Leaf Spring

RP/composite leaf springs constructed of unidirectional glass fibers in a matrix, such as epoxy resin, have been recognized as a viable replacement for steel springs in truck and automotive suspension applications. Because of the material's high specific strain energy storage capability compared with steel, direct replacement of multi-leaf steel springs by mono-leaf composite springs is justifiable on a weight saving basis. Other advantages of RP springs accrue from the ability to design and fabricate a spring leaf having continuously variable width and/or thickness along its length. Such design features can lead to new suspension arrangements in which the composite leaf spring will serve multiple functions thereby providing part consolidation and simplification of the suspension system.

The spring configuration and material of construction should be selected so as to maximize the strain energy storage capacity per unit mass without exceeding stress levels consistent with reliable, long life operation. Elastic strain energy must be computed relative to a particular stress state. For simplicity, two materials are compared, steel and unidirectional glass fibers in an epoxy matrix having a volume fraction of 0.5 for the stress state of uniaxial tension. If a long bar of either material is loaded axially the strain energy stored per unit volume of material is given by

where 8a is the allowable tensile stress and E is Young's modulus for the material.

In Table 4.9 the appropriate E for each material has been used and a conservative value selected for 8A. On a volume basis the RP is about twice as efficient as steel in storing energy; on a weight basis it is about eight times as efficient.

Glass fiber-epoxy RP leaf spring design




U/w* (in)









" w = specific weight

" w = specific weight

The RP has an advantage because it is an anisotropic material that is correcdy designed for this application whereas steel is isotropic. Under a different loading condition (such as torsion) the results would be reversed unless the RP were redesigned for that condition. The above results are applicable to the leaf spring being reviewed because the principal stress component in the spring will be a normal stress along the length of the spring that is the natural direction for fiber orientation.

In addition to the influence of material type on elastic energy storage, it is also important to consider spring configuration. The most efficient configuration (although not very practical as a spring) is the uniform bar in uniaxial tension because the stresses are completely homogeneous. If the elastic energy storage efficiency is defined as the energy stored per unit volume, then the tensile bar has an efficiency of 100%. On that basis a helical spring made of uniform round wire would have an efficiency of 32% (the highest of any practical spring configuration) while a leaf spring of uniform rectangular cross section would be only 11% efficient.

The low efficiency of this latter configuration is due to stress gradients through the thickness (zero at the mid-surface and maximum at the upper and lower surfaces) as well as along the length (maximum at mid-span and zero at the dps). Recognition of this latter contribution to inefficiency led to development of so-called constant strength beams which for a cantilever of constant thickness dictates a geometry of triangular plan-form. Such a spring would have an energy storage efficiency of 33%. A practical embodiment of this principle is the multi-leaf spring of constant thickness, but decreasing length plates, which for a typical five leaf configuration would have an efficiency of about 22%.

More sophisticated steel springs involving variable leaf thickness bring improvements of energy storage efficiency, but are expensive since the leaves must be forged rather than cut from constant thickness plate. However, a spring leaf molded of the RP can have both thickness and width variations along its length. For instance, a practical RP spring configuration having a constant cross-sectional area and appropriately changing thickness and width will have an energy storage efficiency of 22%. This approaches the efficiency of a tapered multi-leaf configuration and is accomplished with a material whose inherent energy storage efficiency is eight times better than steel.

In this design, the dimensions of the spring are chosen in such a way that the maximum bending stresses (due to vertical loads) are uniform along the central portion of the spring. This method of selection of the spring dimensions allows the unidirectional long fiber reinforced plastic material to be used most effectively. Consequendy, die amount of material needed for the construction of the spring is reduced and the maximum bending stresses are evenly distributed along the length of the spring. Thus, the maximum design stress in the spring can be reduced without paying a penalty for an increase in the weight of the spring. Two design equations are given in the following using the concepts described above.

To develop design formulas for RP springs, we model a spring as a

Figure 4,28 RPspring model

Figure 4,28 RPspring model

circular arc or as a parabolic arc carrying a concentrated load 2F„ at mid-length (Fig. 4.28).

The governing equation for bending of the spring

p R ~ El where R = radius of curvature of unloaded spring; p = radius of curvature of deformed spring; M = bending moment, E = Young's modulus; and 7= moment of inertia of spring cross section.

Using the coordinate system shown in Fig. 4.28, equation 4-37 is rewritten as tfy = fv(/-x) _ I Q<x<l (4-38)

where the coordinate y is used to denote the deformed configuration of the spring. Once the maximum allowable design stress in the spring is chosen, equation 4-38 will be used to determine the load carrying capability of the spring. Due to the symmetry of the spring at a: = 0, only half of the spring needs to be analyzed. It should be noted that equation 4-38 is only an approximate representation of the deformation of the spring. However, for small values of A.//, it is expected to give reasonably good prediction of the spring rate. Here X is the arc height and 2?//is the chord length of the spring. Although a nonlinear relation can be used in place of equation 4-38, it would be difficult to derive simple equations for design purposes.

For this particular design, the thicknesses of the spring decreases front the center to the two ends of the spring. Hence, the cross-sectional area of the spring varies along its length. The maximum bending stresses at every cross section of the spring from a; = 0 to a; = a0 are assumed to be identical (Fig. 4.28). The value of a0 is a design parameter that is used to control the thickness and the load carrying capability of the spring. If a0 is the maximum allowable design stress, then the thickness of the spring for 0 <x < a0 is determined by equating the maximum bending stress in the spring to a0> thus:

where v is Poisson's ratio, and b and h are the width and thickness of the spring, respectively.

The factor (I - v2) is introduced to account for the fact that b could be several times larger than h. If b and h are of the same order of magnitude, a zero value of v is suggested to be used with equation 4-

39. This equation shows that the thickness of the spring should be a function of Fv a0,1, and b. Once Fv> a0> and / are fixed, then the value of h is inversely proportional to the square root of the width of the spring.

For x >a0, the thickness of the spring is assumed to remain constant. The minimum value of h is governed by the ability of the unidirectional composite to carry shear stresses. Using equation 4-38 and the appropriate boundary and continuity conditions, the following equation for the determination of the spring rate is obtained,

where ky is the spring rate per unit width of the spring in lb per in. of vertical deflection. In deriving equation 4-40, the maximum bending stress a0 is assumed to develop when y = 0 at which the center of the spring rate has undergone a deflection equal to X. If the actual design value of 2 Fv is less than or greater than bkb, the appropriate value of a0 to be used in equation 4-74 can be determined easily from the maximum design stress by treating a0 as a linear function of 2 Fv.

A constraint on the current fabricating method of the RP leaf spring is that the cross-sectional area of the spring has to remain constant along the length of the spring. This imposes a restriction on the use of variable cross-sectional area design since additional work is required to trim a constant cross-sectional area spring to fit a variable cross sectional area design. Unless the design stresses in the spring are excessively high, it is preferable to use the less labor-intensive constant cross-sectional area spring. This section describes the design formulae for this type of spring design.

Using the same coordinate system and symbols as shown in Fig. 4.28, equations 4-37 and 4-38 remain valid for the constant cross-sectional area spring. The mid-section thickness of the spring h0 is related to the maximum bending stress a0 by:

b0 o0

where b0 is the corresponding mid-section width of the spring. Imposing the constant cross-sectional area constraint, b0h0 = bh (4-42)

the thickness of the spring at any other section is given by:


The corresponding width of the spring is then obtained from equation

4-42. Based on equations 4-42 and 4-43 that the width of the spring will continue to increase as it moves away from the mid-section. In general there is a limitation on the maximum allowable spring width. Using ba, to denote the maximum width, the value of x beyond which tapering of the spring is not allowed can be determined by imposing the constant cross-sectional area constraint. One can use a0 to denote this value of x, then:

Thus, equation 4-43 holds only for x <. Beyond x = a0, the thickness of the spring remains constant and is given by:


An implication of equations 4-43 and 4-44 is that the maximum bending stresses will remain constant along the length of the spring for [a;] >a„. Equation 4-38 with the appropriate boundary and continuity conditions, the spring rate, k, can be shown to be:

where I0 is the moment of inertia of the cross-section of the spring at mid-section. In the design of a spring, the values of ba„ I, R, X, and k are usually given and it is required to determine the values of h0 for a desirable value of a0. The following equation has been obtained for the determination of h,;.

Once ho is determined, the corresponding value of b0 is then obtained from equation 4-37. In equation 4-47, the value of a0 corresponds to a center deflection equal to X. If the actual design value of 2Fv, is less than or greater than kX, the appropriate value of <7U to be used in equation 4-47 can be determined easily from the maximum design stress by treating o0 as a linear function of 2Fv.

Consider, as an example, the design of a pair of longitudinal rear leaf springs for a light truck suspension. The geometry of the middle surface of the springs is given as:

The design load per spring is 2200 lb and a spring rate of 367 lb/in. is required. If a0 is set equal to 53 ksi in equation 4-47, two possible design values of h0 are obtained. Using equation 4-41, the corresponding values of b0 are determined. Thus, there are two possible constant cross-sectional area designs for this particular spring: (S) h0 = 1.074 in, b0 = 2.484 in. and (SS) h0 = 1.190 in., b0 = 2.023 in. A value of Young's modulus of 5.5 x 106 psi is used in the design of these springs. This corresponds to the modulus of a unidirectional RP with 50vol% of glass fibers. If a value of a0 less than 53 ksi is used in the design, negative and complex values of h0 are obtained from equation 4-47.

This indicates that it is impossible to design a constant cross-sectional area spring to fit the given design parameters with a maximum bending stress of less than 53 ksi. If a constant width design is required, it can be shown from equation 4-40 that a spring with a constant width of 2.484 in. and a maximum thickness of 1.074 in. will satisfy all the design specifications. The corresponding value of a0 is 18 in. If a constant width of greater than 2.484 in. is allowed, then a maximum design stress of less than 53-ksi can be obtained.

The above example shows that two plausible constant cross-sectional area designs are obtained to satisfy all the design requirements. If the spring were subjected only to vertical loading, the second design would be selected since it involves less material. However, if the spring is expected to experience other loadings in addition to the vertical load, then it is necessary to investigate the response of the spring to these loadings before a decision can be made.

The effects of these loadings can be determined easily using Castigliano's Theorem, together with numerical integration. For illustration, a comparison summation of the responses of the two constant cross-sectional area spring designs are reviewed:

1. Rotation due to axle torque, MT : The rate of rotation of the center portion of the spring due to the axle torque, MT, is: design (S) = 1.901 x I 05 in-lb/radians and design (SS) = 1.895 x 105 in-lb/ radians.

If an axle torque of 15,000 in-lb is used for Mr, the rotation and the maximum bending stresses for these two springs are in table form:

Rotation, degree Maximum stress design (S) 4.5 15.7

The responses of these two designs to the axle torque are, for all practical purposes, identical. As in the case of transverse loading, the maximum bending stresses are uniform along the springs for [jc]

Effect of longitudinal force, FL: The longitudinal force F¿, will produce a longitudinal and vertical displacement of the spring. Using kL and kv, to denote the corresponding spring rate associated with Fl, results in:

design [S3 design (SS)

Assuming that a maximum value of FL equal to the design load is expected to be carried by the spring, the deflection and the maximum stress experienced by the spring are:

Longitudinal disp., in. 0.83 0.87

Maximum stress, ksl 13.8 13.8

The responses of the two designs to the longitudinal force are essentially identical, The maximum bending stresses are uniform along the springs for [x] < a0.

Effect of twisting torque, ML: In the usual suspension applications, leaf springs may be subjected to twisting, for example, by an obstacle under one wheel of an axle. For the two springs studied here, the rate of twist is: design (S) = 1.47 x IO4 in-lb/radians and design (SS) = 1.23 x 104 in-lb/radians.

In addition, due to the geometry of the spring, the twisting torque Mi will cause the spring to deflect in the transverse direction. The rate of transverse deflection is: design (S) = 3319 in-lb/in.and design (SS) = 2676 in-lb/in.

If a maximum total angle of twist of 10 degrees is allowed, the response of the spring will be:

Twisting torque Lateral deflection Maximum shear design (S) design (SS)

in-lb 2559 2150

stress, ksi 2.55 2.58

In calculating the effect of the twisting torque, the transverse shear modulus of the unidirectional RP has been used. For an RP with

50vol% of glass fibers, the modulus has a value of 4.6 x 105-psi. The maximum shearing stress occurs at [x] = a0. For designs (S) and (SS), the values of a0 are 10.3 in. and 12.66 in., respectively. The values of the bending stresses associated with the twisting torque are negligibly small.

4. Effect of transverse force, FT: As in the case of the twisting torque, the transverse force, FT, will cause the spring to twist as well as to deflect transversely. The spring rates associated with the transverse force are:

Twist (in-lb/radian) 3319 2676

Deflection (lb/in) 600 458

Assuming that a maximum value of FT is equal to 0.5 times the design load expected to be carried by the spring, the deflection and the maximum stress experienced by the spring will be:

Angle of twist (degree)

19 23.6

Transverse deflection (in.)

Max. bending stress (ksi)



Max. shear stress (ksi)

The angle of twist and the maximum shear stresses associated with this lateral force are rather high. In practice, the spring will have to be properly constrained to reduce the angle of twist and the maximum shear stress to lower values. Assuming that a maximum angle of twist of no more than 10 degrees is allowed, the deflection and the maximum stresses experienced by the spring are:

Angle of twist (degree)

10 10

Transverse deflection (in.)

Max. bending stress (ksi)



Max. shear stress (ksi)

The maximum bending stresses occur at the center of the spring while the maximum shear stresses occur at the ends of the spring. Based on the above numerical simulations, it appears that both designs respond approximately the same to all different types of loadings. However, design (S) will be preferred since it provides a better response to the lateral and twisting movement of the vehicle.

The maximum bending stress that will be experienced by the spring is obtained by assuming a simultaneous application of the vertical and the longitudinal forces together with the axle torque. A maximum bending stress of 82.5 ksi is obtained. This bending stress is uniform along the spring for [x] < a0. In view of the infrequent occurrence of this maximum bending stress, it is expect that the service life of the spring is guaranteed to be long in service. However, a maximum shearing stress close to 6.3 ksi can be reached when the spring is subjected to both twisting torque and transverse force at the same time. The value of this shearing stress may be too high for long life application. However, a more complete assessment of the suitability of the design can only be obtained through interaction with the vehicle chassis designers.

Special Spring

As RP leaf springs find more applications, innovations in design and fabrication will follow. As an example, certain processes are limited to producing springs having the same cross-sectional area from end to end. This leads to an efficient utilization of material in the energy storage sense. However, satisfying the requirement that the spring become increasingly thinner towards the tips can present a difficulty in diat the spring width at the tip may exceed space limitations in some applications. In that case, it will be necessary to cut the spring to an allowable width after fabrication. There arc special processes such as basic filament winding that can fabricate these type structures.

A similar post-molding machining operation is required to produce variable thickness/constant width springs. In both instances end to end continuity of the fibers is lost by trimming the width. This is of particular significance near the upper and lower faces of the spring that are subject to the highest levels of tensile and compressive normal stresses. A practical compromise solution is illustrated in Fig. 4.29. Here excess material is forced out of a mid-thickness region during molding that maintains continuity of fibers in the highly stressed upper and

Spring with a practical loading solution

Spring with a practical loading solution

Figure 4.30 Spring has a bonded bracket

lower face regions. A further advantage is that a natural cutoff edge is produced. The design of such a feature into the mold must be done carefully so that the molding pressure (desirable for void-free parts) can be maintained.

An area of importance is that of attaching the spring to the vehicle. Since the RP spring is a highly anisotropic part especially designed as a flexural element, attachments involving holes or poorly distributed clamping loads may be detrimental. For example, central clamping of the spring with U bolts to an axle saddle will produce local strains transverse to the fibers that in combination with transverse strains due to normal bending may result in local failure in the plastic matrix. The use of a hole for a locating bolt in the highly stressed central clamped region should also be avoided.

Load transfer from the tips of the RP spring to the vehicle is particularly difficult if it is via transverse bushings to a hanger bracket or shackles since the bushing axis is perpendicular to all the reinforced fibers. One favorable design is shown in Fig. 4.30. This design utilizes a molded random fiber RP (SMC; Chapter 1) bracket that is bonded to the spring. Load transfer into this part from the spring occurs gradually along the bonded region and results in shear stresses that are conservative for the adhesive as well as both composite parts.

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