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Large deflection of elastic composite circular springs under uniaxial compression
Pergamon
hr. J. Non-Linear Mechanics,
Vol. 29, No. 5, pp. 781-798, 1994 Elwier Science Ltd Printed in Great Britain OOZO-7462194 $7.00 + 0.00
0020-7462(94)EOOOST
LARGE DEFLECTION OF ELASTIC COMPOSITE CIRCULAR SPRINGS UNDER UNIAXIAL COMPRESSION P. C. Tse, T. C. Lai, C. K. So and C. M. Cheng Department of Mechanical Engineering, Hong Kong Polytechnic, Hunghom, Kowloon, Hong Kong (Received 28 October 1992; in revised form 8 January 1994) Abstract-The equivalent flexural rigidity approach for large deflection analysis has been applied to orthotropic midplane symmetric laminated circular rings under uniaxial compression. The analytical approach incorporates the exact expression of the moment+urvature relationship for the bending of symmetric laminated beams. Non-linear spring behaviour is obtained and verified by experimental testings. The circumferential strain distributions around the rings are compared with that obtained by a small deflection strain energy approach and found to provide a more accurate prediction, especially at high loads where the deflection is large. Finite element calculations performed with some of the ring models show good correlation with both analytical and experimental results and the distribution of circumferential strains at increased external load is illustrated.
NOMENCLATURE
CA1
CA’1
CBI
WI CC’1 PDI CD’1 E 11 E
$34) E(P) F(P, F(P) h
4)
CM1 CNI ii u, v, w tl
CEO1
4 Cr”1 8 P
Cl/PI
inplane stiffness matrix inplane compliance matrix coupling stiffness matrix coupling compliance matrix coupling compliance matrix equivalent flexural rigidity per unit width of the shell flexural stiffness matrix flexural compliance matrix Young’s modulus in the major principal direction Young’s modulus in the minor principal direction Legendre’s incomplete elliptic integral of the second kind Legendre’s complete elliptic integral of the second kind Legendre’s incomplete elliptic integral of the first kind Legendre’s complete elliptic integral of the first kind radius of the construction circle for the elastica curves moment resultants stress resultants modulus governing the shape of the elastic line radius of the midplane of the undeformed shell displacements in the x, y, z directions slope of midplane with respect to the z coordinate axis midplane strains limit parameter of integration in the elliptic integrals; parameter of the coordinates of the elastic line midplane shear strains angle of the slope of the elastic line radius of curvature middle surface curvatures INTRODUCTION
Glass reinforced plastic has been widely used for secondary structures of aircraft and vehicles. Recently, the marked excellence in high strength-to-weight ratio, as well as the feasibility of optimum material design, has led to extensive research activities in exploiting NLM 29:5-J
781
P. f. Tse er al.
782
the possibility of proceeding with the application of composite materials up to the primary structure and load bearing components of vehicles [l]. One of these components is the automotive suspension spring. In order to utihse the high tensile strength of the hbre reinforced plastic in spring systems, which must function in the same vertical deflection mode and within the same space of a steel coil spring, the “Sulcated Spring” was introduced and test data showed it to be a very promising concept [2]. At the same time, composite elliptic springs were also being studied [3]. Several elliptic spring elements constructed from unidirectional E-glass fibre reinforced epoxy tapes were mounted in series and tested in static compression. It was observed that the primary failure was due to interfaminar shear [4]_ An analytical study using the energy method to evaluate the spring constants of elliptic rings was undertaken [S]. Spring constants in the directions of the principal axes and for bending-shear and bending-torsion were measured for elliptic rings made of carbon fibre reinforced thermoplastics. Good agreement was obtained between the experimental and the predicted results. More recently, the mechanical behaviour of composite cylindrical springs with midpfane symmetry has been investigated [6]. The theory was developed based on a linear-elastic assumption and the stress-strain distributions around the ring were evaluated. Small deformations were accurately predicted. However, for suspension applications, a large deflection analysis for the spring system is necessary. The development of non-linear springs called upon the large deflection analysis originating from the study of cantilever beams carrying a concentrated load utilising elliptic integrals [7, S]. The basic assumption is that the nodal fine is completely elastic and inextensible thus, albeit that the relationship between force and deflection is non-linear, the stress-strain relationship is still linear. The problem was later solved with the free end load acting in any direction [9J. A similar analysis for initially circular rods was carried out [lo]. The entire problem was eventually tackled in a general manner, whereby not only the free end displacement of a cantilever or a circular rod caused by several concentrated loads was found, but that the actual defIected shape could also be determined [ 1il. Using the results of large deflections of a flexible bar, the deformation of an elastic circular ring subjected to two opposite radial loads was subsequently investigated [12]. The approach is now extended with the development of an equivalent Bernoulli-Euler equation for a laminated composite beam leading onto the exploration of the non-linear load-detection and stiffness characteristics of composite circular rings. Strain distributions along the circumference can then be evaluated. Such information will be essential for the design of composite circular spring systems. EQUIVALENT
FLEXURAL
RIGIDITY
OF ORTHOTROPIC
LAMINATED
BEAMS
Consider a section of laminate in the X-Z plane which is deformed due to some loading, as shown in Fig. 1. Let the x-y plane coincide with the midplane of the beam with the z-axis oriented in the thickness direction, Neglecting shear deformation, displacement u, in the x-direction for an arbitrary point C at a distance z, from the midplane is M, = ug - z,sina. In general, 63W
where n0 and oe are midplane displacements in the x and 9_’ directions and z’ is the distance from the midplane.
Deflection
783
of circular springs
Fig. 1. Bending geometry in the x-z plane.
The strain-displacement
equations can now to be written as
(4)
where
[ 1
-_= PX
l+
-- a2w ax2 01aw
2
312
ax
l[ 01 aw azw a3
-_= PY
2
I+
312
ay
% II
With this strain~isplacement
relationship, the constitutive equations can be written as r
0
0 &Y 0 YXY VPX UP,
l/P,,
=
k
(5)
784
P. C. Tse et
ui
For midplane symmetric laminate, [B] = [0] and therefore [Cl’ = - CD]-’ {[B-J [A]-‘l
= [O].
Hence we have 1
ii
P
= [D’]
CM).
For the bending of any midplane symmetric laminated straight beam of width L by a pure bending moment h4 in the x-z plane, we have 1
M
L
PX
L
D;IP,’
-ED;,-+=>==
If we put IDI
D=
DZZD66 - D&i
=-
I WI
9
then M=qD
(9)
Px ’ LD is designated the “equivalent flexural rigidity”. For specially orthotropic
RING
IN COMPRESSION
BY POINT
material:
LOAD
Consider a symmetrical laminated ring to be acted upon by radial compressive forces as shown in Fig. 2. In view of the symmetry of the ring, only one quadrant is required for the analysis. The bending of this quadrant of arc by force P is equivalent to the analysis of the large deformation of a straight strut of length i = R71/2 and subjected to a vertical load P and bending moment MR = LD/R simultaneously [13]. Initially, P is small, the eccentricity e in Fig. 3 will be very large to maintain the bending moment MR. The deformed curve AR will lie on a nodal elastica ABC as the load P cannot act directly on the beam. Equation (9) now takes the form
I?
t p’=2P
1 B
0 R
I 3
3
3
MI<
A
/
t
1P’
I
= 2P
Fig. 2. Equivalent
analysis
for the ring in compression.
Lkfiection of circular springs
785
Fig. 3. Nodal elastica curve for the ciefonned beam.
and
where k=
J&.
Equation (IO) is integrated with the boundary condition of B = 0 at the fixed end A, this leads to the expression z = h cos Cp[93.
STAGE
1 COMPRESSION
In the initial stage of compression the parameter d, satisfies the relation sin (0/2) = (l/p) sin # and p is a variable less than unity defined by the relation hZ = @I/p2k*). The arc length of the beam AB is [9f
i.e.,
.
786
P. C. Tse et al.
The semi-minor
axis is
~=.,=,=,[E(,,~)-(l-~),(,.~)I
=#P,;)-(1 -;)F(P,;)]. The deflection
of the load P from the original
quadrant
of arc to point
(13) B is
6=-‘=R-;[P(p,$-(I-;)F(p,;)] (14)
The force to produce
this deflection
is p,; ( ) R2n2 ’
4p2F2 P = LDk’ Since both the deflection K = dp = dpldp d6 dS/dp
j
= LD
6 and the force P are functions
of p, the spring
(15) stiffness is
K = 8p4LD R3d
2E PQ F P,; -PE P,: F p,; +pE p,; F P,~ -2F ( “) ( “) I( “) ( “) ( “) I( “)
2(
P,~ “) 1’
(16)
where
The deflection 6 and the load P in the above analysis refer to the quadrant arc only and are equal to half the actual values. By transforming these into the actual deflection and load 6’ and P’ of the system and grouping equations (14)-(16) the dimensionless deflection, load and stiffness are, respectively,
(19)
P’R2 __ = LD
(20)
[F4(~~~)+~F3(~,$o,~)] %=F,
( “) ( “) 2.5 P>; F P>; -pE
,(
“) ( “) ( “) ,( “) P,~ F P,: +pE P,~ F P,~ -2F
2(
“)]’ p,4
(21)
Deflection
of circular
springs
78-l
It can be observed that the variable p is a very important parameter that determines the deflection, load and stiffness of the springs. The value of p starts from zero and increases with the load P. When p = 0, the dimensionless stiffness term approaches a value of 6.7215. When p = 1, the deformed beam will become an infinite beam and P will eventually touch the extension of the nodal elastica. This represents the end of the applicability of the nodal elastica in the compressive range (stage 1 compression). The corresponding values of P and P’ are P=
= 0.3148 $
R2R2
P’ = 0.6296
STAGE
$.
(22)
2 COMPRESSION
For P greater than the value indicated in (22), the force will act on the imaginary extension of the elastica line AB (stage 2 compression). The deformed curve AB will lie on an undulating elastica ABC, as shown in Fig. 4; 4 now satisfies the relation sin (e/2) = p sin 4 and h2 = (4p2/k2). Thus at & = (7r/2), & = sin-’ (sin (~/4)/p) and the arc length of the beam AB is
Rn ‘A’ = 2
1 4s = k o (1 _ s = kF[p,
p2
sin2
~)1/2
=
_
1
sin-ii-)]
PJZ
(23)
’
i.e., (24)
I
IFig. 4. Undulating
elastica
curve for the deformed
beam.
P. c. Tse
7X8
Cl
al.
The semi-minor axis is
WP> &I) -___ F(P,A?) k
b=XJ.$,,9B=
k
= i {2b[P,sin-‘($j]
- F[P,sin-I(&)]}.
(25)
The deflection of the point B from the original quadrant of arc is
The force to produce this deflection is
4Fz~p,sin-l(~)] P=LD
f27f
R2d
The spring stj~~ess is
(1
-
-p sin2 (p dd,p2 sin2f$)“*
Grouping (26)--(28) into dimensionless forms, we have
J-LD
1
P J473
(29)
Deflection of circular springs
KR3 -=LD
789
8 d
(33) With P increasing, it is obvious that P will act on B eventually. When this happens, the moment at B is equal to zero and the presence of a couple is not necessary to secure a horizontal end tangent at B, this marks the end of stage 2 compression. To obtain the arc length AB, the limits of integration must be chosen such that 4 sweeps the quadrant from OA to OB',i.e. dB = 7712 CAB
=
F =; *‘*
I o
df
(1 - p2 sm* +)‘I2
i.e.,
k=&F&
(35)
() * The force to produce this deflection is
P=LD
4F2 5 _ () R27C2 - 1.3932s
P'= 2.7864LD R2'
STAGE
(36)
3 COMPRESSION
The zero bending moment at B implies that the middle surface curvature l/p will be equal to zero as well. At this point, a further increase in P will mean a negative curvature at B and hence the quadrant of arc will have a point of contraflexure at C, as shown in Fig. 5 (stage 3 compression). Making use of the results obtained for stage 2, the arc length of the beam AB in stage 3 is [12]
=1{2FW k where 4B = sin-‘(l/P*),
Fb,hJ) (37)
since OB= 7t/2 and
k=
&-2F(d -
F(P, 4~) .
(38)
The semi-minor axis is -
I[
2&4 4~) F(p, 44 k
-7’
1
(39)
P. C. Tse ef (11.
790
Fig. 5. Deformed
The deflection
of the point
beam with point of contraflexure.
B in stage 3 is R?T
6=R-
$41
2C2F(p) - F(Pt
The force to produce
this deflection
C4HP) - 2fYP) - WP> 6J) + F(P,
KC-
(40)
is - 0~~ ui* R27C2
P = LD 4c2~(~)
The spring
4B)l~
(41)
stiffness is
8LD R37t3 CWP)
x CUE
-
E(P,
ddl
W”(P)
-
dd13 W’(P)
F(P,
F’(P, 4~)1-
-
W(P)
-
F’(P?
ddl
F(P, +B)I CUE’
-
E’(P, 4~11’ (42)
where E’(p, 4B) and F’(p, c#J~)are as defined a:2
E’(P) =
s0 n/2
F’(P) = Grouping
(40)-(42)
into the required
s0
by equations
(29) and (30) and
-psin’b (1 - p*sin*4)“’
d$
(43)
d4.
(44)
p sin’ I#J (1 - p* sin* 4)“’
dimensionless 0, /Fs
forms &J) - 4E(p) + 2E(p, &) (45)
791
Deflection of circular springs
(46)
RING
IN COMPRESSION
BETWEEN
PLATES
the ring is being compressed by load P’ = 2P between two horizontal plates, the first and second stages of compression of the ring by a point load are still applicable as the plates will touch the ring tangentially at one point only. For the stage 3 compression, the deflected shape of the quadrant will consist of a straight portion of length t in contact with the horizontal flat plates and a curved part of R7t/2 - C,as shown in Fig, 6. The arc length for the curved section section AG is When
t-m= !ct=t 2
s
UP
d4-j
k o (1 - pZsinZ #)lfZ
k
i.e.,
The semi-minor axis is
Defiection of the point B from the original quadrant of the arc is
*_R-&+E(-j=)-F(j=)]v
i-
-i-
Fig. 6. Deformed beam for the stage 3 compression between plates.
(51)
P. C. Tse et al.
792
The force to produce this deflection is P > 1.3932 $
P’ > 2.7864 $.
(52)
The spring stiffness is (53)
Transforming
6 and P into 6’ and P’ to form the dimensionless deflection and stiffness:
(54)
KR3
/P’R2\3/2
1
STRESS-STRAIN
(55)
DISTRIBUTIONS
Consider an initially curved beam, as shown in Fig. 7, with the assumptions of neglecting shearing deformation and the length of the midplane AF remains unchanged after the bending moment is applied. Take a plane CE at a distance z, from the midplane. Strain for the arc CE is ECE
=
(P2
+
Zdaz (Pl
+
(PI
+
Z&l
Zc)al
=
Z&l Pz(P1
+
P2)
(56)
Zc)’
Since z, is small when compared with pr, we have (57)
Z
Z
A tier
Fig. 7. Bending
of initially
curved
beam.
Deflection
of circular
springs
793
In general, the expression for the circumferential strain of any surface at a distance z’ from the midplane will be
For specially orthotropic laminated rings with the same material properties in the principal directions, the shear strain is zero while the transverse strain is due to Poisson’s effect of the circumferential strain. Thus the complete picture of the strain distributions can be obtained once the circumferential strains have been detected. Substituting into (58), the corresponding expressions z = h cos 4, the coordinate of the deflecting point and the applied load P for the various stages of compression, the circumferential strains are obtained.
Stage 1 compression
E =< 4pF p,sin-’ R( [ II (,x)]jT
11
Stage 2 compression
(59)
(60)
4p[zW - f’[p, sin-‘( $)]]dq P2
R
Stage 3 compression by point load
(61)
Stage 3 compression between plates for the curved section
(62)
I
E= - i
Stage 3 compression between plates for the flat portion.
DISCUSSION
OF
(63)
RESULTS
Quasi-static compression tests were performed on circular composite springs fabricated by E-glass woven cloth/Ciba-Geigy epoxy resin (XH750A/HY956) with nominal width and Table Spring no.
1. Experimental Thickness
spring
stiffness
Experimental
spring stiffness
H (mm)
K W/mm)
12.13 7.65 5.48 2.38 2.30 1.58 1.00 0.89
2433.10 705.60 259.60 20.43 22.64 4.19 0.94 0.99
794
P. C. Tse et ol.
internal diameter of 5 1 mm and 114 mm respectively. The spring stiffnesses, as tabulated in Table 1, are determined from the initially linear portion of their respective load-deflection curves. By substituting these experimental spring stiffnesses into the expression derived in reference [6], the Young’s modulus and, hence, the equivalent flexural rigidity of the test specimen are obtained. Two predictions were compared with experimental results for specially orthotropic midplane symmetric laminated springs in compression, The first was calculated by the large deflection analysis and the second by the general purpose finite element program ANSYS [ 143. Eight-node, isoparametric axisymmetric quadrilateral elements with compressive line loading to simulate the actual testing situation were employed in the finite clement analysis. The theoretical results as shown in Figs 8 and 9 are computed by Simpson’s numerical integration algorithm. The comparison between the characteristics of the springs in compression by point load to that between flat plates are also shown. The first two stages of compression are common to both types of loading and deviation begins at stage 3 onIy. The first stage of compression extends to the range where the value of the dimensionless load P'R'/LDis around 0.63, while for the second stage is around 2.79. The springs exhibit non-linear behaviour and the dimensionless load-deflection curve is only linear over a very small range of initial deflection. It should also be pointed out that the dimensionless deflection 6'lRis 2 when the dimensionless load P'R21LDequals about 8.04 for point load compression, This corresponds to the case where the top and bottom surfaces of the ring are brought into contact. However, this will not occur for compression between plates, as
Fig. 8. Theoretical
dimensionless
load~eflection
characteristic
for rings in compression
Compressron betweenplates 15 -
Compressmnby pomt load 0 0
Fig. 9. Theoretical
1
2
dimensionless
3
6
load-stiffness
characteristic
7
8
9
for rings in compression.
795
Deflection of circular springs
fracture by crushing will obviously take up the failure process well in advance. By taking the limit of p approaching zero for the dimensionless stiffness term in stage 1 compression, the spring stiffness for a specially orthotropic midplane symmetric laminated spring is K = 6.7215 $
= 0.5061=$$
This spring stiffness expression is comparable to that being found by the strain energy approach [6] and that being employed for estimating the elastic modulus of filament wound ring [15]. The present analysis is thus consistent with the linear elastic small deformation theory for the initial stage of compression. The stiffness of the springs decreases with the deflection and then increases for values of P'R',/LD at around 4.7 well into stage 3 for compression by point load and at the junction of the second and third stages of compression between plates. The springs compressed by a point load are softer than those between plates, since the stiffness is much lower for the former during stage 3 compression. Figures 10 and 11 show the theoretical, finite element and experimental dimensionless load-deflection and stiffness characteristics of composite springs compressed between flat
0
0.5
I
I
I
I
I
1
1.5
2
2.5
3
P’R*iLD Fig. 10. Dimensionless Ioad-deflection characteristic of ring in compression between horizontal flat plates.
KR3/LD 8j
2t
-
Largedsnscfion theory 0 springA
A
. SpringE
0.5
SpringA FEM SpringB
-. . SpringF FEM 0 Springc
a .spnngF
0
0
--
1
I 1.5
2
2.5
I 3
P!?fLD Fig. 11. Dimensionless load-stiffness characteristic of ring in compression between horizontal flat plates.
796
P. C. Tse et al.
plates. As most of the thick springs fractured within the initial part of the second stage of compression and thus failed to reach the third stage, relatively thin specimens are selected to verify the dimensionless load-deflection and stiffness characteristics of rings compressed by point load, as shown in Figs 12 and 13 respectively. Good agreement exists among the theoretical, finite element and experimental results for both stage 1 and 2 compression under both sets of loading conditions. Non-linear effect for these two stages is progressive and weak and the spring stiffness decreases with the deflection and hence exhibits soft-spring characteristics. Discrepancies are generally acceptable, except for very high loads when plastic deformation has already taken place. From the dimensionless load-stiffness characteristic, the stiffness decreases initially with the deflection and increases slightly afterwards. This phenomenon occurred for both loading conditions. As the high load region is difficult to reach in practice, the springs are still classified as soft. The circumferential strains obtained by the present analysis for the specially orthotropic midplane symmetric laminated ring are compared with the experimental results; calculations from the strain energy approach and from finite element analysis. They are shown in Figs 14 and 15 for the inner and outer surfaces respectively. The theoretical
1
0
0.5
1
1.5
2
2.5
3
P’R?LD
Fig. 12. Dimensionless load-deflection
0
0.5
t
characteristic of ring in compression by point load.
1.5
2
2.5
3
P’R*/LD
Fig. 13. Dimensionless load-stiffness characteristic of ring in compression by point load.
Deflection _
Strain (1000 ps)
Spring
., +,I,
10
797
springs
D
0, X, 0 Expcrimentalrcsults Strain energy theoretical result.3 Large deflection theoretical rewits Fink element prediction (line loading)
---
0
of circular
20
30
40
50
60
70
_:.
80
90
Angle 0 (degree) Fig. 14. Circumferential
strain
distributions for the inner surface. (0) 2OON,(x) 250Nand(O)
Spring
Strain (1000 pe)
Load of (H ) 50 N, ( +) 100 N, (*) 300N.
150 N,
D
4
., +, D,O,
-4
x, 0 Expsimentalresults Strain energy theoretical ~CSUI~ Large deflection theoretical results Finite element prediction (line load&
---
-6 -8
0
10
20
30
40
50
60
70
80
90
Angle t3 (degree) Fig. 15. Circumferential
strain
distributions for the outer surface. Load of (m) 50 N, (Cl) 200 N, ( x ) 250 N and (0) 300 N.
( +) 100 N, (*) 150 N,
circumferential strain distributions around the ring predicted by the present analysis and the finite element predictions are in good consistence with those predicted by the strain energy approach. However, a closer examination of Figs 14 and 15 reveals that the large deflection theories are able to provide more accurate predictions, especially at high loads where the deflection of the spring is large. The realistic modelling of line loading at the 90” position of the composite ring in the finite element analysis, as in actual testing, provides a better prediction of strain variation, especially in the loading position. Strain-free locations were also observed on both the inner and outer surfaces of the ring by the present analysis. These strain independent locations are of great potential to the future development of the composite springs, as they can be used to facilitate attachment of additional accessories or discontinuities, such as holes or fitments. NLM 29:5-K
798
P. C. Tse et al. CONCLUSIONS
A theoretical analysis of the bending of symmetric laminated composite rings by considering the exact expression of the moment-curvature relationship for beams based on the equivalent flexural rigidity is presented. Large deflection study has been conducted and the deformed shapes of the springs are classified into stages according to the load and are governed by the theories of nodal and undulating elastica. The springs exhibit non-linear behaviour and the dimensionless load-deflection curve is only linear over a very small range of initial deflection with the dimensionless load values below about 0.5. This analysis provides basic information for the design of composite springs. The circumferential strain distributions obtained through the present analysis resemble those from the small deflection strain energy approach, with the added capability of providing more accurate prediction at high loads. Tests on composite rings have shown that the use of present large deflection analysis or finite element analysis to predict mechanical behaviour of composite rings in the third or final stage can be misleading without considering plastic deformation. Composite material with improved toughness and strength are thus needed to verify the final stage of deflection. Acknowledgments-This work was carried 0341/040/430 and 0341/084/430.
out with the support
of the Hong
Kong
Polytechnic
research
funds
REFERENCES 1. G. D. Scowen, Transport applications for libre reinforced composites. IMechE C49/86, 245-255 (1986). 2. G. D. Scowen and D. Hughes, The sulcated spring. Int. Seminar, Autotech 85 Congress, The Institution of Mechanical Engineers, Automobile Division, Nov 1985. 3. P. K. Mallick, Design and development of composite elliptic springs for automotive suspensions. 40th Annual Co& Reinforced Plastics/Composites Institute, The Society of the Plastics Industry, Inc. 28 January-l February, 1985. of composite elliptic springs. ASME J. Engng Mater. Technol. 4. P. K. Mallick, Static mechanical performance 109,22-26 (1987). T. Nakakura and H. Sakai, Spring constants of elliptic rings made of carbon5. T. Akasaka, M. Masutani, fiber-reinforced thermonlastics. Proc. 33rd Int. SAMPE Svmo.. 7-10 March. 1988. 670-680. 6. C. K. So, P. C. Tse, T. 6. Lai and K. M. Young, Static mechanical behaviour of composite cylindrical springs. Composite Sci. Technol. 40(3), 251-263 (1991). 7. M. J. Barten, On the deflection of a cantilever beam. Quarterly J. Applied Mathematics 2, 168-171 (1944). 8. K. E. Bisshopp and D. C. Drucker, Large deflection of cantilever beams. Quarterly J. Applied Mathematics 3, 272-275 (1945). 9. R. Frisch-Fay, On large deflections. Australian J. Applied Sci. 10(4),418-432 (1959). 10. H. D. Conway, The non-linear bending of thin circular rods. ASME J. Applied Mechanics 23, 7-10 (1956). bending of thin rods. ASME J. Applied Mechanics 26, 40-43 (1959). 11. T. P. Mitchell, The non-linear 12. R. Frisch-Fay, The deformation of elastic circular rings. Australian J. Applied Sci. 2(3), 329-340 (1960). 13. R. Frisch-Fay, Flexible Bars, Butterworths, London (1962). 14. ANSYS Engineering Analysis System User Manual, version 4.3. Swanson Analysis System Inc. (1987). 15. D. V. Rosato and C. S. Grove Jr, Filament Winding: its Deuelopmenf Manufacture, Applicarions and Design. Interscience Publishers, New York (1964).
hr. J. Non-Linear Mechanics,
Vol. 29, No. 5, pp. 781-798, 1994 Elwier Science Ltd Printed in Great Britain OOZO-7462194 $7.00 + 0.00
0020-7462(94)EOOOST
LARGE DEFLECTION OF ELASTIC COMPOSITE CIRCULAR SPRINGS UNDER UNIAXIAL COMPRESSION P. C. Tse, T. C. Lai, C. K. So and C. M. Cheng Department of Mechanical Engineering, Hong Kong Polytechnic, Hunghom, Kowloon, Hong Kong (Received 28 October 1992; in revised form 8 January 1994) Abstract-The equivalent flexural rigidity approach for large deflection analysis has been applied to orthotropic midplane symmetric laminated circular rings under uniaxial compression. The analytical approach incorporates the exact expression of the moment+urvature relationship for the bending of symmetric laminated beams. Non-linear spring behaviour is obtained and verified by experimental testings. The circumferential strain distributions around the rings are compared with that obtained by a small deflection strain energy approach and found to provide a more accurate prediction, especially at high loads where the deflection is large. Finite element calculations performed with some of the ring models show good correlation with both analytical and experimental results and the distribution of circumferential strains at increased external load is illustrated.
NOMENCLATURE
CA1
CA’1
CBI
WI CC’1 PDI CD’1 E 11 E
$34) E(P) F(P, F(P) h
4)
CM1 CNI ii u, v, w tl
CEO1
4 Cr”1 8 P
Cl/PI
inplane stiffness matrix inplane compliance matrix coupling stiffness matrix coupling compliance matrix coupling compliance matrix equivalent flexural rigidity per unit width of the shell flexural stiffness matrix flexural compliance matrix Young’s modulus in the major principal direction Young’s modulus in the minor principal direction Legendre’s incomplete elliptic integral of the second kind Legendre’s complete elliptic integral of the second kind Legendre’s incomplete elliptic integral of the first kind Legendre’s complete elliptic integral of the first kind radius of the construction circle for the elastica curves moment resultants stress resultants modulus governing the shape of the elastic line radius of the midplane of the undeformed shell displacements in the x, y, z directions slope of midplane with respect to the z coordinate axis midplane strains limit parameter of integration in the elliptic integrals; parameter of the coordinates of the elastic line midplane shear strains angle of the slope of the elastic line radius of curvature middle surface curvatures INTRODUCTION
Glass reinforced plastic has been widely used for secondary structures of aircraft and vehicles. Recently, the marked excellence in high strength-to-weight ratio, as well as the feasibility of optimum material design, has led to extensive research activities in exploiting NLM 29:5-J
781
P. f. Tse er al.
782
the possibility of proceeding with the application of composite materials up to the primary structure and load bearing components of vehicles [l]. One of these components is the automotive suspension spring. In order to utihse the high tensile strength of the hbre reinforced plastic in spring systems, which must function in the same vertical deflection mode and within the same space of a steel coil spring, the “Sulcated Spring” was introduced and test data showed it to be a very promising concept [2]. At the same time, composite elliptic springs were also being studied [3]. Several elliptic spring elements constructed from unidirectional E-glass fibre reinforced epoxy tapes were mounted in series and tested in static compression. It was observed that the primary failure was due to interfaminar shear [4]_ An analytical study using the energy method to evaluate the spring constants of elliptic rings was undertaken [S]. Spring constants in the directions of the principal axes and for bending-shear and bending-torsion were measured for elliptic rings made of carbon fibre reinforced thermoplastics. Good agreement was obtained between the experimental and the predicted results. More recently, the mechanical behaviour of composite cylindrical springs with midpfane symmetry has been investigated [6]. The theory was developed based on a linear-elastic assumption and the stress-strain distributions around the ring were evaluated. Small deformations were accurately predicted. However, for suspension applications, a large deflection analysis for the spring system is necessary. The development of non-linear springs called upon the large deflection analysis originating from the study of cantilever beams carrying a concentrated load utilising elliptic integrals [7, S]. The basic assumption is that the nodal fine is completely elastic and inextensible thus, albeit that the relationship between force and deflection is non-linear, the stress-strain relationship is still linear. The problem was later solved with the free end load acting in any direction [9J. A similar analysis for initially circular rods was carried out [lo]. The entire problem was eventually tackled in a general manner, whereby not only the free end displacement of a cantilever or a circular rod caused by several concentrated loads was found, but that the actual defIected shape could also be determined [ 1il. Using the results of large deflections of a flexible bar, the deformation of an elastic circular ring subjected to two opposite radial loads was subsequently investigated [12]. The approach is now extended with the development of an equivalent Bernoulli-Euler equation for a laminated composite beam leading onto the exploration of the non-linear load-detection and stiffness characteristics of composite circular rings. Strain distributions along the circumference can then be evaluated. Such information will be essential for the design of composite circular spring systems. EQUIVALENT
FLEXURAL
RIGIDITY
OF ORTHOTROPIC
LAMINATED
BEAMS
Consider a section of laminate in the X-Z plane which is deformed due to some loading, as shown in Fig. 1. Let the x-y plane coincide with the midplane of the beam with the z-axis oriented in the thickness direction, Neglecting shear deformation, displacement u, in the x-direction for an arbitrary point C at a distance z, from the midplane is M, = ug - z,sina. In general, 63W
where n0 and oe are midplane displacements in the x and 9_’ directions and z’ is the distance from the midplane.
Deflection
783
of circular springs
Fig. 1. Bending geometry in the x-z plane.
The strain-displacement
equations can now to be written as
(4)
where
[ 1
-_= PX
l+
-- a2w ax2 01aw
2
312
ax
l[ 01 aw azw a3
-_= PY
2
I+
312
ay
% II
With this strain~isplacement
relationship, the constitutive equations can be written as r
0
0 &Y 0 YXY VPX UP,
l/P,,
=
k
(5)
784
P. C. Tse et
ui
For midplane symmetric laminate, [B] = [0] and therefore [Cl’ = - CD]-’ {[B-J [A]-‘l
= [O].
Hence we have 1
ii
P
= [D’]
CM).
For the bending of any midplane symmetric laminated straight beam of width L by a pure bending moment h4 in the x-z plane, we have 1
M
L
PX
L
D;IP,’
-ED;,-+=>==
If we put IDI
D=
DZZD66 - D&i
=-
I WI
9
then M=qD
(9)
Px ’ LD is designated the “equivalent flexural rigidity”. For specially orthotropic
RING
IN COMPRESSION
BY POINT
material:
LOAD
Consider a symmetrical laminated ring to be acted upon by radial compressive forces as shown in Fig. 2. In view of the symmetry of the ring, only one quadrant is required for the analysis. The bending of this quadrant of arc by force P is equivalent to the analysis of the large deformation of a straight strut of length i = R71/2 and subjected to a vertical load P and bending moment MR = LD/R simultaneously [13]. Initially, P is small, the eccentricity e in Fig. 3 will be very large to maintain the bending moment MR. The deformed curve AR will lie on a nodal elastica ABC as the load P cannot act directly on the beam. Equation (9) now takes the form
I?
t p’=2P
1 B
0 R
I 3
3
3
MI<
A
/
t
1P’
I
= 2P
Fig. 2. Equivalent
analysis
for the ring in compression.
Lkfiection of circular springs
785
Fig. 3. Nodal elastica curve for the ciefonned beam.
and
where k=
J&.
Equation (IO) is integrated with the boundary condition of B = 0 at the fixed end A, this leads to the expression z = h cos Cp[93.
STAGE
1 COMPRESSION
In the initial stage of compression the parameter d, satisfies the relation sin (0/2) = (l/p) sin # and p is a variable less than unity defined by the relation hZ = @I/p2k*). The arc length of the beam AB is [9f
i.e.,
.
786
P. C. Tse et al.
The semi-minor
axis is
~=.,=,=,[E(,,~)-(l-~),(,.~)I
=#P,;)-(1 -;)F(P,;)]. The deflection
of the load P from the original
quadrant
of arc to point
(13) B is
6=-‘=R-;[P(p,$-(I-;)F(p,;)] (14)
The force to produce
this deflection
is p,; ( ) R2n2 ’
4p2F2 P = LDk’ Since both the deflection K = dp = dpldp d6 dS/dp
j
= LD
6 and the force P are functions
of p, the spring
(15) stiffness is
K = 8p4LD R3d
2E PQ F P,; -PE P,: F p,; +pE p,; F P,~ -2F ( “) ( “) I( “) ( “) ( “) I( “)
2(
P,~ “) 1’
(16)
where
The deflection 6 and the load P in the above analysis refer to the quadrant arc only and are equal to half the actual values. By transforming these into the actual deflection and load 6’ and P’ of the system and grouping equations (14)-(16) the dimensionless deflection, load and stiffness are, respectively,
(19)
P’R2 __ = LD
(20)
[F4(~~~)+~F3(~,$o,~)] %=F,
( “) ( “) 2.5 P>; F P>; -pE
,(
“) ( “) ( “) ,( “) P,~ F P,: +pE P,~ F P,~ -2F
2(
“)]’ p,4
(21)
Deflection
of circular
springs
78-l
It can be observed that the variable p is a very important parameter that determines the deflection, load and stiffness of the springs. The value of p starts from zero and increases with the load P. When p = 0, the dimensionless stiffness term approaches a value of 6.7215. When p = 1, the deformed beam will become an infinite beam and P will eventually touch the extension of the nodal elastica. This represents the end of the applicability of the nodal elastica in the compressive range (stage 1 compression). The corresponding values of P and P’ are P=
= 0.3148 $
R2R2
P’ = 0.6296
STAGE
$.
(22)
2 COMPRESSION
For P greater than the value indicated in (22), the force will act on the imaginary extension of the elastica line AB (stage 2 compression). The deformed curve AB will lie on an undulating elastica ABC, as shown in Fig. 4; 4 now satisfies the relation sin (e/2) = p sin 4 and h2 = (4p2/k2). Thus at & = (7r/2), & = sin-’ (sin (~/4)/p) and the arc length of the beam AB is
Rn ‘A’ = 2
1 4s = k o (1 _ s = kF[p,
p2
sin2
~)1/2
=
_
1
sin-ii-)]
PJZ
(23)
’
i.e., (24)
I
IFig. 4. Undulating
elastica
curve for the deformed
beam.
P. c. Tse
7X8
Cl
al.
The semi-minor axis is
WP> &I) -___ F(P,A?) k
b=XJ.$,,9B=
k
= i {2b[P,sin-‘($j]
- F[P,sin-I(&)]}.
(25)
The deflection of the point B from the original quadrant of arc is
The force to produce this deflection is
4Fz~p,sin-l(~)] P=LD
f27f
R2d
The spring stj~~ess is
(1
-
-p sin2 (p dd,p2 sin2f$)“*
Grouping (26)--(28) into dimensionless forms, we have
J-LD
1
P J473
(29)
Deflection of circular springs
KR3 -=LD
789
8 d
(33) With P increasing, it is obvious that P will act on B eventually. When this happens, the moment at B is equal to zero and the presence of a couple is not necessary to secure a horizontal end tangent at B, this marks the end of stage 2 compression. To obtain the arc length AB, the limits of integration must be chosen such that 4 sweeps the quadrant from OA to OB',i.e. dB = 7712 CAB
=
F =; *‘*
I o
df
(1 - p2 sm* +)‘I2
i.e.,
k=&F&
(35)
() * The force to produce this deflection is
P=LD
4F2 5 _ () R27C2 - 1.3932s
P'= 2.7864LD R2'
STAGE
(36)
3 COMPRESSION
The zero bending moment at B implies that the middle surface curvature l/p will be equal to zero as well. At this point, a further increase in P will mean a negative curvature at B and hence the quadrant of arc will have a point of contraflexure at C, as shown in Fig. 5 (stage 3 compression). Making use of the results obtained for stage 2, the arc length of the beam AB in stage 3 is [12]
=1{2FW k where 4B = sin-‘(l/P*),
Fb,hJ) (37)
since OB= 7t/2 and
k=
&-2F(d -
F(P, 4~) .
(38)
The semi-minor axis is -
I[
2&4 4~) F(p, 44 k
-7’
1
(39)
P. C. Tse ef (11.
790
Fig. 5. Deformed
The deflection
of the point
beam with point of contraflexure.
B in stage 3 is R?T
6=R-
$41
2C2F(p) - F(Pt
The force to produce
this deflection
C4HP) - 2fYP) - WP> 6J) + F(P,
KC-
(40)
is - 0~~ ui* R27C2
P = LD 4c2~(~)
The spring
4B)l~
(41)
stiffness is
8LD R37t3 CWP)
x CUE
-
E(P,
ddl
W”(P)
-
dd13 W’(P)
F(P,
F’(P, 4~)1-
-
W(P)
-
F’(P?
ddl
F(P, +B)I CUE’
-
E’(P, 4~11’ (42)
where E’(p, 4B) and F’(p, c#J~)are as defined a:2
E’(P) =
s0 n/2
F’(P) = Grouping
(40)-(42)
into the required
s0
by equations
(29) and (30) and
-psin’b (1 - p*sin*4)“’
d$
(43)
d4.
(44)
p sin’ I#J (1 - p* sin* 4)“’
dimensionless 0, /Fs
forms &J) - 4E(p) + 2E(p, &) (45)
791
Deflection of circular springs
(46)
RING
IN COMPRESSION
BETWEEN
PLATES
the ring is being compressed by load P’ = 2P between two horizontal plates, the first and second stages of compression of the ring by a point load are still applicable as the plates will touch the ring tangentially at one point only. For the stage 3 compression, the deflected shape of the quadrant will consist of a straight portion of length t in contact with the horizontal flat plates and a curved part of R7t/2 - C,as shown in Fig, 6. The arc length for the curved section section AG is When
t-m= !ct=t 2
s
UP
d4-j
k o (1 - pZsinZ #)lfZ
k
i.e.,
The semi-minor axis is
Defiection of the point B from the original quadrant of the arc is
*_R-&+E(-j=)-F(j=)]v
i-
-i-
Fig. 6. Deformed beam for the stage 3 compression between plates.
(51)
P. C. Tse et al.
792
The force to produce this deflection is P > 1.3932 $
P’ > 2.7864 $.
(52)
The spring stiffness is (53)
Transforming
6 and P into 6’ and P’ to form the dimensionless deflection and stiffness:
(54)
KR3
/P’R2\3/2
1
STRESS-STRAIN
(55)
DISTRIBUTIONS
Consider an initially curved beam, as shown in Fig. 7, with the assumptions of neglecting shearing deformation and the length of the midplane AF remains unchanged after the bending moment is applied. Take a plane CE at a distance z, from the midplane. Strain for the arc CE is ECE
=
(P2
+
Zdaz (Pl
+
(PI
+
Z&l
Zc)al
=
Z&l Pz(P1
+
P2)
(56)
Zc)’
Since z, is small when compared with pr, we have (57)
Z
Z
A tier
Fig. 7. Bending
of initially
curved
beam.
Deflection
of circular
springs
793
In general, the expression for the circumferential strain of any surface at a distance z’ from the midplane will be
For specially orthotropic laminated rings with the same material properties in the principal directions, the shear strain is zero while the transverse strain is due to Poisson’s effect of the circumferential strain. Thus the complete picture of the strain distributions can be obtained once the circumferential strains have been detected. Substituting into (58), the corresponding expressions z = h cos 4, the coordinate of the deflecting point and the applied load P for the various stages of compression, the circumferential strains are obtained.
Stage 1 compression
E =< 4pF p,sin-’ R( [ II (,x)]jT
11
Stage 2 compression
(59)
(60)
4p[zW - f’[p, sin-‘( $)]]dq P2
R
Stage 3 compression by point load
(61)
Stage 3 compression between plates for the curved section
(62)
I
E= - i
Stage 3 compression between plates for the flat portion.
DISCUSSION
OF
(63)
RESULTS
Quasi-static compression tests were performed on circular composite springs fabricated by E-glass woven cloth/Ciba-Geigy epoxy resin (XH750A/HY956) with nominal width and Table Spring no.
1. Experimental Thickness
spring
stiffness
Experimental
spring stiffness
H (mm)
K W/mm)
12.13 7.65 5.48 2.38 2.30 1.58 1.00 0.89
2433.10 705.60 259.60 20.43 22.64 4.19 0.94 0.99
794
P. C. Tse et ol.
internal diameter of 5 1 mm and 114 mm respectively. The spring stiffnesses, as tabulated in Table 1, are determined from the initially linear portion of their respective load-deflection curves. By substituting these experimental spring stiffnesses into the expression derived in reference [6], the Young’s modulus and, hence, the equivalent flexural rigidity of the test specimen are obtained. Two predictions were compared with experimental results for specially orthotropic midplane symmetric laminated springs in compression, The first was calculated by the large deflection analysis and the second by the general purpose finite element program ANSYS [ 143. Eight-node, isoparametric axisymmetric quadrilateral elements with compressive line loading to simulate the actual testing situation were employed in the finite clement analysis. The theoretical results as shown in Figs 8 and 9 are computed by Simpson’s numerical integration algorithm. The comparison between the characteristics of the springs in compression by point load to that between flat plates are also shown. The first two stages of compression are common to both types of loading and deviation begins at stage 3 onIy. The first stage of compression extends to the range where the value of the dimensionless load P'R'/LDis around 0.63, while for the second stage is around 2.79. The springs exhibit non-linear behaviour and the dimensionless load-deflection curve is only linear over a very small range of initial deflection. It should also be pointed out that the dimensionless deflection 6'lRis 2 when the dimensionless load P'R21LDequals about 8.04 for point load compression, This corresponds to the case where the top and bottom surfaces of the ring are brought into contact. However, this will not occur for compression between plates, as
Fig. 8. Theoretical
dimensionless
load~eflection
characteristic
for rings in compression
Compressron betweenplates 15 -
Compressmnby pomt load 0 0
Fig. 9. Theoretical
1
2
dimensionless
3
6
load-stiffness
characteristic
7
8
9
for rings in compression.
795
Deflection of circular springs
fracture by crushing will obviously take up the failure process well in advance. By taking the limit of p approaching zero for the dimensionless stiffness term in stage 1 compression, the spring stiffness for a specially orthotropic midplane symmetric laminated spring is K = 6.7215 $
= 0.5061=$$
This spring stiffness expression is comparable to that being found by the strain energy approach [6] and that being employed for estimating the elastic modulus of filament wound ring [15]. The present analysis is thus consistent with the linear elastic small deformation theory for the initial stage of compression. The stiffness of the springs decreases with the deflection and then increases for values of P'R',/LD at around 4.7 well into stage 3 for compression by point load and at the junction of the second and third stages of compression between plates. The springs compressed by a point load are softer than those between plates, since the stiffness is much lower for the former during stage 3 compression. Figures 10 and 11 show the theoretical, finite element and experimental dimensionless load-deflection and stiffness characteristics of composite springs compressed between flat
0
0.5
I
I
I
I
I
1
1.5
2
2.5
3
P’R*iLD Fig. 10. Dimensionless Ioad-deflection characteristic of ring in compression between horizontal flat plates.
KR3/LD 8j
2t
-
Largedsnscfion theory 0 springA
A
. SpringE
0.5
SpringA FEM SpringB
-. . SpringF FEM 0 Springc
a .spnngF
0
0
--
1
I 1.5
2
2.5
I 3
P!?fLD Fig. 11. Dimensionless load-stiffness characteristic of ring in compression between horizontal flat plates.
796
P. C. Tse et al.
plates. As most of the thick springs fractured within the initial part of the second stage of compression and thus failed to reach the third stage, relatively thin specimens are selected to verify the dimensionless load-deflection and stiffness characteristics of rings compressed by point load, as shown in Figs 12 and 13 respectively. Good agreement exists among the theoretical, finite element and experimental results for both stage 1 and 2 compression under both sets of loading conditions. Non-linear effect for these two stages is progressive and weak and the spring stiffness decreases with the deflection and hence exhibits soft-spring characteristics. Discrepancies are generally acceptable, except for very high loads when plastic deformation has already taken place. From the dimensionless load-stiffness characteristic, the stiffness decreases initially with the deflection and increases slightly afterwards. This phenomenon occurred for both loading conditions. As the high load region is difficult to reach in practice, the springs are still classified as soft. The circumferential strains obtained by the present analysis for the specially orthotropic midplane symmetric laminated ring are compared with the experimental results; calculations from the strain energy approach and from finite element analysis. They are shown in Figs 14 and 15 for the inner and outer surfaces respectively. The theoretical
1
0
0.5
1
1.5
2
2.5
3
P’R?LD
Fig. 12. Dimensionless load-deflection
0
0.5
t
characteristic of ring in compression by point load.
1.5
2
2.5
3
P’R*/LD
Fig. 13. Dimensionless load-stiffness characteristic of ring in compression by point load.
Deflection _
Strain (1000 ps)
Spring
., +,I,
10
797
springs
D
0, X, 0 Expcrimentalrcsults Strain energy theoretical result.3 Large deflection theoretical rewits Fink element prediction (line loading)
---
0
of circular
20
30
40
50
60
70
_:.
80
90
Angle 0 (degree) Fig. 14. Circumferential
strain
distributions for the inner surface. (0) 2OON,(x) 250Nand(O)
Spring
Strain (1000 pe)
Load of (H ) 50 N, ( +) 100 N, (*) 300N.
150 N,
D
4
., +, D,O,
-4
x, 0 Expsimentalresults Strain energy theoretical ~CSUI~ Large deflection theoretical results Finite element prediction (line load&
---
-6 -8
0
10
20
30
40
50
60
70
80
90
Angle t3 (degree) Fig. 15. Circumferential
strain
distributions for the outer surface. Load of (m) 50 N, (Cl) 200 N, ( x ) 250 N and (0) 300 N.
( +) 100 N, (*) 150 N,
circumferential strain distributions around the ring predicted by the present analysis and the finite element predictions are in good consistence with those predicted by the strain energy approach. However, a closer examination of Figs 14 and 15 reveals that the large deflection theories are able to provide more accurate predictions, especially at high loads where the deflection of the spring is large. The realistic modelling of line loading at the 90” position of the composite ring in the finite element analysis, as in actual testing, provides a better prediction of strain variation, especially in the loading position. Strain-free locations were also observed on both the inner and outer surfaces of the ring by the present analysis. These strain independent locations are of great potential to the future development of the composite springs, as they can be used to facilitate attachment of additional accessories or discontinuities, such as holes or fitments. NLM 29:5-K
798
P. C. Tse et al. CONCLUSIONS
A theoretical analysis of the bending of symmetric laminated composite rings by considering the exact expression of the moment-curvature relationship for beams based on the equivalent flexural rigidity is presented. Large deflection study has been conducted and the deformed shapes of the springs are classified into stages according to the load and are governed by the theories of nodal and undulating elastica. The springs exhibit non-linear behaviour and the dimensionless load-deflection curve is only linear over a very small range of initial deflection with the dimensionless load values below about 0.5. This analysis provides basic information for the design of composite springs. The circumferential strain distributions obtained through the present analysis resemble those from the small deflection strain energy approach, with the added capability of providing more accurate prediction at high loads. Tests on composite rings have shown that the use of present large deflection analysis or finite element analysis to predict mechanical behaviour of composite rings in the third or final stage can be misleading without considering plastic deformation. Composite material with improved toughness and strength are thus needed to verify the final stage of deflection. Acknowledgments-This work was carried 0341/040/430 and 0341/084/430.
out with the support
of the Hong
Kong
Polytechnic
research
funds
REFERENCES 1. G. D. Scowen, Transport applications for libre reinforced composites. IMechE C49/86, 245-255 (1986). 2. G. D. Scowen and D. Hughes, The sulcated spring. Int. Seminar, Autotech 85 Congress, The Institution of Mechanical Engineers, Automobile Division, Nov 1985. 3. P. K. Mallick, Design and development of composite elliptic springs for automotive suspensions. 40th Annual Co& Reinforced Plastics/Composites Institute, The Society of the Plastics Industry, Inc. 28 January-l February, 1985. of composite elliptic springs. ASME J. Engng Mater. Technol. 4. P. K. Mallick, Static mechanical performance 109,22-26 (1987). T. Nakakura and H. Sakai, Spring constants of elliptic rings made of carbon5. T. Akasaka, M. Masutani, fiber-reinforced thermonlastics. Proc. 33rd Int. SAMPE Svmo.. 7-10 March. 1988. 670-680. 6. C. K. So, P. C. Tse, T. 6. Lai and K. M. Young, Static mechanical behaviour of composite cylindrical springs. Composite Sci. Technol. 40(3), 251-263 (1991). 7. M. J. Barten, On the deflection of a cantilever beam. Quarterly J. Applied Mathematics 2, 168-171 (1944). 8. K. E. Bisshopp and D. C. Drucker, Large deflection of cantilever beams. Quarterly J. Applied Mathematics 3, 272-275 (1945). 9. R. Frisch-Fay, On large deflections. Australian J. Applied Sci. 10(4),418-432 (1959). 10. H. D. Conway, The non-linear bending of thin circular rods. ASME J. Applied Mechanics 23, 7-10 (1956). bending of thin rods. ASME J. Applied Mechanics 26, 40-43 (1959). 11. T. P. Mitchell, The non-linear 12. R. Frisch-Fay, The deformation of elastic circular rings. Australian J. Applied Sci. 2(3), 329-340 (1960). 13. R. Frisch-Fay, Flexible Bars, Butterworths, London (1962). 14. ANSYS Engineering Analysis System User Manual, version 4.3. Swanson Analysis System Inc. (1987). 15. D. V. Rosato and C. S. Grove Jr, Filament Winding: its Deuelopmenf Manufacture, Applicarions and Design. Interscience Publishers, New York (1964).