1

Dr Andrei Lozzi

Design II MECH 3460

School of Aerospace, Mechanical and Mechatronic Engineering

Dynamically Loaded Bolts

Refs: R L Norton, Machine Design …, Preston. Bolts lecture AL 017

J E Shigley et al, Mechanical Machine Design, ed 4 to 10, McGraw Hill.

P Orlov, Fundamentals of Machine Design, Vol 5, MIR Pub Moscow.

D N Reshetov, Machine Design, MIR Pub Moscow.

G Pahal & W Beitz, Engineering design, Springer

1. An overview: the figure below shows the locations of typical tensile fatigue failures that occur to

machine bolts. That is, bolts that have been preloaded (torqued up) and which have been subjected to

external alternating tensile forces. These failures predominantly occur because of one or more of the

following three conditions: excessive preload, excessive alternating loads transmitted to the bolts by the

flanges, and higher than necessary stress concentrations. Many dynamically loaded bolted joints that have

failed could have been corrected by improving some or all the above conditions. The situation is not

simple, and there are no completely ideal designs, but there are some that clearly are better than others.

The aim of this set of lectures is to turn you into the engineer that can design a bolted joint like this that

will suit a bridge, a truck, a high performance engine, an aircraft or even possibly a space vehicle.

Fig 1. Shown is a distribution of fatigue failures in plain shank bolts that have been preloaded and subjected to

alternating external tensile loads. The likelihood of failure is indicated by the high level of stress at 3 locations. The

percentages show that fatigue failures are random in nature and are best described statistically. Each dark and light

photoelastic contour represent areas of uniform stress, crossing a contour reflects a change in stress level. Locations

where the contours are packed closer together than in others, reveal areas of stress concentration.

2

Fig 2 a) b) c) d) e) f)

In Fig 2 a) above is shown the tensile stress distribution in a poorly designed bolt stem. Note that the stress level at

locations 1 & 2 do not have the benefit of there being a fillet adjacent to the head or at the first thread under the nut,

therefore the fillets that are there are less effective than they could have been. From b) to e) the stem is reshaped so

that the shoulders in the stem are removed away from the ends allowing more effective fillets to be placed there.

Note that the areas of enlarged diameter along the stem, is used to center the bolt in its clearance hole and maybe for

the bolt to act as a dowel pin. Finally f) (from a German text!) the top of the external thread of the bolt, is above

the bottom of the internal thread of the nut. By this means the stress concentration in the first threads of the bolt is

reduced. From the upper f) it appears that load at the first thread can be reduced from 35 to 25% and 35 to 15% in

lower f) if using a special nut. That is the concentration can be reduced by ~28% and even by ~56% over the use of a

simple nut like that in 2 a).

2. No shear force. Bolted flanges in machines should be designed so that the flanges are clamped

together with sufficient force that no expected external transverse force (perpendicular to CL of bolt) can

cause the flanges to slip laterally with respect to each other. In other words, the clamping force multiplied

by the coefficient of friction at the flange’s interface, must be larger than the transverse force, which

would place the bolts in shear. The bolts discussed here must always be only in tension. Pins, keys,

serations and steps are some of the features employed to prevent slippage, when its occurance would be

particularly detrimental to machine operation and these will be treated separately.

a) b) c) d)

Fig 3 Design features employed to prevent shear or transverse forces from displacing flanges laterally. These may

be used when the clamping force and friction at the interface cannot guarantee that no displacement will take place

and or where such displacement would be very detrimental. From left to right we have a rectangular key, cylindrical

sleeve, a transverse pin (which may have been drilled for after the flanges are assembled), and best but most

expensive – serrations. Other common features are steps, such that a bearing cap may be trapped laterally between

them, and interference fitted dowel pins, drilled parallel to bolts.

3

Pitch p

D

F cos()

F cos()

F sin()

F

p

D

3. No bending moment. Preloaded bolts must also not be subjected to bending. Bending moments may

be transferred to bolt shanks if the flange faces are not sufficiently parallel, or if the external load causes

one flange to pry (lever up) with respect to the other. You have to look out for these conditions and either

eliminate them or deal with them.

Fig 4. Spherical washers used under a screw at

left and at right on one side or both sides of the

flanges. The spherical radius of the washers

can be chosen so that the relocation of the bolt

can be done relatively easily. The washer at left

also allows for a large fillet radius under the screw,

as compared to the use of a flat washer.

4. Screws as wedges. Fig 2 below represents threads as helixes that slide in contact, while rotating with

respect to each other, about their common CL (Centre Line).

a) b) c)

Fig 5. a) An angled view of a single helix. b) A side view of 2 helixes rotating and sliding along each other,

representing an external thread sliding on an internal thread. D is the representative pitch diameter, p the pitch or the

spacing between succeeding turns (measured parallel to the CL) and the helix angle. If a single circuit of the

helixes in b) is rolled against a flat plane we get the wedges shown on c) where the axial force between the threads F

is indicated. The tangential and normal components of F at the threads’ interface are: Fsin() and Fcos(). The

frictional resistance to the sliding of one helix on the other is of course: Fcos().

Fig 5 c) indicates that the 2 screws (or wedges) )sin()cos( FF Eq 1

would unscrew (or slide) under the influence of

the axial force F, if the tangential component of F is )tan(

larger than the frictional force resisting it (Eq 1). For small : (in rads) Eq 2

4

p/4

R/cos30

R

p

p/8

p/4

Flat crest

60°

Fillet at

root

Dmajor

Dminor

Dstress

5. Self loosening of bolts. Taking ISO metric thread M20 as an example, it can have 2 pitches: 1.5 and

2.5 mm. From Fig 2c) we then have p=1.5 or 2.5, diameter D=20, giving helixes angles ~0.02 and

~0.04. Now, the coefficient of static friction between oiled smooth steel surfaces is expected to be in

the order of 0.4 to 0.6. Therefore from Eq 2 we should never expect an M20 bolt to come undone by

itself, because friction in the threads should be about 20 times too large. Yet without taking special

precautions bolts do come undone, and simple observations indicate that machine vibrations contribute to

this. We will return to this, but meanwhile note that finer threads are more resistant to coming undone.

6. Standard threads. Figs 6 a) & b) indicate some features of modern screw threads. When selecting

fasteners you will quickly notice that not all possible diameters and pitches are made available, for

general purposes over-the-counter sales. Round numbers are preferred such as M10 & M20 (and 2” &

2½”), with in-between diameters selected so that the thread cross-sectional area varies by about 20 to 30%

between adjacent sizes. Universally two standard series of threads are offered for any one diameter, one

with relatively small pitches or fine threads and the other with large pitches or coarse threads. Non

standard extrafine or extracourse are also possible. Mass produced threads are cold rolled onto the stem,

which provides about 10% better fatigue strength than if the threads are cut.

Coarse threads are more robust when handled and more resistant to thread crossing. They are preferred on

the outside of machinery. Fine threads provide greater cross-sectional area to withstand loads and as we

have seen are more resistant to undoing. Fine threads are seen predominantly on the inside of machines.

The stress diameter: Dsress is the dia of a cylindrical bar of the same tensile strength as a threaded bar.

This dia is about 25% of the distance from the minor to the major dia.

a) b)

Fig 6. Shown is the form typical of modern threads, the 60° included angle, fillets at the roots and flat crests. The

root fillet is very important in limiting stress concentration, in particular applications this fillet is made larger and

better finished than standard. Concave outlines give rise to stress concentration, whereas convex corners such as the

crests have low stress but if left sharp may cause injury and damage. In b) we see that in a real threads the normal

forces between thread faces is larger than the Fcos() shown in Fig 5 c), by a factor of 1/cos30°, or about 15%

larger.

7. Specialised fasteners. For modern industrial machinery, and even for many consumer products, the

standard shapes, diameters, pitches often and even thread forms do not provide sufficient choice. Because

the size of the fasteners influence the size of joints and these in turn influence the size of the whole

machine, manufacturers often define specialised fasteners. Bolted joints are becoming more varied and

5

-Fi

Fi mi

bi p

p

Fb

-Fm

Fm= Fb=0 Fi =-Fm=Fb P

P

Q

Q

Fb

Fb=Fi+rP

Fm=-Fi+(1-r)P

Fm=0

sophisticated. Today one sees bolted joints in trucks that one would only have seen on aircrafts a few

years ago. Nothing is quite ‘agricultural’ any more. Many manufacturers intentionally make critical

fasteners in their machine difficult to mistake and almost impossible to replace with cheaper over-the-

counter items. This trend has lead to better performing machinery, making manufacturers safer from

litigation and possibly more profitable.

8. Stress & Strain in a bolted flange. The bolt and flanges on Fig 7 a) to d) are shown in four stages,

progressively from an unloaded to an overloaded state.

Fig 7 a) Here the flanges, bolt and nut are closed together, that is removing all clearances, but the bolt is

not torqued or pretensioned. In b) the bolt is pretensioned. In c) the flanges are subjected to an

appropriate external load P. Finally in d) the external load Q is excessive, the bolt pretension is overcome

and the flanges are parted. These figures exaggerate real deformations but not as much as you may think.

Bolts are at times referred to as ‘elastic elements’, because the extension of a reasonably preloaded 100

mm bolt, can be detected using a steel rule, by eye.

Fig 7 b) The bolt has been pretensioned. The preloaded tension Fb= Fi in the bolt is balanced by an

equal an opposite compression in the members Fm=-Fi. These loads stretch the bolt by δbi and compress

the flanges by δmi. Since the bolts as a rule have smaller cross-sections than the members, the bolts

extend more than the members contract, ie δbi > δmi

Fig 7 c) An external load P is transmitted to the whole flange and bolt assembly. While this assembly

remains clamped together, the preload Fi is effectively no more than some residual stresses buried within

the assy. P will be resisted in part by incresed tension in the bolts and in part by decreasing the

compression in the members. The beauty of a properly performing preloaded joint is that the bolts are

subjected to only part of any external loads. More significantly, if the external load alternates the bolts

will be exposed to a fraction of that alternating load, P. That fraction: r will be the ratio of the stiffness

of the bolt to the stiffness of the whole flange assy. As shown on figure c) the bolt load is Fb=Fi+rP

where r < 1. Note that δp is the (additional) extension common to both the bolt and of the flanges, when

external force P is applied, and also that δp is the same for the bolt and flange.

Fig 7 a) b) c) d)

6

FT

P

a b

c

FT

S

a

c

b

Fig 7 d) Finally, the external load has become excessive for the preload initially imparted in the bolt, to

the extent that the clamping force between the members has been completely relieved, ie Fm=0 and the

flanges have separated. This separation is itself detrimental to the most machine, because leakages can

take place and foreign material can enter, but almost certainly a worse effected than that, is that the whole

load Q will now act just on the bolt, drastically reducing its potential fatigue life.

9. Springs in parallel and in series. In order to arrive at a model of how the external load is divided

between incresed bolt tension and decreased member’s compression, we will examine how springs in

parallel share a load, we can then arrive the fractions r of the external load experienced by the bolt.

Fb=Fi+rP Eq 3

Assuming that the bolt and its surrounding flange members

both behave as linear springs elements, which is quite

reasonable while they are stressed within the yield

condition:

kF Eq 4

In Fig 8a) The FT is equal to the sum of individual forces

provided by the springs, and since the deflection P is the

same for all springs:

cbaT FFFF Eq 5

Fig 8 a) elements in parallel PcPbPa kkk a

iiPT kF b

PPT kF c

where: iP kk Eq 6

We conclude that the stiffness of elements in parallel is

equal to the sum of their individual stiffnesses.

Fig 5 b) shows springs in series, where now the whole

force FT is transmitted undiminished through each spring.

The total deflection is the sum of each individual spring’s

deflection:

c

cbaS Eq 7

iiST kF a

cba

T

S

Fk

b

iS kk

11 c

Fig 8 b) Elements in series i

Sk

k/1

1

Eq 8

We conclude that for elements in series, it seems easiest to state that the inverse if the total stiffness is the

sum of the inverses of the individual elements’ stiffnesses, ie Eq 7 c.

7

10. A few comments on stiffness. For a system of elements in parallel, the total stiffness will be greater

than the stiffness of any one element. If there is a big disparity between the stiffness of the elements, the

stiffest element will effectively dominate. With elements in series, the opposite is true, the total stiffness

will be less than the least stiff element, and if there is a significant disparity in their stiffness, then the

softest element will dominate. You can probably come up with a few everyday examples of these

situations, like if you put your pilow on a solid floor, it will not matter much if the floor is from

hardwood, concrete or steel, the softness of the pillow will dominate.

11. Bolted Flange. Returning to Fig 4 c) we note that when the external force P was applied to the

flange assembly, the bolts and flanges both expanded by the same additional amount P. The bolt and

flange here acted in parallel. Therefore the force on the bolt is the preload plus that required to

extend the bolt by P bPib kFF Eq 9

since P was applied to both the flanges and bolt, b

P

ib kk

PFF a

and the stiffness of the parts in parallel kP using Eq 6. b

mb

ib kkk

PFF

b

The fraction of the external load that the bolt experiences, Eq 3 is: mb

b

kk

kr

Eq 10

12. Stretchy bolts & stiff memebers. An obvious observation of Eq 10 is that when dealing with an

alternating external force ( P), the way to reduce the alternating stresses on the bolt, is to design a joint

with relatively more elastic bolts (low kb) and relatively stiffer members (high km).

13. Stiffness of bolts. Assuming that the shank of the bolt, that exists between the faces of the flanges,

can be characterised as a bar of one diameter, then the elongation of that

bar will vary proportionally to the force and the bar’s length, AE

Flb Eq 11

inversely with its cross-sectional area and its modulus of elasticity.

Rearranging equation 4 as applied to the bolt: b

b

bk

F

substituting into Eq 11, we get that the stiffness of the shank of the bolt: l

AEkb Eq 12

14. Stiffness of Flange members. The stiffness of flanges is a much more interesting. Before so much

science was applied to stress analysis, there was a simple rule that said, that within a solid body the stress

distribution coming away from a point load, is contained within a cone of 60° included angle.

Fig 9. Finite Element Analysis and

experimental observations support that

rule. The bolt only compresses, somewhat

nonuniformly, the material contained

within 2 cones of about 30°, surrounding

the CL, starting from both ends of the bolt.

This 30° angle is slightly conservative,

possibly 32° would be more accurate for

most materials. In the past 25°, 33° and

45° have been used in calculations.

30° cone

8

Clearance

bolt hole

Washer dia

Max clamped circle

dia at midplane

Fig 10. The volume of the flanges that is compressed

by a bolt can be described as 2 hollow truncated cones,

sometimes referred to as frustrums. This is shown in

side view on Fig 9 and in pictorial view here at right.

Apart from the conical surfaces the boundaries are

the clearance bolt hole and the washer diameters

under the head of the bolt and of the nut.

If common bolts are used, the effective washers

diameters are the circular sections under the hexagonal

bolt head and nut, being about 1.5 the bolt shank dia.

Larger diameter washers may be used, but to contribute to

stiffness of the truncated cones they have to be thicker than

the usual washers, see Fig 12.

15. Stiffness of Flanges. The stiffness of these hollow cones can be calculated, but one must be carefull

that these values be used only if the flanges are sufficiently wide to contain the full cones. Note that there

are situations where lighter and more economic assemblies are obtainable by using narrower flanges and

bigger bolts. The stiffness of the cones on Fig 10 can be evaluated by applying the differential version of

Eq 11, applied per unit force

where dx is the thickness of a circular disc slice through )(xAE

Fdxd

Eq 13

the cone and A(x) is the area of that slice.

d is integrated for one cone, then summed for

For two cones, as in Fig 10, using Eq 8, we get:

))(tan(

))(tan(ln2

tan

ddddl

ddddl

Edk

ww

ww

m

Eq 14

using l as the bolt grip length,

the cone angle of 30°, d bolt hole dia and

for common bolts, washer dia dw = 1.5d, we get:

))5.25774.0(

)5.05774.0(5ln(2

5774.0

w

w

m

dl

dl

Edk

Eq 15

16. Fraction of external load added to the bolt. Eq 10 gave the ratio of the external load transmitted to

the bolt. As noted this ratio can be made small if we can design a joint with bolts of relatively low

stiffness and flanges of relatively high stiffness. We must also keep in mind that we will need sufficient

cross-section of bolt or bolts to provide a preload (with a safety margin) that cannot be overcome by the

largest reasonably expected external load, as represented by Fig 7 d).

Eq 12 indicates that we can decrease the stiffness of bolts if we make them longer and or reduce their

diameter. The flanges will also become less stiff as we increase their thickness (a fact that may not be

obvious from Eq 14), but their stiffness will fall off much less quickly then the stiffeness of the bolts.

Consequently if we use more but thinner bolts in thicker flanges, r of Eq 10 will decrease, improving the

bolts’ fatigue life. These have been the sort of joints one has become accustomed of seeing in aircraft

machinery but which has of late been more common in well made ground based vehicles as well.

9

Fraction of an

external load applied to a preloaded steel bolts

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0 2 4 6 8 10 12

Bolt length / diameter

par

t o

f ex

tern

al lo

ad o

n b

olt

Steel members

Aluminium members

Cast iron

The graph on Fig 11 below evaluates Eq 10 using Eqs 12 & 15, to provide an overview of the advantages

of relatively long bolts. It also shows the detrimental effect of using flanges material less rigid than steel,

ie aluminium and cast iron. Since km varies as the modulus of elasticity of the flange material, there are

deliterious consequences when softer materials are used for the flanges with steel bolts. In rare and smart

designs, light alloy bolts are used in aluminium alloy assemblies (Porsce I believe).

Using steel a bolt of length equal to its diameter, it will experience ~33% of the external load. A 10 times

longer bolt only sees 9% of that load. If cast iron or aluminium flanges are used, the bolt sees about 2 to

2.5 times those percentages.

Fig 11. Graphs of the fraction of the external load, that is transmitted to the bolts, versus flange thickness, where the

flange thickness is given as a ratio to the bolt diameter. Graphs for flanges made from steel, cast iron and aluminium

are shown, all using steel bolts with washer dia = 1.5 of the bolt shank dia.

Fig 12. a) b) c)

17. Gaskets and flat washers. Since compressive stress exits only within the hollow cones, the clamping

pressure at the flanges interface in Fig 12 a) cannot be uniform. In fact after some preload is applied to

those bolts the flanges will separate and a gap between the flanges will open, between the compression

cones. This indicates that if the bolted joint contains high fluid pressure and a compression gasket is used,

then the cones may have to merge, by using more bolts or thicker flanges.

10

Fig 12 b) highlights the fact that the normal steel washers do not expand the top cone diameter, because

they are too thin. Normal steel washers protect the flanges from scoring and provide smooth surfaces and

relatively low coefficient of friction for the nut to turn on. In c) a thick steel washer effectively expands

the cone and at the same time extends the bolt.

18. Force deflection characteristics of a bolted joint. Fig 13 below represents diagramatically the 4 states

of the bolted joint shown on Fig 7. This diagram is a little unusual because it is made up of 2 diagrams,

almost mirror images of one another. On the left of the vertical line through T is the tension extension

diagram for the bolt, on the right of T is the compression contraction diagram for the members. The

broken line from the left origin Ob through T shows the linear response of the bolt, on the right of T

the broken line from origin Om is the response of the flanges. As you should expect from Eq 7, the line

Om T has a gradient of kb and the line Om T has a gradient of km.

On the vertical axes are plotted forces, at far left bolt tension Fb, at the right flange members compression

Fm. On the horizontal axis, from the origin Ob to the right, the bolt extension is plotted b. from the origin

Om moving to the left, the member’s contraction m.

Fig 7 a), that is the state when the joint is just closed but not loaded, is represented here by the points Ob

for the bolt and Om for the members. The joint in a preloaded state, Fig 7 b), is indicated by the point T.

At T the bolt tension and member’s compression are balanced, equal and opposite to the preload Fi.

Fig 13. A force deflection graph for a bolted joint in the states portrayed in Fig 7 a) to d). The above diagram is in

principle 2 similar graphs mirrored about the vertical line through T. On the left of T are the bolt tension-extension

axes, on the right is the flanges compression-contraction axes.

Fb

Fm

Fi

FbP

Fi

FmP

mi bi Ob Om

p

P Q

mQ=0

a

b

T

11

The Line P is proportional to the external force that is transmitted to the flanges. P increses the bolt

tension and decreases the members compression. Note that P together with the new bolt tension FbP and

members compression FmP are of course in equilibrium. Because of the gradients of the dotted lines we

can argue that portion a of P is proportinal to kb, and portion b of P is proportinal to km, therefore:

For the bolt: Pkk

kFP

ba

aFF

mb

b

iibP

Eq 16

and for the members: Pkk

kFP

ba

bFF

mb

m

iimP

Eq 17

Note that of course Eq 16 is in agreement with Eq 9 b), while derived by analogous means. In paragraph

11 above it was pointed out that if P is a variable load then to improve the fatigue life of the bolt, aside

from just making everything bigger and heavier, we may design a relatively more elastic bolt and stiffer

flanges. This is exactly what Fig 10 and Eq 16 show that the fraction of P on the bolts is proportional to

kb and inversely to km. If a high preload is required then it can be achieved by simply using more bolts.

19. Reduced Clamping Force The down side of this phenomenon is that improving conditions for the

bolt leads to decreasing the clamping force between the flanges, as given by Eq 17. This has to be

considered in your designs, some residul clamping force is absolutely necessary. Clamped mated faces

tend to wear and embed together when subjected to vibration, thereby reducing the bolt elongation and

consequently of course their preload. Furthermore mating faces tend to oxidise, undergo electrolitic

reactions and otherwise erode. If an open gasket is used, then be warned that they tend to pack, crush or

otherwise loose stiffness thereby becoming thinner, of course also reducing bolt preload. For better

longetivity it is better to use a compression gasket that is contained in a grove. Then changes in the gasket

will have minimal effect on bolt tension.

20. Stresses and strains. We are going to analyse the fatigue life of a bolt under the assumption that the

load imposed on the bolt can be represented as a combination of a mean load (constant in time) and an

alternating sinusoidal load that is added to it, as pictured in Fig 14.

The external load could vary sinusoidally between limits eg: -P to +P, 0 to +P, or –P to 0. Eq 18

The upper and lower limits substituted in Eq 16 to give: Fbmax and Fbmin Eq19

The mean and alternating force on the bolt stem is then: 2

minmax bb

bm

FFF

Eq 20

2

minmax bb

ba

FFF

Eq 21

The mean and alternating stresses at the threads, shown on Fig 11: s

bm

mA

F Eq 22

s

bt

aA

F Eq 23

Where As is the stress cross sectional area of the thread, based upon Dstress shown on Fig 6. As is available

from tables, can be calculated or can be estimated.

12

a

m min

max

a

Time

Stress Fig 14. Shows the stress - time variation in a preloaded bolt, to

which a sinusoidal external load is applied. Note that m may be

+ve, zero or –ve but a is always +ve. A more complex a(t)

can be approximated by a number of sinusoidal components, ie

by a Fourier transformation. Each of these components may be

in turn examined in a similar manner to that outlined here.

21 Preload. It is suggested that a preload of 75% of proof stress be used for the typical reusable

connection and up to 90% for those joints that will remain permanent connected or rebuilt very few times,

when possibly the fasteners will discarded. Practical methods of preloading or pretensioning bolts are:

a) Torque wrench. Preload or pretensioning of bolts is most commonly done with the use of torque

wrenches. Unfortunately studies of the resulting tension induced in bolts stems, with unlubricated threads,

exhibit about a 15% standard deviation from the mean tension. This reduces to about 8% for lubricated

bolts using oil. This leaves us realistically at best with an uncertainty of about 20% (2.5 sd) in the

preload of the bolts if a torque wrench is used. The use of graphite greases or surface binding lubricants

like molybdumenn disulphide can greatly reduced this uncertainty. Obviously torque wrenches are very

practical but, only give a moderate level of confidence unless used by knowledgeable personel.

b) Turn angle. A more reliable means of determining preload can be achieved when the components

that are clamped together are well made, with flat parallel faces and with no foreign material trapped

between them. Here the nuts are turned by hand until all clearance is removed and a rapid rise in

resistance to further turning can be detected. The nuts are usually then torqued to a low preliminary level,

followed by turning the nuts by a prescribed angle (~180 to 360) to provide the required bolt stem

elongation.

c) Direct length measurements. Possibly the most accurate means to determine the bolt preload is by

direct measurement of the bolt length before and while the nut is turned. The procedure usually begins

with the low preliminary torqued condition as described above. This is taken to be 0 preload and the

length of the bolt is measured. The nut is then turned progressively while the length of the bolt is checked,

when the required extension is reached the bolt is deemed preloaded. This procedure can best be done if a

bolt has ground faces at each end, or if a conical recess is machined at the ends. With the use of steel balls

clamped into these recesses, the overall length can be measured with a gauge quite precisely. Obviously

this technique is very time consuming and is carried out on only the more critical fasteners in relatively

critical machines.

Fig 15. At right are 2 common means of directly measuring

bolt length and consequently bolt elongation. These diagrams

show the nut’s and the stem’s lower threads to begin at the same

place. This is impossible to ensure since tolerances have to be

applied to all dimensions. The only safe means is to have a longer

than necessary nut and locate the stem’s threads always within it.

13

Fatigue diagram

0

100

200

0.0 200.0 400.0 600.0 800.0 1000.0

mean stress

alt

ern

ati

ng

str

ess

Su

Sp

Gerber line

Goodman line

Proof line

Se

22 Bolts Strengths. Shown below are the material properties for a range of ASTM high grade

commercially bolts and screws. This American Society for Testing of Metals standard (A574 M) assures

us that 99% of fasteners exceed the stated strengths. Individual fasteners manufactures provide properties

within and beyond the range shown below, eg 6.6, 14.9, 18.8 etc.

Class

(bolt grade)

Min proof

strength SP

N/mm2

Min tensile

strength Su

N/mm2

Min yield

strength Sy

N/mm2

Endurance

strength Se

N/mm2

4.8 310 420 340 65

5.8 380 520 420 81

8.8 600 830 660 129

9.8 650 900 720 140

10.8 830 1040 940 162

12.9 970 1220 1100 190

Table 1 Data from Shigley et al, showing minumun strengths for a range of commercial grade bolts. The

relatively high strength steel used in these bolts do not show a clear yield condition, resulting is some

deformation at stresses levels below but near yield. Therefore the proof stress (SP) is used which is about

85% of SY. The fully corrected endurance limit takes into account all the normal mass production

manufacturing processes for fasteners with rolled thread. Note also that Se is approximately 15.5% of Su.

23. Fatigue diagrams. We will use here 3 lines or curves to determine safe bolt designs under fatigue

conditions: the alternating and mean stresses have to be below

the proof line

FSSS P

m

P

a 1 Eq 24

and either the Gerber parabola: 1

2

u

m

e

a

S

FS

S

FS Eq 25

or the ASME ellipse: 1

22

u

m

e

a

SFS

SFS

Eq 26

The functions in Eq 24, 25 and 26 are plotted below for a hypothetical material of Su = 1000 Nmm-2

, and

Sp and Su as given by paragraph 22 and Table 1 above.

14

Cross-sectional area lost

0.70

0.72

0.74

0.76

0.78

0.80

0.82

0.84

0.86

0.88

0.90

0.92

0 2 4 6 8 10 12 14 16 18 20 22 24 26

Major dia

fra

ctio

n lo

st

Coarse threads

Fine threads

Extra fine threads

Fig 16. Shown above are plots of Goodman, Gerber and Proof relations. Any combination of a and m stresses

represented by a point to the left of the Proof line indicates that no yielding will take place. Any combination of a

and m below the Gerber line, indicates that any bolt made to the ASTM standard described in paragraph 22, will

have a likelihood of less than 1% of experiencing a fatigue failure. Hence any bolt that can be represnetd by a point

below both the proof and Gerber lines can be deemed safe. Some texts propose that the ASME ellipse may be used

alternatively to the Gerber parabola.

The Goodman line has often been used in the past to indicate the safe design condition as we now use the

Gerber parabola, prior to the adoption of PCs, simply because it has been easier to calculate, being just a

straight line. But, one can see at a glance, that using the Goodman line as a criterion discounts 10 to 30%

of available fatigue strength for no good reason, when one has available the sort of numerical solvers that

built currently into engineering programs and even Excel.

24 Loss of stem cross sectional area due to thread depth.

Fig 12 - above. Fraction of cross sectional lost to the thread

for coarse, fine and extra fine threads. At right is a table

of the gradients and Y intercepts for the straight line

regressions through the points above.

Fig 13 - below, FEA results of the compression between two flat plates clamped by a bolt. Note that the

faces of the plates have stretched under compression and have become convex at their interface.

Extrafine fine coarse

m 0.00363 0.00531 0.00406

yintercpt 0.703 0.741 0.817

Plot of - cross-section of stem remaining, allowing for depth of threads, plotted against major diameter in mm.