Static mechanical behaviour of composite cylindrical springs

Composites Science and Technology 40 (199I) 251-263

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Static Mechanical Behaviour of Composite Cylindrical Springs

C. K. So, P. C. Tse, T. C. Lai & K: M. Y o u n g Department of Mechanical and Marine Engineering, Hong Kong Polytechnic, Hunghom, Kowloon, Hong Kong (Received 6 November 1989; accepted 22 March 1990)

ABSTRACT The spring rate, stress and strain distrib'utions of a mid-plane symmetric laminated circular ring under uniaxial loading are analysed using a simple strain energy approach. Theoretical analysis suggested that for a specially orthotropic mid-plane symmetric laminate the spring stiffness is linearly proportional to a spring parameter. The experimental strain distributions around the ring werefound to agree in general with the theoretical predictions at various loadings. Several circular springs were fabricated from bidirectional E-glass fibre reinforced epoxy resin and were tested hr static compression.

NOTATION [A], [-B], [-D] E~I, E22 F G~2 h L M, MB I-M] n IN]

Constitutive matrices of a laminate Young's moduli in the principal directions Uniaxial load Shear modulus with reference to the principal axes Thickness of the shell Width of the shell Bending moment Moment resultants Number of layers of a laminate Stress resultants

251 Composites Science and Technology 0266-3538/90/$03"50© 1990Elsevier Science Publishers Ltd, England. Printed in Great Britain

252

[Q] [Q] R Ro [S] [oe] U v V W W' _g leO] 0,] v~2 o [ ]r ] I

C. K. So, P, C. Tse, T. C. LaL K. M. Young

Lamina stiffness matrix referring to the principal axes Lamina stiffness matrix referring to an arbitrary set of axes Radius of the mid-plane of the shell Internal radius of the shell Lamina compliance matrix referring to the principal axes Lamina compliance matrix referring to an arbitrary set of axes Strain energy Vertical displacement of the spring Volume of the shell Strain energy density function Strain energy density function for a cross-sectional area Distance from mid-surface Strain Mid-plane strains Surface area of the mid-plane Middle surface curvatures Major Poisson's ratio Stress Transpose of a matrix Determinant of a matrix

INTRODUCTION Although glass reinforced plastic (GRP) has had extensive usage for many years, its use as a material for load-bearing structures or components, such as springs for vehicles, is more recent. Basically there are several potential advantages that could arise from the replacement of steel in automotive suspensions by fiber composite materials. The foremost advantage is the higher specific strength of continuous fiber composites, which can yield significant weight reduction. Another possible advantage is the inherent damping characteristic in fiber reinforced plastics, which is potentially useful in that the shock absorbers of a conventional system can be eliminated. Although fiber reinforced plastic composites have higher energy storage capacity than steel, they have, in general, poor resistance to shear stresses. The unique feature of the cylindrical construction of the composite spring is that the fibers are utilized in tension or bending instead of shear, thus avoiding the inherent weakness of a composite material in coil spring type applications. Study of the uniaxial elastic loading response of composite

Composite cylindrical springs

253

circular rings usually involves the solving of coupled partial differential equations and is thus difficult. 1 In this paper we show how a simpler but explicit analysis can be applied to the prediction of the uniaxial loading response of mid-plane symmetric laminated circular rings.

THEORETICAL CONSIDERATIONS

Constitutive equations For an anisotropic laminated cylindrical shell with n layers as shown in Fig. 1, the constitutive equations are

,1, where

k=l n

B,~=

12

5

(Q,A(h~ -

hL 1)

k=l n

12 (Q,y)k(h~ -- h~_ ,)

D,i = ~

k=l

or, by rearrangement,

=LC, ID,A(Mj where [A'] = [A*]

-- [ B * ] [ D * ]

- ~[C*]

[A*]

-- [ A ] - i

[B'] = [B*] [D*] - '

[B*] = -- [A] - ' [ B ]

[C'] = - [D*]

[C*]

[D'] = [D*] - '

- '[C*]

= [S] [A] - '

[D*] = [D] - [B] [A] - ' [B]

(2)

C. K. So, P. C. Tse, T. C. Lai, K. M. Young

254

Nx

Z

h

Fig. 1.

Mid-plane stress and moment resultants system.

Strain energy expression Strain energy U=

crqe;~d V =

Wd V

(3)

As a plane stress state is assumed for individual laminae, and we have [el =

[e°l + z[~] and [~]k = [ff]k([dl + zFK]), W reduces to

1

1

~

w=g~xex+5~o~o+Gxoexo= ([e°]+z[K])~[O]~([e°]+z[K])

(4)

H e n c e the strain energy density function for a cross-sectional area with width L o f a shell is

W' =

(5)

WL dz k-I k=l

As all matrices are independent o f z, n

w' =

~- (& _ & _ ,)[~o]~[Q]~[~o] + ~ (& _ h~_ , ) [ d ] ~ [ Q ] ~ [ ~ -] k=l

+ ~1 ( & 2 - h~_,)[~]~[Q]~[~°] + ~1( & 3 _ h~_,)[K]~[Q]~[~]

t .J

L [ e o ] r [ N ] + L [~c] r E M ] 2

(6)

Compositecylindricalsprings

255

Hence the strain energy for a segment from 0 t to 02 is

U = Jot ( "~[~°]r[N]

=

LR f "VNlTA'i lTNl 2 LMJ LC-9~J LMJ d0

(7)

,

Spring stiffness

In view of the symmetry of the shell, only one q u a d r a n t need be considered, as shown in Fig. 2. For a mid-plane symmetric laminate, [B] = [0], [D'] = [D] - ~ and M o = M/L, with all other stress and m o m e n t resultants equal to zero. U=

LR ['":~ ' r/:M'~dO 2 1o

R In/2DllD66-D26[MB-~(1-cosO)]2dO

i~f

=~j0

(8)

By Castigliano's Second Theorem, the bending moment, M, for any crosssection of a mid-plane symmetric laminated slender ring under uniaxial load F is

(1 1)

M = FR -~cos 0 -

(9)

The vertical displacement

OU RDxlD66_D~ 6 f]/2 OM v = O(F/2~)- L [DI .~o M~-~-/-~dO FR3DtlD66-D26(~ = T

IDI

1) ~- - ~-

(10)

Hence the spring stiffness K = (F/2)

v

L IDI

R3(DllD66 _D~6)(4 = (~2

2)

32rcL IDI __ 8)(D 1ID66 _ D~6)(2Ro + h)3

(11)

256

C. K. So, P. C. Tse, T. C. Lai, K. M. Young

" A

vv Lo

Fig. 2.

Forces and m o m e n t s on a quadrant.

If the same material is used for each lamina and each layer is orientated in its principal direction, [A] =

h[Q]

(12)

h3

[D] = i 2 [Q]

(13)

[e°] = ~[Q]-~[N] = ~[S][N]

(14)

12

12

It<] = ~ [Q] - ' [?v/] = ~-X [S] [ M ]

(15)

If we consider further that the properties in the 1 and 2 directions are the same, then from eqns (11) and (13),

8rr K

ElILh 3

3(rr 2 - 8)(2R o + h) 3

(16)

S t r e s s - s t r a i n distributions

F r o m eqns (9), (14) and (15), the stress and strain terms for a specially o r t h o t r o p i c laminate with the same properties for the 1 and 2 principal directions are as follows: [t] = [t o] + z [ x ] = [0] +

[S][M] =

~cos0-

[,7] = [Q] [~] = - Z - ~ ~5 cos 0 - X / [ o J

(17)

(18)

Composite cylindrical springs

257

EXPERIMENTAL TESTING All specimens were manufactured by circumferential winding of E-glass woven cloth impregated with Ciba-Geigy epoxy resin (MY750/HY956) on a PVC mandrel of 114-mm nominal diameter. After gelation at ambient temperature, the G R P tube was post-cured at 80°C for 24 h and cut into spring elements for compression tests and subsequent burn-off tests 2 for the determination of glass content. Tensile tests were also performed on rectangular strips of unreinforced epoxy resin according to the standards set by ASTM, 3 to determine the engineering properties of the resin. The elastic modulus and Poisson's ratio of fibres were taken to be 75-9 GPa and 0.22 respectively3 ,h, modified 'rule of mixtures '4 was used as a mathematical model to predict the engineering properties of the composite rings. Load 0~)

S~'ing Ho.9

5

Interlarnlnor crack i I

4

Intedaminar crack

C~hlng

I

2

I

0

0

A

m

m

l

t

I

I

5

10

t5

20

25

30

35

Vertical deflection (mm) Fig. 3. Typicalload--deflectiondiagram in compression tests. Figure 3 shows a typical load-deflection curve, indicating an initially linear portion from which the spring rates were calculated. As the load increases, the curve becomes non-linear, with the appearance ofinterlaminar shear cracks, until finally, crushing of the spring element occurs. The experimental and theoretical spring rates are tabulated in Table 1 for comparison purpose. Some spring elements were then compressively loaded to failure and Table 2 shows the value of the failure loads.

258

C. K. So, P. C. Tse, T. C. Lai, K. M. Young Spring stiffness K(N/u] tOO0 80O

4OO

2OO

0

I

I

I

I

I

0

5O

I0o

isO

2O0

Spring parameter E,,Lh31(2Ro+h)3 (Nlmm) Fig. 4.

Spring stiffness versus spring parameter. ( - ) Theoretical, ( + ) group A, (U]) group B, ( ~ ) group D and (/k) group E.

Equation (16) indicates that the spring stiffness is linearly proportional to a spring parameter (E l ~Lha)/((2Ro + h)a). Figure 4 shows the close agreement between the experimental data and the theoretical prediction and thus confirms the usefulness of the spring parameter in validating the experimental results relative to the theoretical prediction. Figure 5 shows a plot of the ratio of spring stiffness to modulus ratio as a function of the spring thickness. The stiffness to modulus ratio was TABLE 1 Experimental and Theoretical Spring Rates Group

Spring no.

Thickness h (mm)

Volume fraction Vr

Experimental spring rate (N/mm)

Theoretical spring rate (N/mm)

A

3 1 2

1"53 1"58 1'59

0"251 0 0"232 0 0"265 3

4"42 4"79 5.00

5-73 5.92 6'73

E

30 32

2-38 2.49

0-326 9 0'291 7

20.43 21"48

25"99 27'28

B

10 l1 9

5"30 5"33 5"38

0"349 1 0-349 9 0"349 7

274-90 259-90 263"80

280-96 276-05 294'87

D

29 23

7"59 8"20

0-308 4 0"313 4

70 !'40 903.70

704-93 887.60

C

16 20

12'16 12-18

0"3190 0-321 3

2243-60 2 500-00

2671-96 2 768"49

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259

TABLE 2 Compression Test Data

Group

Spring no.

Thickness h (mm)

Width L (ram)

Weight (g)

Volume fraction

Experimental spring rate (N/mm)

Failure load (N)

1° 2° 3" 4 5 6 7

1.58 _+ 0"18 1"59 -I- 0-14 1"53-1-0"17 1"55+0'16 1-39 +__0-22 1'35 + 0"21 1.34 +_ 0-24

50-76 51'12 51'17 51"16 53-73 51'68 52-06

46"7 48-1 46-3 47.0 49.4 48"0 45-3

0'232 0 0'265 3 0"2510 -----

4-79 5"00 4"42 -6'34 5"56 --

315 330 230 -400 400 --

B

8 9 10 11 12 13 14

5'48 + 0"20 5"38+0-21 5-30 +_ 0"21 5"33+0"23 5"48 + 0'23 5.37 + 0-20 5.28 + 0.22

50"85 51"14 51"02 49"23 50-63 50-84 50-36

172"3 171.4 168-7 164-0 170-0 170.8 167.0

-0-3497 0'349 1 0-3499 ----

259'60 263"80 274-90 259"90 -308"90 299"30

5 250 4880 4 600 4750 ----

C

16 17 18 19 20 21 22

12-16+0-62 12"20 + 0-50 12"11 + 0"68 12'13+0"49 12" 18 +__0'67 12'14+0"67 12"12 +__0"40

51-08 50"92 50"71 50-91 52"44 51.64 50"77

390'2 389'3 385-5 390-5 403-5 400-0 389-5

0"3190 ---0"321 3 ---

2243"60 2 433-20 2 399"00 2433"10 2 500"00 2451'00 --

19000 --18600 19 700 16550 --

D

23 24 25 26 27 28 29

8"20 7-65 8"04 8"08 8'02 7"89 7'59

+ 0'40 + 0"45 + 0"41 + 0-52 + 0'35 + 0-39 ___ 1'41

51 "04 51-16 50-85 51'06 50-82 50"65 51-13

243"8 237"4 245"8 246-0 247-5 243-0 242"0

0-313 4 -----0"3084

903'70 705"60 866"30 905-80 905'80 -701"40

9 840 -------

E

30 31 32 33 34 35 36

2"38 +__0"28 2'30+0"31 2"49 + 0"30 2"23 + 0"33 2'27 ___0'26 2"50 ___0"32 2"51 ___0"34

51"31 51'25 51-37 51'34 51"53 51"21 51"31

76-7 72"6 70-5 74-8 73-0 77"3 74-7

0'3269 -0"291 7 -----

20"43 22"64 21-48 19"47 19"18 21'83 --

910 1 120 1 110 990 950 980 --

A

° N o p o s t curing. F o r all o t h e r springs, p o s t c u r i n g w a s p e r f o r m e d at 80°C for 24 h.

260

C. K. So, P. C. Tse, T. C. Lai, K. M. Young

0.3

Stiffness over Young'smdulus K/E,=(ml

0,20

0.t0

~'~-'--"'-~

I

0.00

I

I

I

I

I

I

2

4

6

8

I0

12

14

Spring thickness h (mm) Fig. 5. Spring stiffness to modulus ratio versus sp6n8 thickness. ( - ) Theoretical, (+ group A, (0) group B, ( x ) group C, (O) group D and (/X) group E.

Strain (1000ue)

Exl~fimental results

/

//Y.~

Theoretical results

-6 "

//,~./~ f l c ~

I

I

I

I

I

I

I

I

I

I

0

I0

20

30

40

50

60

70

B0

90

Angle (deg) Fig. 6.

Strain distributions for the inner surface. Load of ( I ) 50 N, (+) 100 N, (~) 150 N, ([-1) 200N, (x) 250N and (O) 300N.

Composite cylindrical springs

261

Strain (lO00ue}

6 ~

IL-.----

-4

Experimental results

-5

Theoretical results

\x~, ~

-6 -7 -8 -9

~-

\

\ \

I

I

i

I

I

i

i

I

i

0

I0

20

30

40

50

60

70

80

\ \ ",.

90

Angle (deg) Fig. 7.

Strain distributions for the outer surface. Load of ( I ) 50 N, ( + ) I00 N. ( ~ ) 150 N, (l-l) 200N, ( x ) 250N and ( ~ ) 300N.

employed to rationalize the base for comparison purpose because the Young's modulus of the test specimens could not be kept constant. A similar trend is expected for the relationship between the spring stiffness and the spring thickness. Electrical resistance foil strain gauges were mounted on the inside and outside surfaces of spring No. 30 to measure the strain distributions in the circumferential direction at various locations in uniaxial compression test. Figures 6 and 7 show the theoretical and experimental strain distributions in the circumferential direction for the inner and outer surfaces of the spring subjected to different uniaxial loadings.

DISCUSSION By rearranging the energy expression for a general laminate developed in the theory section, we have [" l [ t ; x ] r [ o ' x ]

ood

1~ ] ( 130 1) fT N

,19,

where ( is the surface area of the mid-plane. This energy expression thus demonstrates the fact that the strain energy of an elastic laminated shell is

262

C. K. So, P. C. Tse, T. C. Lai, K. M. Young

equal either to half the algebraic sum of the product of the stress and strain terms over the whole volume, or to half the algebraic sum of the product of the mid-plane strain and curvature terms multiplied by the mid-plane stress and moment resultant terms over the whole mid-plane surface. The strain energy expression of an elastic laminated shell is therefore subjected to the same rules as the stress and m o m e n t resultants. As the elastic properties of the circular rings were estimated based on the modified rule of mixtures, Table 1 shows the theoretical spring rates to be generally higher than the experimental values. This discrepancy can be attributed to the fact that the engineering properties estimated by the modified rule of mixtures also tend to be higher than the actual values. For the strain distributions, basically the general trends for the experimental results are consistent with those theoretical results and the discrepancies are attributed to the approximation of elastic properties. For both the theoretical and experimental results, load-independent locations were observed. Theoretically, all stress and strain terms will be independent of the load and equal to zero when 0 is equal to _+50.5 ° or 180 ~_+ 50.5 °, or when z is equal to zero (i.e. the mid-plane). These load independent locations are of great importance to the development of the composite spring as its performance will not be affected by fixing attachments such as highdamping elastomer 5 at these locations. CONCLUSIONS A theoretical analysis of the composite cylindrical spring based on the principle of minimum potential energy has been presented and is validated by experimental data obtained for specially orthotropic composite springs. A spring parameter, which is linearly proportional to the spring stiffness, is found to be useful as a normalized bases for accessing the experimental results. Strain distributions indicated the existence of load-independent points on both the inner and outer surfaces of the shell. These points free of stress and strain are of great interest for the development of the composite spring and can be used for fixing additional accessories such as high-damping elastomer, etc. ACKNOWLEDGMENTS This project was supported by the Hong Kong Polytechnic research fund 340/370/A3/430. The authors thank the Department of Mechanical and Marine Engineering, Hong Kong Polytechnic, for the facilities provided in connection with the experimental work.

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263

REFERENCES I. Halpin, J. C., Primer on Composite Materials: Analysis. Technomic Publishing Co., Lancaster, PA, 1984. 2. British Standards Institution, Glass reinforced plastics--determination of loss on ignition. BS 2782 Part I0. BSI, London, 1977. 3. American Society for Testing of Materials, Standard test method for tensile properties of fibre-resin composite. Standard Method D3039. ASTM, Philadelphia, 1989. 4. Pilkington Reinforcement Limited, Fibreglass composites design data. 1985. 5. Scowen, G. & Hughes, D., The sulcated spring. Paper presented at International Seminar. Autotech 85 Congress, The Institution of Mechanical Engineers, Automobile Division, Birmingham, UK, November 1985.