1

Dr Andrei Lozzi Design IIA, MECH3460

School of Aerospace, Mechanical and Mechatronic Engineering University of Sydney, NSW 2006 Australia

DESIGN of SPRINGS Spring Lecture 017.doc

Refs: P Orlov, Fundamentals of Machine Design, 1977 Vol 5 Ch 4, MIR Moscow

Reshetov, Machine Design, 1977, MIR Moscow.

R L Norton, Machine Design, an integrated approach, 2000 Prentice Hall.

1 General overview of ‘stress analysis’ methods. The traditional approach to the modelling the stresses in any

particular type of machine component is to firstly determine what are the most significant loads that those type of

components are subjected to. We then derive mathematical relationships, that given the size and shape of a

particular component and the loads acting on it, we can arrive at equations that estimate of its stresses and

deflections. This is all rational provided the loads that we ignore are truly significantly small, and that the

components that we so analyse are then not subjected to loads that were never conceived in the original analysis.

The simplest models are obtained if there is only one significant load. For example, preloaded bolts-in-tension

(torqued up bolts), lead us to just the tensile stresses in the bolts. For bolts-in-shear, it is just the shear stress in the

bolts. For shafts, the loads are typically combinations of torques and moments, both steady and alternating. As a

consequence the ‘equations’ for shafts are more elaborate than for simpler machine elements. For most elements,

over the years a number of these relationships have been derived, the newer ones more accurate but usually more

elaborate than the older methods. With some care and with the use of modern computer tools, we can apply the

more faithful of these analyses to arrive at better and better designs.

Computer can be used to iterate through a range of variables, preselected by us, to arrive at that combination that

may give, for example, the lightest or cheapest design. The computer software has not invented the type of design

that is being analysed. It may only be able find the combination of dimensions and materials that will meet our

prescribed conditions. Sometimes with some enthusiasm we refer to these as ‘optimal’ designs, but they are not

truly optimal, because someone (hopefully not ever a computer) may think of a yet better design.

2 Overview of spring design. Springs are peculiar among the family of machine components is so far that their

deflection is necessary and relatively large. Often in other type of components we try to minimise deflection or we

may even be able to ignore it, but with springs we need it, we require it and we calculate for it. Deflection is as

fundamental to the design of a spring as is the applied force. An associated variable that is used as a yardstick in

comparing springs, is the ratio of these two, the force divided by deflection, which is the stiffness of the spring.

A good example of these variables is provided by suspension springs used in cars. The springs have to carry the

weight of the car at the wheels, that is cope with a force: F. They have to provide sufficient travel, to cope with

the sort of road the vehicle will traverse, ie a deflection: Δ. And, the ratio of these F/Δ together with the mass of

the car, determines the resonant frequency of the vehicle-spring system, that is a measure of the ride quality. Next

in the order of significance may be the volume available for the spring’s installation, the number of cycles that is

required in its service life, and also the cost allowed for it. The volume of springs assembled inside machines will

affect the overall size of the machines, furthermore although springs themselves may be cheap, if they prove

unreliable they will cause expensive breakdowns. For these reasons springs used inside machines are usually well

made, from high quality materials and compact in size. On the outside we may see cheaper and more voluminous

spring assemblies.

2

3 Some uses of springs in machines:

produce a near constant force take up clearances in mechanisms

store energy and act as a short term motor dissipate shocks and control oscillations

limit and calibrate forces

4 Atlas of spring types

Fig 1. Shown above is the wide range of constructions of springs, or more generally we may say, of machine

elements that provide smoothly increasing deflection - with an increasing load.

a) We have extension springs, which allow elongation proportional to tension. Coiled compression springs in

b) respond to compression loads. For both a) and b) the spring wire is stressed essentially just in torsion.

The concentric and conical Belleville springs e) & f) stress their elements in tension and or compression.

The coiled torsion and the spiral springs d) & g) stress the wire in bending, while

the torsion bar h) stresses the bar again in torsion. The leaf springs i), as seen in most trucks and older

vehicles, stress the leaves in bending.

Under the figure of the coiled compression spring – c) a number of wire cross-sections are shown. These begin

at left with the most common section ie a circular wire, then square and rectangular, and at right a multi-strand

section, ie as seen on steel ropes.

3

α

F

θ Δ

l=πDN

D/

2

5 Coiled compression springs. To begin with we are going to deal with just one type of spring, the coil

compression springs. If such a spring is well made and properly loaded the centre of the force will be aligned

with the CL (Centre line) of the spring. If we conceptually move the force from the CL of the coil to the centre

of the wire we retain the shear force F and develop the torque T=FD/2. Note that there is no bending moment

transmitted to the coiled wire, if the force or the centre of pressure at the ends of the spring, is directly over the

CL of the coil, as shown on Fig2.

Fig 2. Cross-section of coiled

compression spring with applied

force, centred on coil CL. Note Fig 3. The force is shown applied to

the free body analysis of the loads cranked end of the wire, resulting in a torque

at a section of the wire. down the length of the wire, causing an angular

rotation of θ and linear displacement of Δ.

A spring made as shown on Fig 3 would experience some bending deflection, only because just the

two cranked ends are subjected to bending loads. Most machine springs are not made like that,

hence bending deflection is not included in the calculations.

6 Shear stress in wire. The stress in the wire is

simply given by eq 1. Where J is the polar moment

of area of the wire = πd4/32. If the wire is not

circular in section then J is replaced by a ‘torsional

constant’ which will be numerically less than the

polar moment of area calculated for that section.

We use here the combined variable C =D/d - the

spring index, beloved by the spring industry. The

value of this index indicates the manufacturing

difficulty in making the spring and hence its cost.

J

dDF 22 eq 1

3

8

d

FD

eq 2

2

8

d

FC

eq 3

D

d

s + d

F

T = F D/2

F

4

7 Stress raising factors. Fig 4 at right is a view looking

down the CL of a spring. If the coils are displaced along

the axis of the spring by some distance, then the inner

fibres of the wire would be subjected to greater strain ε,

and hence greater stress then the outer fibres. If section b-c

is lifted up by δ above section n-m then fibre c-m would

have to strain or stretch proportionally more than fibre b-n

because it is shorter.

Fig 5 at left indicates the shear stress τa due to

torsion, together with the stress concentration

effect mentioned above. Note that the stress is

larger on the inside than the outside of the coil.

Shown also is the transverse shear stress τd, due

to fact that the applied force F is transverse to the

wire. We can apply a correction factor to eq 3, for

the transverse shear stress alone, ie:

)5.01(1 Ck eq 4

or we can have a correction factor that allows for

both the transverse shear stress and the stress

concentration, namely the Wahl factor:

CC

Ck

615.0

44

142

eq 5

from eq 3 221

8

d

FCk or

eq 6

Eq 6 then gives us the maximum shear stress in the circular wire of a coil spring, subjected to an axial load,

using a correction factor k1 or k2 as defined by either eq 4 or 5, depending whether stress concentration is

otherwise allowed for or not. The transverse shear stress will always be there and should always be allowed

for. That is: k1 should be used for ‘set’ springs, in which the negative magnitude of stress is trapped in the

wire as residual stress (see sect 9) and k2 for all other springs.

Fig 6. The Wahl factor is dependent on the helix angle α (Fig

3) at which the spring is wound. Fig 6 at left indicates the

effect of this angle. A spring with a great deal of space

between coils (s in Fig 2), suffers about 7% less stress

concentration than one that has its coils spaced close together.

This tells us that as a spring is increasingly loaded so will the

stress concentration increase. The effect is greatest for spare

open springs, increasing k1or2 by up to about 10%.

5

Spring wire strength

0

500

1000

1500

2000

2500

3000

0.1 1 10 100

Wire dia mm

Te

ns

ile

str

en

gth

N/m

m2

A227

A228

A229

A401

A323

8 Deflection. Based upon the model and loading

conditions defined on Figs 2 & 3 we can calculate the

angular rotation - θ of the length of wire - l due to the

applied torque – FD/2, and the linear displacement of the

cranked end of the coil Δ. θ Increases proportionally with

the torque T, the length of the wire l, and inversely with

the material stiffness G (the modulus of rigidity) and the

polar moment of area of the wire J. Δ Is of course the total

axial contraction or deflection of the spring, and δ is the

deflection per coil. We see from eq 9 that deflection is

proportional to the number of coils.

Note that J = π D4/32 only if the wire is circular.

GJ

lT

eq 7

2D eq 8

dG

NFC 38 eq 9

dG

FC 38 eq 10

33

4

88 C

Gd

D

GdFk

eq 10a

9 Spring wire static strengths. Spring wire is usually manufactured by pulling the steel bar cold

through dies of progressively decreasing diameters. With large wire diameters the stock may be rolled

between plates to reach the required sizes. The tensile strength of the material in the wire can increase by

a factor of 3, by the time it is reduced from about 100 to 0.2 mm diameter. The wire may then be left as

drawn, or it may be turned and ground to precise diameters, with a fine surface finish, and finally it may

be shot peened.

Fig 7. The graphic presentation of the strengths of typical wire material can leave us with a useful but

subjective impression of their differences. Graphs like Fig 7 can help us in making the decision of

selecting material, for the strengths and diameters required. But, when it comes to calculating for a

suitable solution to some design requirements, then we need equations that relate the steel grade to the

wire size and its strength. The costs of the material may vary by a factor of 4, considerably more than

their strengths (eg A401 ~ 2×A232, ~ 4×A227), but the choice of material is usually influenced by the

need for reliability as well as costs. The dearer steels have a better history.

6

0.5Sfw

0.5Sfw

Sfs Goodman line

Gerber parabola

τa

τm

The ultimate tensile strength Sut, may be estimated for a given diameter d, using eq 11, using the values of

coefficient A and exponent b provided in Table 1.

b

ut AdS eq 11

Cold drawn A227

Music wire

A228

Oil tempered

A229

Chrome Vanadium

A232

Chrome Silicon A401

A - coefficient 1753.3 2153.5 1831.2 1909.9 2059.2

b - exponent -0.1822 -0.1625 -0.1833 -0.1453 -0.0934

Table 1 Coefficients for evaluating the ultimate tensile strength of steel alloys spring wire.

Testing has established that for spring wire the ultimate shear strength in torsion can be reliably estimated

from its strength in tension, by eq 12. The yield strengths in shear on the other hand fits in with

expectations supported by the distortion energy theory and test results, eq 13:

utuS SS 67.0 (R L Norton eq 4.4) eq 12

ytyS SS 577.0 (Shigley eq 5-21) eq 13

8 Spring wire fatigue strength. Compression springs are tested by SAE and others, by applying a

repeated compressive load, from a maximum force to almost 0. Spring compression tests vary the shear

stress from a maximum to 0 with a mean half way. Fig 8 a) shows the type of results obtained from an

SAE test, giving the maximum stress that the spring may fail as Sfw, giving mean and alternating stresses

that are equal to half of Sfw. Fig 8 b) show those values plotted on a Goodman diagram. If one uses the

Goodman line to estimate the fatigue strength in shear Sfs of the wire, the value so obtained would be

higher than the true Sfs. Statistical data indicates that the scatter of fatigue results are better represented by

the Gerber parabola than by the Goodman line. Hence a straight line through the point (0.5Sfw, 0.5Sfw)

would give an exaggerated value for Sfs. I suggest that for a first and reasonable approximation we may

take Sfs~0.5Sfw but more precisely from Eq 15 of the notes on fatigue, with FS=1:

))/(1/(5.0 2usfwfwfs SSSS eq 14

Fig 8 At left in a), the shear stresses from the repeated compressive load on a coil spring are shown,

and at right in b) , the resulting stresses are plotted on a Goodman or mean-alternating stresses diagram.

Note that if the fatigue strength has been determined for 106 cycles or more, Sfs should be renamed Ses .

τm

τa

Time

Stress

Sfw

0

7

Ssy

τ

Ta Tb Td

τ’max Ssy Ssy

9 Prestressing, residual stresses and the Setting of a compression spring. Residual stresses may be

imparted to springs, by overloading them in the direction of the loads expected in service. This will

result in a better stress distribution during normal use. Fig 9 below shows the shear stress on the section

of the circular wire of a colied compression spring, being subjected to an axial load. In a) the resulting

torsion Ta in the wire causes shear stress that is below the yield condition Ssy. In b) the torsion Tb is

sufficient to cause significan yielding. c) Shows the wire when torsion Tb is removed. The yielded

material just by itself cannot return to its original location because the remaining elastic strain in it is

insufficient. The material at the inner section of the wire, which has been under only elastic strain, on

the other hand can return to its original location, ie at the vertical axis. For the section to reach a

balanced state in torsion some inner stresses will be trapped to balance stresses in the outer section.

About 10–30% improvement in the effective ultimate and yield strengths of the wire is possible by

‘setting’ or overloading the wire during manufacture. For a compression spring this is achieved by

making the spring with sufficiently large spaces between the coils, that it may be compressed to a

height where the wire begins to yield - Fig 9 b). This has to be arranged so that when the overload is

released the spring will settle at the desired free length. Some residual stresses then reamain in the wire

– 9 c). For springs that have been preset the stress concentartion factor k1 is used in eq 6 to calculate τ.

a) b) c) d)

Fig 9. Shown in a) is a torsion that causes only elastic stresses. In b) the torsion causes the yield

condition to be exceeded, resulting in some plastic strain to take place in the outer section of the wire.

In c) the ‘setting’ torque Tb is removed and since the material cannot now everywhere return to its

original locations, some residual stresses shown are trapped in the section. In d) A new torque Td < Tb is

applied that would have otherwise caused τ > Ssy had no setting have taken place. This torque now

produces a more uniform stress distribution within the wire, comparing a) and d).

10 Endurance limit. Shot peening is one of the few additional options available to improve the

fatigue life of a spring, but there are limits. High strength steels show a clear trend towards an upper

boundary to their endurance strengths independent of their static strengths. That trend particularly

applies to spring wire because the material in them ia some of the strongest available:

Unpeened wire 300esS Nmm-2 eq 15

Peened wire 460esS “ eq 16

Some manufacturers of modern engine valve springs claim values about 10% higher than these.

Technology marches on but higher values should be adopted only after adequate verification.

8

FT

P

a

b

c

FT

S

a

c

b

11 Stiffness. One of the remaining critical variable of springs are their stiffness. In section 6 above we

calculate total deflection simply by eq 9 and deflection per coil by eq 10. From these we can arrive at

the stiffness:

NC

GdFk

38

, where G = 80.8 103 N/mm-2 , for all steels eq 17

This equation shows that the stiffness does not depend on material strength Su, that it is the same for all

steels and that it is inversely proportional to the number of coils. If you were to cut a coil or two from

the front suspension springs in your car (to make your Torana or PJ Holden look mean) you will reduce

the front ground clearance but simultaneouly also will stiffen the front suspension thus limiting the

chances of the car bottoming. I strenuously advise you not to do that, but some UWS students have

been seen in cars that look like that.

12. Springs in parallel and in series. In Fig 10a) force FT is

equal to the sum of the forces provided by the individual springs,

and since the deflection P is the same for all springs: .

cbaT FFFF eq 20

PcPbPaT kkkF a

iPT kF b PPT kF c

iP kk eq 21

Fig 10 a) elements in parallel, upper left. Where kp is the

stiffness of the set of springs in parallel, that is: the stiffness of

springs in parallel is equal to the sum of the individual spring

stiffnesses

Fig 10 b) elements in series, the figure at left shows springs in

series, where now the whole force FT is transmitted through

each spring and where the total deflection is the sum of each

individual spring’s deflection:

iiST kF a

cba

T

S

Fk

b

T

cba

S Fk

1 c

iS kk

11 eq 22

We may say that springs act in an inverse manner to that of resistances in

electrical circuits. We may also conclude that for a set of springs in paraller, the

total stiffness will be greater than the stiffest spring. For springs in series, the

total stiffness will be less than the softest spring.

9

13 End preparion. Coiled springs formed as just simple helical coils (Fig 11 a), can only be used in

the least demanding applications. To ensure that the centre of the pressure, or the axis of the force,

applied to the end of the coil is aligned with the CL of the coil, the circular end of the spring has to be

uniformely loaded. On the other hand Fig 11 shows increasingly effective spring end preparations, used

to provide improved uniform force distribution around the end of the coil.

a) b) c) d) e) f)

Fig 11. a) Shows a simple cropped hellical coiled spring, a force applied to the cropped end would

produce a considerable amount of bending stress, as well as the expected shear stresses. Style b) is

similar to a) except the end is now ground to the full depth of the wire. For the wire diameter d and the

spacing s used in this figure, one quarter of the end of the coil shown here could shares the applied load.

But in effect the thinned down wire tends to easily bend and possibly break. In c) the wire at the end is

closed or bent for about ¾ of a winding, to touch the next lower coil and is better able to share the

force. In d) the end wire has been closed as in c) but in addition it has been wound touching the adjacent

coil for about an additional half turn. The top half of the additional winding has been ground off, as

shown by the ghosted outline in d). For e) the same preparation is applied as in d) with the addition that

the helix angle has been progressively reduced as it approaches the end of the spring, and in f) that has

been extended further.

14 A good spring In calculating the length and mass

of a spring it is necessary to estimate the number of

active and inactive coils. The latter are those that play

no part in determining the stiffness of the spring, but

add to its length. The end preparations shown in Fig 11

may be said to have the following number of inactive

coils per end: a) & b) 0, c) ½, d) 1, e) 1½ & f) 2 or

more. Fig 12 at right provides a side view of a spring

that has initially 1½ inactive coils per end. Because of

the varying helix angle the stiffness will increase as

the spring closes under load. The front springs of some

motorcycles employ this feature to the extent that

about ½ of the coils will close in a severe bumps,

increasing stiffenss by a factor of 2.

10

15 Spring index – C. Springs of 6

11

Fig 15. Greater force can be

generated with the use of

concentric parallel springs.

Greater travel by concentric

Series springs

Fig 16. Methods employed to

dampen vibration in the coils.

Variable helix angle dampens by

coils touching, variable C spreads

the resonances, and an external

clip, dampens by rubbing. The

first and the last require

lubrication to minimise wear.

Fig 17. Tapered springs may provide variable

stiffness for variable loads, as in cushions.

For rectangular wire section, large forces and

Internal space for other components to fit.

Fig 18. Springs tend to grow in outside diameter

as they are compressed, hence space has to be

provided to prevent jamming.

Fig 19 above, a number of saddles are shown which reduce the

end preparation required to load compression springs adequately.

These saddles may be made from semi rigid rubber or plastic.

Fig 20 at right demonstrates that the work of a compression

spring can be preformed by other elastic members, here a valve

compression spring is replaced by a balanced torsion spring.

12

A method for designing a compression spring

Select the following variables according to the application:

1 Select an appropriate spring index – C : a small index < 5 means a compact but

difficult spring to make, a large > 10 will be cheap, large and imprecise.

2 Select the end preparation giving the no of inactive coils : Ni ≥ 3 is very good, Ni

~1 is the minimum for flat end surfaces. Use better end-preparation for the more

demanding installations.

3 The minimum (decimal) space between wire coils – s : to ensure that the wires

never touch, when a spring is fully loaded, s ≥ 0.2 for high quality springs, s > 0.5 for

cheaper springs. Springs should never be loaded to a packed condition, a buffer, possibly

made from rubber, should take the load before a spring closes solid. And, the minimum

load should not go to 0, but a certain small preload should prevent movement under

vibration.

4 For wire material quality - Sus and Ses : use the top quality for springs that are

assembled within machines and have high number of cycles, use lower grades wire

material for external uses,

We need to know 3 of the following conditions:

5 Forces : maximum, minimum, average, preload.

6 Stiffness: constant or variable.

7 Deflection : or change in length between specified forces.

8 Length of spring : at max force, at preload, or when unloaded.

We calculate for:

9 We must know some of the variables in 5 to 8 above, we calculate for the others.

10 We may have to meet boundary conditions eg max height, min inside or max

outside dia., fundamental frequency etc

11 Our objective function may be minimise mass, volume or cost.

12 Effect of production tolerances on design variables to ensure proper functions.

We use the following equations, only 2 of the first 3 are independent:

11 Shear stress due to the torque in the wire: 221

8

d

FCk or

12 Deflection due to force F: Gd

NFC a38

13 Stiffness (note no reference to Sus or Ses): aNC

Gdk

38

14 Min. spring length, or length at max. load: dNsddNL ia )(min

15 Spring free length k

FLL minmax