simplified stress calculation method for helical spring

Simplified Stress Calculation Method for Helical Spring

Koutaro WATANABE TAMA SPRING CO.,LTD, 2-5-14 Oyamagaoka2Choume, Machida-city, Tokyo, 194-0215 Japan TEL:+ 81-42-798-5511 FAX:+81-42-798-5515 e-mail:[email protected]

Hideo YAMAMOTO  NHK SPRING CO.,LTD, CO.,LTD, 3-10 Fukuura, KAnazawa-ku, KAnazawa-ku, Yokohama, 236-0004 Japan TEL:+ 81-45-786-9657 FAX:+81-45-786-7582 e-mail:[email protected]

Yuichi Ito CHK SPRING INDUSTRY CO.,LTD, 503 Niibaru-mati, Hita-city, Oita, 877-8501 Japan TEL:+ 81-973-22-1112 FAX:+81-973-23-4190 e-mail:[email protected]

Hisao Isobe TOGO SEISAKUSYO CORPORATION, Togo-cho, Aichi-gun, Aichi, 470-0162 Japan TEL:+ 81-561-38-5384 FAX:+81-561-38-5855 AX:+81-561-38-5855 e-mail:[email protected] e-mail:[email protected] o.jp

Various formulae are devised for calculating the stress correction factor of helical springs. However, different formulae are commonly used in different countries, such as Wahl’s formula in Japan and Bergsträsser’s formula in Europe. Therefore, the difference of values obtained by various formulae was compared with after clarifying the assumptions introduced in each formula. Those formulae for calculating stress correction factor have limitations when applied to suspension helical springs, which have increasingly larger initial pitch angles as the strength of materials for spring wire increases,  because those formulae omit the effect of the initial pitch angle. Furthermore, as the pitch angle increases, it  becomes important to consider not only the maximum shear stress stress at the inside of the coil but also the maximum  principal stress at the outside of the coil, because b ecause of the effect of the bending action on the wire. However, there are no formulae that calculate the maximum principal stress in a simple way. Accordingly, in order to solve this problem, a simplified calculation formula and a chart of the maximum shear  stress and maximum principal stress that take initial pitch angles into consideration were devised using the design of experiments and FEM analysis. This paper is a summary of activities in the collaboration research committee of the Japan Society of Spring Engineers. Keywords: Helical spring, Stress correction factor, Maximum shear stress, Maximum principal stress, FEM analysis, Design of experiments

1. INTRODUCTION Traditionally, various formulae have been devised to calculate the stress correction factor of helical compression springs. However, different formulae are

commonly used in different countries, such as Wahl’s formula in Japan and Bergsträsser’s formula in Europe. Therefore, the difference of values obtained by various formulae was compared after clarifying the assumptions

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introduced in each formula. Also, helical springs are now designed in detail by using FEM analysis that considers the contact of  adjacent wire surfaces. However, traditional stress correction formulae are often used during the earlier  development stage of the preliminary design. In recent years, the initial pitch angle of suspension helical springs tends to increase as the strength of materials for  spring wire increases. Consequently, the stress correction factor calculated by those traditional formulae, which were derived by leaving out the effect of the initial pitch angle, is showing a limitation for  application. Furthermore, as the pitch angle increases, the effect of the bending action on the wire also increases, and the maximum principal stress on the outside of the coil must be considered. However, there is no formula that calculates the maximum principal stress on the outside of the coil in a simple way. Accordingly, a simplified formula for calculating the maximum shear stress and maximum principal stress that consider initial pitch angles were devised by combining the design of the experimental method and FEM analysis. There are many kinds of helical springs depending on the type of spring, wire cross-section and form of  springs, etc. In this paper, the discussion is limited to helical springs with a circular cross-section. This paper is a summary of results obtained by the collaboration research committee of the Japan Society of Spring Engineers1). 2. NOMENCLATURE : Wire diameter  d  : Mean coil diameter   D : Spring index (c= D/d ) c : Load  P  : Deflection δ δm : Total Deflection (difference between free height and solid height) : Uncorrected shear stress (τ 0=8 DP /(π d 3) τ 0 κ  : Stress correction factor  τ  : Shear stress (τ =κ ・τ 0) : Young’s modulus ( E =206.0GPa)  E  : Shear modulus (G=78.5GPa) G : Initial pitch angle α0

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τ in σ in τ out σ out κ τ in κ σ out

: Maximum shear stress at the inside of the coil : Maximum principal stress at the inside of  the coil : Maximum shear stress at the outside of the coil : Maximum principal stress at the outside of  the coil : Stress correction factor for τ in (κ τin=τ in/τ 0) : Stress correction factor for σ out (κ σout=σ out/σ 0)

3. OUTLINE OF THE FORMULA FOR CALCULATING THE STRESS CORRECTION FACTOR 

3.1 Formula for calculating the stress correction factor  Many formulae have been devised to correct the stress at the inside of the coil. The review of the derivation process of various correction factor formulae showed that they are generally divided into three techniques; 1) strength of materials method; 2) theory of elasticity method; and 3) method to approximate the correction factor formula published previously. (1) Correction factor formula derived by the strength of  materials method Wahl2) ， 3) ， Röver 4) ， Wood1) and Honegger 5) have derived their final formulae on the basis of shear stress, induced by the torsional moment acting on the curved-bar of the torus at a zero pitch angle and by adding the direct shear stress induced by external force to the same. In these cases, different correction factor  formulae are derived, depending on whether the offset of a rotation center necessary to balance the stress distribution due to torsional moment generated by the curved coil is considered or not, and the difference of  factors in direct shear stress against the average shear  stress 4 P /(π d 2). These assumptions in deriving the formula are summarized in Table 1. Among these formulae, Wahl’s formula seems to best meet the actual conditions. These formulae for calculating the stress correction factor are shown below. Wahl：

4c − 1 0.615 + c 4c − 4

･･････

(1)

R őver ： Wood： Honegger ：

1 c − 1 4c c

･･････

+

c c −1

･･････

+

+

and Wahl’s formula respectively. Bergsträsser ：

1 c − 1 2c c

(2) (3)

･･････

(4)

c

Table1 Stress correction factor formula and derivation conditions First term Second term Author  Curved-bar  Offset of  Shearing name theory a rotation stress center  factor  Wahl 1.23 ○ ○ Röver  0.50 ○ × Wood 1.00 ○ × Honegger  1.23 ○ × (2) Correction factor formula derived by the theory of  elasticity Göhner 6)，Henrici7)，and Ancker & Goodier 8) have derived the stress correction factor formula on the basis of the theory of elasticity. Their formulae for  calculating the stress correction factor  κ  are shown  below. These three formulae give virtually identical values for the correction factor. 5 7 1 Göhner ： ･･････ (5) 1+ + 2 + 3 4c 8c c Henrici： 5 7 155 11911 (6) 1+ + 2 + + +K 4c 8c 256c 3 24576c 4 ･･･ Ancker & Goodier ： 1 + 5 + 7 (7) 4c 8c 2

(3) Correction factor formula derived by the approximation of the previous stress correction formulae Bergsträsser 9) and Sopwith10) have obtained a stress correction factor formula that can simplify the calculation by approximating the stress correction factor formulae derived by their predecessors. The formula by Bergsträsser is considered to be an approximation of the Göhner formula, and likewise, the formula by Sopwith is that of the Wahl’s. Accordingly, the correction factors obtained by these formulae correlate positively to the values obtained by Göhner’s

･･････

(8)

･･････

(9)

c − 0.75

Sopwith：

0.615

c + 0.5

c + 0. 2 c −1

(4) Stress correction factor obtained by FEM The various stress correction factor formulae  proposed all introduce certain assumptions to derive the formula, and there are no strict solutions. Therefore, the stress correction factor was calculated, based on the condition of making the least possible number of  assumptions using the FEM. During the analysis, a 90 degree section of a curved-bar of torus with zero pitch angle is cut out and one side of the bar cross-section is firmly fixed to the wall. Subsequently, a rigid beam is fixed to the center  of the coil on the other cross-section, and a small displacement (0.01mm) is applied to the coil center. 20 nodes hexahedral elements are used and all calculations are performed to double precision, with MARC used as an FEM solver. The model used for the analysis is shown in Fig. 1. Analyses were done while changing c from 3.0 to 12.0. Uncorrected stress is calculated from the reaction force induced at the coil center, and the stress correction factor is derived by dividing the maximum shear stress on the inside of the coil at a  point 45 degrees from the bordering end-surface by this uncorrected stress. The comparison of FEM analysis and correction factors calculated by the typical stress correction factor  formulae traditionally used by Wahl ， Göhner and

Fig.1 1/4 circle beam analysis model Bergsträsser is shown in Table 2 and an example of  maximum shear stress contour in Fig. 2.

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Table 2 Comparison between FEM analysis result and theoretical value Stress correction factor  Spring FEM BergWahl Göhner  index c Results strässer  3.0 1.553 1.580 1.551 1.556 3.5 1.452 1.476 1.452 1.455 4.0 1.382 1.404 1.383 1.385 4.5 1.331 1.351 1.332 1.333 5.0 1.293 1.311 1.293 1.294 5.5 1.262 1.278 1.262 1.263 6.0 1.237 1.253 1.237 1.238 7.0 1.199 1.213 1.199 1.200 8.0 1.172 1.184 1.172 1.172 9.0 1.151 1.162 1.151 1.152 10.0 1.135 1.145 1.135 1.135 11.0 1.122 1.131 1.122 1.122 12.0 1.111 1.119 1.111 1.111 3.2 Comparison of correction factors Fig. 3 is a graphical representation of the result shown in Table 2. As clearly seen from the results shown in Table 2 and Fig. 3, the values obtained by Göhner’s formula, derived from the theory of elasticity and by Bergsträsser’s formula, an approximate solution of the former, coincide very well with the result of  FEM. However, when the stress correction factors calculated following Göhner and Bergsträsser are compared with that calculated by Wahl’s formula, the difference is only about 1.5% at c=3. Also, the difference diminishes as c increases, and, whichever  formula is used, there seems to be relatively little difference from an engineering standpoint.

Fig.2 Maximum shear stress contour figure

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（c =5）

1/4 Curved Beam, Comparison of Theory and FEM

1.6

Wahl Gohner Bergstrasser FEM Results

1.5

  r   o    t   c 1.4   a    F   n   o 1.3    i    t   c   e   r   r 1.2   o    C

1.1 1

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Spring Index

Fig.3 result

Comparison

of

FEM

analysis

4. STRESS CORRECTION FACTOR CONSIDERING THE PITCH ANGLE During the calculation of stress correction factor  considering the pitch angle, a formula for estimating the stress correction factor is derived by combining FEM analysis and the design of experiments (D.O.E.) methodology. 4.1 FEM analysis When calculating the stress correction factor by FEM analysis considering the pitch angle, one turn of the effective part of the coil is cut out. Subsequently, a rigid surface is fixed to its border surface and displacement is applied to that rigid surface. A nonlinear analysis was  performed to consider the effect of significant deformation, and the displacement δ of up to 80% of  the total deflection δm was applied. This analysis was  performed for the nine conditions shown in Table 3 for  an assumed value of 10mm for the wire diameter, the value range of 5-12.5 for the spring index and 5-15° for  the initial pitch angle. One example of the model used in the analysis is shown in Fig. 4. As a result of the analysis in the point of 180 degrees from the boundary edge, the maximum shear stress on the inside and outside of the coil and the maximum  principal stress were obtained. For the case No. 6 for the value of  δ=0.8δm, as illustrated in Fig. 4, the result of the analysis is shown in Table 4, and the distribution of maximum shear stress and maximum principal stress in Fig. 5.

spring index c, 1/c. Accordingly, 1/c and α0 were taken Table3 Spring index, the initial pitch angle 12.5 12.5 12.5 7.14 7.14 7.14 5.0 5.0 5.0

1 2 3 4 5 6 7 8 9

Fig.4

α0

c

 No.

One

de 5 10 15 5 10 15 5 10 15

turn

analysis

model

Table 4 Maximum shear stress ，maximum  principal stress (δ=0.8δm) Inside Outside  No. 1 2 3 4 5 6 7 8 9

τ max

(MPa) 344.1 823.2 1285.5 451.3 1361.1 2237.5 389.1 1805.4 3167.2

σ max

(MPa) 317.8 703.9 1014.6 415.1 1154.9 1752.4 356.0 1518.5 2458.8

τ max

(MPa) 282.4 678.0 1065.4 318.8 967.8 1608.6 235.7 1104.2 1968.0

σ max

(MPa) 307.0 782.4 1301.2 352.0 1129.6 1986.0 263.9 1301.8 2453.3

4.2 Stress correction factor formula derived by the design of experiments methodology In order to obtain an estimation formula for the stress correction factor, the design of the experiments methodology based on an orthogonal array was used. (1) Design of experiments11), 12) (a) Design factor and level Design factors affecting the stress correction factor  are considered to include the spring index c and initial  pitch angle α0. The traditional stress correction factor  formulae are also often expressed by the reciprocal of 

Fig.5 Stress contour figure (Analysis No.6) as the factor in the design of experiments, and the number of levels for each factor was set to three so that the estimation formula could be expressed by quadratic equations. The values of level are shown in Table 5. Table 5 Level of design factors Desi n Factor 1st Level 2nd Level 1 /c 0.08 0.14 (deg) 5 10 α0 12.5 7.14 c

3rd Level 0.20 15 5.0

(b) Orthogonal array L9 of the three-level system was used, with the value of levels for each experiment shown in Table 3. (c) Characteristics for evaluation The stress correction factors for the maximum shear  stress at the inside and outside of the coil and maximum principal stress were obtained by dividing each stress obtained from the FEM analysis by the uncorrected stress τ 0. Two of the key characteristics are the correction factor for the maximum shear stress at the inside of the coil κ τin and that of the maximum  principal stress at the outside of the coil κ σout.

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(2) Results of FEM analysis The values of the stress correction factor obtained from FEM analysis at deflections of 0.2, 0.4, 0.6 and 0.8 times the total deflection δm are shown in Table 6. (3) Results of variance analysis As for the result of variance analysis for each correction factor of  κ τin and κ σout, as shown in Table 6, the case of  δ=0.8δm is shown in Tables 7 and 8. In any of the evaluation characteristics, the error ratio is very small and highly precise values are obtained.

Table 6 Stress correction factor  δ δm

0.2

0.4

0.6

0.8

38

Stress correction factor   No. 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

κ τin

κ σin

κ τout

κ σout

1.142 1.140 1.147 1.253 1.232 1.233 1.365 1.328 1.323 1.124 1.129 1.143 1.220 1.217 1.227 1.325 1.312 1.319 1.118 1.128 1.148 1.211 1.215 1.232 1.314 1.310 1.324 1.116 1.129 1.153 1.207 1.215 1.237 1.308 1.311 1.330

1.040 0.910 0.796 1.167 0.990 0.859 1.294 1.071 0.924 1.023 0.921 0.829 1.119 0.992 0.889 1.222 1.068 0.952 1.025 0.942 0.869 1.111 1.010 0.928 1.204 1.083 0.991 1.031 0.965 0.910 1.110 1.031 0.969 1.197 1.102 1.033

0.937 0.939 0.950 0.885 0.876 0.886 0.827 0.812 0.822 0.922 0.930 0.948 0.862 0.865 0.882 0.802 0.803 0.819 0.918 0.929 0.951 0.855 0.864 0.886 0.796 0.801 0.823 0.916 0.929 0.955 0.853 0.864 0.889 0.792 0.802 0.826

1.054 1.144 1.253 1.021 1.077 1.176 0.977 1.004 1.095 1.022 1.111 1.217 0.970 1.042 1.141 0.917 0.973 1.065 1.007 1.091 1.192 0.952 1.024 1.119 0.898 0.957 1.047 0.996 1.073 1.167 0.941 1.009 1.098 0.887 0.945 1.030

In the variance analysis chart, the degree of effect of  each factor upon the evaluation characteristics can be evaluated by the contribution ratio or risk ratio. The trend of each of the evaluation characteristics is summarized as follows: - The primary component of 1/ c significantly affects the correction factor for the maximum shear stress at the inside of the coil κ τin. - The primary component of  α0 and then the primary component of 1/c significantly affects the correction factor for the maximum principal stress at the outside of the coil κ σout. (4) Formula for estimating the correction factor  Through variance analysis, the formula for  estimating the evaluation characteristics can be obtained as a form of orthogonal polynomial expression, with the factors as variables. The formula for estimating the stress correction factor is given by the equation (10). Its coefficients  β 1～ β 7 for the correction factor for the maximum shear stress at the inside of the coil κ τin and that for the maximum principal stress at the outside of  the coil κ σout are shown in Table 9. κ= f (1/ c,α 0 ) = β 1 + β 2 (1 c) + β 3 (1 c)  β 4α 0 + β 5α 0

+

2

1

1

 β 6 α 0 + β 7 α 0 c c

+

2

(10)

2

Table 7 Analysis of variance for  κ τin Design Factor  DOF ContribuF Ratio De ree tion Ratio 1/c 1 1 4.6E+05 ** 97.1% 2 1 3.8E+02 ** 0.08% α0 1 1 1.2E+04 ** 2.48% 2 1 9.2E+02 ** 0.19% 1/c×α0 2 5.4E+02 ** 0.11% Error 2 0.00% Total 8 100% F 0.05 =18.5 F 0.01 =98.5 Table 8 Analysis of variance for κ σout Design Factor  DOF ContribuF Ratio de ree tion Ratio 1/c 1 1 1.3E+06 ** 38.2% 2 1 9.3E+00 0.00% 1 1 2.1E+06 ** 61.0% α0 2 1 1.5E+04 ** 0.42% 1/c×α0 2 1.2E+04 ** 0.34% Error 3 0.00% Total 8 100% F 0.05 =18.5 F 0.01 =98.5

Table 9 The coefficient of assumed formula δ κ τin 0.2δ 0.4δ 0.6δ 0.8δ κ σout 0.2δ 0.4δ 0.6δ 0.8δ

 β 1

 β 2

 β 3

β 4

β 5

β 6

 β 7

9.734E-01 9.929E-01 9.953E-01 9.971E-01

2.230E+00 1.597E+00 1.449E+00 1.373E+00

4.306E-01 1.060E+00 1.213E+00 1.273E+00

6.263E-03 2.570E-03 1.842E-03 1.375E-03

-1.407E-04 5.333E-05 1.210E-04 1.663E-04

-1.192E-01 -5.233E-02 -3.775E-02 -2.958E-02

4.033E-03 1.567E-03 1.083E-03 8.500E-04

9.501E-01 9.805E-01 9.871E-01 9.900E-01

5.124E-01 -3.868E-01 -5.954E-01 -6.779E-01

-9.074E-01 -3.868E-01 3.704E-02 7.870E-02

2.948E-02 2.149E-02 1.771E-02 1.477E-02

-2.157E-04 6.200E-05 1.603E-04 2.113E-04

-2.184E-01 -1.072E-01 -7.525E-02 -5.767E-02

7.550E-03 3.367E-03 2.250E-03 1.700E-03

(5) Correction factor diagram for κ  If the change in the shape of helical springs is taken into account, the stress correction factor will change as the load changes. In Table 9, coefficients in the equation (10) for the cases when the deflection δ equals 0.2, 0.4, 0.6 and 0.8 times the total deflection δm are shown. Using these coefficients and the equation (10),

Wahl

αo= 5°

αo=7°

αo=11 °

αo= 13°

αo=15 °

αo= 9°

1.4 1.3    n      i     τ

1.2

    κ

1.1 1.0 5

6

7

8

9

10

Spring index

11

12

c 

Fig.6 Diagram of stress correction factor  κ τin （δ=0.8δm） αo= 5°

αo=7°

αo= 9°

αo= 11 °

αo=13 °

αo= 15 °

1.2 1.1     t    u    o     σ

1.0

    κ

0.9 0.8 5

6

7

8

9

Spring index

10

11

c 

Fig.7 Diagram of stress correction factor κ σout （δ=0.8δ ）

12

the correction factor can be estimated for an arbitrary spring index and initial pitch angle at the above deflection values. The diagram of the stress correction factor drawn by using these estimated values for  δ=0.8δm are shown in Figs. 6 and 7. 5. Comparison between the measured data and FEM analysis data According to the conditions described in Section 4, the stress becomes identical anywhere in the effective  part of the coil. However, in the actual helical compression springs, the center of the coil axis and the load axis never coincide. Accordingly, the phenomenon of the load off-center occurs and the distribution of  stress in the effective coil section becomes wavelike. As there is no theoretical method to precisely calculate this wavelike distribution, it is inevitable that the detailed design must rely on the FEM that considers the actual  boundary conditions. Therefore, in order to verify the stress obtained by the estimation formula of Section 4 against the actual stress distribution, it was compared with the result of  FEM analysis for the actual suspension helical springs. The configuration of the suspension helical spring used for this verification is shown in Table 10. 5.1 Method of analysis Based on the measured data of the spring shape, the coil was divided into 60 elements per turn. The element used is of a three-dimensional, two-node beam. The seat of the coil is simulated as rigid, and the contact condition between the wire and seat and between the wires was defined. The assumed condition of the

39

constraint is shown in Fig. 8. Analysis was performed Table 10 Spring specification S ecification Wire diameter mm 13.5 Mean coil diameter (mm) 159.5 Spring index 11.81  Number of active coils 3.01 Total number of coils 4.26 Free height (mm) 350 Pitch Angle (deg) 12 Spring rate (N /mm) 23.5  Normal height (mm) 190 Maximum height (mm) 106 Upper (A-side) Tapered Coil Ends Lower (B-side) Open end for significant deformations and the updated Lagrange method was used. In the calculation of stress, formulae for calculating the stress from the strength of the member, described in the “FEM for Springs13)” edited by the Japan Society for Spring Research, were used. In addition, the stress was actually measured by applying three-dimensional strain gages to the test sample.

described previously, the stress obtained by this formula is constant within the effective length of the coil and it emerges that its value is approximately the average of  the wavelike stress distribution. Spring No.1; Comparison of FEM results Measurements and D.O.E. (Normal Height)     )   a    P    M     (   s   s   e   r    t    S   r   a   e    h    S   m   u   m    i   x   a    M

FEM Results

850

Measurements

D.O.E.

750 650 550 450 300

500

700

900

1100

(deg) θ1300

Fig.9 Comparison of stress distribution inside at normal height Spring No.1; Comparison of FEM results Measurements and D.O.E (Normal Height)     )   a    P    M     (   s   s   e   r    t    S    l   a   p    i   c   n    i   r    P   m   u   m    i   x   a    M

FEM Results

850

Measurements

D.O.E.

750 650 550 450 300

500

700

900

1100

(deg) θ1300

Fig.10 Comparison of stress distribution outside at normal height

Fig.8

Analysis model

5.2 Analytical results Analytical results of the maximum shear stress at the inside of the coil and maximum principal stress at the outside of the coil at working spring height, together  with measured values, are shown in Figs. 9 and 10. The transverse of the graph shows the angle from the B-side (lower) in the cylindrical coordinate and the analytical results and measured values generally show good agreement. The stress obtained by the stress estimation formula discussed in Section 4 is also shown in Fig. 10. As

40

6. CONCLUSION (1) As the formula for stress correction factor of the helical spring, Wahl’s formula is used in Japan while Europe uses Bergsträsser’s. The Bergsträsser’s formula, which is an approximation of Göhner’s formula, is more accurate, but the difference is relatively small, as shown by the fact that it is about 1.5% when compared at c=3. Furthermore, the difference diminishes as c increases. There seems to be no problem in using Wahl’s formula, as in the past, because it is derived theoretically and easy to understand, as shown by the fact that it is divided into torsion and shear terms. (2) By combining the design of experiments and FEM analysis, a formula for estimating the stress correction factor of helical springs, which takes into account the initial pitch, was obtained. During the design of 

experiments, the spring index and initial pitch angle are set as the factors. The maximum shear stress at the inside and outside of the coil and the stress correction factor  κ  were calculated during the FEM analysis, and the estimation formula for each correction factor was obtained by variance analysis. This method is effective for evaluating helical springs with a large initial pitch angle, particularly for the maximum principal stress at the outside of the coil. (3) The results of FEM analysis on the beam element of  the actual helical spring shapes and experimental stress measurements were compared. In the actual helical springs, both maximum shear stress and maximum  principal stress show a wavelike distribution, due to the load off-center, and the results of FEM analysis and measurement correlated fairly well. It is confirmed that the corrected stress, calculated by using the stress correction factor formula derived in this study, coincides fairly well with the average value of the wavelike distribution of stress.

ACKNOWLEDGEMENTS The authors greatly acknowledge the efforts of  members of the research committee of the Japan Society of Spring Engineers in the literature research and collection, review of the derivation process of the correction factor formula and FEM analysis, as well as for their valuable suggestions. Presently, detailed analyses of the helical springs are  performed by FEM. However, in the design of helical springs, a preliminary design is created to obtain rough values for the wire and coil diameter and the number of  effective turns in the first place. Wahl’s formula and Bergsträsser’s formula are then used to calculate the shear stress at the inside of the coil. As the initial pitch angle of the suspension helical springs shows a tendency to increase in recent years, the formula to estimate stress correction factor that considers the pitch angle is derived as a substitute for the traditional formula, which is derived by assuming a pitch angle of  zero. In addition, the estimate of maximum principal stress at the outside of the coil is made obtainable. In literature1), the formulae for estimating correction factors for deflection in the helical springs with an

initial pitch angle are obtained. Also, please refer to the diagram of the stress correction factor, which presents each correction factor as a contour by taking the spring index c and initial pitch angle α0 as coordinates, for the values of δ=0.2δm, 0.4δm, 0.6δm and 0.8δm. The authors will be happy if this report will be of  help for the design of helical springs. REFERENCES 1)Research Committee on the Analysis of Helical Spring ， Transactions of Japan Society for Spring Research， No.49，35(2004) 2) A.M.WAHL ， Stresses in Heavy Closely Coiled Helical Springs ， Trans.ASME ， APM-51-17 ， 185 (1929)． 3) A.M.WAHL ， Mechanical Springs 2nd edition ， McGraw-Hill ，1963， pp241． 4) A.RÖVER  ， Beanspruchung zylindrischer  Schraubenfedern mit Kreisquerschnitt，Z.VDI，Bd57， 1906 (1913)． 5) E.HONEGGER  ， Zur Berechnung von Schraubenfedern mit Kreisquerschnitt，P.Appl.Mech.， Vol.12，99 (1930)． 6) O.GÖHNER  ， Die Berechnung zylindrischer  Schraubenfedern，Z.VDI，Bd76，269 (1932)． 7) P.HENRICI，On Helical Springs of Finite Thickness， Quat.J.Appl.Mech.，Vol.XⅢ， No.1，(1955)，106． 8) C.J.ANCKER.JR. ， J.N.GOODIER ， Theory of  Pitch and Curvature Corrections for the Helical Spring-1(Tension)，J.Appl.Mech.， 471 (1958)． 9) M.BERGSTRÄSSER ，Die Berechnung zylindrischer  Schraubenfedern， Z.VDI，Bd77，198 (1933)． 10) Notes on the design of helical compression springs ， Spring Design Memorandum No.1 ， (1942) ， 5 ．， The Department of Controller Genneral of Research and Development Minsrty of Supply． 11) G.Taguchi ， Design of Experiments 3rd edition ， Maruzen，1976， pp48． 12) Y.Kashiwamura, M.Shiratori and Y.Qiang ， Optimization of Non-linear Probrem by Design of  Experiments，Asakura Pub. Co. Ltd.，1998， pp18． 13) Japan Society of Spring Engineers ， FEM for  Springs， Nikkankogyo Shinbun Ltd.，1997，

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