fatigue and reliability analysis of unidirectional gfrp composites under rotating bending loads

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2003 37: 317Journal of Composite MaterialsU. A. Khashaba

Bending LoadsFatigue and Reliability Analysis of Unidirectional GFRP Composites under Rotating

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Fatigue and Reliability Analysis ofUnidirectional GFRP Composites under

Rotating Bending Loads

U. A. KHASHABA*Mechanical Design and Production Engineering Department

Faculty of Engineering

Zagazig University, Zagazig, Egypt

(Received October 9, 2001)(Revised June 6, 2002)

ABSTRACT: Rotating bending fatigue tests have been conducted on unidirectionalglass fiber reinforced polyester (GFRP) composites. Standard test specimens weremanufactured in form of circular rods with various fiber volume fraction (Vf ) ratios.Failure modes of the composite rods have been examined using scanning electronmicroscope. The two-parameter Weibull distribution function was used to investi-gate the statistical analysis of the experimental fatigue life results. Safe design lifebased on time to first failure (TTFF) concept was calculated at high confidence level�¼ 0.99 and at two values of reliability, R¼ 0.368 and 0.99.S–N diagrams of mean life, 50% survival life, lower bound life and safe design

fatigue life have been constructed for GFRP rods with various values of Vf ratios.These diagrams are of considerable value to the designer specially when the structurecontains a critical component where any failure is catastrophic. No rigorous fatiguelimit was observed within 107 cycles in these S–N diagrams. The fiber volumefractions have insignificant effect on the slope of the power function that fits themean fatigue life. At the same number of cycles the stress amplitude required tofracture the specimens with Vf¼ 44.7% was increased by an order equal to 1.58 and1.25 than specimens with Vf¼ 15.8 and 31.8% respectively. The widest scatter wasobserved at the life range of 105 and 106 cycles for GFRP rode with different Vf

ratios. This tendency in the dispersion of fatigue life at varying stress levels isextremely important and deserves much attention for the design and application ofGFRP composites.

KEY WORDS: unidirectional composite rods, rotating bending fatigue, Weibulldistribution, reliability analysis, mean life, survival life, lower bound life, safedesign life.

*E-mail: [email protected]

Journal of COMPOSITE MATERIALS, Vol. 37, No. 4/2003 317

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INTRODUCTION

RECENTLY, FIBER REINFORCED composite materials are brought into greater use in theindustries of aircraft, automobiles and especially solar vehicles, because of their light

density and high specific strength. In these applications the fatigue loads are usuallyunavoidable. For this reason resent designs do not specify static strength alone as aprimary design criterion but also include fatigue analysis. The demand for improvedperformance of structural materials in transportation industries, particularly in aircraft,makes fatigue analysis an important consideration.

FRP composite materials exhibit very complex failure mechanisms under static andfatigue loading due to the high degree of anisotropy which creates a local triaxial stressstate inside the composite (resulting from differences in Poisson’s ratio and elasticmodulus values between the fiber and the matrix) upon uniaxial loading [1]. The completeunderstanding of fatigue failure sequence and corresponding mechanisms is most essentialto be able to modify the composite design for better performance as well as to uncover thefatigue life limiting parameters. Fatigue failure is usually accompanied with extensivedamages, which are multiplied through specimen volume instead of a predominant singlecrack, which is often observed in most isotropic brittle materials [2]. Fatigue failure incomposite laminates involves combination of several damage modes including matrixcracking, delamination, interfacial debonding and fiber fracture [3–6]. The order in whicheach type of damage occurs may vary depending on the constituent materials, materialproperties, stacking sequence, type of fatigue loading, etc.

The mechanical properties of FRP composites have a remarkable scatter even when thespecimens are prepared and tested under assumed identical conditions. In the past, thisvariability of mechanical properties was relatively unimportant, since large safety factorwas used. With the advance of high performance aircraft, however, this variability ofmechanical properties takes on new significance. In particular, if the structure containsa critical component where any failure is catastrophic, the following items are neededfor successful structure reliability analysis [7]: (a) improve load sequence representation,(b) describe failure conditions more realistically, (c) define statistical variation of load andresistance (strength) with more confidence and (d) define design criteria taking reliabilityconcepts into account.

In assessing the reliability of composite structures, Weibull distribution function hasproved to be a useful and versatile means of describing composite material properties. Thisbecause the probability density function of the Weibull distribution has a wide variety ofshapes. For example when the shape parameter (�) equal to 1, it becomes the two-parameter exponential distribution. For � value near 3, its coefficient of skewnessapproaches zero and the function is capable of approximating a normal distribution [7,8].Also the two-parameter Weibull function has the following advantages [9]: (a) thedistribution is expressed in a simple function form and is easy to apply, (b) the distributionaccurately describes composite static strength and fatigue life and has been widelyaccepted for composite statistical data analysis, (c) standard tables and computingroutines are available, (d) data can be interpreted on a sound physical basis, so thatA-Basis and B-Basis allowable determined for static strength and fatigue life are morereliable and (e) hypothesis testing methods for statistical significance are available andverified.

The objective of the present paper is to study the fatigue behavior of unidirectional glassfiber reinforced polyester (GFRP) composites with various values of fiber volume

318 U. A. KHASHABA



fractions (Vf). Standard composite rods with circular cross-section are manufactured usingpultrusion technique. Fatigue tests are performed on rotating bending fatigue machine(cantilever type). The fractured surfaces are examined using scanning electron microscope.The two-parameter Weibull function is used to investigate the statistical analysis of fatiguelife results. Safe design fatigue life was calculated using time to first failure (TTFF)concept at high confidence level and different reliabilities. S–N diagrams of 50% survivallife, characteristic life, lower bound life, safe design fatigue life have been constructedfor GFRP composites.

EXPERIMENTAL WORK

Specimen Preparation

Standard unidirectional glass fiber reinforced polyester GFRP composites have beenmanufactured in form of circular rods using the pultrusion technique according to BS3691and ISO 3605. The details about this technique are illustrated elsewhere [10] and is onlyoutlined here. The constituent materials of the composite rods are shown in Table 1.

The required number of bundles for certain fiber volume fraction Vf have beendetermined from the following equation:

n ¼�

4d2Vf

�f

�Lð1Þ

where d is the specimen diameter, �f is the glass fiber density¼ 2.56� 106 gm/m3 and �L isthe linear density of glass fiber¼ 1.15 gm/m.

These bundles were cut to size and soaked in a trough of catalyzed resin. When thebundles are completely impregnated, they are pulled vertically by a metal wire into a glasstube treated by release agent; care must be taken to ensure that no twisting of the bundleswas occurred. This pultrusion process squeezed out excess resin and trapped air especiallyat high Vf . Therefore, the pultruded rods appeared to be clear and free from voids andother faults. The mold has been matured at room temperature for at least 24 h in a verticalposition. Then, the rod (in glass mold) was cured in the oven for 4 h at 80�C. The rodspecimens produced by this technique have 300mm long and 7.9� 0.05mm averagediameter with various values of fiber volume fractions (Vf ¼ 15.8, 31.8 and 44.7%).

The composite rods were subject to a series of investigations in order to characterizetheir mechanical properties as affected by the fiber volume fraction, Khashaba and others[10–15]. Such investigations included uniaxial tension, uniaxial compression, three-pointbending [10], shear (direct and torsion) tests [11], pure bending fatigue tests [12–14],torsion fatigue tests and combined (torsion/bending) fatigue tests [15].

Table 1. Composition of GFRP rods.

Matrix Orthophthalic polyester resin (QL 8520 A)Reinforcement E-Roving glass; linear density¼ 1150 g/km, d¼ 17�2 mm)Catalyst Methylethyl ketone peroxide (0.8% of matrix volume)Hardener Cobalt naphthenate (0.5% of matrix volume)

Fatigue and Reliability Analysis of Unidirectional GFRP Composites 319



In the present work, a rotating bending fatigue machine was designed and manufacturedto perform the fatigue tests on GFRP composite rods. The specimen was fixed as acantilever by three-jaw self-centering chuck connected by electric motor through flexiblecoupling. The speed of the motor is 1500 rpm (25Hz) which is recommended, JIS K7119,for the flexure fatigue tests of rigid plastics. The load (dead weights) is imposed at thespecimen end through fluctuating ball bearing. The specimens were machined on turningmachine to the dimensions shown in Figure 1.

RESULTS AND DISCUSSIONS

Failure Mode

The failure sequence of GFRP composite rods during rotating bending fatigue tests canbe summarized as follows:

. After few cycles (depending on the stress level) a visible longitudinal cracks, parallel tothe fiber direction, were found at the specimen waist close to the machine grip, Figures2 and 3. These cracks occurred due to the machining process on unidirectional GFRPcomposite rods to the dimensions illustrated in Figure 1, which result in a discontinuousspalled ended fibers, Figure 4(a).

. The longitudinal cracks are joining up at the surface between the discontinuous andcontinuous fibers. Therefore, longitudinal shear cracks were propagated at this surfaceleading to an increase in the parallel portion, which extended from point 1 to 2 underthe specimen waist, Figures 2 and 3.

. The length of the extended parallel portion under the specimen waist was increasedfrom1 to 4mm with increasing fiber volume fraction from 15.8 to 44.7%. This behaviorwas due to the insufficient matrix between the adjacent fibers for GFRP rods with highfiber volume fractions, Vf 44.7%. The matrix in GFRP composites is the responsibleelement for transmitting the load to the reinforcing fibers. Therefore, the lack of matrixbetween adjacent fibers, at high Vf, leads to failure of fiber–matrix interface bond at thesurfaces between the continuous and the discontinuous fibers due to the repeatingtension–compression stresses during the rotating bending fatigue tests.

. Shear cracks between the continuous and discontinuous fibers leaves irregularsurface with several peaks and valleys on the circumference of the continuous fibers,Figure 4(b), at portion 1–2, Figure 2. The diameter of the specimen at portion 1–2

Figure 1. Dimensions of rotating bending fatigue parallel specimen, (mm).

320 U. A. KHASHABA



was less than that of the parallel portion diameter (5.2mm). The decrease in thespecimen diameter and increasing the bending moment at point 2 leads to higherstresses, at this point, than the initial values (at point 1). Increase in the stresses at point2 leads to matrix cracks propagation in both longitudinal and transverse (shear)directions, Figure 4(c). Coalescence of transverse and shear cracking leads to completefiber shear failures, Figure 4(d). This figure also indicates no fiber pull-out in thefractured cross-section.

. The clean fiber surfaces in Figures 4(b) and (c) was due to the friction between the lowersurface of the discontinuous fibers and the upper surface of continuous fibers, wherea white powder was observed on the specimen surface during the fatigue tests specially

Figure 2. Bending moment and stress distribution diagrams of GFRP specimen under rotating bendingfatigue loads.

Parallel Length withcontinuous fibersExtend length of

the parallel portion

The white area due to the peeling of the discontinuous fibers.

Longitudinal cracks at the waist closed to machine grip

Figure 3. Photograph illustrates the failure of GFRP rod, Vf¼44.7%.




that has long life. The heat generated due to the friction between the lower surface of thediscontinuous fibers and the upper surface of the continuous fibers reduces the fatigue lifeof the tested composites that have low thermal conductivity.

Statistical Analysis of Fatigue Life Data

A good understanding of statistical aspects of fatigue properties is essential for thesuccessful application of composite materials due to the nonuniformity and the anisotropyof these materials as compared with conventional materials. The statistical analysis offatigue life results was investigated using a two-parameter Weibull distribution functionwhich characterized by a probability density function f (x) and the associated cumulativedistribution functions Pf (x) and Ps(x) as follows [16]:

f ðxÞ ¼�

�

x

�

� ��1

exp �x

�

� �� ð2Þ

Shearing the fibers at plane normal to the fibers direction.

Longitudinal cracks

57 µm

Valleys Peaks

340 µm

Continuous fiberswith clean surfaces

Longitudinal splitting in the matrix

c

49 µm

Fibers with spalling surfaces due tomachining the waist

a b

d

57 µm

Figure 4. Scanning Electron Microscopic (SEM) examination of failed GFRP rod with Vf¼ 44.7%: (a) Spallingthe fibers at the upper surface of specimen waist; (b) and (c) surface of extended continuous fibers under thespecimen waist; (d) fractured surface normal to the fiber direction.

322 U. A. KHASHABA



Pf ðxÞ ¼ 1� exp �x

�

� �� ð3Þ

PsðxÞ ¼ 1� FðxÞ ¼ exp �x

�

� ��

ð4Þ

where Pf (x) is the probability of failure, Ps(x) is the probability of survival, � is the shapeparameter which is the inverse measure of the dispersion in the fatigue life results and � isthe scale parameter that locates the life distribution. The values of � and � are determinedby rewriting Equation (3) in the form;

LnðxÞ ¼1

�Ln Ln

1

1� Pf ðxÞ

� �� þ Lnð�Þ ð5Þ

Equation (5) is an equation of straight line in the form Y¼ bzþ a with Y¼Ln(x),b¼ 1/�, a¼Ln(�), and z¼Ln(Ln[1/(1�Pf (x))]). The two variables in Equation (5) is theexperimental data of fatigue life x Nf’’ which sort ascending and the mean rank, Pf (x),which calculated from the following equation:

Pf ðxÞ ¼i

n þ 1ð6Þ

where i is the failure order number and n is the total number of samples in each test. Thevalues of � and � have been determined using the least squares fit and the results areillustrated in Table 2.

Mean life, M[X], variance V[X] and coefficient of variation, CV, of the two-parameterWeibull distribution were calculated from the following equations [9,17],

M½X ¼

Z 1

0

x � f ðxÞ dx ¼ � � � 1þ1

�

� �ð7Þ

Table 2. Weibull parameters and coefficient of variation (CV) of fatigue liferesults.

Vf (%) Sa (MPa)

Weibull Parameters

CV

Normalized WeibullParameters

� � �n �n

15.8 161.87 2.649 8.039�103 0.4062 2.391 1.014126.25 3.864 2.194�104 0.2894577.19 1.232 3.354�105 0.8163253.47 1.889 2.185�106 0.550239.94 6.926 1.118�107 0.16949

31.8 126.25 1.681 8.061�104 0.61184 2.152 1.004100.68 1.380 2.831�105 0.7331957.53 2.126 1.345�106 0.4947950.96 6.530 1.029�107 0.17907

44.7 156.13 3.780 9.706�104 0.2952 2.369 0.994125.81 1.816 2.754�105 0.5695985.13 1.423 1.413�106 0.7123961.90 2.454 1.331�107 0.43506




V ½X ¼ M½X2 � ðE ½X Þ2¼ �2 � 1þ

2

�

� �� 2 1þ

1

�

� �� ð8Þ

CV ¼

ffiffiffiffiffiffiffiffiffiffiffiV ½X

p

M½X ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ð1þ 2=�Þ � �2ð1þ 1=�Þ

p�ð1þ 1=�Þ

ð9Þ

where � is the gamma function.

S–N Diagram

Figure 5 shows a series of S–N diagrams for GFRP composite rods with various valuesof fiber volume fractions. The constants (A and B) of the power function, Equation (10),which fits the mean fatigue life data are illustrated in Figure 5. It is interesting to notethat the values of fiber volume fractions have insignificant effect on the slope of the fittingcurve, (B¼� 0.19 to � 0.191).

Sa ¼ AðNf ÞB

ð10Þ

The values of constant A in Figure 5 indicate that at the same number of cycles the stressamplitude required to fracture the specimens with Vf ¼ 44.7% was increased by an orderequal to 1.56 and 1.27 than specimens with Vf ¼ 15.8 and 31.8% respectively. This resultsbecause the fatigue strength increases with the increase of static strength, and the staticflexural strengths increase with increasing Vf (15.8–44.7%) as investigated by Abdin andothers [10]. Figure 5 also indicates that no rigorous fatigue limit was observed for GFRProds with various Vf ratios within 107 cycles.

Figure 5. S–N diagram of GFRP rods with various values of Vf ratio.

324 U. A. KHASHABA



Scatter in the Fatigue Life Results

The values of coefficient of variation are calculated for fatigue lives of composite rodsusing Equation (9) and the results are illustrated in Table 2. Figure 6 shows the coefficientof variation, CV, versus mean lives, M[X], which is calculated from Equation (7). Theresults in this figure indicate that the scatter in the fatigue life become widest at 105 to 106

cycles for GFRP rode with different Vf ratios. Similar behavior was observed by Tanimotoand others [16] for different composite materials tested under rotating bending fatigue test.This tendency in the dispersion of fatigue life at varying stress levels is extremely importantand deserves much attention for the design and application of GFRP composites.

Reliability Analysis of Fatigue Life Results

The term ‘‘reliability’’ in engineering refers to the probability that a product, or system,will perform its designed functions under a given set of operating conditions for a specificperiod of time. It is also known as the ‘‘probability of survival’’.

Figures 7(a), 8(a) and 9(a) show the fatigue life distributions (probability of survival,Ps(x), Equation (4) for GFRP rods with Vf ¼ 15.8, 31.8 and 44.7% respectively.

These figures are of considerable value to the designer where the fatigue life can be easilydetermined at any survival percent. For example the 50% survival life was determinedfrom these figures by drawing a horizontal line that intersected with the distributioncurves. The lives at the intersected points are the 50% survival life at various stress levels.These values of survival lives also can be determined from Equation (4) and the values of �and � in Table 2.

Safe Design Life Based on Time to First Failure (TTFF) Concept

Rotating bending fatigue results on GFRP composite rods, tested under identicalconditions, showed a remarkable experimental scatter due to their quasi-brittle nature.

Figure 6. Effect of fatigue life on the coefficient of variation, CV.




Figure 8. (a) Fatigue life distribution, Vf ¼31.8%, S1 ¼ 126.25, ^ S2 ¼ 100.68, g S3 ¼ 67.53,œ S4¼ 50.962 MPa; (b) Normalized fatigue life distribution, ^ Nf/�, for each stress amplitude.

Figure 7. (a) Fatigue life survival probability, Vf¼ 15.8%, S1¼ 161.87, � S2¼126.25, � S3¼ 77.19,� S4¼53.47, � S5¼ 39.94 MPa; (b) Normalized fatigue life distribution, f Nf/�, for each stress amplitude.

Figure 9. (a) Fatigue life distribution, Vf ¼44.7%, B S1 ¼ 156.13, 4 S2 ¼125.81, n S3 ¼85.13,r S4¼61.90 MPa; (b) Normalized fatigue distribution, m Nf/�, for each stress amplitude.

326 U. A. KHASHABA



The design criterion for high performance applications becomes not one selecting stresslevels to guarantee that the object will never fail but to insure that the body survives for thetime period required for its intended usage.

Safe design strength or life based on TTFF concept has a wide application on FRPcomposites [9,14,16–19]. This method illustrates how the required degree of reliability canbe achieved through a careful characterization of the composite. Safe design fatigue life(NR) was calculated as follows:

NR ¼�̂�

S¼

�̂�

SM � SN � SR¼

��

SN � SRð11Þ

where S is the scatter factor which subdivided into three scatter factors, SM, SN, and SR.SM is the sample size factor. This scatter factor indicates the penalty paid to gain

confidence � from a finite sample size (m). In this paper m¼ 2 (the first two lowest fatiguelives). The value of SM was calculated from the following equation,

SM ¼1

2mX2

� ð2mÞ

� �1=�n

ð12Þ

where X2� ð2mÞ is the chi-square distribution with 2m degree of freedom. The value of

X2� ð2mÞ ¼ 13.277 at confidence level �¼ 99% and m¼ 2, [20]. �n is the normalized shape

parameter. The values of �n have been determined for each Vf ratio by normalizing thefatigue lives by their scale parameter (�). For example the fatigue lives of GFRP rods withVf¼ 15.8%, Figure 7(a), that determined at initial stress levels 161.87, 126.25, 77.19, 53.47and 39.94MPa were normalized by their scale parameters, �¼ 8.039� 103, 2.194� 104,3.35� 105, 2.185� 106 and 1.118� 107 respectively, Table 2. Similarly the fatigue lives inFigures 8(a) and 9(a) for GFRP composite rods with Vf¼ 31.8 and 44.7% respectivelyhave been normalized by their scale parameters. Figures 7(b), 8(b) and 9(b) illustrate thenormalized fatigue life distribution for GFRP rods with Vf¼ 15.8, 31.8 and 44.7%,respectively. In these figures, the Weibull function is represented by straight line because itis drawn in the Weibull probability paper; the abscissa is Ln(Nf/�) and the ordinate isLn(Ln[1/(1�Pf(x))]). The slope of the straight line in these figures is the normalized shapeparameter (�n).SN is the fleet size factor, which depend on the size of the fleet (n). The value of SN iscalculated from the following equation:

SN ¼1

n

� ��1=�n

ð13Þ

SR is the reliability factor, which indicates the penalty paid to gain reliability R in a onemember fleet. The value of SR was calculated from the following equation:

SR ¼ Ln1

R

� �� 1=�n

ð14Þ




�̂� and �� are the characteristic life and the lower bound life respectively. The values of �̂�and �� are calculated from Equations (15) and (16) respectively.

�̂� ¼x�n

i

m

� �1=�n

ð15Þ

�� ¼�̂�

SMð16Þ

In the present paper the safe design fatigue life (NR) was calculated at two values ofreliability, R. The first value of R¼ 0.368, is the probability that a part will survive thecharacteristic strength or large (x¼ �). This value of R can be determined fromEquation (4) by substituting x¼ �. Therefore, the value of Ps(x)¼R¼ exp(� 1)¼ 0.368;in this case SR¼ 1. The second value of R¼ 0.99, for high fleet reliability.

Safe design fatigue life based on TTFF concept was calculated at high confidence level,� ¼ 0.99, for GFRP rods with Vf¼ 15.8, 31.8 and 44.7% and the results are illustrated bythe S–N diagrams, Figures 10–12 respectively.

These figures also show the characteristic life (�̂�), 50% survival life, and lower bound life(��) of the composite rods. The constants (A and B) of power function which used topredict the different fatigue lives have been determined for GFRP rods with various Vf

ratios. Figures 10–12 are of considerable value to the designer for the practicalapplications depending on the degree of the reliability. For example, if the structurecontains a critical component where any failure is catastrophic, the safe design life(NR¼0.99) calculated at high confidence level � ¼ 0.99 and high reliability R¼0.99 can beused for this purpose.

Figure 10. S–N diagrams of GFRP composites, Vf¼ 15.8%, at various fatigue lives.

328 U. A. KHASHABA



CONCLUSIONS

1. After few cycles a longitudinal splitting was observed at the specimen waist close to themachine grip. Therefore, the parallel portion of test specimen was extended under thespecimen waist leading to increases the applied stress amplitude than the initial value.The final failure of these composite rods was due to longitudinal and transverse cracksin the matrix followed by shearing the fibers at plain normal to the fiber direction, i.e.parallel to the loading direction.

2. The S–N diagrams for GFRP rods with various Vf ratios show no rigorous fatiguelimit within 107 cycles. The fiber volume fractions have insignificant effect on theslope of the power function that fitting the mean fatigue life. At the same number of






cycles the stress amplitude required to fracture the specimens with Vf¼ 44.7% wasincreased by an order equal to 1.56 and 1.27 than specimens with Vf¼ 15.8 and 31.8%,respectively.

3. Fatigue life distribution diagrams have been constructed for the composite rods usingthe two-parameter Weibull cumulative function (probability of survival) for GFRProds with various Vf ratios. From these diagrams the fatigue life can be easilydetermined at any survival percent.

4. The widest scatter was observed at the life range of 105 and 106 cycles for GFRP rodewith different Vf ratios. This tendency in the dispersion of fatigue life at varyingstress levels is extremely important and deserves much attention for the design andapplication of GFRP composites.

5. S–N diagrams of 50% survival life, characteristic life, lower bound life and safedesign fatigue life have been constructed for GFRP rods with various values of Vf

ratios. Safe design life based on TTFF concept was calculated at high confidence level�¼ 0.99 and at two values of reliability, R¼ 0.368 and 0.99. These diagrams are ofconsiderable value to the designer specially when the structure contains a criticalcomponent where any failure is catastrophic.

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