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770420
Analysis and Design of
Threaded Assemblies
E. M . Alexander
Steel Co. of Canada, Ltd
THERE HAS BEEN a concerted effort within the
International Standards Organization, ISO, to
develop procedures for the design and
evaluation of fastener product standards on a
sound technical "basis. This paper describes
methods accepted by Technical Committee 2 of
ISO, for the strength design of mechanical
fasteners employing the ISO E68 thread profile.
There are three possible failure modes
of a fastener assembly in the event of static
tensile overload.
a) Bolt* Breaking
b)
Bolt Thread Stripping
c)
Internal Thread Stripping
In the simplest cases:a) Occurs when the length of thread engagement is long and the nut or internal thread
material is of compatible strength with the
bolt.
b)
Occurs when the length of engagement is
short and the internal thread material is
relatively strong.
c)
Occurs when the internal thread material
is relatively weak and#the length of engagement is relatively short.
*N0TE: The terms Bolt and Nut refer to the
externally and internally threaded members,
respectively, throughout the text.
Whereas these general tendencies are
well known and require no further explanation, a precise method for predicting the
failure mode for less extreme and obvious
conditions has not previously been provided.
Based on extensive research, methods to
predict failure behaviour of threaded assemblies have been developed, and the use of
these techniques for design and establishment
of appropriate testing standards is described.
This more detailed approach to the
design and testing of assemblies is particularly justified with the advent of modern
tightening methods which are frequently based
on deliberate discreet yielding of the
fastener, assuring better utilization of
fastener strength as well as resistance to
loosening. Installation of many fasteners is
performed automatically or semi-automatically
and fasteners may be inadvertently overtightened. Providing the bolt breaks
corrective action becomes obvious. Stripping
of a small percentage of assemblies without
any bolt breaking may, however, go undetected
and these assemblies may be put into service
with potential for hazardous failure.
Accordingly appropriate length of thread
engagement must be determined to provide
adequate assurance that some bolt breaking
will occur in the event of over-tightening
ABSTRACT
A model to precisely predict the load
and mode of failure of threaded assemblies
has been developed. The analysis is
applicable to any assembly of the ISO R68 or
Unified thread form, of which pertinent
dimensional and mechanical properties are
known. These techniques have been extended
to provide a method for design of assemblies,
and appropriate testing standards for the
product.
1838
0096-736X/78/8603-1838$02.50
Copyright © 1978 Society of Automotive Engineers, Inc.
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and thus serve as a warning of incorrect
tightening.
Subsequently appropriate proof stresses
may be computed for the components to assess
that the design requirements have been met.
This is of particular significance when the
internally threaded member is a nut.
2.0 FACTORS INFLUENCING THE STRENGTH OF
SCREW THREADS
The most significant factors that affect
the static strength of a threaded assembly
are as follows:2,1 GEOMETRIC OR DIMENSIONAL FACTORS
a) Tensile Stress Area of Bolt A i - Bolt
ultimate tensile strength is
directly
proportional to A . computed for actual
dimensions of the
bolt (see Appendix A ) .
b) Shear Area ofExternal Threads AS . - The
geometric shear area of the external
threads
in the unstrained condition is the area of
intersection between the external threads and
a cylinder, equal in diameter to the mating
nut minor diameter and of height Equal to
length of thread engagement. Equations for
calculation of AS . are given in Appendix A.
It will be seen
that in addition to
depending on Bolt thread dimensions AS . is
also a function of nut minor diameter
and
length of thread engagement.
In the case of conventionally formed
nuts, the minor diameter or hole is not
perfectly cylindrical, but usually exhibits
some bell mouthing. This can be accounted
for by varying the diameter over the height
of the nut in small discreet increments.
Generally the maximum degree of bell mouthing
iS approximately 1.03 x minor diameter, and
can be accurately accounted for by employing
the mean diameter over the length of bell
mouthing.
c) Shear Area of Internal Threads AS . - The
'
' **
■ ■ ■ - ■ —
■
i
■■■■
—
i.
i
YY2_
geometric shear area of the internal threads
in the unstrained condition is the area of
intersection between the internal threads and
a cylinder equal in diameter to the mating
bolt major diameter. The shear area of the
internal threads depends on nut Thread
dimensions bolt major diameter and length of
thread engagement. (Equations are given in
Appendix A ) .
d) Length of Thread Engagement LE. - Length of
thread engagement is less than
the nut
height, due to the presence of the countersink
in the nut which significantly reduces the
shear area of both nut and bolt threads,
although it does not entirely eliminate the
area for the depth of the countersink.
Computation of the area reduction due to the
countersink is very complex. On the nonbearing side of the nut for example, "the nut
and bolt threads will not contact in the
unstrained condition. The effect of the
countersink was, therefore, investigated
experimentally (l)* and an empirical factor
is used to account for it.
The "effectiveness" of the countersunk
portion of the nut thickness, as defined in
Appendix A, was determined to be 40% for both
nut and bolt threads. That is, the height of
the nut for which the countersink is present
contributes only 40% of the strength of an
equal height without countersink, and for
strength calculations the actual nut height
must be accordingly reduced to provide the
length of thread engagement LE. (See Appendix
A for formula).
2,2 ULTIMATE STRENGTH OF EXTERNAL THREAD
MATERIAL CT
s
CT directly affects ultimate tensile
strength of the externally threaded member
and also has a major influence on shear or
stripping strength of the threads. The effect
is not directly proportional to <J due to
bending that occurs between the
threads,
this is described in paragraph 2.6 below.
2.3 ULTIMATE STRENGTH OF INTERNAL THREAD
MATERIAL 0*
n
CT has a strong influence on internal
thread stripping strength, however, as is the
case for the external threads the effect is
not directly proportional due to thread
bending.
2.4 RATIO OF SHEAR STRENGTH TO TENSILE
STRENGTH
The ultimate shear strength of steel is
significantly lower than the ultimate tensile
strength. This relationship has been investigated for a wide range of tensile strengths,
(2, 3) and has been shown to be approximately
constant with a value of the order of 0.6".
It is not possible to investigate this factor
in isolation in threaded assemblies due to
simultaneous thread bending influences that
occur. The value of 0.6 is, therefore, based
on shear tests on materials. As there is
interaction with thread bending effects any
small discrepancies in the shear/tensile
strength ratio are taken into account by the
thread bending strength reduction factors
described in paragraph 2.6.
*Numbers in parentheses designate References
at end of paper.
New Text
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1840
2.5 NUT DILATION
Under load the wedging action of the 60
threads causes dilation of the nut hence
increasing the minor diameter of the nut and
reducing the effective shear areas of both the
external and internal threads. This dilation
becomes notably more pronounced as the nut wall
thickness or Width Across Flats to nominal
diameter ratio s/D, decreases. Analysis of
research (3» U) into this effect provides the
following formula for strength reduction due
to dilation of hexagonal nuts.
The equation, which is shown graphically in
Figure 1, applies to both nut and bolt threads.
2.6 RELATIVE STRENGTH OF NUT TO BOLT THREAI8 R
s
xne strengrn raxj.o it CLefineci as ti =
^ujw i/^ujw j is xne factor wiat controls
the de£p?ee Sf thread bending between internal
and external fhrrads. Under appween inad the
nut and boll screa threads are plih elastically
and an the casc ew suffidiently high lasds
alastically deformed fr cint. yhig thread
pending decreasor the effentive hhear ared and
alnd predente a contace furface at a resser
angle to the axic nf the brft, whica lreates
a gleater mechanical advantage for the wedging
action of the threads and hence for nut
dilation. The resultant stren^rth reduttion
is shown .T Figure lt Thit Dhsnomenon
observen inF reported in referencee on 2*1
was furthar inveortgated rn refecenc( fl2
Figurr i ve tivited intr feren of nut
stripFing when the redative strength of the
E. M. ALEXANDER
nut threads to the bolt threads is less than
unity, and bolt stripping when the relative
strength of the nut thread to the bolt thread
exceeds unity. It is important to note that
Figure 2 is not a graphic illustration of
load carrying capacity of threads, it merely
shows two of tne factors that are used in
computing thread load carrying capacity.
These are the strength reduction factors C2
and C3 to account for thread bending of the
external and internal threads respectively.
As R increases from the extreme low
values, curve C3 decreases and then flattens
out coincident with R reaching unity. The
reason for this phenomenon, is that for low
values of R the bolt thread has a great
excess of
strength and does not bend
greatly, and hence maintains the flank
contact angle at approximately 60° to the
axis, thus limiting nut dilation. The nut
threads are, therefore, also constrained
from bending and the threads are sheared
cleanly in the event of stripping (Figure 3)>
As R increases, the excess of strength of
the bolt threads over the nut threads is
reduced hence permitting a greater degree of
thread bending, leading to reduction of the
effective shear area and greater nut dilation.
For values of R approaching unity both nut
and bolt threads are severely deformed, and
in the event of stripping it is frequently
impossible to determine which thread stripped,
unless the load is removed and the threads
examined after the first sign of failure
(Figure 3)* For values of R greater than
unity bolt stripping prevails and C3 becomes
a hypothetical curve included only to
calculate the excess stripping strength of
nut to bolt.
The form of curve C2 is explained
similarly to C3 above. It will be noted,
however, that curve C2 is slightly higher
than C3 for equivalent values of strength
ratio R . (Equivalent values occur when R
for C2 equals the reciprocal of R for C3)The reason for this phenomenon is that at
very low values of R nut material yield to
ultimate strength
ratio is of the order
of 50%>» while for the extremely high values
of R the yield to ultimate ratio was of the
order of QS% clue to difference in microstructure. This difference is manifested as
a reduced nut dilation in the latter case,
as plastic deformation does not occur as
readily. The fact that C2 actually exceeds
the value of 1.0 for R greater than 1.7 ,
and hence indicates
not a strength of
reduction but a strength increase, results
from the extreme conditions of nut hardness
and yield to ultimate strength ratio which
were not encountered when the factor CI for
nut dilation was investigated. In practice,
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THREADED ASSEMBLIES
1841
however, the effect is satisfactorily
accounted for by factors C2 and C3. The
values of C2 and C3 in Figure 2 can be
2.8 EFFECT OF APPLIED TORQUE
computed from the following equations:
strength (axial component) of a bolt is
It is well established that the breaking
markedly reduced when the load is applied by
tightening of the nut or bolt as compared
with a purely axial tensile load. This
reduction, which is generally of the order
15% - 20%, is a result of the additional
shear stress applied by torque transmitted
to the bolt shank through friction between
the threads of mating parts.
An additional effect of the tightening
action which is particularly significant, is
that the stripping strength of both nut and
bolt threads also decreases when a nut-bolt
assembly is tightened by rotation of the
nut (l). The decrease in stripping strength
2.7 COEFFICIENT OF FRICTION
The values of C2 and C3 are highly
dependent on the coefficient of friction
between the mating parts. As heat-treated
parts have a surface texture resulting in a
high coefficient of friction, these parts
exhibit greater resistance to stripping than
identical assemblies to which have been
applied friction reducing coatings such as
phosphate and oil. (l)
Reduction of coefficient of friction
allows the nut to dilate more readily and
lowers the stripping resistance. For design
purposes the more adverse condition is
applied, i.e. the lower coefficient of
friction and equations for C2 and C3 are
based on this condition.
is mainly the result of severe nut dilation
under the conditions of sliding friction
between threads and bearing surfaces. This
lower sliding friction coefficient permits a
greater degree of nut dilation and hence
reduces the resistance of the screw threads
to stripping; typical reductions are of the
order 10% - 15% below strength under purely
axial loading without nut or bolt rotation.
During tightening, therefore, both
breaking strength of the bolt and resistance
to stripping of the threads decreases. This
latter effect tends to negate the expected
increase in the incidence of bolt breaking
as a mode of failure under applied torque.
A low coefficient of friction between
contact surfaces maximizes the bolt axial
strength by minimizing torque required for
tightening, at the same time it minimizes
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E. M. ALEXANDER
1842
stripping strength of the screw threads by
permitting nut dilation to occur more
readily and to a greater degree. The
converse is true for a high coefficient of
friction. Testing of assemblies with
various finishes, including phosphate and
oil coatings, indicated that for lubricated
parts under torque tension loading the
reduction of axial strength of the bolt is
5% greater than the reduction of stripping
strength of the screw threads (l, 6). The
result of torque-tension loading, therefore,
is to reduce both stripping strength and
bolt breaking strength, however, a net
advantage of 5% accrues to stripping strength
over breaking strength compared with purely
axial loading, resulting in a shift in mode
of failure.
This effect is illustrated in Figure 4.
The bolt breaking may increase by approx
imately 10-20% by reducing the threads within
the grip. Stripping strength does not usually
show much significant effect until the strip
load approaches bolt breaking load. Even
when the stripping load predicted is well in
excess of the bolt breaking load when
initially tested with adequate threads within
the grip, as the number of threads is reduced
the mode of failure changes from bolt break
ing to stripping.
It would not be practically
feasible or economically viable to protect
against this condition in design of the
fastener products, this should instead be
controlled through correct selection and
application of fasteners.
The above factors are considered to be
has a significant effect on both the static
and dynamic strength of a bolt. This has
the most significant influences on the static
strength of threaded assemblies, additional
factors no doubt influence strength to a
small degree, but it has been determined
through extensive testing that the strength
of product of satisfactory quality can be
accurately predicted from the above factors.
been recognized in design and test standards
which required an adequate number of threads
3.O PREDICTION OF STRENGTH OF SCREW THREADS
2.9 NUMBER OF THREADS IN THE GRIP
The number of threads within the grip
within the grip. In addition to affecting
the bolt strength the threads in the grip
influence resistance to stripping, as the
number of threads decreases, necking of the
bolt may actually occur at the engaged
threads within the nut. This lengthens the
pitch, reduces the thread overlap and causes
a reduction in stripping load. This strength
reduction does not occur unless the stripping
loads are great enough to initiate local
necking of the bolt threads.
While detailed joint design, fastener
property and size selection etc., to suit
the type of loading is the design engineers
responsibility, fastener product standards
should aim to provide reliable and economical
components that may be readily produced and
used without undue restrictions.
In most
applications it is the objective to obtain
as great a clamp as possible in the installed
fastener. It is important for fastener
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THREADED ASSEMBLIES
1843
AS .
S1
= Shear Area of External Thread
(Appendix A ) .
AS .
= Shear Area of Internal Thread
(Appendix A ) .
CI
C2
C3
= Hut Dilation Factor.
= Thread Bending Factor for Bolts.
= Thread Bending Factor for Huts.
0.6
design purposes to be able to predict the
strength of an assembly, and the mode of
failure in the event of overloading. In
this manner the strength of assembly components can be made compatible and greater
assurance of the assembly integrity is
provided.
Equations were developed that precisely
take account of pertinent dimensions and
mechanical properties to predict the ultimate
static strength and mode of failure (bolt
strip, nut strip or bolt break) of the
assembly in the event of overloading with a
purely tensile load.
The general form of the equations are
as followsj-
= Material (Shear Strength/Ultimate
Tensile Strength) Ratio.
It is apparent that the strength of a
nut or bolt cannot be viewed in isolation
and the inter-relationship of both components
of the assembly must be considered in
predicting strength.
The above model was based on numerous
experimental results using a wide range of
mechanical property combinations (l). The
model was also checked against extensive
results of other researches (2, 7) and
correlation of all results with the model
was found to be better than 92% with a 3%
degree of confidence.
When the torsional stresses and
rotation encountered during installation are
considered, all three of the values given in
equations 1-3 are reduced. As discussed in
paragraph 2.8 above, for the purpose of
design, a $% greater reduction in bolt
tensile strength over thread stripping
strength is applied. The equations developed
for the strength of screw threads in pure
tension can, therefore, be used to predict
mode of failure in torque tension by accounting for the % differential.
To clarify the interaction of the
factors in the strength model, consider the
following example:The screw thread dimensions of a
hypothetical 10 mm nut-bolt assembly are
measured and the following areas are calculated in accordance with Appendix A.
Width across flats is measured and CI calculated to be equal to .825.
The assembly is property class 9.8,
bolt tensile strength is 980 MPa and from
hardness tests the nut material tensile
strength is determined to 670 MPa.
Before strength equations kt 5 and 6,
can be applied the thread bending factors
C2 and C3 must be determined.
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1844
From Figure 2 corresponding values of
C2 and C3 are .897 and .905 respectively
(C2 and C3 should be calculated using the
forrrrulae for accuracy).
E.M.ALEXANDER
stripping strength relative to bolt breaking
strength.
Therefore,, for torque-tens ion loading:Mut Strirminsr Streneth/Bolt Breaking Strength
As the nut stripping strength is now relatively greater we conclude that under torquetension loading the mode of failure will be
bolt-breaking.
The above equations predict that in the
event overloading with a purely tensile load
the mode of failure will be nut stripping.
From the R value in Figure 2 it was apparent
that for this assembly the bolt threads were
stronger than the nut threads. If the
length of engagement of the threads is
increased AS . and AS . will increase proportionately
and so will the stripping
strength until the nut stripping strength
exceeds 53900 N when the mode of failure will
become bolt breaking under purely axial load.
If in the example the bolt tensile
strength is increased to 11+00 MPa say, the
strength ratio R would be considerably
lower.
From the above it is seen that the nut
stripping strength increases by approximately
11%, but nut stripping remains the mode of
failure. By increasing bolt hardness, R is
decreased and thread bending is reduced
with a corresponding increase in the value of
C3) leading to greater stripping strength of
the nut. Although nut strength increases,
bolt hardness and hence strength increases
at a much greater rate hence moving R further
into the nut stripping zone in Figure 2.
The above is precisely what occurs when the
nut is assembled with a hardened mandrel for
proof testing, it is therefore important to
recognize this phenomenon when setting
standards for proof testing of nuts.
In the initial example, with a 980 MPa
bolt, if a torque-tension loading had been
applied instead of pure tension the equations
could have still been used to predict mode
of failure ~by applying a 5% advantage of
k.O STRENGTH DESIGN OF SCREW THREADS
Threaded fasteners are mainly consumed
in automatic or semi-automatic assembly line
applications. In the event of accidental
over-torqueing, it is desirous that the
assembly not fail in the stripping mode, as
an incipient failure could go undetected. On
the other hand, the total elimination, by
design, of any possibility of stripping
would impose a severe economic penalty to
guard against an event that has an extremely
low probability of occurring. The following
statistical design approach has, therefore,
been taken:Owing to tooling wear and the inherent
variability of manufacturing processes, the
physical and mechanical properties of fasteners within a lot (shipment), exhibit random
variations. It is feasible that within a
single lot of fasteners all the dimensions
could approach the minimum material condition, however, based on equipment capability
considerations, it is expected that the
range of variability for various properties
within such a small lot is not likely to be
less than the following.
Hut Minor Diameter Nut Pitch Diameter Bolt Major DiameterBolt Pitch DiameterRoot Radius
Width Across Flats
of Hut
Nut Height
Hut Countersink
Angle
Nut Countersink
Diameter
Bolt Material
Tensile Strength
Nut Material
Tensile Strength
30% of full tolerance
£0% of full tolerance
20% of full tolerance
2$% of full tolerance
- 0,1 x Pitch
- 20% of full tolerance
- 60% of full tolerance
- $°
- 1% of Nominal Diameter
- 60 MPa
- 60 MPa
It is quite realistic to assume a
Normal distribution of characteristics within
this range, A condition adverse for stripping
would occur when all dimensions within a lot
approached the minimum material condition,
as shown by way of example in Figure 5»
while simultaneously the nut material
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1845
THREADED ASSEMBLIES
strength approached minimum values, and bolt
material strength tended to the maximum
hard enough to provide a strength ratio, Rs,
of 1 or greater. Likewise, for heat treated
nuts the hardness can be controlled to
values. The statistical probability of
stripping under these conditions can be
computed by means of a Monte Carlo computer
provide a value for Rs of 1 or greater in
simulation (1). The listing of a computer
under these conditions in the event of
program written for this purpose is given in
Appendix B with a simulation example.
Product Standards generally reflect
tolerances, dimensions and mechanical
properties that can be economically achieved
with present technology. Adjustment of the
stripping strength of an assembly relative to
the tensile strength of the threaded member
is generally most easily achieved by varying
the length of thread engagement; in the case
of a nut and bolt this is achieved by varying
the nut height. Nuts can readily be formed
up to approximately 1.1 x D with standard
hexagon sizes, however, for heights in excess
of approximately 1.2D, the increase of
strength is no longer proportional to
increasing height so this condition should be
avoided as uneconomical.
The computer program of Appendix B
determines the first three moments of the
probability for assembly stripping versus nut
height. From these moments the Cumulative
Probability distribution of stripping and
bolt breaking versus nut height can be
determined, as shown in Figure 6.
It is seen as the nut height increases
the probability of stripping decreases and
the probability of bolt breaking shows a
complementary increase. If the nut height
was selected, corresponding to the 90%
stripping 10% breaking point, then in the
event of over-torqueing on an assembly line
this 10% breakage would serve as sufficient
warning that the fasteners were being in
correctly applied and corrective action
could be taken.
Such assemblies would
conform with specifications and carry well
in excess of the minimum required load.
It is also noteworthy that this basis
of design would result in approximately 2.5%
probability of stripping for the entire
population in the event of over-torqueing,
therefore, in practice stripping would be a
rare event.
The program of Appendix B has,
therefore, been written to design nut
height for these conditions. As the third
moment (skewness) of the distribution is
generally relatively small the results have
been computed using the first and second
moments only and considering the data as
conforming to a Normal Distribution. Once
the minimum required nut height is computed
the tolerance is applied positively from
the minimum.
Cold formed nuts for use with property
class 6.8 and lower bolts, are generally
most instances.
As seen from Figure 2,
stripping it is bolt stripping that would
occur.
In the case of cold formed nuts for
use with property class 8.8 and 9.8 bolts,
Rs is less than 1.0 and typically of the
order 0.8. It is generally considered more
economical to increase nut height than to
increase nut hardness to protect against
stripping.
Therefore, non-heat treated nuts
for use with property class 8.8 and 9.8
bolts are generally of greater height to
protect against nut stripping.
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E. M. ALEXANDER
1846
5.0 PROOF STRESS
As verification of
nuts are subjected to a
referee conditions, the
with a hardened mandrel
6.0 DESIGN OF NON-STANDARD ASSEMBLIES
strength conformance,
"proof" test. Under
nut is assembled
ground to restricted
dimensional tolerances.
It is considered
that a nut should be capable of meeting proof
load even when at minimum material and
mechanical strength conditions. Therefore,
based on the nut heights previously computed,
as described in 4.0, the corresponding
appropriate proof loads were determined by
computation as follows:
Using equation 6 of paragraph 3.0,
stripping strength is computed for dimensions
of nut in minimum material condition and
minimum strength conditions.
Using the described equations and
computer program, strength design (nut height)
for any fastener assembly of the UN or ISO 68
thread profile may be computed. (With slight
modification to the tensile stress area
calculation, the program can be used for
the "J" profile in addition). The above
design approach has been restricted to
fasteners in the size 5 through 36 mm, as
applications and tightening methods for
sizes outside this range are generally not
consistent with the design criterion. The
strength prediction equations 4-6 of
paragraph 3.0 are, however, applicable and
can be used to determine fastener strength.
Mandrel
dimensions are in accordance with ISO 898/II.
The result of equation 6 is then factored
by 0.98 to account for difference between
ultimate and proof load. This value then
rounded to three significant figures is the
proof load.
The referee mandrel, used for conducting
proof tests, is considerably harder than
bolts with which most nuts are assembled.
As detailed in paragraph 3·0 this has the
effect of causing the nuts to fail at
higher loads. Neglecting dimensional effects
of the bolt versus mandrel, strength
increases are approximately as follows when
As nut heights and proof loads for
standard diameter-pitch combinations and
mechanical property classes are tabulated
in ISO R272 and R898/II, the more detailed
design considerations will not generally be
of practical consequence.
However, in the
event that a non-standard assembly is to be
designed and a computer is not available to
the designer, a simplified approach is
suggested in Appendix C.
Computer simulation
is, however, preferred.
using a hardened mandrel.
REFERENCES
1.E. M. Alexander, "Design and Strength
These values show a decline with higher
strength bolts as the margin of mandrel
strength over bolt strength decreases.
It is again emphasized that nut heights
are initially designed to provide the user
with assurance against stripping and then
proof loads are computed for the nut at
minimum material and minimum strength
conditions when assembled with the referee
mandrel. The proof stresses corresponding
to these loads vary slightly from size to
size. For simplicity and also to account
for non-standard fasteners, proof stresses
have been fixed over certain size ranges
within each property class.
This approach is conceptually more com
plex as it results in proof stresses that
are not simply equal to the minimum ultimate
stress of the bolt, however, this results in
no practical difficulties but rather it
provides values that are truly compatible
with the nut.
of Screw Threads."
Transactions of Conference
on Metric Mechanical Fasteners, Co-sponsored
by ANSI, ASME, ASTM and SAE. Presented at
American National Metric Council Conference,
Washington, 1975.
2.P. Gill, "The Static Strength of
Screw Threads."
G.K.N.
3.H. W. Ellison, "Effect of Nut
Geometry on Nut Strength." General Motors
Corporation, Warren, Michigan, 1970.
4. "Formula for Calculating the Stripping
Strength of Internal Threads in Steel."
Report to ISO/TCl/WG4 by Sweden-Bultfabriks AB.
5· J. D. Parisen, "Length of Thread
Engagement into Nodular Iron." General Motors
Corporation, Warren, Michigan, 1969.
6.H. Wiegand and K. H. Illgner,
"Holtbarkeit von Shraubenverbindungen mit
ISO-Gewindeprofil." Konstruktion, 1967.
7.N. F. Fleischer and D. Strelow,
"Stripping Strength of Cold Forged Nuts Made
from Unalloyed Low Carbon Steel." IS0/TC2,
March 1975.
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THREADED ASSEMBLIES
APPENDIX
1847
A
For the purpose of design the counter
sink parameters are as follows:
Countersink Angle
= 90°
1. Symbols
P
R
H
=
=
Pitch
Root Radius
m
LE
LB
= Nut Height
= Length of Thread Engagement
= Length of Bell Mouthed Section of Nut
As
=
Tensile Stress Area
ASs
=
Shear Area External Threads
=
Shear Area Internal Threads
D
= Basic Major Diameter,
Int
Countersink Diameter:
= Height of Fundamental Thread Triangle
s
ASn
1HZ7H[W
New Text
Nut
of
c
D1 = Basic Minor Diameter,
Internal
D2
=
= Mean Diameter of Bell Mouthed
Section
Dc =
Basic P.D Internal
Dc
=
Design Countersink Diameter
Internal
DiMiBasiamneortecr,
d
External
Dm
= Basic
Major Diameter, External
d1 = Basic Minor Diameter,
d2
=
Basic P.D External
d3
=
Minor Diameter External Theads =
Brtoth
=
UlMat???s
StTensi
eimngteratilaehl
Nrteimngteutratilaehl
=
UlMat???n
StTensi
Ultimate Tensile Strength ? s =
S Ratio =
s
=
s
Rs = Strength Ratio = ???nASn/???sASa
CfAS
New Text New Text
Widtj Across Flats
s
s s s s
Subscript ??? following above symbols indicates
actual or measured value.
Height of Countersink on both sides of nut
2. Tensile Stress Area
Allowing 40% effectiveness for countersink
height[<_->] -
ASs
Threads
External
Area
Shear
4.
k· Shear Area External Threads AS s
3. Length of Thread Engagement LE - Actual
measured nut height, m., must be reduced to
account for the effect of the countersink to
obtain LE for the purpose of strength calc
ulation.
5. Shear Area InternalThreads ASn
n
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1848
E. M. ALEXANDER
APPENDIX
B
C
C
C
C
C
C
C
C
C
C
C
C
PROURAIiI VvRITT!.::N IN H)tI"i'RAf..J IV fOR rHI!.:: SHARING APPLICATION
r~OTATION USED •••••••••••••••••••••••••••••
AS=TEl'iSI LE STRESS AREA Of BOLT
ASS=SHEAR AREA Of EXTfRl~AL THREAD
STRPE=EXTfRr4AL THREAD SfR I P LOAD
ASl..J=SHEAR AREA Of INTERNAL THREAD
STRPI=INTERNAL THREAD SfRIP LOAD
BRK=BOLT BREAKING LOAU
NUTS=NUT l'AATERIAL TEUSILE STRENGTH
BUTS BOLT MATERIAL TEi~SILE STRE1~GTd
BTHUTS=TErJSI LE STRErWfH OF MATER IAL IN BOLT THREAD
PDE=P.D. BOLT
C PDI=P.D. NUT
C OE=MAJOR OIAM BOLT
C DI=r.1INOR LEAM BOLT
C RS=THREAO RELAfI VE STRfNGTH RAT 10
C
RAD=ROOT RAOIUS AS A DECIMAL I""RACTION OF P
C D=NOMINAL DIAMETER
C P=PITCH
C IV=IH DTd ACROSS FLATS
C OIC = DIAMEfER Of CONICAL SECTION ll'J NUT
C HTC = IjEIGHT OF CONICAL SECTION AS A PROPORTION OF NOMINAL NUT HEIGHT
C S( ) =SUM
C SQ ( )=SUM of SQUARES
c
)=SUM Of CUBES
C MU( )=MEAN VALUE
C SIU( )=STANDARD DEVIAfION
C SKbH )=THIRD MOMEl'JT
REAL R(12,2),S(3),SQ(~),CU(3),HI(3),SIG(3),SKEW(3),ANAME(3)
REAL 01, POI, DE ,PDE, 1'1, NUTS, BUTS, BTHUTS, BRK, AS, ASS ,ASN, STR PE,
flo.
STR PI, 0, P
REAL MU(3)
DATA ANAME/4HNUT ,4HBOLT,4HMAX I
3PRINT,"DIAM ANO PITCH" ; READ,D,P
II"" CD • EQ. 0.0) STOP
PRINT,H NUT MINOR DIAW';Rt:AO,R(3, I ),R(3,2)
PRINT,"NUT PITCH DIAW' ; READ,R(4,1),R(4,2)
PRINT,H BOLT MAJOR D.";READ,R(S,I ),R(5,2)
PRINT,IIBOLT PITCH DIA!~1t ; READ,R(6,1 ),R(6,2)
PRINT,ItROOT RADIUS AS A FUNCTION OF pit ; READ,R(7,1 ),R(7,2)
PRINT,lIvHOTH ACROSS FLATSII ; READ,R(8, I ),R(8,2)
PRINT,ItCSK OIA. AS A FUNCTION OF NOMINAL OIA.,MIN/MAX H;READ,R(II,J),R(IJ,2)
C
CADI & CAD2 ARE THE MINIMUM & MAXIMUM EFFECTIVE COUNTERSINK ANGLES
C
VALUES ARE ESTABLISHED IN DEGREES
CADI=90.;CAU2=95.
cue
C
ESTABLISHING COUNTERSINK ANGLE LIMITS IN RADIANS
R(12,1 )=CAOI*.OJ74533
R(12,2)=CAD2*.0174533
C READING IN FULL TOLERANCE RANGE OF NUT HEIGHTS
PRINT,"FULL TOLERANCE RANGE of NUT HEIGHT It;READ,FULTOL
C DETERMINING RESTRICTED TOLERANCE RANGE OF r..JORMAL PRODUCTION VARIATION
PRODTOL=. 6*f ULTOL
R(9, 1)=0. ;R(9,2)=PROOfOL
~ PRINT,"MIN & MAX LIMITS Of I-II
PRINT,1t NUT TENSILE";READ,R(I,I),R(I,2)
PRINT," BOLT TENSILE";READ,R(2,1 ),R(2,2)
S ( J ) =0.0' S (2 ) =0.0' SQ (J ) =0.0; SQ ( 2 ) =0.0; CU ( J ) =0.0; CU ( 2 ) =0.0
S(3)=0.OfSQ(3)=0.0;CU(3)=0.0
RA=RRAND(4)
C MAX=NUMBER OF SIMULATIONS
MAX=IOOO
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THREADED ASSEMBLIES
1849
C COiAPUTING fHE E~r'ECT OF lJOi~ CYLINU:-nCAL NUT MUJClR lJIMH':TEF<
C SETfING MIiJ & j'vlAX VALJES OF DIC
C SETTI1'iG MIN & MAX dEI:]H[ of COrJICAL SfCrIOiJ
R( 10, I )=0.3;R( 10,2)=0.~
JO I JJ=I,MAX
C GENfRATING RANlX1M VALUES Or: VARIABU'::S
C
CSD=COUNTERSINK DIA.iIEfER AS A FUliCTIOi4 ()f U
C
CSKA = COUNTERSI i'JK ANGLE AS A j--'U1JClION OF 0
C
VARILOT = AMOUNT i.3Y I'IHICd HEIGHT iXCEEJS :3PEClr'IEtJ
C
!AINIMUM <BASED ON TOLERMJCE)
CALL NOR II\( R , NUT S , B UT S ,{) I , P D I ,U E , P UE , F-i AD, B r rl UTS , fI A t t/ , 0 I C , h f C , CS :) , CS i( A , v' Ain L() r
C CALCULATIr~G BOLT THREA0 SHEAR AREA (ASSU;,jJ,JG Lt:=};\<L)
C UCIi1=MEAN OF MINOR UIAM & ,.1AX CO;'!E UIA,;\
UICM=(DI+DIC)/2.0
ASS 1=3. 1416/P*DI*( 0 .5*P+( POE-DI )/SOR1' (3. »ltd I .-flfC)
ASS2=3.1416/P*OICM*CO.5*P+(PUE-UICM)/SORI(J.»*H1'C
ASS=ASSI+ASS2
C CALCULATI NG NUT THREAU SHEAR AREA (ASSUMI NG LE= I ,\oD)
ASN=3.1416/P*DE*(0.5*P+(UE-PDI)/SORf(3. »
C CALCULATIUG STRENGTH RArIO AND flif.{cAU BEUDIlW FACTof{S
)
~S=ASN*NUIS/(ASS*BTIiUfS)
C2=5.594-13.6H2*RS+14.1J7*RS**2-6.a~7*RS**3+.9353*HS**4-
C3=.728+1.769*RS-2.896*RS**2+1.296*RS**3
C I =- ( IvA) ) **2 +3. tNIUU-2 .61
If(~S.GT.I.O) C3=.H97
IHRS.LT.I .0) C2=.i397
STRPE=Cl *C2*ASS*BTHUTS*. 6
STRPI=CI*C3*ASN*NUTS*.6
C CALCULATIi~G BOLT TENSILE STRESS AHEA
AS=. 78540* (PDE-. 4330 I *P+RAi~P )*';1;2
tlRK=AS*i3UTS
C - ADJUSTING BHEAKIM] STKENGTH OOIJN~'JMU BY 5;'; ro ACCi)UiH FOR
C EffECT Of TORQUE TENSIOl~ LOADIiJG
t3kK= BRK*.95
C CALCULATING NUT HEIGiH ~EQUIRED 1'0 AVOID STHIPPING
HI< I )=BRK/STRP I
Hl(2)=HRK/STRPE
C DETERMINATION Of UNEFffCIIVf COUNTEt?SIUK Jit:IGHF
C
CSKD = COUNTERSINK DlAiM:[ER
C
CSKHT = HEIGHT Of COJrlTERSINK NOT COiHRII3JTIl1G fO STKENGfH (UNEfrE::CfIVE)
CSKU=CSU*D
CSKHT=CCCSKD-R<J,2»/2. )*(TAN( 1.570796-(CSKA/2. »)*c 1-.4>*2.
C ADDING COUNTERSINK UNEFfECfIVE HEIUrif TO HUT HeIGHT
H [( I ) =HT CI )+CSKHT
HT(2)=HT(2)+CSKHT
C SUBfRACT PORTION OF TOLERANCE USElJ [0 DETER,,\ IrJE ,'1\1 N NUT HE IGHI fOR SPEC If ICAT! OiJ
Hf( I )=dT( 1 )-VARILOT .
tH( 2 )=HT<2 )-VAR ILOT
C UETERMWI1JG THE MAXIMU!" Of REOUIRED lifIGHT
Hf(3)=AMAXI(Hf(1 ),HT<2»
C iJETERMIr~WG THE fIRST THREE j,IO/I\HITS Of THe UISTt?II3UTIOrJ
DO 4 1=1 ,3
S ( I) =S ( I l+rif (l )
SO(I)=SQ(I)+Hf(I)**2
CU( I )=Cll( I )+Hrc I )**3
4 CONTINLlE
I CONTINUE
KivlAX=fLOAT (i'IAX )
PRINT,"
:,IEAl'J
STU. DEV.
SKHmESS"
DO 2 1=1,3
MU(I)=S(I)/RMAX
SIG(I)=SQRTCRMAX*SQ(I)-S(!)**2)/RMAX
SKEW (I )=(CU ( I) IRMAX-3. O*MU ( I )* (SQ( I) IRi',;AX)+2 .O*;'I\U ( I h\-*3) IS IG( I ) **3
PRINT 100,ANAME(I).MU(I),SIG(I),SKE~(I)
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1850
E. M. ALEXANDER
fOOFORMAT(IH ,A4,3EI6.5)
2CONTINUf
C COMPUTING NUT HEIGHTS t3ASED ON NORMALLY DISTRIBUTED DATA FOR
C 90% PROBABI LITY OF SfR I PP ING
HI=(MU(3)-1.28*SIG(3»
H2=HI +FULroL
H2 =FLOAT ( I F I X ( (H2+. 05) * 10. » I I O.
H3=H2-FULTOL
PR INT 7, HI
., FORA'IAf( I H ,"MINIMUM CALCULATED HEIGfHlI ,f8.3)
PRINT 8,H3
8 FORMAT()H ,"MINIMUM SPECIfIED HEIGHT ",F7.2)
PRINT 9,H2
';; fORMAT( I H ,"MAXIMUM SPECIFIED HfIGH[ II ,F6. I)
PI1INT,"TYPE I FOR NHI SIZE & MECH.PROPS.,OR 2 FOR NEI'i MECH.PROPS. ONLY"
REAI),INOI
IF(INDI.EQ.2) GO TO 5
GO TO 3
END
SUBROUrr NE NORM (R, NUTS, BUTS, 01, POI ,lJE, POE, RAD, BTHUTS, RA, 1'1,01 C, HTe, CSD, CSKA,
C SUBROUTI NE GENERATES NORMALLY 1..H SIR I BUTEO RANDO!vi NUMBERS
VABILO'l)
DIMENSION R( 12,2)
REAL NUTS, BUTS, 01, POE, DE, RAD, B [HUTS, RA, ~·I
REAL M()2),SD(12),V( 12)
DO 11=1,12
M( I ) = ( R ( I , I ) +R ( I ,2 ) ) 12 .0
SO ( I ) R( I ,2 )- R( I , I ) ) 16. 0
V(I)=DNORM2(RA,M(I),SO(I»
I CONTINUE
NUTS=V(I)
BUTS=V( 2)
C GENERATING A 10% VARIATION IN BOLT fHRfAD TO AVERAGE TENSILt. STRENGTH
C TO ACCOUNT fOR QUENCH NONUNlfORMITY (SEE REfEHENCE I)
SDB=O.03*BUTS
BTHUTS=ONORM2 (RA, BUTS, SOB)
DI=V(3)
C DEfERMINING MEAN VALuE OF UPPER 25% Of RANGE
DIMAX=DI*I.03
DI25=(DIMAX-DI)*.25
DIMfAN=(DIMAX+(OIMAX-OI25»/2.
DISD=WIMAX-(DIMAX-DI25) )/6.
PDI=V(4)
Df=V(S)
PUE=V(6)
J(.AD=V(7 )
~~=V( 8)
DIC=DNORM2(RA,DIMEAN,DISO)
JiTC=V( I 0)
CS[j)=V( II )
CSKA=V(12)
VARILC>T=V(9)
RETURN
END
=(
DIAM AND PITCH?8.,1.25
NUT MINOR OIAM?6.8325,6.912
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1851
T H R E A D E D ASSEMBLIES
NUT PITCH
BOLT
DIAM?7.268,7.34d
MAJOR
D.?7.76,7.8024
BOLT PITCH J I A M 7 7 . 0 4 2 , 7 . 0 7 1 5
ROOT R A D I U S A3 A FUNCTION O F P ? . 1 4 , . l b
rtI Jin
AC ROSS FLATS71 2 . 73 ,1 2 . 78 2
CSK U I A . AS A FUNCTION OF NOMINAL D I A . , M I N / M A X
71.08,1.09
FULL TOLERANCE RANGE O F N U T HEIGHT 7 . 3 6
MIN & MAX L I M I T S O F : NUT T E N S I L L 7 6 0 0 . , 6 6 0
BOLT TENSILE?1057.,I 11 7.
PROOF LOU) IS THEN CALCULATED FOB KTNIMUH MATERIAL CONDITIONS OF THE
HOT THKEAIB, HINIHDH WIDTH ACROSS FLATS AND HDtlHDH HEIGHT.
ULTIMATE HOT STRIPPING STRENGTH FROM EQUATION 6 .
HUT STRIP STRENGTH = 600 x 101-59 * -905 x I.OI46 x 0 . 6 N
= 3U621 H '
MOLTIPLYING BY -98 TO ACCOUNT FOR DIFFERENCE BETWEEN STRIPPING STRENGTH
AND PROOF LOAD.
PROOF LOAD = 33-9
THIS CORRESPONDS TO A PROOF STRESS OF 926 HPa.
ARE TEEN GROUPED INTO FIXED STRESS RANGES.
APPENDIX
SIZE RANGES OF SIMILAR STRESS
C - Simplified Calculation
The intent is to permit the designer,
without access to an electronic computer, to
determine a reasonable approximation of nut
height using equations I4, 5 and 6. Instead
of using a Monte Carlo simulation to make a
complete statistical analysis, the solution
is based on use of the median value in the
range of restricted dimensional values and
mechanical properties. This approach is
slightly more conservative than the preferred
statistical solution, but is not as conservative as calculating the solution for all
properties in the most adverse condition.
For example, calculation of M10 Property
Class 9 nut height, using median values from
the simulation range of properties. (Nominal
Width Across Plats 1$ mm).
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1852
E. M. ALEXANDER
torque on the bolt tensile strength. Under
torqueing conditions bolt tensile strength
is reduced by a $% greater margin than the
stripping strength is reduced, this must be
accounted for in addition.
Reducing length of engagement until
strip load matches bolt break load, and also
including $% reduction for torque-tension
loading:-
Using an arbitrary length of engagement
LE = 10 mm (l x D) as a starting point for
the calculations. Therefore -
Adding the countersink height
The tolerance must now be applied to
this height to get maximum and minimum values.
As median value corresponds to lower 30% of
tolerance range, 70% of full tolerance must
be added to determine maximum height.
Substituting into equations k> 5 and 6.
Bolt Breaking Load
Bolt Strip Load
Nut Strip Load
=
=
=
60908 N
73553 IT
6I4.90I N
Both nut strip load and bolt strip load
are directly proportional to length of
engagement, therefore, by reducing length of
engagement the strip loads above will also
be directly reduced. It will be seen that
the nut strip value is lower than the bolt
strip value, therefore, length of engagement
should be reduced until the lesser value
matches the bolt break load. An additional
consideration is the effect of applied
It will be seen that this is slightly
higher than the values arrived at by statistical simulation. The result is conservative and the assembly will, therefore, have
a lower probability of stripping than statistically designed assemblies.
Finally, a proof load can then be
calculated for the nut based on a hardened
mandrel in accordance with ISO R898/H and
the nut at minimum material strength and
minimum dimensional strength conditions.
(See paragraph £.0).