finite element based leaf spring design

265

9

Leaf Springs

9.1 INTRODUCTION

The spring is a machine part used to absorb sudden loads and to accumulate elasticenergy. There are different mechanical designs and forms of springs. The springunder consideration is called a

leaf spring.

This type of spring has an advantageover other kinds of springs because of its compact design and essential structuralrole. Its importance, first and foremost, comes from the part’s unique role, utilizedin motor vehicles to provide the absorption of irregular loads caused by unevenroads. The leaf spring is also used in other machines such as heavy presses thatoperate under loads at large displacements. Since the displacements undergo inter-mittent absorptions and releases, a sturdy design of the part must be provided—thedesign that combines optimum strength with a needed elasticity. This is achievedby an assembly of narrow leaves acting in unison as bending beams.

A typical leaf spring is shown in Figure 9.1. Figure 9.1a shows the leaves holdtogether by a center bolt and clamp. Figure 9.1b and c show different spring endsused in practice. The top leaf is designated as the main leaf. The leaves are bentwith the ends facing upward. When a spring is designed to be used in a reversed

FIGURE 9.1 Leaf spring: (a) spring (1, center bolt; 2, clamp), (b) eye spring end, and (c)plain spring end.

Ch09 Page 265 Tuesday, November 7, 2000 2:40 PM

266

Nonlinear Problems in Machine Design

position, the main leaf is at the bottom. The load is applied simultaneously at eachend of the main leaf, while the reaction forces concentrate in the center of the spring,or vice versa. All leaves are subjected to significant deflections and change incurvature. The normal working load is a vertical load that engages the leaves, bendingthem in a direction that relieves the curvature. The change in geometry causes themto slide while the interleaf friction hinders the motion. This phenomenon causes apart of applied energy to be transformed into irreversible work and to dissipate. Themagnitude of dissipated work depends on several factors such as the condition ofleaf surfaces, applied load, and speed of sliding.

The exposure to varying loads subjects leaf springs to hazardous stresses thatresult in fatigue. The best known deterrent against fatigue is the surface treatmentof the metal, namely shot peening, done prior to assembly. Processing with shotpeening produces residual compressive stresses in the surface layer. Consequently,the tensile stresses at the surface provide protection against fatigue.

1,2

The following design analysis of a leaf spring presents two approaches. Oneuses a simplified theory, and the other, for more complex problems, uses the finiteelement method.

9.2 DESIGN FUNDAMENTALS

9.2.1 T

HEORETICAL

S

TRESS

D

ISTRIBUTION

Originally, the leaf spring was conceived as a beam of uniform strength.

3

Ourdiscussion begins with this concept to help lay down the basic theoretical principles.Consider Figure 9.2. Due to symmetry, only one-half of the leaf spring is analyzed,representing a cantilever beam. The beam has the form of a flat triangle loaded atits apex, where the maximum bending stresses are identical throughout. Dividingthe triangle into parallel leaves and stacking them on top of each other, an ideal leafspring is achieved (see Figure 9.2b).

The maximum bending stress at the fixed end of the spring is determined fromthe correlation of a beam of a rectangular cross-section,

(9.1)

where

m

is the number of leaves,

and

W

is the section modulus of a single leaf. Thedeflection of the free end of the spring is assumed to be the same as the deflectionof a beam of the constant cross-section of a width that equals two-thirds of the baseof the triangle. The deflection equals

(9.2)

where

I

is the moment of inertia of a single leaf. A different equation, presented byWahl,

3

considers a trapezoidal beam (Figure 9.3). Here, the deflection becomes

SPl

mW----------=

f13--- Pl3

23---EmI

-------------------- Pl3

2EmI--------------= =

Ch09 66 Tuesday, November 7, 2000 2:40 PM

Leaf Springs

267

(9.3)

K

is a correction factor that is a function of the number of leaves

m

(Figure 9.4).Equations (9.1) to (9.3), in their simplified form, are presented here to help

understand the more accurate derivatives included in the standard design formulaeused in practice today.

FIGURE 9.2 Triangular beam theory: (a) bending triangle and (b) leaf spring.

FIGURE 9.3 Trapezoidal beam.

f KPl3

3EmI--------------=


268


SAE Design Formulae

The formulae that form standards for practical application are based on widelyaccumulated professional experience. The published source for designing leafsprings is the

SAE Spring Design Manual,

issued by the Spring Committee of theSociety of Automotive Engineers.

1

Introducing the varying leaf thickness, the sim-plified formulae above become as follows. The expression for bending stresses,derived from Equation (9.1), is

(9.4)

The deflection, derived from Equation (9.2), equals

(9.5)

SF

denotes a stiffening factor, which is a function of leaf engagement as explainedbelow. Derived from Equation (9.5) is the load rate,

(9.6)

Leaf Engagement

The above triangular beam conjecture implies that the leaves are engaged throughouttheir lengths and are bearing on each other. This, however, is contrary to reality.

FIGURE 9.4 Correction factor for deflection of leaf spring.

Slt

2 I∑------------ P⋅=

fPl3

2E I∑---------------- 1

SF-------⋅=

kPf---

2E I∑l3

---------------- SF⋅= =


Leaf Springs

269

Only a small area near the tip is engaged, while the rest of the leaf is free.

1

To proveit analytically, consider two adjoining leaves of a leaf spring under load (Figure9.5a). The leaves are subject to bending as follows: Face 1 of the upper leaf iscompressed, while face 2 of the lower leaf is subject to tension. The curvatures ofboth faces are expressed by the differential equation of a beam.

(9.7)

Equation (9.7) is of a curve that has no turning points, i.e., curvature d

2

y

/

dx

2

nowhere equals zero. Two curves of this kind with different radii

ρ

will cross at twopoints (see Figure 9.5b). Therefore, according to behavior of one-half of the leafspring, any pair of leaves whose patterns are defined by bending moment

M

canhave one bearing point only—point A in Figure 9.5.

In practice, as stated above, the leaves are engaged in a small area near the tip.The size of the area depends on the form of leaf ends, which may be square, tapered,or trimmed. The area affects spring deflection

f

through the stiffening factor

SF

. Forleaves with square ends,

SF

= 1.15 while, for tapered ends,

SF

= 1.10.

1

Example

Let us derive more accurate stress distribution using the following example.

4

Con-sider the three-leaf spring shown in Figure 9.6. For simplicity, assume the three

FIGURE 9.5 Concerning leaf engagement: (a) leaf separation under load and (b) engage-ment of two adjoining leaves.

MEI------ 1

ρ---

d2y

dx2--------

1dydx------

2

+

--------------------------- d2y

dx2--------≅= =

FIGURE 9.6 Three-leaf spring.


270


leaves to be of the same thickness. The contact between the leaves takes place atbearing points

A

and

B,

while the rest become disengaged. The stress distributionis based on the assumption that the deflections of adjoining leaves at the bearingpoints are equal, i.e.,

(9.8)

The deflections are expressed as functions of contact forces

P

1

and

P

2

as follows(see Figure 9.7). The deflection of the main leaf 1 at point

A

is

(9.9)

The deflection of the middle leaf 2 at the same point is

(9.10)

The deflection of the middle leaf 2 at point

B

is

(9.11)

yA1 yA2= ; yB2 yB3=

FIGURE 9.7 Load distribution in a three-leaf spring: (a) forces and (b) bending moments.

yA1l3

3EI--------- 14P 8P1–( )=

yA2l3

3EI--------- 8P1

52---P2–

=

yB2l3

3EI--------- 5

2---P1 P2–

=


Leaf Springs

271

The deflection of the bottom leaf 3 equals

(9.12)

It follows from Equations (9.8) that

(9.13)

(9.14)

from which one gets

(9.15)

The bending moments produced by forces

P

1

,

P

2

, and

P

are shown in Figure 9.7b.One finds the critical bending stresses to be at points

A

,

B,

and

C. (Point C representsthe support of the bottom leaf, as shown in Figure 9.6.) Comparing the resultingstresses, we get

(9.16)

where indices 1, 2, and 3 refer to the respective leaves. One notes that the smallestbending stress occurs in the main leaf, while the greatest is in the bottom leaf.

9.2.2 CURVED LEAVES

In practice, the leaves are designed in form of arcs, with each leaf having a differentradius, scaled down accordingly.1 As a result of this geometry, there are spacesbetween the leaves. See Figure 9.8. At assembly, the center bolt and clamp act topull the leaves together, changing the curvatures and causing mechanical prestress.

Let us consider prestressing of leaves in the spring as shown in Figure 9.8. Theleaves are of equal thickness, and stresses and deflection can be determined fromEquations (9.1) and (9.2). Tying the leaves together is explained schematically inFigure 9.9. For simplicity, assume that the tying is accomplished by tightening thecenter bolt, while ignoring the center clamp. Figure 9.9a presents the condition priorto bolt tightening, and Figure 9.9b shows the local leaf deformation due to the tying.The local compression of the leaves affects only the vicinity of the center bolt andhave negligible influence on the bending stresses in them.

As the tightening force P rises, the leaves draw closer together. The distributionof forces between the leaves is shown in Figure 9.10. The forces necessary to bringthe leaves together, ignoring the leaf curvatures, may be expressed as follows

yB3l3

3EI---------P2=

32P1 5P2– 28P=

5P1 4P2– 0=

P1 1.087P= ; P2 1.359P=

S2, max 1.087S1,max= ; S3,max 1.359S1,max=


272 Nonlinear Problems in Machine Design

FIGURE 9.8 Curved leaves in a leaf spring.

FIGURE 9.9 Tightening of leaves by a center bolt: (a) condition before tightening and (b)deformation caused by tightening.

FIGURE 9.10 Forces acting on the leaves during bolt tightening.


Leaf Springs 273

(9.17)

(9.18)

To bring leaves 1 and 2 together, the tightening force must equal ; for leaves2 and 3, it must be .

The compressive stresses induced in the main leaf through tightening equal

(9.19)

See Equations (9.1) and (9.2). The external load applied upon the leaf spring inducesbending stresses that are superimposed on the preliminary ones (i.e., created atassembly). In consequence, the maximum tensile stresses in the main leaf decrease,and those in the bottom leaf increase. Meanwhile, in the intermediate leaves, thestresses vary proportionally to the leaf location. Because the bottom leaves aresubjected to a greater stresses, the spaces between stay small, thus lessening theinfluence of the preliminary stresses.

9.3 FE ANALYSIS OF LEAF SPRINGS

The finite element method of analyzing displacements and stresses is known to givebetter results than any other approach. Furthermore, it allows analysis of frictionaleffects that cannot be considered by standard machine design equations. In thefollowing analysis, the leaves are represented by a two-dimensional FE model, andthe contact between them is simulated by contact elements using the penalty method.The friction factor is assumed to be µ = 0.45.

9.3.1 PROBLEM DEFINITION

The leaf spring under consideration is adapted from an example given by the SAEManual on Leaf Springs.1 The spring has five leaves, the same as in the example.However, contrary to the SAE example, here the spring is assumed to be symmetric.The leaves are made from alloy steel SAE 9260 with the following properties:

Two kinds of leaves are analyzed: leaves that were not processed by shot peening,and leaves that were.

The relevant information is summarized in tables and figures. Figure 9.11 showsthe leaves tied by a center clamp and the spring’s end supports. The load is applied

Brinell hardness BHN = 400

Tensile strength Su = 1560 MPa

Yield point Syp = 1350 MPa

s12

P′a2

48EI------------ 3l2

2 4a22–( )

P′l23

48EI------------–=

s23P′′l3

3

48EI------------

P′′a2

48EI------------ 3l2

2 4a22–( )+=

P P′=P P″=

SP′lmW---------- s12

Eh

l2-------= =



at the bottom of the center clamp and is directed upward. At assembly, the bolts aretightened, prestressing the leaves and affecting the form of arcs. The design data areas follows:

Table 9.1 presents the dimensions of free leaves, before tying. Table 9.2 showstheir position in relation to each other. The nomenclature used in the tables isexplained in Figure 9.12.

Equations used to derive the dimensions, listed in Table 9.2, are as follows:

(9.20)

Design load 4150 N

Maximum load 6432 N

Length of center clamp 100 mm

Bolts M 8 × 1.25, class 4.8

Tightening force 10 kN

TABLE 9.1Spring Leaves–Main Dimensions

Leaf no.Half Length

l (mm)Thickness

t (mm)Width

w (mm)Radiusr (mm)

1 570.0 6.7 63.0 1603.0

2 454.6 6.3 63.0 1481.0

3 352.4 6.3 63.0 1399.0

4 245.4 6.3 63.0 1359.0

5 135.7 6.0 63.0 1300.0

FIGURE 9.11 Design details of a leaf spring: (a) center clamp and (b) end support.

αi

li

ri 0.5+ ti

---------------------180π

---------=


Leaf Springs 275

(9.21)

(9.22)

(9.23)

9.3.2 FE SOLUTION

Let us analyze the deformation of the spring and the stress distribution. Two FEmodels are developed: a coarse model for spring deformation and a precise modelfor the stress distribution.

The stress distribution is derived first for leaves that were not processed by shotpeening, and then for the processed leaves.

TABLE 9.2Spring Leaves–Derived Dimensions

Leaf No. α (degrees) h (mm) x (mm) c (mm)

1 20.330933 99.864506 556.949945

2 19.549885 68.933472 446.57488 134.447283

3 14.400052 43.952472 349.91838 90.9863898

4 10.337378 22.026573 243.50555 49.0688909

5 5.9670278 9.0435508 135.14296 63.6287013

FIGURE 9.12 Leaf geometry nomenclature.

xi ri αisin=

hi ri 1 αicos–( )=

ci ri 1= ti 1–+( )2 xi2– hi ri–+=



Coarse Model

The leaf spring is represented by a plane model, using plane strain relations ofelasticity. Because of symmetry, only half of the spring is being modeled. The leavesare made of quadrilateral elements with height that equals the leaf’s thickness. Thecontact between the leaves is simulated by contact elements. The center clamp issimulated by link (rod) elements. Figure 9.13 shows the leaves in the loose condition.The spring ends are supported by contacts with two quadrilateral elements repre-senting a mounted body (see Figure 9.14). The upper quadrilateral restricts theupward displacement of the spring, while the lower one restricts the downwarddisplacement. To achieve a tightening force of 10 kN in each of four bolts in thecenter clamp, it is necessary to specify an initial strain as part of the input data ofthe link elements. An initial strain of 0.2791 was derived by a preliminary FE solution(see below).

FIGURE 9.13 Finite element model of a leaf spring.

FIGURE 9.14 Finite element representation of spring end support.


Leaf Springs 277

The FE model is created and solved using the ANSYS program. Two non-linearities are present: varying contact surfaces between the leaves and large dis-placements. A solution process based on the Newton–Raphson method is used,proceeding with small incremental loading steps.

Preliminary Step: Tightening of LeavesThe first computational step pertains to deformation of leaves before loading, dueto tying. It is performed to evaluate the initial strain in the link element. The resultconfirms a tightening force of 10 kN in each of the clamp bolts. By trial and error,the initial strain is found to be 0.22791. The deformation of the model due totightening is illustrated in Figure 9.15 and shows the leaves before and after tying.Because of tightening, the center clamp moves up 1.433 mm.

Final Solution: Incremental Loading. The maximum normal load is 6432 N. A two-dimensional plane-strain model isused, 1 mm thick. The load per millimeter of spring width, applied to half-spring, is

The load acting at the bottom of the center clamp is applied as a pressure on twobottom elements, with a length equal to the length of half the clamp. The length ofthe two corresponding elements is 50 mm. The pressure upon the elements equals

FIGURE 9.15 Deformation of leaves due to tightening, computed by the ANSYS program.

F /w 0.5643263.0------------× 51.048 N/mm= =



The loading process is divided into 40 equal increments of ∆p = 0.0255 MPa each. The computational results are shown in Figure 9.16. It follows that the spring

deflection at maximum load equals 142.70 mm. In a similar way, we find thedeflection with design load equaling 94.228 mm. The net deflection under the appliedload is smaller by 1.433 mm, due to the shrinkage caused by initial tightening (seeFigure 9.15). The load rate, based on the maximum load, equals

while the load rate that is based on the design load equals

The difference in rates is explained by nonlinearity due to friction and large dis-placements. Neglecting friction and bolt tightening, the load rate equals, respectively,

p51.04850.0

---------------- 1.021 MPa= =

FIGURE 9.16 Deflection of the leaf spring computed using the ANSYS program: (a) atdesign load, F = 4150 N, and (b) at maximum load, F = 6432 N.

k6432

142.70 1.433–------------------------------------ 45.531 N/mm= =

k4150

94.228 1.433–------------------------------------ 44.722 N/mm= =


Leaf Springs 279

Comparing SAE design results, we note lower values, due to considering nonlineareffects, neglected by SAE formulae. For computation of the load rate, we useEquation (9.6) multiplied by a factor of two (the equation presented above is for acantilever spring). The result is

Hysteresis

The loading history of the spring under consideration is presented in Figure 9.17.To determine the energy dissipation due to friction, the load is plotted against springdeflection, Figure 9.18. Energy dissipated during a loading cycle is expressed bythe area within the hysteresis loop.

k6432

160.47---------------- 40.08 N/mm= =

k4150

160.26---------------- 39.06 N/mm= =

k 22 20700 63 6.73×

12--------------------- 3

63 6.33×12

--------------------- 63.603

12---------------++××

570.03--------------------------------------------------------------------------------------------------------------------- 1.15×× 34.19 N/mm= =

FIGURE 9.17 Loading history of the leaf spring.



Precise Model

The most common failures in leaf springs are fatigue failures. To derive the lifeexpectancy of a leaf spring by means of a fatigue analysis, an accurate stressdistribution must be determined. For this purpose, a precise FE model of the leafspring is prepared. The model comprises small plane-strain quadrilateral elementscreated by subdividing the elements of the coarse model (see Figure 9.19). To

FIGURE 9.18 Hysteresis loop of a leaf spring computed using the ANSYS program.

FIGURE 9.19 Precise FE model of a leaf spring.


Leaf Springs 281

facilitate the stress analysis, elements bordering with the leaf surface are consideredto be thinner than those inside the leaf. (They are made 0.8 mm thick, correspondingto the depth of the layer of residual stresses discussed below.) The applied load inour computation equals the designed load, 4150 N.

The precise model of the leaf-spring is created and solved using the MSC.NAS-TRAN program. For contact simulation, it uses the penalty method. As before, thesolution proceeds in steps applying incremental loading. The resulting bending stressdistribution is presented in Figure 9.20. Figure 9.20a shows preliminary stressescreated at assembly due to bolt tightening, while Figure 9.20b presents the finalstresses caused by loading. One notes that maximum bending stresses differ fromone leaf to another. The bottom leaf has the highest stresses, similar to the examplediscussed above, Equation (9.16).

It is interesting to note how the FE results compare with those based on the SAEformula, Equation (9.4). According to SAE, the highest stress appears in the mainleaf, where it equals

Table 9.3 presents a comparison of stresses in all leaves computed by the FE methodwith those resulting from Equation (9.4).

One notes the large difference of stress values in leaf no. 1. Part of the differencestems from preliminary stresses due to tying of the leaves, a factor disregarded inthe SAE formula.

Shot Peening and Stress Peening

There is a special surface process, prior to assembly, used to increase the fatiguelife of leaf springs. In its simpler application, it is referred to as shot peening. Shotpeening induces compressive residual stresses near a surface of the leaf. Uponsuperposition with working stresses, the maximum bending stresses at the surfacedecrease, increasing the fatigue life.2,5 (See Chapter 7.) When shot peening is donewhile the leaf is subject to tensile stresses, the process is called the stress peening

TABLE 9.3Maximum Bending Stresses in Leaves

Leaf no.Max. bending stress, MPa

FE methodMax, bending stress, MPa

Eq. (9.4)

1 334.56 595.71

2 383.62 560.15

3 421.21 560.15

4 448.84 560.15

5 615.07 533.47

S570.0 67×

2 63 6.73×12

--------------------- 363 6.33×

12---------------------

63 6.03×12

---------------------++×------------------------------------------------------------------------------------------------------ 4150

2------------× 595.71 MPa= =



process. The magnitude of the residual stresses is then considerably larger. Stresspeening is usually done on one side of the leaf, while shot peening is applied toboth sides.

Figure 9.21 shows residual stresses in a leaf caused by both processes. The depthof the compressed layer is assumed to be 0.8 mm. The residual stresses caused bystress peening reach 550 MPa, while those caused by shot peening equal 400 MPa.

FIGURE 9.20 Bending stresses in a leaf spring computed using MSC.NASTRAN program:(a) after assembly and bolt tightening and (b) final stresses after loading.


Leaf Springs 283

To insert residual stresses into the FE model, we compute thermal stresses inducedby an imaginary thermal loading of the leaves; the thermal stresses produce the sameeffect as shot peening.

The distribution of stresses due to loading of a leaf spring processed by stresspeening and shot peening is shown in Figure 9.22. The figure presents a superpositionof working stresses upon the residual stresses in leaf no. 5, illustrating the beneficialeffects of both processes. One notes a substantial decrease of tensile stresses on thesurface of the leaf, which renders the desired prolongation of fatigue life.

The introduction of residual stresses changes stress equilibrium in leaves. Thestress peening process causes a nonsymmetrical deformation, resulting in a decreasedcurvature (see Figure 9.23a). The shot peening, when done on both sides of the leaf,causes a symmetrical deformation without any change in curvature (see Figure9.23b). Table 9.4 lists new and old curvature radii in leaves processed by stresspeening.

Windup Torque and Longitudinal LoadLeaf springs in vehicles are occasionally loaded by horizontal forces and torques inaddition to standard vertical loading. Leaf springs designed for this purpose must

FIGURE 9.21 Distribution of residual stresses in leaf no. 5 computed using MSC.NAS-TRAN program: (a) caused by stress peening on one side of the leaf and (b) caused by shotpeening on both sides of the leaf.



be provided with eye ends, shown in Figure 9.1b, to accommodate the horizontalloading.

Consider the deformation and stress distribution of a leaf spring subject toloading as follows:

Vertical load Fn = 4150 N

Horizontal load Ft = 3780 N

Windup torque T = 2,765,700 mmN

FIGURE 9.22 Superposition of working stresses upon residual stresses in leaf no. 5, com-puted using the MSC.NASTRAN program: (a) in a leaf with stress peening on one side ofthe leaf and (b) in a leaf with stress shot peening on both sides of the leaf.


Leaf Springs 285

The FE model is shown in Figure 9.24a. For the purpose of analysis, since thereis no symmetry, the half-spring model is extended to cover the whole spring. Thelocation of the left-end support is fixed, while the right-end support is sliding. (TheFE model is without the eye-end since deformation, and stresses in the eye end werenot computed.) The horizontal load and windup torque are applied at the centerclamp. The resulting spring deformation is shown in Figure 9.24b. The computedstress distribution is presented in Figure 9.25.

9.4 CONCLUSIONS

The design of a leaf spring includes consideration of geometric and frictional phe-nomena that present a nonlinear numerical problem. The problems are solved twicein the text, first using the finite element method, followed by the simplified design

TABLE 9.4Leaf Deformation Due to Stress Peening on One Side

Leaf no.Radius before

Stress Peening (mm)Radius after

Stress Peening (mm)

1 1603.0 1830.5

2 1481.0 1782.5

3 1399.0 1659.0

4 1359.0 1571.5

5 1300.0 1471.5

FIGURE 9.23 Deformation of a leaf caused by residual stresses computed using MSC.NAS-TRAN program: (a) in a leaf with stress peening on one side of the leaf and (b) in a leaf withstress shot peening on both sides of the leaf.



formulae. The comparison of the two distinctly different solutions confirms theanticipated discrepancies, explained by the inaccuracies of the linear equations. TheFE method is therefore more reliable.

The FE solution illuminated the following properties of spring leaf:

• The stress distribution depends on the leaf’s location, as the bendingstresses at the bottom leaves are higher than at the top leaves. The FE

FIGURE 9.24 Leaf spring subjected to vertical load, windup torque, and longitudinal loadcomputed using MSC.NASTRAN program: (a) loading of leaf spring and (b) deformation ofleaf spring.

FIGURE 9.25 Stress distribution in the leaves due to vertical load, windup torque, andlongitudinal load computed using the MSC.NASTRAN program.


Leaf Springs 287

analysis shows a very large difference between the stresses in the first andlast leaves (about 100 percent). See Table 9.3.

• Interleaf friction and tying of leaves, due to increased resistance, causeshigher spring rates, contrary to the simplified formula, which provided aconstant spring rate.

• Shot peening produces lower tensile stresses at the surface, enhancingfatigue life. Stress peening (shot peening, while the leaf is subject totensile stresses) produces better results; however, it causes a change ofcurvature of the leaf, since it is done on one side only.

• A comparison of results obtained by simplified equations and FE analysisshows a difference of about 30% of spring deflection.

The advantage of FE computation over that of simplified equations, such as SAEdesign formulae, is due to the ability to consider the effects of the following phe-nomena:

1. The leaves are engaged at the tips only.2. The friction between the leaves causes resistance to spring’s deflection.3. The applied load causes considerable geometric changes.4. Tying the leaves by center bolt and clamp makes the spring less flexible.

In addition, the FE method is able to analyze what, until recent developments, couldhave been evaluated only through tests, namely: (1) the energy dissipation andhysteresis caused by friction, and (2) the positive influence of shot peening and stresspeening.

However, the solutions to complex problems that deal with receding contact andsliding friction are not infallible. The accuracy of FE solutions must be criticallyexamined with regard to the following considerations:

1. precision of the FE model2. exactness of the iterative process3. awareness of penalty parameters and the resulting numerical errors (as

discussed in Chapter 6.)

Topic 3, above, was discussed in Chapter 6, explaining that evading the errors mayproduce an unstable solution that, by failing to converge, renders the probleminsoluble.

REFERENCES

1. Design and Application of Leaf Springs, Spring Design Manual, SAE AE-11, Part1, Society of Automotive Engineers, Warrendale, PA, 1990.

2. Almen, J.O., and Black, P.H., Residual Stresses and Fatigue in Metals, McGraw-Hill,New York, 1963.

3. Wahl, A. M., Mechanical Springs, McGraw-Hill, New York, 1963.


finite element based leaf spring design

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