- Email: [email protected]
Stress-strength interference theory for a pin-loaded composite joint
compositesEngineering,Voi. 5, No. 8, pp. 975-982, 1995
Copyright @ 1995ElsevierScienceLtd Printed in Great Britain. All rights reserved 0961-9526/95 $9.50+ .00
Pergamon
0961-9526(94) 00063-8
STRESS-STRENGTH INTERFERENCE PIN-LOADED COMPOSITE
THEORY JOINT
FOR A
Paul D. Herrington Department of Mechanical Engineering, University of New Orleans, New Orleans, LA 70148, U.S.A.
(Received 17 May 1994; final version accepted 25 July 1994) Abstract--Considering the increasing applications of composites as structural components, the reliability of these materials is an important issue. In this paper, stress-strength interference theory is used to determine the reliability of a component that is characterized having a Weibull strength distribution and is under the effects of an applied load that is not deterministic, but follows a probability distribution. The assumption that failure is caused by the maximum of a sequence of applied loads is the basis for describing the applied load distribution as a Gumbel Type 1 extreme value distribution. Reliability plots are given for a class of strength parameters that are typical o f graphite/epoxy laminates. A numerical example for a pin-loaded composite laminate is shown using experimentally obtained data.
INTRODUCTION
Composite materials are increasingly being used as primary load bearing structures. Due to previous experience of joining structures composed of isotropic members and due to the need for disassembly, the mechanical fastening of composites is an often used method of joining. While exhibiting a weight penalty over other joining methods such as adhesive bonding, it is important that analytical techniques are developed in order to predict the structural response and reliability of pin-loaded composite joints. The study of pin-loaded and bolted composite joints is an active field of interest for many investigators including CoUings (1987), Crews and Naik (1986), De Jong (1987), Eriksson (1987), Hart-Smith (1987), Hyer and Chastain (1985), Murthy et al. (1991), Ramkumar et al. (1987) and others. Particularly important is the study of the bearing strength of laminates. The procedure often used for designing composite structures incorporating pin or mechanically fastened joints is to provide sufficient edge and width dimensions such that net tension and shear out failures are avoided. These types of failures are generally catastrophic and provide little warning prior to occurrence. On the other hand, bearing type failures may result only in structural geometry shifts and are therefore more tolerable than complete failure. Since the appropriate design criterion is based upon the bearing mode of failure, the focus of this paper is to determine the reliability of a composite laminate based on its bearing strength and the applied loading. It is widely acknowledged that composite material strength data exhibit significant scatter. This scatter requires the use of statistical methods for properly characterizing the material strength and the reliability of composite materials and structures. Studies that have analyzed the statistical nature of polymer composite strength data include Hwang and Han (1987), Sun and Yamada (1978), Talreja (1981), Tenn (1981), Wetherhold (1986) and others. Recent studies concentrating on the reliability of composites include Cederbaum et al. (1990) and Thomas and Wetherhold (1991) who determined the bounds on laminate reliability under multiaxial stress states. In this study, the reliability of a pin-loaded composite will be determined based on the theory of stress-strength interference under the assumption that the composite strength data follow a Weibull distribution and that the applied stress is a random variable that is adequately characterized by a Gumbel Type 1 extreme value distribution. A numerical example and plots of the reliability versus distribution parameters are given for values that are typical of graphite/epoxy laminates. 975
976
P. D. Herrington INTERFERENCE THEORY
A major contributor to interference theory and its application to structural safety is Freudenthal et ai. (1966). Interference theory is based on the concept that the structural response depends upon a loading variable (applied stress) and a resistance variable (strength), both of which can be described by a probability density function. The probability of failure of a structure is defined as the probability that the loading variable exceeds the resistance variable. Numerous examples exist of the stress/strength distribution being characterized by normal/normal, log-normal/log-normal [Haugen (1968), Lewis (1987) and others] yet few exist for cases where the convolution integral cannot be solved explicitly. The contribution of this paper is to provide results for the case where stress/strength random variables are described by Type I e.v.d./Weibull (1951), respectively. To the author's knowledge, the only other case where this combination of distributions is considered is that of Kapur and Lamberson (1977) who provided tabular results. However, the table they provided does not extend to parameters that are typical of composite materials. In this paper, the results are extended to incorporate parameters characteristic of polymer composites. The following outlines the development of interference theory in general, and then considers the specific case of Type I extreme value and Weibull distributions. The probability of a component failing under the applied stress o 1 equals the probability that the strength S is less than or equal to a l , or, P(S -< o l)
P(S <_ al) =
f °'
(1)
g(S) dS
(2)
--oo
where g(S) is the probability density function for the component strength. If the applied loading is not a deterministic value, but varies according to a frequency distribution denoted by f(a), then for all possible values of applied stress, the probability of failure becomes,
17/=
l ,[t f(tr
-oa
]
g(t~) d~ de
-oo
(3)
where t~ is a dummy variable of integration. If the strength probability density function follows a Weibull distribution, then g(S) is given by,
g(S; O, fl) = ~ ~ /
exp[-
(4)
where fl and 0 are known as the shape and scale parameters, respectively. The cumulative form of the Weibull distribution is written as,
G(S; O, fl) = I - e x p [ - ( S ) ~]
(5)
with the mean given by,
and the variance defined as, (7) where F(-) denotes the gamma function. Now considering that the failure of the joint depends on the largest value of a sequence of applied stresses, the use of an extreme value distribution for the applied stress would be reasonable. If it is assumed that the parent
Interference theory for a pin-loaded joint
977
cumulative distribution function of the applied stress has the form,
F(s) = 1 - e -g~x)
(8)
where g(t) is an increasing function of x, then for a large number of stress applications, the distribution of extreme values is classified as a Type 1 extreme value distribution as discussed by Gumbel (1958). Therefore for parent density functions that are of an exponential form, such as the normal, gamma, and others, the Type 1 largest extreme value distribution is applicable. Type 1 asymptotic distributions of the largest value have a cumulative form of a double exponential, namely,
F(X)=expl-exp[-(X~na~-)]
1
(9)
and a density function given by, 1
-
a n
f ( X ) =----exp On I - ( X n-if
X o~n ) - exp[ - ( - - ~ ) 1 1
(10)
where X is the extreme random variable and an and 6n are the parameters of the distribution. These parameters may be determined by maximum likelihood methods applied to a sample of extreme values. They may also be related to the parent distribution via the equations (Gumbel, 1958), 1
F~(an) = 1 - -N f~(a.) =
(11)
1
(12)
6.N
where F~ and fx are the parent cumulative and probability distribution functions, respectively, and N is the number of samples. Measures of central tendency and dispersion of the Type 1 largest extreme value distribution are given by, g = C~n + Y6n
(13)
and 7r
cr = 6 . . ~
(14)
which correspond to the mean and standard deviation, respectively, and y is Euler's constant (y = 0.577216). Assuming that the pin bearing strength is well characterized by a Weibull distribution and that a maximum extreme value distribution is applicable for the applied loading, then the probability of failure given by eqn (3) becomes,
pf=
l_~61n--expI-(t7 \-------~-] - an~ - e x p [ - ( ~ - ~ a ~ ) ] l
I 1 - e x p l - ( O ) ~ 1 1 dtT. (15)
By a change of variables, this result can be expressed in a more convenient form, namely in terms of the reliability R, where R = 1 - py, as given by Kaput and Lamberson (1977),
R= f~expI-y-exp[-fO
~nnY 1/¢- ~ann ) ] l dy
(16)
where
Therefore the reliability is expressed as a definite integral whose evaluation is unobtainable in closed form and thus requires numerical integration. Hence, part of this work has been the development of a stable computer algorithm to determine the reliability R.
978
P.D. Herrington RESULTS
The purpose of this paper is to determine and present the reliability of a composite laminate when its strength is characterized by a Weibull distribution and the applied loading is characterized by a Type I extreme value distribution. In physical terms, this is an important and useful result since the strength of composite materials is often well defined by the Weibull distribution and that failure in use is often the result of an extreme value of independently applied loads. Therefore, based on interference theory, the reliability (where reliability is defined as 1 - probability of failure) for a wide range of parameters for the Weibull distributed strength and type 1 e.v.d, of applied stress was determined using eqn (16). The parameters used for the Weibull distributed strength fall within the range that is typical of graphite/epoxy and boron/epoxy composite materials. [Jones (1975) compares Weibull parameters for graphite/epoxy and boron/epoxy laminates tested in tension and gives values for the shape parameter ranging from 7.54 to 27.9.] Figure 1 shows the reliability versus - c t , / J , (the parameters of the extreme value distribution) for a Weibull distributed strength shape parameter of/~ = 2.5 and values of a ranging from 5 to 40. Note that a is defined as the ratio a = 0 / J , and hence is the parameter that couples the Weibull and extreme value distributions. Figures 2-5 are plots of the reliability versus the parameters of the extreme value distribution for Weibull shape parameters ranging from/~ = 5 to p = 30. In all figures, several important trends are apparent. One trend is that the reliability decreases as the ratio [- or,/~, I increases. This ratio is a rough approximation of the mean of the applied load to the variance of the applied load, using the parameters of the extreme value distribution. Therefore for a fixed 6,, corresponding to a constant variance in the applied load distribution, an increase in the magnitude of a , implies increasing the applied load which results in a decrease in reliability. Another apparent trend is that the reliability increases as the parameter a, which couples the strength and stress parameters (a = 0/d~,), increases. This is an expected result since 0 is the scale parameter of the composite strength and J , is a measure of the variability of the applied stress. The variation in parameters shown in Figs 1-5 allows the determination of the reliability for a wide range of composite laminates. An example using the stress-strength interference theory to determine the reliability o f a pin-loaded composite is given below.
Numerical example As discussed by Herrington and Sabbaghian (1992), tests were performed on a quasi-isotropic laminate, [ 0 / 9 0 / 4 5 / - 4 5 ] s , constructed from the graphite-epoxy prepreg
I
- -
~.:$.
1
I]=2.5
j --
. . . •
.
a=5
--*'-- a=lO ...... o.-.. a = l 5
I
q_.
0.8
--x--
a=20
-- + .
a=40
0.6 a
#
0.4
0.2.
....
o 5
10
I " f ",", -,-', 15
20
,
=
,":
25
- an/8 n
Fig. 1. Reliabilityfor Weibulldistributed strength and maximumextremevaluedistributed stress, B= 2.5.
Interference theory for a pin-loaded joint
979 I
1
-~-5.:'-~i:. •
~ .......
a=5
- - * - - a=lO
==5 4-
r
-..
I .... e--.. a = 1 5
'-
a=20
•-
0.8
..
I +
x,
a=30
0.6 J~
.$ 0.4
n,-
0.2
""
0
,
, ",-',-~
0
N~
""t"-,
. . . .
15 -
~
,
,
20
25
Ctnl5n
Fig. 2. Reliability for Weibull distributed strength and maximum extreme value distributed stress, /7=5.
13=-10 T
. . . . . .
• -L,
.
[ "+.
"
%.
X+,
0.8
" -~
a=5 ,
0.6
',
-.$ n-
', 0.4
',
--o-.... e-.-. - - x-' ÷
a=lO a=15 a=20 a=30
0.2 0
'
5
I 0
"'
I, "?"~", I . 15 20 - COn/5n
,%'-." 25
Fig. 3. Reliability for Weibull distributed strength and maximum extreme value distributed stress, B = 10.
J
a=5 - - *" - a = l O ...... o--- a = l 5
I~=20 •
÷-
a=25
0.8 ¸
0.6 ®
n:
0.4
..
0.2
0 5
10
15 -
20
25
an/8n
Fig. 4. Reliability for Weibull distributed strength and maximum extreme value distributed stress, # = 20.
980
P.D. Herrington
~=30 ,
1
,
I
,
,
,
,
I
"%...
,
,
,
,
Ii,
,
~ t ",
t
a=5 --*-a=lO ..... ~--a=15 --~- a=20 "+ --÷- a=25
0.8 L "q
0.6 .0
o~
i
0.4 L
\
0,2
0
i
,,,
5
,
,
15
lO
20
25
- ct n / S n
Fig. 5. Reliabilityfor Weibull distributed strength and maximum extremevalue distributed stress, ,8= 30. system AS4/3501-6. The specimen dimensions had sufficient edge and width to diameter ratios ( E / D -- W / D = 6, D = 6.35 mm) such that failure occurred only in the bearing mode. A steel pin with an outside diameter of 6.25 m m (0.246 in) was used to transfer the load to the laminate. This pin/hole combination results in a bolt-hole clearance of 0.1 m m (0.004 in), which is typical of m a n y fastening systems. The data needed to determine the reliability of a laminate are the parameters of the Weibull strength distribution and the parameters of the extreme value stress distribution. For illustrative purposes, a wide range of parameters for the stress distribution were chosen and the strength parameters were obtained from experimental tests. Nine pin bearing strength tests were performed which resulted in Weibull shape and scale parameters of 25 and 288 MPa, respectively. In order to test the hypothesis that the pin bearing strength indeed follows a Weibull distribution, several "goodness of fit" tests were employed. The first test considered was the classical K o l m o g o r o v - S m i r n o v test. For the assumption that the pin bearing strength follows a Weibull distribution, the null hypothesis was not rejected by the K o l m o g o r o v - S m i r n o v test at a level of significance of 10%. In this case, the null hypothesis claims that the sample data obtained follow the tested distribution. Distribution specific tests were expected to be more powerful than general goodness of fit tests, and therefore further statistical tests were performed using the methods of Mann et al. (1971) To test the goodness of fit for the assumption that the bearing strength data follow a Weibull distribution, the method as described by Mann et al., developed specifically for this distribution, was used. Defining xi as xi = In Si, where i = 1, 2 . . . . . r is the number of tests performed, the test statistic is, U =
r-I ~i=(r/2)+l [(Xi+l r-1 l [(Xi+l --
El=
xi)/Mi] xi)/Mi]
(18)
where U and M are tabulated in Mann et ai. This test statistic is then compared to critical values of U to determine if the sample data follow a two parameter Weibull distribution. Using this test at a significance level of 10%, the assumption that the sample population follows a Weibull distribution was not rejected. After performing the described goodness of fit tests on the experimental data, the resulting Weibull parameters were substituted into eqns (16) and (17). Equation (16) was solved (numerically) for a wide range of applied loading parameters. The results were obtained in the form of a central safety factor n and a coefficient of variation V~. In this case, the central safety factor is defined as the ratio of the mean of the Weibull distribution to the mean of the extreme value distribution, n = / ~ / g , where/a and g are given
981
Interference theory for a pin-loaded joint 1 0.95
0.9
0.85 .~ -$ n-
0.8 0.75 0.7 0.65 0.6 1
1.25
1.5
1.75
2
2.25
2.5
I1
Fig. 6. Reliability for Weibull distributed strength, (1~ = 25, 0 = 288) and maximum extreme value distributed stress. by eqns (6) and (13), respectively. The coefficient o f variation o f the applied stress, /Io, is given by the ratio o f the standard deviation o f the extreme value distribution to the mean, namely, Vo = ¢r/2, where a is the standard deviation as given by eqn (14). The plot o f reliability versus central safety factor n is shown in Fig. 6. This figure represents the reliability for a pin-loaded AS4/3501-6 laminate with the layup [ 0 / 9 0 / 4 5 / - 4 5 ] s , under a wide range o f loading parameters. For example, if the parameters o f the extreme value stress distribution were or, = 166.72 and 6, = 36.62, then by eqns (13) and (14), we have = 187.86 and ~r = 46.965. F r o m eqn (6), /z = 281.79, and therefore n = 1.5 and Vo = 0.25. With these values o f n and Vo, the reliability m a y be obtained f r o m Fig. 6, namely R = 0.95. As can be expected, the reliability o f the pin-loaded composite increases with an increasing safety factor and with a decreasing coefficient o f variation o f the applied stress. CONCLUSION The reliability o f a pin-loaded composite laminate has been determined based u p o n stress-strength interference theory. The probabilistic nature o f the strength o f composite materials has been accounted for by assuming that the bearing strength follows a Weibull distribution. Variations in applied loading have also been considered by assuming that the loading is adequately represented by an extreme value distribution. This is an important practical issue since the strength o f composite materials is often well defined by the Weibull distribution and laminate failure can often be the result o f an extreme value o f independently applied loads. The values o f the reliability for a wide class o f strength and loading parameters were calculated and presented. Results were also presented f r o m a set o f experimentally obtained data that highlighted the effect o f the central safety factor and coefficient o f variation on the reliability o f a laminated plate. REFERENCES Cederbaum, G., Elishakoff, I. and Librescu, L. (1990). Reliability of laminated plates via the first-order second-moment method. Composite Structures 15, 161-167. Collings, T. A. (1987). Experimentally determined strength of mechanically fastened joints. In Joining o f FibreReinforced Plastics (Edited by F. L. Matthews). Elsevier Applied Science Publishers, London. Crews, J. H. and Naik, R. V (1986). Failure analysis of graphite/epoxy laminate subjected to bolt bearing loads. In Composite Materials: Fatigue and Fracture, ASTM STP 907 (Edited by H. T. Hahn), pp. 115-133. American Society for Testing and Materials, Philadelphia, PA. De J ong, T. (1987). Stresses in pin-loaded anisotropic plates. Behaviour and Analysis o f Mechanically Fastened Joints in Composite Structures, AGARD-CP-427. April, pp. 5/1-5/17. Eriksson, I. (1990). On the bearing strength of bolted graphite/epoxy laminates. Journal o f Composite Materials 24 (Dec.), 1246-1269. Freudenthal, A. M., Garrelts, J. M. and Shinozuka, M. (1966). The analysis of structural safety. Journal o f the Structural Division, ASCE 92 (Feb.), 267-325. Gumbel, E. J. (1958). Statistics o f Extremes. Columbia University Press, New York.
982
P . D . Herrington
Hart-Smith, L. J. (1987). Design and empirical analysis of bolted or riveted joints. In Joining Fibre-Reinforced Plastics (Edited by F. L. Matthews). Elsevier Applied Science, London. Hangen, E. B. (1968). Probabilistic Approaches to Design. John Wiley, New York. Herrington, P. D. and Sabbaghian, M. (1992). Effect of radial clearance between bolt and washer on the bearing strength of composite bolted joints. Journal of Composite Materials 26(12) (Dec.), 1826-1843. Hwang, W. and Han, K. S. (1987). Statistical study of strength and fatigue life of composite materials. Composites 18(1), 47-53. Hyer, M. W. and Chastain, P. A. (1985). The effect of bolt load proportioning on the capacity of multiple-hole composite joints. NASA-CR-178019. Jones, B. H. (1975). Probabilistic design and reliability. In Structural Design and Analysis, Part II, ,Vol. 8 (Edited by C. C. Chamis). Academic Press, New York. Kaput, K. C. and Lamberson, L. R. (1977). Reliability in Engineering Design. John Wiley, New York. Lewis, E. E. (1987). Introduction to Reliability Engineering. John Wiley, New York. Mann, N. R., Fertig, K. W. and Scheuer, E. M. (1971). Tolerance Bounds and a New Goodness-of-Fit Test for Two-Parameter WeibuU or Extreme-Value Distribution, AF-ARL 71-0077, May. Murthy, A. V., Dattaguru, B., Narayana, H. V. L. and Ran, A. K. (1991). Stress and strength analysis of pin joints in laminated anisotropic plates. Composite Structures 19, 299-312. Ramkumar, R. L., Saether, E. S. and Appa, K. (1987). Strength analysis of mechanically fastened composite structures. Behaviour and Analysis of Mechanically Fastened Joints in Composite Structures, AGARD-CP427. April, pp. 7/1-7/22. Sun, C. T. and Yamada, S. E. (1978). Strength distribution of a unidirectional fiber composite. Journal of Composite Materials 12, 169-176. Talreja, R. (1981). Estimation of Weibull parameters for composite material strength and fatigue life data. In Fatigue of Fibrous Composite Materials, ASTM STP 723, pp. 291-311. American Society for Testing and Materials, Philadelphia, PA. Tenn, L. F. (1981). Statistical analysis of fibrous composite strength data. In Test Methods and Design Allowables for Fibrous Composites (Edited by C. C. Chamis), ASTM STP 734, pp. 229-244. Thomas, D. J. and Wetherhold, R. C. (1991). Reliability analysis of composite laminates with load sharing. Journal of Composite Materials 25, 1459-1475. Thomas, D. J. and Wetherhold, R. C. (1991). Reliability analysis of continuous fiber composite laminates. Composite Structures 17, 277-293. Weibull, W. (1951). A statistical distribution function of wide applicability. Journal of Applied Mechanics 18, 293-297. Wetherhold, R. C. (1986). Statistical distribution of strength of fiber-reinforced composite materials. Polymer Composites 7(2), 116-123.
Copyright @ 1995ElsevierScienceLtd Printed in Great Britain. All rights reserved 0961-9526/95 $9.50+ .00
Pergamon
0961-9526(94) 00063-8
STRESS-STRENGTH INTERFERENCE PIN-LOADED COMPOSITE
THEORY JOINT
FOR A
Paul D. Herrington Department of Mechanical Engineering, University of New Orleans, New Orleans, LA 70148, U.S.A.
(Received 17 May 1994; final version accepted 25 July 1994) Abstract--Considering the increasing applications of composites as structural components, the reliability of these materials is an important issue. In this paper, stress-strength interference theory is used to determine the reliability of a component that is characterized having a Weibull strength distribution and is under the effects of an applied load that is not deterministic, but follows a probability distribution. The assumption that failure is caused by the maximum of a sequence of applied loads is the basis for describing the applied load distribution as a Gumbel Type 1 extreme value distribution. Reliability plots are given for a class of strength parameters that are typical o f graphite/epoxy laminates. A numerical example for a pin-loaded composite laminate is shown using experimentally obtained data.
INTRODUCTION
Composite materials are increasingly being used as primary load bearing structures. Due to previous experience of joining structures composed of isotropic members and due to the need for disassembly, the mechanical fastening of composites is an often used method of joining. While exhibiting a weight penalty over other joining methods such as adhesive bonding, it is important that analytical techniques are developed in order to predict the structural response and reliability of pin-loaded composite joints. The study of pin-loaded and bolted composite joints is an active field of interest for many investigators including CoUings (1987), Crews and Naik (1986), De Jong (1987), Eriksson (1987), Hart-Smith (1987), Hyer and Chastain (1985), Murthy et al. (1991), Ramkumar et al. (1987) and others. Particularly important is the study of the bearing strength of laminates. The procedure often used for designing composite structures incorporating pin or mechanically fastened joints is to provide sufficient edge and width dimensions such that net tension and shear out failures are avoided. These types of failures are generally catastrophic and provide little warning prior to occurrence. On the other hand, bearing type failures may result only in structural geometry shifts and are therefore more tolerable than complete failure. Since the appropriate design criterion is based upon the bearing mode of failure, the focus of this paper is to determine the reliability of a composite laminate based on its bearing strength and the applied loading. It is widely acknowledged that composite material strength data exhibit significant scatter. This scatter requires the use of statistical methods for properly characterizing the material strength and the reliability of composite materials and structures. Studies that have analyzed the statistical nature of polymer composite strength data include Hwang and Han (1987), Sun and Yamada (1978), Talreja (1981), Tenn (1981), Wetherhold (1986) and others. Recent studies concentrating on the reliability of composites include Cederbaum et al. (1990) and Thomas and Wetherhold (1991) who determined the bounds on laminate reliability under multiaxial stress states. In this study, the reliability of a pin-loaded composite will be determined based on the theory of stress-strength interference under the assumption that the composite strength data follow a Weibull distribution and that the applied stress is a random variable that is adequately characterized by a Gumbel Type 1 extreme value distribution. A numerical example and plots of the reliability versus distribution parameters are given for values that are typical of graphite/epoxy laminates. 975
976
P. D. Herrington INTERFERENCE THEORY
A major contributor to interference theory and its application to structural safety is Freudenthal et ai. (1966). Interference theory is based on the concept that the structural response depends upon a loading variable (applied stress) and a resistance variable (strength), both of which can be described by a probability density function. The probability of failure of a structure is defined as the probability that the loading variable exceeds the resistance variable. Numerous examples exist of the stress/strength distribution being characterized by normal/normal, log-normal/log-normal [Haugen (1968), Lewis (1987) and others] yet few exist for cases where the convolution integral cannot be solved explicitly. The contribution of this paper is to provide results for the case where stress/strength random variables are described by Type I e.v.d./Weibull (1951), respectively. To the author's knowledge, the only other case where this combination of distributions is considered is that of Kapur and Lamberson (1977) who provided tabular results. However, the table they provided does not extend to parameters that are typical of composite materials. In this paper, the results are extended to incorporate parameters characteristic of polymer composites. The following outlines the development of interference theory in general, and then considers the specific case of Type I extreme value and Weibull distributions. The probability of a component failing under the applied stress o 1 equals the probability that the strength S is less than or equal to a l , or, P(S -< o l)
P(S <_ al) =
f °'
(1)
g(S) dS
(2)
--oo
where g(S) is the probability density function for the component strength. If the applied loading is not a deterministic value, but varies according to a frequency distribution denoted by f(a), then for all possible values of applied stress, the probability of failure becomes,
17/=
l ,[t f(tr
-oa
]
g(t~) d~ de
-oo
(3)
where t~ is a dummy variable of integration. If the strength probability density function follows a Weibull distribution, then g(S) is given by,
g(S; O, fl) = ~ ~ /
exp[-
(4)
where fl and 0 are known as the shape and scale parameters, respectively. The cumulative form of the Weibull distribution is written as,
G(S; O, fl) = I - e x p [ - ( S ) ~]
(5)
with the mean given by,
and the variance defined as, (7) where F(-) denotes the gamma function. Now considering that the failure of the joint depends on the largest value of a sequence of applied stresses, the use of an extreme value distribution for the applied stress would be reasonable. If it is assumed that the parent
Interference theory for a pin-loaded joint
977
cumulative distribution function of the applied stress has the form,
F(s) = 1 - e -g~x)
(8)
where g(t) is an increasing function of x, then for a large number of stress applications, the distribution of extreme values is classified as a Type 1 extreme value distribution as discussed by Gumbel (1958). Therefore for parent density functions that are of an exponential form, such as the normal, gamma, and others, the Type 1 largest extreme value distribution is applicable. Type 1 asymptotic distributions of the largest value have a cumulative form of a double exponential, namely,
F(X)=expl-exp[-(X~na~-)]
1
(9)
and a density function given by, 1
-
a n
f ( X ) =----exp On I - ( X n-if
X o~n ) - exp[ - ( - - ~ ) 1 1
(10)
where X is the extreme random variable and an and 6n are the parameters of the distribution. These parameters may be determined by maximum likelihood methods applied to a sample of extreme values. They may also be related to the parent distribution via the equations (Gumbel, 1958), 1
F~(an) = 1 - -N f~(a.) =
(11)
1
(12)
6.N
where F~ and fx are the parent cumulative and probability distribution functions, respectively, and N is the number of samples. Measures of central tendency and dispersion of the Type 1 largest extreme value distribution are given by, g = C~n + Y6n
(13)
and 7r
cr = 6 . . ~
(14)
which correspond to the mean and standard deviation, respectively, and y is Euler's constant (y = 0.577216). Assuming that the pin bearing strength is well characterized by a Weibull distribution and that a maximum extreme value distribution is applicable for the applied loading, then the probability of failure given by eqn (3) becomes,
pf=
l_~61n--expI-(t7 \-------~-] - an~ - e x p [ - ( ~ - ~ a ~ ) ] l
I 1 - e x p l - ( O ) ~ 1 1 dtT. (15)
By a change of variables, this result can be expressed in a more convenient form, namely in terms of the reliability R, where R = 1 - py, as given by Kaput and Lamberson (1977),
R= f~expI-y-exp[-fO
~nnY 1/¢- ~ann ) ] l dy
(16)
where
Therefore the reliability is expressed as a definite integral whose evaluation is unobtainable in closed form and thus requires numerical integration. Hence, part of this work has been the development of a stable computer algorithm to determine the reliability R.
978
P.D. Herrington RESULTS
The purpose of this paper is to determine and present the reliability of a composite laminate when its strength is characterized by a Weibull distribution and the applied loading is characterized by a Type I extreme value distribution. In physical terms, this is an important and useful result since the strength of composite materials is often well defined by the Weibull distribution and that failure in use is often the result of an extreme value of independently applied loads. Therefore, based on interference theory, the reliability (where reliability is defined as 1 - probability of failure) for a wide range of parameters for the Weibull distributed strength and type 1 e.v.d, of applied stress was determined using eqn (16). The parameters used for the Weibull distributed strength fall within the range that is typical of graphite/epoxy and boron/epoxy composite materials. [Jones (1975) compares Weibull parameters for graphite/epoxy and boron/epoxy laminates tested in tension and gives values for the shape parameter ranging from 7.54 to 27.9.] Figure 1 shows the reliability versus - c t , / J , (the parameters of the extreme value distribution) for a Weibull distributed strength shape parameter of/~ = 2.5 and values of a ranging from 5 to 40. Note that a is defined as the ratio a = 0 / J , and hence is the parameter that couples the Weibull and extreme value distributions. Figures 2-5 are plots of the reliability versus the parameters of the extreme value distribution for Weibull shape parameters ranging from/~ = 5 to p = 30. In all figures, several important trends are apparent. One trend is that the reliability decreases as the ratio [- or,/~, I increases. This ratio is a rough approximation of the mean of the applied load to the variance of the applied load, using the parameters of the extreme value distribution. Therefore for a fixed 6,, corresponding to a constant variance in the applied load distribution, an increase in the magnitude of a , implies increasing the applied load which results in a decrease in reliability. Another apparent trend is that the reliability increases as the parameter a, which couples the strength and stress parameters (a = 0/d~,), increases. This is an expected result since 0 is the scale parameter of the composite strength and J , is a measure of the variability of the applied stress. The variation in parameters shown in Figs 1-5 allows the determination of the reliability for a wide range of composite laminates. An example using the stress-strength interference theory to determine the reliability o f a pin-loaded composite is given below.
Numerical example As discussed by Herrington and Sabbaghian (1992), tests were performed on a quasi-isotropic laminate, [ 0 / 9 0 / 4 5 / - 4 5 ] s , constructed from the graphite-epoxy prepreg
I
- -
~.:$.
1
I]=2.5
j --
. . . •
.
a=5
--*'-- a=lO ...... o.-.. a = l 5
I
q_.
0.8
--x--
a=20
-- + .
a=40
0.6 a
#
0.4
0.2.
....
o 5
10
I " f ",", -,-', 15
20
,
=
,":
25
- an/8 n
Fig. 1. Reliabilityfor Weibulldistributed strength and maximumextremevaluedistributed stress, B= 2.5.
Interference theory for a pin-loaded joint
979 I
1
-~-5.:'-~i:. •
~ .......
a=5
- - * - - a=lO
==5 4-
r
-..
I .... e--.. a = 1 5
'-
a=20
•-
0.8
..
I +
x,
a=30
0.6 J~
.$ 0.4
n,-
0.2
""
0
,
, ",-',-~
0
N~
""t"-,
. . . .
15 -
~
,
,
20
25
Ctnl5n
Fig. 2. Reliability for Weibull distributed strength and maximum extreme value distributed stress, /7=5.
13=-10 T
. . . . . .
• -L,
.
[ "+.
"
%.
X+,
0.8
" -~
a=5 ,
0.6
',
-.$ n-
', 0.4
',
--o-.... e-.-. - - x-' ÷
a=lO a=15 a=20 a=30
0.2 0
'
5
I 0
"'
I, "?"~", I . 15 20 - COn/5n
,%'-." 25
Fig. 3. Reliability for Weibull distributed strength and maximum extreme value distributed stress, B = 10.
J
a=5 - - *" - a = l O ...... o--- a = l 5
I~=20 •
÷-
a=25
0.8 ¸
0.6 ®
n:
0.4
..
0.2
0 5
10
15 -
20
25
an/8n
Fig. 4. Reliability for Weibull distributed strength and maximum extreme value distributed stress, # = 20.
980
P.D. Herrington
~=30 ,
1
,
I
,
,
,
,
I
"%...
,
,
,
,
Ii,
,
~ t ",
t
a=5 --*-a=lO ..... ~--a=15 --~- a=20 "+ --÷- a=25
0.8 L "q
0.6 .0
o~
i
0.4 L
\
0,2
0
i
,,,
5
,
,
15
lO
20
25
- ct n / S n
Fig. 5. Reliabilityfor Weibull distributed strength and maximum extremevalue distributed stress, ,8= 30. system AS4/3501-6. The specimen dimensions had sufficient edge and width to diameter ratios ( E / D -- W / D = 6, D = 6.35 mm) such that failure occurred only in the bearing mode. A steel pin with an outside diameter of 6.25 m m (0.246 in) was used to transfer the load to the laminate. This pin/hole combination results in a bolt-hole clearance of 0.1 m m (0.004 in), which is typical of m a n y fastening systems. The data needed to determine the reliability of a laminate are the parameters of the Weibull strength distribution and the parameters of the extreme value stress distribution. For illustrative purposes, a wide range of parameters for the stress distribution were chosen and the strength parameters were obtained from experimental tests. Nine pin bearing strength tests were performed which resulted in Weibull shape and scale parameters of 25 and 288 MPa, respectively. In order to test the hypothesis that the pin bearing strength indeed follows a Weibull distribution, several "goodness of fit" tests were employed. The first test considered was the classical K o l m o g o r o v - S m i r n o v test. For the assumption that the pin bearing strength follows a Weibull distribution, the null hypothesis was not rejected by the K o l m o g o r o v - S m i r n o v test at a level of significance of 10%. In this case, the null hypothesis claims that the sample data obtained follow the tested distribution. Distribution specific tests were expected to be more powerful than general goodness of fit tests, and therefore further statistical tests were performed using the methods of Mann et al. (1971) To test the goodness of fit for the assumption that the bearing strength data follow a Weibull distribution, the method as described by Mann et al., developed specifically for this distribution, was used. Defining xi as xi = In Si, where i = 1, 2 . . . . . r is the number of tests performed, the test statistic is, U =
r-I ~i=(r/2)+l [(Xi+l r-1 l [(Xi+l --
El=
xi)/Mi] xi)/Mi]
(18)
where U and M are tabulated in Mann et ai. This test statistic is then compared to critical values of U to determine if the sample data follow a two parameter Weibull distribution. Using this test at a significance level of 10%, the assumption that the sample population follows a Weibull distribution was not rejected. After performing the described goodness of fit tests on the experimental data, the resulting Weibull parameters were substituted into eqns (16) and (17). Equation (16) was solved (numerically) for a wide range of applied loading parameters. The results were obtained in the form of a central safety factor n and a coefficient of variation V~. In this case, the central safety factor is defined as the ratio of the mean of the Weibull distribution to the mean of the extreme value distribution, n = / ~ / g , where/a and g are given
981
Interference theory for a pin-loaded joint 1 0.95
0.9
0.85 .~ -$ n-
0.8 0.75 0.7 0.65 0.6 1
1.25
1.5
1.75
2
2.25
2.5
I1
Fig. 6. Reliability for Weibull distributed strength, (1~ = 25, 0 = 288) and maximum extreme value distributed stress. by eqns (6) and (13), respectively. The coefficient o f variation o f the applied stress, /Io, is given by the ratio o f the standard deviation o f the extreme value distribution to the mean, namely, Vo = ¢r/2, where a is the standard deviation as given by eqn (14). The plot o f reliability versus central safety factor n is shown in Fig. 6. This figure represents the reliability for a pin-loaded AS4/3501-6 laminate with the layup [ 0 / 9 0 / 4 5 / - 4 5 ] s , under a wide range o f loading parameters. For example, if the parameters o f the extreme value stress distribution were or, = 166.72 and 6, = 36.62, then by eqns (13) and (14), we have = 187.86 and ~r = 46.965. F r o m eqn (6), /z = 281.79, and therefore n = 1.5 and Vo = 0.25. With these values o f n and Vo, the reliability m a y be obtained f r o m Fig. 6, namely R = 0.95. As can be expected, the reliability o f the pin-loaded composite increases with an increasing safety factor and with a decreasing coefficient o f variation o f the applied stress. CONCLUSION The reliability o f a pin-loaded composite laminate has been determined based u p o n stress-strength interference theory. The probabilistic nature o f the strength o f composite materials has been accounted for by assuming that the bearing strength follows a Weibull distribution. Variations in applied loading have also been considered by assuming that the loading is adequately represented by an extreme value distribution. This is an important practical issue since the strength o f composite materials is often well defined by the Weibull distribution and laminate failure can often be the result o f an extreme value o f independently applied loads. The values o f the reliability for a wide class o f strength and loading parameters were calculated and presented. Results were also presented f r o m a set o f experimentally obtained data that highlighted the effect o f the central safety factor and coefficient o f variation on the reliability o f a laminated plate. REFERENCES Cederbaum, G., Elishakoff, I. and Librescu, L. (1990). Reliability of laminated plates via the first-order second-moment method. Composite Structures 15, 161-167. Collings, T. A. (1987). Experimentally determined strength of mechanically fastened joints. In Joining o f FibreReinforced Plastics (Edited by F. L. Matthews). Elsevier Applied Science Publishers, London. Crews, J. H. and Naik, R. V (1986). Failure analysis of graphite/epoxy laminate subjected to bolt bearing loads. In Composite Materials: Fatigue and Fracture, ASTM STP 907 (Edited by H. T. Hahn), pp. 115-133. American Society for Testing and Materials, Philadelphia, PA. De J ong, T. (1987). Stresses in pin-loaded anisotropic plates. Behaviour and Analysis o f Mechanically Fastened Joints in Composite Structures, AGARD-CP-427. April, pp. 5/1-5/17. Eriksson, I. (1990). On the bearing strength of bolted graphite/epoxy laminates. Journal o f Composite Materials 24 (Dec.), 1246-1269. Freudenthal, A. M., Garrelts, J. M. and Shinozuka, M. (1966). The analysis of structural safety. Journal o f the Structural Division, ASCE 92 (Feb.), 267-325. Gumbel, E. J. (1958). Statistics o f Extremes. Columbia University Press, New York.
982
P . D . Herrington
Hart-Smith, L. J. (1987). Design and empirical analysis of bolted or riveted joints. In Joining Fibre-Reinforced Plastics (Edited by F. L. Matthews). Elsevier Applied Science, London. Hangen, E. B. (1968). Probabilistic Approaches to Design. John Wiley, New York. Herrington, P. D. and Sabbaghian, M. (1992). Effect of radial clearance between bolt and washer on the bearing strength of composite bolted joints. Journal of Composite Materials 26(12) (Dec.), 1826-1843. Hwang, W. and Han, K. S. (1987). Statistical study of strength and fatigue life of composite materials. Composites 18(1), 47-53. Hyer, M. W. and Chastain, P. A. (1985). The effect of bolt load proportioning on the capacity of multiple-hole composite joints. NASA-CR-178019. Jones, B. H. (1975). Probabilistic design and reliability. In Structural Design and Analysis, Part II, ,Vol. 8 (Edited by C. C. Chamis). Academic Press, New York. Kaput, K. C. and Lamberson, L. R. (1977). Reliability in Engineering Design. John Wiley, New York. Lewis, E. E. (1987). Introduction to Reliability Engineering. John Wiley, New York. Mann, N. R., Fertig, K. W. and Scheuer, E. M. (1971). Tolerance Bounds and a New Goodness-of-Fit Test for Two-Parameter WeibuU or Extreme-Value Distribution, AF-ARL 71-0077, May. Murthy, A. V., Dattaguru, B., Narayana, H. V. L. and Ran, A. K. (1991). Stress and strength analysis of pin joints in laminated anisotropic plates. Composite Structures 19, 299-312. Ramkumar, R. L., Saether, E. S. and Appa, K. (1987). Strength analysis of mechanically fastened composite structures. Behaviour and Analysis of Mechanically Fastened Joints in Composite Structures, AGARD-CP427. April, pp. 7/1-7/22. Sun, C. T. and Yamada, S. E. (1978). Strength distribution of a unidirectional fiber composite. Journal of Composite Materials 12, 169-176. Talreja, R. (1981). Estimation of Weibull parameters for composite material strength and fatigue life data. In Fatigue of Fibrous Composite Materials, ASTM STP 723, pp. 291-311. American Society for Testing and Materials, Philadelphia, PA. Tenn, L. F. (1981). Statistical analysis of fibrous composite strength data. In Test Methods and Design Allowables for Fibrous Composites (Edited by C. C. Chamis), ASTM STP 734, pp. 229-244. Thomas, D. J. and Wetherhold, R. C. (1991). Reliability analysis of composite laminates with load sharing. Journal of Composite Materials 25, 1459-1475. Thomas, D. J. and Wetherhold, R. C. (1991). Reliability analysis of continuous fiber composite laminates. Composite Structures 17, 277-293. Weibull, W. (1951). A statistical distribution function of wide applicability. Journal of Applied Mechanics 18, 293-297. Wetherhold, R. C. (1986). Statistical distribution of strength of fiber-reinforced composite materials. Polymer Composites 7(2), 116-123.