto the Np plane with (1). Transpose each point Sap in the Np plane along the CL lines onto the Sm = 0 axis using (2a) or (2b). The average of all individual points will be used as the SAp in the model. The parameters Np, m0, D, aT and aC will now be optimised until the best fit is obtained of the model through the data points. The least square method is followed for the shortest distance of each measuring point to the S–N line for its particular mean stress Sm. The shortest distance is composed of the deviation in stress amplitude DSa and the deviation in the number of cycles Dn. The reason to do so is as follows: On the tensile side of the mean values the S–N slope is rather steep and on the compressive side more gentle. This will cause that measuring points on the compressive side will become more dominant than those on the tensile side when the least squares of the number of cycles is used for determination of the best fitting regression line. Likewise can be seen that points on the tensile side will become more dominant than those on the compressive side when the least squares of the stress is used for the regression analysis. To balance the influence of both directions with each other the regression can best be performed by minimising the combination of both deviations. First the deviation on the Stress amplitude DSa is calculated for each point as the difference between the logarithm of the projected amplitude in the Np plane and the logarithm of the amplitude value following from the model for this Sm, using (2a) or (2b). DSa ¼ lnðS ap Þ lnðS ap;mod Þ
ð4Þ
Secondly the deviation on the number of cycles Dn is calculated for each measured point as the difference between the realised number of cycles and the predicted number of cycles Ne Fig. 3. The complete Multislope model.
Dn ¼ lnðN Þ lnðN e Þ
ð5Þ
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G.K. Boerstra / International Journal of Fatigue 29 (2007) 1571–1576 Table 2 Influence of Np on the model parameters
Table 1 Model sophistication and obtained accuracy
Np m0 aT aC D SDt SA1
Traditional Goodman diagram
Constant slope, curved CL lines
Variable slope, straight CLD for Np
All parameters free
n.a. 10.10 1 1 1 0.173 471
1 10.08 1.09 2.90 1 0.126 389
2245 10.47 1.00 1.00 234 0.0945 358
100 10.54 2.06 1.04 244 0.0787 389
Since DSa and Dn were both calculated from logarithms they have become dimensionless and may now be combined to the shortest distance to the S–N line Dt: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð6Þ Dt ¼ signðDSaÞ ð1=ð1=DSa2 þ 1=Dn2 ÞÞ The standard deviation of all values of Dt will now be called the total standard deviation SDt. The best fit will be reached by minimising the SDt by varying the estimated parameters m0; D; aT and aC. 5. Analysis of existing data set To test the efficiency of the expanded model a set of existing data is taken of fatigue tests on glassfibre-polyester coupons measured at ECN and TU Delft. These data are part of the database FACT and already analysed in [9– 11]. It comprised material with 50% UD and 50% ± 45 fibres known as GP 0/45. The static strength of this material was: UTS = 370 MPa and UCS = 286 MPa The S–N line for R = 1 showed a slope of 1:10.
Np
10
100
1000
m0 aT aC D SDt
10.51 3.37 0.94 268 0.0798
10.54 2.06 1.04 244 0.0787
10.57 1.36 1.22 250 0.0797
Starting with the traditional Goodman triangular model with the apex on Sm = 0 apex and the same slope for all values of Sm now the influence of the various parameters in the new model is investigated by optimising to a lowest value for SDt. The most sophisticated expanded model with all parameters free is compared with the more simple models with restricted parameters for aT, aC and D. The result can be summarised as follows (Table 1). As can be seen from Table 1, giving up the triangular shapes already gives a remarkable reduction of the scatter just as introducing a variable slope. Combination of both expansions gives the best results, as was expected. The choice of Np, the reference number for the projection plane, is not very critical for the achieved SDt values (Table 2). For all values between 1 and 104 the SDt was about 0.08 with almost the same values for m0, and D. Only the values for aT and aC adapted themselves to the chosen number Np. From the point of view of the lowest standard deviation it is best to take Np equal to 100. Furthermore with this value the bundle of CL lines showed the best resemblance with Fig. 1 (Fig. 4). The effectiveness of the model can be illustrated when all measuring points are projected along the straight SN slopes onto the Np = 100 plane together with the CL line that follows from the model (Fig. 5).
100
400
1000
Sa 350
1E+4
300
1E+5
250
1E+6 1E+7
200 1E+8 150
Static
100 50 0 -300
-200
-100
0
100
200
Fig. 4. Constant life lines generated by the model.
300
Sm 400
G.K. Boerstra / International Journal of Fatigue 29 (2007) 1571–1576
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350 Sa 300
250
200
150
100
50
0 -400
-300
-200
-100
0
100
200
Sm
300
400
Fig. 5. CL diagram for Np and the projected data points.
The dashed line represents the calculated CL line and the continuous line is the line for 95% probability with 95% confidence drawn as a reference. This so called 95/ 95 line was calculated from the standard deviation on the stress amplitude using the equation according DIN 55303: 1 S 95=95 ¼ S 1 1:645 m 1 þ pffiffiffi n in which n = number of data points, m = coefficient of variation. A second way for illustration of the effectiveness is the transposition of all measuring points along the CL lines to the Sm = 0 plane and to plot them together with the calculated S–N line for R = 1 (Fig. 6): Again the 95/95 limit is drawn as a reference. From Figs. 5 and 6 can be seen that the fit of the model is remarkably well. Even the test results with failure cycles 1000.0 SA
100.0
R2 = 0.96146
10.0 1.E+00
1.E+02
1.E+04
1.E+06
1.E+08
N
1.E+10
Fig. 6. SN line for Sm = 0 and the transposed data points.
below 100 can still be described accurately in spite of the fact that for higher mean stresses the model predicts an amplitude larger than the Goodman line for one cycle: the static limit (see Fig. 4). 6. Conclusions and discussion The proposed Multislope model gives a very good fit with measured data points. The standard deviation of all points is more than half the standard deviation in the traditional model with triangular CL lines and a constant S–N slope. The model is relatively insensitive for ‘‘good’’ or ‘‘bad’’ static values. The aT and aC can simply flex the CL curves to the best fit whatever the static values are. This certainly is an advantage considering the difficulty to perform reliable compressive tests. The model uses 7 parameters for its description. One can argue that this lack of simplicity or elegance is proof that it can never be a true description of reality. The model however does not pretend to give a theory explaining why failure takes place but is merely a practical description for engineering purposes. A drawback of the model could be that it looses accuracy for very low numbers of cycles and ignores the static strength. This could be solved by introducing S–N lines shaped as Sendeckyj lines but it is well justified to state that fatigue has nothing to do with static strength and it is allowed to split the problem in two. The fatigue model can be declared valid for numbers of cycles larger than 100 and will be based on measurements with this minimum lifetime. A static strength verification must be performed separately using the static values. The model fitting is based on reducing the scatter in stress as well as in the number of cycles. Normally for
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the fit of the S–N line for one R-value a regression calculation of N on S is used since S is the independent variable and N is the dependent. This is however not completely true since only the average stress is known. Internal stress distribution can be different for various test coupons so local stresses can cause scatter as well. Another reason is that also the calculation of the 95/95 limit is performed on stress level using the SD on the stress.
[4]
[5]
[6] [7]
References [1] Goodman J. Mechanics applied engineering. London: Longman, Green and Co.; 1899. [2] Morrow J. Fatigue properties of metals. In: Proceedings of Division 4 of the SAE Iron and Steel Technical Committee. November 4, 1964. [3] Walker K. The Effect of Stress Ratio During Crack Propagation and Fatigue for 2024-T3 and 7075-T6 Aluminum, Effects for Environment and Complex load History on Fatigue Life. ASTM
[8] [9] [10]
[11]
STP 462. West Conshohocken (PA): American Society For Testing and Materials; 1970. Schu¨tz W. Gesichtspunkte zur Werkstoffauswahl fu¨r Schwingbeanschpruchte Konstruktionen. LBF Kolloquium, Darmstadt, 1968. Lab. Fu¨r Betriebsfestigkeit, Bericht Nr. TB-80; 1968. Sutherland HJ, Mandell JF. Updated Goodman diagrams for fibreglass composites materials using the DOE/MSU fatigue database. Global Windpower 2004, Paper #18983 , AWEA/EWEA; 2004. Brønstedt P, Lilholt H, Lystrup A. Composite materials for wind power turbine blades. Annu Rev Mater Res 2005;35:505–38. Beheshty MH et al. An empirical fatigue life model for highperformance fibre composites with and without impact damage. Composites: Part A 1999;30:971–87. Gerber WZ. Bayer Architect Ing 1874;6:101. Bach PW. Fatigue lifetime of glass–polyester laminates for wind turbine blades. ECN-C-94-020, Petten; 1994. Bach PW, De Smet BJ. Database FACT, Fatigue of composites for wind turbines. ECN-C-94-045, Petten; 1994.