fatiga multiaxial

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Article

Multiaxial fatigue analysis of stranded-wire helical springs

Hossein Darban1, Mostafa Nosrati1 and

Faramarz Djavanroodi1,2

Abstract

In this paper, finite element method is implemented to model stranded-wire helical springs under differentloading conditions. Finite element results are coupled with multiaxial fatigue criteria such as Fatemi–Socie

and Kandil–Brown–Miller together with a uniaxial fatigue criterion, Coffin–Manson, to predict fatigue lifeof the stranded-wire helical springs. It is shown that due to damping effects between wires, stranded-wirehelical springs have longer fatigue life compared to their equivalent single-wire helical springs at a similarcondition. It is also demonstrated that fatigue life is longer for loadings with higher initial displacement of spring head. As practical examples, fatigue life of stranded-wire helical springs with 9 and 15 wires areestimated and compared. It is shown that the spring with 15 wires gives longer fatigue life. It is alsoobserved that Kandil–Brown–Miller and Fatemi–Socie criteria give the least and the highest fatigue lifeprediction, respectively.

Keywords

Stranded-wire helical spring, multiaxial fatigue criterion, frictional force, finite element method

Introduction

A stranded-wire helical (SWH) spring is constructed from twisting of several wires together. When

an axial load is applied at the end of a SWH spring, the strand will be subjected to a twisting

moment. In the case of tensile loading, if both of the strand and the coil have the same turn of helix,

the resulting twisting moment tends to tightly stick the wires together (Peng et al., 2012).

Consequently, if the helix of the strand be opposite in direction to the helix of the spring, the

twisting moment tends to unwind the strand adversely. The outstanding characteristic of the

SWH spring is the inherent tendency for damping of high velocity displacement of its coil due to

friction between wires (Min and Wang, 2007; Phillips and Costello, 1979). As a result, SWH springs

have longer fatigue life in comparison with single-wire helical springs (Min and Wang, 2007;

International Journal of Damage

Mechanics

0(0) 1–13

! The Author(s) 2014

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DOI: 10.1177/1056789514560914

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1Department of Mechanical Engineering, Iran University of Science & Technology, Tehran, Iran2Department of Mechanical Engineering, Prince Mohammad Bin Fahd University, Al Khobar, Saudi Arabia

Corresponding author:

Hossein Darban, Department of Mechanical Engineering, Iran University of Science & Technology, Tehran 16887, Iran.

Email: [email protected]


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Phillips and Costello, 1979). Hence, SWH springs are used frequently in many industrial cases (Clark,

1961; Costello and Phillips, 1979; Phillips and Costello, 1979). On the other hand, due to manufactur-

ing difficulties for SWH springs with more than three wires, the central wires must firstly be con-

structed and other wires have to be wound around this core. Moreover, the shape of this core depends

on number of wires (Clark, 1961). It is to be noted that, several works have been carried out to

calculate the geometric parameters of SWH springs (Wang et al., 2008, 2010; Zhang et al., 1999).Wang et al [7] proposed two mathematical models for determination of the twist angle and diameter of

the strands. Kunoh and Leech (1985) showed that for the case of small helix angle, the curvature plays

major role in the cross section shape of a strand and also in the contact position of adjacent wires.

During the past few decades, various fatigue criteria have been proposed to predict the fatigue life

of mechanical elements with assumption of uniaxial stress state. However, in many practical cases,

complicated geometries and loading types do not permit the researchers to analyze the problem in

the framework of uniaxial stress state. Therefore, various methods and criteria have been suggested

to describe and predict fatigue life of different materials under practical condition (Brown and

Miller, 1973; Eyercioglu et al., 1997; Fatemi and Socie, 1988; Glinka et al., 1995; Hanumanna

et al., 2001; Jan et al., 2012; Kandil et al., 1982; Kim and Kang, 2008; Lee et al., 2009;

McDiarmid, 1994; Macha and Sonsino, 1999; Navid Chakherlou and Abazadeh, 2011;Papadopoulos et al., 1997; Varvani-Farahani, 2000; Wang and Yao, 2004, 2006; You and Lee,

1996). These criteria use different approaches for prediction of fatigue life. From one aspect, multi-

axial fatigue criteria are categorized to stress, strain, or energy based (Kandil et al., 1982). As

another classification, some of them are formulated based on critical plane concept while others

are not (Kandil et al., 1982). The principles of critical plane theory are firstly proposed by Brown

et al. for multiaxial fatigue problems (Eyercioglu et al., 1997). The critical plane is a plane in which a

specific parameter in a fatigue criterion meets its maximum value and other parameters of the

criterion must be calculated in this plane (Chen et al., 1999; Del Llano-Vizcaya et al., 2006; Kim

and Park, 1999; Muralidharan and Manson, 1988; Pan et al., 1999; Wang and Brown, 1993). Del

Llano-Vizcaya et al. (2006) applied several multiaxial fatigue criteria such as Fatemi–Socie (FS)

criterion (Fatemi and Socie, 1988) and Wang–Brown criterion (1993) to a single-wire helical springand found good agreement between predicted fatigue lives and experimental results.

In the present study, the well-known multiaxial fatigue criteria such as, Fatemi–Socie (FS) and

Kandil–Brown–Miller (KBM) together with a uniaxial fatigue criterion, Coffin–Manson (CM), are

used to investigate fatigue life of SWH springs with 9 and 15 stranded wires with the complete and

comprehensive discussion presented for differences between the results. Comparison of the results

showed excellent advantages for application of a SWH spring against its equivalent single-wire

helical (ESWH) spring under severe harmonic loadings. Despite the fact that there is no experimen-

tal data about fatigue lives of SWH springs in the literature, the reliability of the present method is

discussed by comparing obtained results with experimental data on the fatigue lives of single-wire

helical springs (Del Llano-Vizcaya et al., 2006). The given numerical results as discussed in this

paper would be useful for interpretation of future experimental tests.

Problem definition

Because of extensive application of SWH springs with 9 and 15 wires in industrial equipments under

external harmonic loadings, their fatigue problems are fully studied and discussed in this work.

Figure 1 illustrates the initial configuration of the cross sections and wires diameter used in this

work. As shown in Figure 1, for the case of spring with 9 wires, 3 wires are placed at the center and

other wires are wounded around the core but for SWH spring with 15 wires, the core is constructed

2 International Journal of Damage Mechanics 0(0)


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of only 1 wire. Furthermore, it can be seen that the outer wires have larger diameter in comparison

with the inner ones for stability considerations. It should be noted that, constructing SWH springs

without the little gap between outer wires is very difficult and practically impossible. But when

applied loading reaches a specific value, the gap disappears and outer wires will be in contact

with each other. Not only all of the dimensions are derived from the actual SWH springs which

are frequently used in industry, but also some other effective parameters such as material properties,

boundary and loading conditions and contact between wires are modeled very similar to the prac-

tical situation. Geometric parameters of the springs with 9 and 15 wires are also given in Table 1.

The wires are made of steel CK101 with mechanical and strain-life properties that are given in

Table 2. In this table, 0 f and "0

f are axial fatigue strength and ductility coefficients while b and c are

axial fatigue strength and ductility exponents, respectively. Respectively, 0

f

, 0

f

, b0, and c0 have the

same definition as 0 f , "0

f , b, and c but for torsional loading.

The mesh patterns used for finite element analysis of SWH springs with 9 and 15 wires are shown

in Figure 2. Since the actual SWH springs are quite long which makes their modeling difficult and

lengthy process, only two coils of each spring have been modeled and other parameters such as

stiffness and displacement amplitude have been chosen accordingly. The critical points for the

springs are also illustrated in this figure. Critical point is a point that experiences highest equivalent

stress during a loading cycle. From Figure 2, it can be observed that the critical points are located

in the inner surface of the strands, which is consistent with the experimental reports

Figure 1. Initial configuration of the cross sections of SWH springs with (a) 9 and (b) 15 wires.

Table 1. Geometric parameters of the SWH springs with 9 and 15 wires.

Spring type

Number of

coils

Length

(mm)

Average coil

diameter (mm)

Strand

diameter

(mm)

Pitch of the

spring (mm)

Pitch of the

strand (mm)

9 wires 24 374 29.5 7.6 22 88

15 wires 40 797.2 29.5 6.5 35 155

Darban et al. 3


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(Del Llano-Vizcaya et al., 2006). Nearly 850,000 eight-node linear brick elements are used to obtain

appropriate results for SWH spring with 15 wires. The models are created using ABAQUS CAE and

dynamic explicit procedure has been selected for the analysis. In order to justify using this element

type and finite element procedure, it must be pointed out that the levels of equivalent stresses at

critical points for all models are much lower than the yield stress of steel CK101. For instance, as

shown in Figure 2(b), the equivalent stress at critical point is approximately half of the yield stress.

Assuming frictional contact between the wires, penalty approach of contact and friction is applied.

Friction coefficients for steel CK101 are equal to s¼ 0.78 and d ¼ 0.42. It is assumed that the

spring is placed inside of a cylindrical shield and therefore it is not permitted to buckle.

From Figure 2, it is seen that both the strand and the coils of the modeled springs have the same

turn of helix, which makes the springs suitable for tensile loading. It should be pointed out that the

Figure 2. Finite element models of SWH springs with (a) 15 and (b) 9 wires using 3D brick elements.

Table 2. Mechanical and strain-life properties of steel CK101 (Muralidharan and Manson, 1988).

Mechanical properties

Strain-life properties

(M method)

Yield stress (MPa) 1275.000 0f (MPa) 2983.000

Ultimate tensile strength (MPa) 1570.000 b¼ b0 0.120Reduction in area 0.200 "0f 0.309

Fracture strain 0.223 c ¼ c 0 0.600

E (Young’s modulus (GPa)) 205.000 0f (MPa) 1722.000

G (shear modulus (GPa)) 79.500 0f 0.535



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springs are stretched before the loading. This puts the springs generally under tension during the

loading cycles. As one of the most important results of the present analysis, due to higher resistant

frictional force as a source of energy dissipation, the SWH spring with 15 wires experiences lower

stresses compared to the spring with 9 wires. In order to support this finding, the friction effect

between the wires is magnified by increasing the friction coefficient of wires. It is observed that

higher friction coefficient results in bigger difference between maximum stress in SWH springs with9 and 15 wires. This point confirms that the frictional force between wires is one of the main reasons

that SWH springs experience lower maximum stress when they are constructed with more wires.

It must be noted that, the fretting effect has not been considered in the modeling. Fretting occurs

when two bodies in contact undergo a small relative displacement, typically in the range of 10–30 mm

(Lee et al., 2004). This phenomenon can be affected by many parameters such as contact pressure,

displacement amplitude and coefficient of friction that make it difficult to model (Amiri et al., 2011;

Cruzado et al., 2013; Naidu and Sundara Raman, 2005; Shariyat, 2010).

Multiaxial fatigue criteria

The CM criterion, which has been widely used to estimate fatigue life of mechanical elements underuniaxial loading, is given as follows

"

2 ¼

0 f

E ð2N f Þ

b þ "0 f ð2N f Þc For axial loading

2 ¼

0 f

Gð2N f Þ

b0 þ 0 f ð2N f Þc0 For torsional loading

ð1Þ

where ", , and N f are the strain range in axial fatigue, the shear strain range in torsional fatigue

and the number of cycles to failure respectively. The other parameters in this equation are already

clarified. Although CM criterion is very frequent in use, it leads to incorrect fatigue life for mech-

anical elements under complicated harmonic loadings. For this reason, several multiaxial fatiguecriteria have been proposed (Brown and Miller, 1973; Eyercioglu et al., 1997; Fatemi and Socie,

1988; Glinka et al., 1995; Hanumanna et al., 2001; Jan et al., 2012; Kandil et al., 1982; Kim and

Kang, 2008; Lee et al., 2009; McDiarmid, 1994; Macha and Sonsino, 1999; Navid Chakherlou and

Abazadeh, 2011; Papadopoulos et al., 1997; Varvani-Farahani, 2000; Wang and Yao, 2004, 2006;

You and Lee, 1996). Among these criteria, those, which are based on critical plane concept, are more

attracted by the researches. Having suitable definitions for equivalent stress and strain factors for

simulation of complicated problems by their equivalent uniaxial problems is the main goal of these

criteria. This obviously helps to reduce the mathematical complication and the amount of numerical

analysis. Therefore, major differences between different multiaxial fatigue criteria are the definition

of the equivalent parameters that affects the location and orientation of the corresponding critical

planes.

The KBM criterion introduces critical plane as a plane in which the maximum shear strain range

max occurs. As it can be observed from equation (2), KBM criterion is strain based in which the

maximum shear strain range max and normal strain range "n play major role and have great

influences on the results (Kandil et al., 1982)

max

2 þ S "n ¼

0 f

E ð2N f Þ

b þ "0 f ð2N f Þc ð2Þ

Darban et al. 5


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where S is the material constant which has been suggested to be equal to 1 for KBM criterion.

Values of max and "n in this equation can be calculated from equations (3a) and (3b) as follows

(Varvani-Farahani, 2000)

max

2 ¼

"1 "3

2

1

"1 "3

2

2

ð3aÞ

"n

2 ¼

"1 þ "3

2

1

"1 þ "3

2

2

ð3bÞ

In the above relations, "1 and "3 are first and third principal strains, respectively. Moreover, 1and 2 show the cyclic loading limits. It is obvious that positive normal stress opens microcrack faces

and consequently leads to early fatigue failure. From equation (4), in the FS criterion maximum

normal stress is used instead of normal strain (Fatemi and Socie, 1988). As it can be seen, for

positive normal stresses, the left hand side of equation (4) increases and therefore the criterion

predicts smaller fatigue life. Furthermore, additional hardening and mean stress effects in the FS

criterion are considered through the normal stress. In addition, the FS criterion has the sameapproach as the KBM criterion in obtaining the critical plane. The FS criterion is given below

max

2 1 þ K

maxn y

¼

0 f

E ð2N f Þ

b þ "0 f ð2N f Þc ð4Þ

where maxn is the maximum normal stress, y is the yield strength and k is a material constant which

can be obtained from uniaxial and torsion fatigue tests, but for most of the materials, it can be

considered equal to 0.6 (Fatemi and Socie, 1988). On the other hand, since the fatigue tests are very

difficult and costly, Muralidharan and Manson (1988) have respectively presented M and MM

methods to approximately attain the material fatigue constants from monotonic tension test data.

Table 3 gives the relations between the strain-life and monotonic properties in which Rm and e f arethe ultimate tensile strength and the true fracture strain, respectively.

On reliability of the method

In this study, finite element method is firstly used to find the critical point. In this manner, a loading

cycle is divided to several increments, then critical points of increments are compared with each

other and finally, the largest equivalent stress is introduced as the critical point. At the next step,

stress information of the critical point is introduced to the computer to predict the fatigue life.

Table 3. The M and MM methods for calculation of the strain-life properties Muralidharan and Manson (1988).

Strain-life parameters

Constants

Axial case M method MM method

Constants

Torsional case

Fatigue strength coefficient 0f 1.9 Rm 0.623 Rm0.823 E0.168 0f ¼ 3

0.5 0f

Fatigue strength exponent b 0.12 0.09 b0¼ b

Fatigue ductility coefficient "0f 0.76 ef 0.6 0.0196 ef

0.155 (Rm E1)0.53 0f ¼ 3

0.5"0f

Fatigue ductility exponent c 0.6 0.56 c 0¼ c



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It must be noted that, because of using the shear strain increment to find the critical plane in FS and

KBM criteria, the computer program calculates this parameter in all directions around the critical

point for 1 rises in and as shown in Figure 2. Hence, in this procedure, the shear strain range is

calculated and compared for 360 sequential planes around the critical point. In order to assess the

accuracy and applicability of the described method in fatigue problems, this method is used to

predict fatigue life of a single-wire helical spring with specific geometry and material parameterswhich is experimentally investigated by Del Llano-Vizcaya et al. (2006).

Figure 3 depicts the single-wire helical spring used as sample by Del Llano-Vizcaya et al. (2006) in

experimental tests. Number of coils, length, wire diameter, and outside coil diameter were N ¼ 9.5,

L¼ 153.6 mm, d ¼ 5.7 mm, and D¼ 44.4 mm, respectively. Wires of the spring were made of high

carbon steel AISI MB, with mechanical and strain-life properties which are tabulated in Table 4.

The tests were carried out under mean stress m¼ 254.9 MPa with variable stress amplitude a. The

strain-life properties obtained using M method, since this method provides better results than the

MM method (Del Llano-Vizcaya et al., 2006).

Figure 4 shows good agreement between the experimental results and those predicted by the

present method. It must be noted that despite the FS criterion, the CM criterion overestimates

the fatigue life of the spring due to the positive normal stress at the critical point. In addition,the KBM criterion gives conservative results for all loading conditions.

As an important outcome, deviation of the KBM criterion results from the real situations

increases at higher stress amplitudes so that for a¼ 141 MPa the deviatoric error is equal to 0.18

but for a¼ 148 MPa is equal to 0.37. It is also observed that the FS criterion gives the most reliable

results for most of the loading conditions.

Table 4. Mechanical and strain-life properties of high carbon steel AISI MB (Del Llano-Vizcaya et al.,

2006).

Mechanical properties

Strain-life properties

(M method)

Yield stress (MPa) 1350.000 0f (MPa) 3173.000

Ultimate tensile strength (MPa) 1670.000 b¼ b0 0.120

Reduction in area 0.325 "0f 0.434

Fracture strain 0.393 c ¼ c 0 0.600

Young’s modulus (GPa) 177.000 0f (MPa) 1832.000

Shear modulus (GPa) 68.600 0f 0.751

Figure 3. Geometric parameters of a single-wire helical spring.

Darban et al. 7


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Results and discussions

As a comprehensive and comparative study, Figures 5 and 6 depict the number of cycles to failure

with respect to the displacement amplitude for SWH springs with 9 and 15 wires subjected to initial

head displacement of 10 mm and 15 mm, respectively. Generally, it is seen in these figures that for

most of the loading conditions, the KBM criterion gives the lowest fatigue lives. As it is shown

previously, the SWH spring with 15 wires has lower stress magnitude and therefore would have

longer fatigue life in comparison with the SWH spring with 9 wires. Consequently, at similar loadingcondition, the spring with 15 wires has longer fatigue life than a spring with 9 wires. For example,

according to the FS criterion, fatigue life of a SWH spring with 15 wires subjected to initial residual

displacement and additional displacement amplitude of 10 mm is approximately equal to 3e5 cycles

but for a SWH spring with 9 wires the fatigue life is approximately equal to 1e4 cycles. Hence,

considering this fact that the frictional force between the wires decreases the stress intensity in SWH

springs, it is concluded that using more wires to construct SWH springs, increases resistance of the

springs against the fatigue failure.

Figures 5 and 6 illustrate that the FS criterion predicts higher fatigue life than CM criterion. This

is because, for FS criterion, sign of normal stress on the critical plane has significant influence on the

estimated fatigue life.

Also, it can be seen from Figures 5 and 6 that the initial head displacement of the spring exten-

sively changes spring fatigue life. For illustration of this fact, Figure 7 shows the influences of initial

head displacement on fatigue life of SWH spring with 15 wires for different displacement amplitudes.

It is shown that increase in the initial head displacement causes substantial rise in fatigue life of the

spring. For example, at displacement amplitude of 7.5 mm, fatigue life of the spring subjected to

15 mm of initial head displacement is 20 times greater than the spring with 10 mm of initial head

displacement. It is to be noted that the SWH springs in this study are designed to sustain tension and

therefore, are stretched before the loading. It is understood that for higher amount of initial dis-

placement, the spring experiences higher level of tension during the loading. It means that the wires

105

106

141

142

143

144

145

146

147

148

149

Number of cycles to failure

S t r e s s a m p l i t u d e ( M P a )

Experimental results [31]

FS criterion results

CM criterion results

KBM criterion results

Figure 4. Verification of the results for different criteria with previous experimental data.FS: Fatemi–Socie; CM: Coffin–Manson; KBM: Kandil–Brown–Miller.



9/13

are stuck together more firmly and consequently, more energy is dissipated due to friction between

the wires. This energy dissipation reduces the intensity of stress in the wires and as a result, the

spring gives longer fatigue life. However, the spring experiences both tension and compression

during a loading cycle in case the displacement amplitude is bigger than the initial displacement

of spring head. In this case, the wires are unwound during the compression and each of them

sustains a portion of the loading separately without any contact with the other wires. This reduces

the efficiency and fatigue life of the SWH spring. Hence, it seems that introducing initial head

displacement is beneficial for the fatigue lives of SWH springs. This finding can be considered for

installation of the SWH springs in practical applications.

102

103

104

105

106

107

0

2

4

6

8

10

12

14


D i s p l a c e m e n t a m p l i t u d e ( m m )




104

105

106

7.5

10

12.5

15

17.5

20

22.5

Number of c ycles to failure

D i s p l a c e m e n t a m p l

i t u

d e ( m m )




(a)(b)

Figure 5. Illustrative plots for fatigue life of SWH springs with (a) 9 and (b) 15 wires subjected to initial displace-

ment of 10 mm.FS: Fatemi–Socie; CM: Coffin–Manson; KBM: Kandil–Brown–Miller.

102

103

104

105

2

3

4

5

6

7

8

9

10

11

D

s p l a c e m

e n t a m p l

t u d e ( m m

)





103

104

105

106

107

108

6

8

10

12

14

16

18

20






(a) (b)

Figure 6. Illustrative plots for fatigue life of SWH springs with (a) 9 and (b) 15 wires under initial displacement of

15 mm.

FS: Fatemi–Socie; CM: Coffin–Manson; KBM: Kandil–Brown–Miller.

Darban et al. 9


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Additionally, to show the advantages of the springs constructed from more than one wire over the

ordinary springs in fatigue analysis, an ESWH spring is defined. The wire diameter is chosen as a

dominant parameter for definition of an ESWH spring. The wire diameters of the corresponding

ESWH springs were calculated based on equality of its stiffness with that of a SWH springs. Other

geometric and material parameters for the ESWH spring are equal to those of SWH spring. Stiffness

of single-wire helical springs K can be calculated from equation (5) in which all parameters are

predetermined. Experimental work has been performed to obtain SWH springs stiffness. In this

manner, different tensile force is applied to the ends of SWH springs with 9 and 15 wires and the

corresponding deflections were measured. Figure 8 shows the obtained force–deflection plots for

SWH springs with 9 and 15 wires in which slope of each interpolated line indicates the stiffness of the

104

105

106

107

108

6

8

10

12

14

16

18

Number of cycles to failure (FS criterion)

D i s p l a c e m e n t a m p l i t u

d e ( m m )

initial displacement of 10 mm

initial displacement of 15 mm

Figure 7. Influence of initial displacement on the fatigue life of SWH springs with 15 wires.

FS: Fatemi–Socie.

0 50 100 150 200 250 300 3500

500

1000

1500

Deflection (mm)

F o r c e ( N )

y = 3.9893*x - 98.799

Experimental results

Interpolated line

0 100 200 300 400 500300

400

500

600

700

800

900

Deflection (mm)

F o r c e ( N )

y = 1.1341*x + 366.97

Experimental results

Interpolated line

(a) (b)

Figure 8. Experimental results of the stiffness test for the SWH springs with (a) 9 and (b) 15 wires.



11/13

corresponding spring. As it can be observed from this figure, stiffness of SWH spring with 9 and 15

wires are equal to 3.9893 N/mm and 1.1341 N/mm, respectively.

By substitution of K , G, D, and N into equation (5), the wire diameters of the corresponding

ESWH springs are obtained equal to 3.97 mm and 3.26 mm for SWH springs with 9 and 15 wires,

respectively.

K ¼ d 4G

8D3N ð5Þ

Predictably, under similar conditions, ESWHS springs show higher equivalent stresses at thecritical points so that it is about 2 and 6 times greater than the SWH springs with 9 and 15

wires, respectively.

Figure 9 shows the fatigue life for SWH spring made of 15 wires and its ESWH spring with respect

to the displacement amplitude under 3 mm of initial head displacement. This figure reveals advantage

of SWH spring in fatigue problems. Unlike single-wire springs, the fatigue life of SWH springs can be

infinite in some cases. As an illustrative example, for displacement amplitude of 1 mm, fatigue life of

ESWH spring is approximately 4e5 cycles but this value for SWH spring is 3.8e9 cycles.

Summary and conclusions

In the presented work, two multiaxial fatigue criteria together with a uniaxial strain-life fatigue

criterion are applied to the SWH springs and their ESWH springs. Loading conditions, geometry,

and material parameters are selected near to practical industrial situation. Finite element method is

employed to find the stress distribution and its related critical points in the spring’s body. Finite

element results are coupled with multiaxial fatigue criteria such as Fatemi–Socie (FS) and Kandil–

Brown–Miller (KBM) together with a uniaxial fatigue criterion, Coffin–Manson (CM) to predict

fatigue life of the SWH springs. Afterward, an ESWH spring is defined with consideration of the

stiffness equality. It is demonstrated that the friction force between the wires in SWH springs act as

resistance force and reduces the stress magnitudes.

109

1010

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


Number of c cles to failure




105

106

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6






(a) (b)

Figure 9. Fatigue life of SWH springs with (a) 15 wires and (b) their ESWH springs subjected to 3 mm of initial

displacement.

FS: Fatemi–Socie; CM: Coffin–Manson; KBM: Kandil–Brown–Miller.

Darban et al. 11


12/13

It is shown that longer spring fatigue life is achieved with increasing the number of strands and

this enhancement in fatigue lives of the SWH springs is directly related to the frictional force

between the wires. As another finding, it is concluded that fatigue life is increased for higher initial

head displacement. This result can be applied for installation of the SWH springs in practical

applications. It is also shown that the KBM criterion gives more conservative results in comparison

with other criteria. It is also observed that the KBM criterion is not suitable for single-wire springsunder severe loading. In addition, it is seen that the sign of normal stress plays a significant role in

FS criterion so that for positive normal stresses, the predicted fatigue life reduces while the trend

inverses for negative normal stresses. Comparing the results, fatigue life predicted by FS criterion is

closer to real situations.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit

sectors.

Conflict of interest

None declared.

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