ME3180
ME 3180 - Mechanical Engineering Design
Stresses in Threads
Lecture Notes
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Stresses in Threads
When nut engages thread, theoretically all threads in engagement
should share load
In actuality, due to inaccuracies in thread spacing, first pair of threads
takes virtually all load
Conservative approach for calculating bolt stresses is to assume worst
case of one thread-pair taking entire load
Other extreme approach for calculating bolt stresses is to assume that all
of engaged threads share load equally
Better compromise is to assume that true stress lies between these two
extremes, but most likely is closer to one thread-pair assumption.
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Stresses in Threads Cont’d
Power screws and fasteners for high-load applications are usually made of
hardened high-strength steels.
Power screw nuts may also be of hardened material for strength and
wear resistance.
Fastener nuts, on the other hand, are often made of soft materials, and
thus, are typically weaker than screws (i.e. – regular fastener and nut).
 This promotes local yielding in nut threads when fastener is tightened,
which can improve thread fit and promote load sharing between threads.
 Hardened nuts are used on hardened high-strength bolts.
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Axial Stress
While power screw can see either tensile or compressive axial load,
threaded fastener sees only axial tensile load
• This equation can be used to compute axial tensile stress in screw.
F

At
Eq. 14-2
For power screws loaded in compression, possibility of column buckling
must be investigated. Use screw’s minor diameter to compute
slenderness ratio.
Slenderness ratio is factor that determines if column is short or long.
• For short column,
Sr 
l

d
 10
where  is radius of gyration.
• If it is short column, use its compressive yield strength as limiting
stress (Page 200, Norton), if it is long column, then use buckling to
perform failure analysis.


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Shear Stress
Possible shear-failure mode involves stripping of threads:
• Out of nut
• Off of screw
Possibility of either of these scenarios occurring depends on relative
strengths of nut and screw materials
If nut material is weaker, it may strip its threads at its major diameter
If screw is weaker, it may strip its threads at its minor diameter
If both materials are of equal strength, assembly may strip along pitch
diameter
In order to calculate stresses, we must assume some degree of load
sharing among the threads
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Shear Stress Cont’d
Since complete failure requires all threads to strip, all of threads can be
considered to share load equally
 This is good approach as long as nut or screw (or both) is ductile, allowing
each thread to yield as assembly begins to fail
If both nut and screw are brittle (e.g., high-hardness steels or cast iron)
and thread fit is poor
 One can envision each thread taking entire load in turn until it fractures
and passes job to the next thread.
 The reality is again somewhere between these extremes.
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Shear Stress Cont’d
Stripping-shear area for each screw thread is area of cylinder of its minor
diameter dr:
As  d r wi p
where
p = thread pitch
wi = factor that defines percentage of pitch occupied by metal at minor
diameter (see Table 14-5)
This area can be multiplied by number of threads in engagement based on
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designer’s judgment.
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Shear Stress Cont’d
For nut stripping at its major diameter, shear area for one screw thread is:
As  dwo p
wo is the factor found in Table 14-5
Shear stress for thread stripping is:
F

As
Minimum Nut Length – If the nut is long enough, the load required to strip the
threads will exceed the load needed to fail the screw in tension. The equations
for both modes of failure can be combined and a minimum nut length computed for
any particular screw size. For any UNS/ISO threads or Acme threads of d ≤ 1in, a
nut length of at least 0.5d will have a strip strength in excess of the screw’s tensile
strength. For larger diameter ACME threads, strip-strength of nut with length ≥
0.6d will exceed the screw’s tensile strength. These figures are valid only if the
screw and nut are of the same material, which is usually the case.
Minimum Tapped-Hole Engagement – When a screw is threaded into a tapped
hole rather than a nut, a longer thread engagement is needed. For same
material combinations, a thread-engagement length at least equal to the nominal
thread diameter d is recommended. For a steel screw in cast iron, brass or bronze,
use 1.5d. The
For steel
screws
in aluminum,
useof2d
of minimum
thread-engagement
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W. Woodruff
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length.
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Torsional Stress
Torsional stress will develop when:
• Nut is tightened on screw
• Torque is transmitted through power screw
Torque that twists screw is dependent on friction at screw-nut interface
If screw and nut are well lubricated, less of applied torque is transmitted
to screw, and more is absorbed between nut and clamped surface
If nut is rusted to screw, all applied torque will twist screw, which is why
rusty bolts usually shear even when you attempt to loosen nut
For power screw, if thrust collar has low friction, all applied torque at
nut will create torsional stress in screw (since little torque is taken to
ground through low-friction collar).
In order to accommodate worst case of high thread friction, use total
applied torque in equation for torsional stress in round section (page 183,
Norton)
Tr 16T
  3
J d r
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School of Mechanical Engineering
for this calculation:
dr Woodruff
= minor diameter
ME3180
Types of Screws/Fasteners
Fasteners can be classified in different ways: by their intended use, by
thread type, by head style, and by their strength.
These fasteners are available in variety of materials including steel,
stainless steel, aluminum, brass, bronze, and plastics.
Classification by Intended Use
Bolts and Machine screws:
Same fastener may take on different name for particular application.
Bolt is fastener with head and straight threaded shank intended to be used
with nut to clamp assembly together. See Fig 14-10a
However, same fastener is called machine screw or cap screw when it is
threaded into tapped hole rather than used with nut. See Fig. 14-10b
Studs:
Headless fastener, threaded on both ends and intended to be semipermanently threaded into one-half of assembly. See Fig. 14-10c
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FIGURE 14-10
Bolt and Nut, Machine Screw and Stud
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Types of Screws/Fasteners
Classification by Thread Type
All fasteners intended to make their own hole or make their own threads
are called tapping screws, as in self-tapping, thread-forming, thread cutting,
and self-drilling screws. See Fig. 14-11
These are used in sheet metal or plastic
Classification by Head Style
Slotted Screw:
Many different types of head styles are made, including: straight-slot,
cross-slot (Phillips), hexagonal, hexagonal socket and others. Head shape
can be round, flat (recessed), filister, pan,etc. See Fig. 14-12
Socket-Head Cap Screw:
Typically made of high-strength, hardened steel, stainless steel or other
metals, and are used extensively in machinery. See Fig. 14-13
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FIGURE 14-13
Various Styles of Socket-Head Cap Screws
Courtesy of Cordova Bolt Inc., Buena Park, Calif.
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Nuts and Washers
Nuts: Please read up on nuts on pp 897 Norton. See Fig. 14-14, & Fig. 1415 on next slide
Washers:
• Plain washer is doughnut -shaped part that serves to increase area of
contact between bolt head or nut and clamped part. See Fig. 10.
• Hardened -steel washers are used where bolt compression load on
clamped part needs to be distributed over larger area than bolt head or
nut provides
• Any plain washer also prevents marring of part surface by nut when it
is tightened
• Softer washer will yield in bending rather than effectively distribute
load
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FIGURE 14-14
Various Styles of Standard Nuts
Courtesy of Cordova Bolt Inc.,
Buena Park, Calif., 90621
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Nuts and Fasteners Cont’d
Lock Washers:
• Help prevent spontaneous loosening of standard nuts (as opposed to
lock nuts)
• Can be used under nut of bolt or under head of machine screw. See Fig.
14-16
SEMS:
• Are combinations of nuts and captive lock washers that remain with nut
• Their main advantage is to ensure that lock washer will not be left out
at assembly or reassembly. See Fig. 14-17
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FIGURE 14-16
Types of Lock Washers
Courtesy of Cordova Bolt Inc.,
Buena Park, Calif., 90621
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Bolts and Fasteners
Strengths of Standard Bolts and Machine Screws
Bolts and screws are selected based on their proof strength Sp.
Proof strength is quotient of proof load and tensile-stress area
Proof Load Fp is maximum load (force) that bolt can withstand
without acquiring permanent set.
Preloaded Fastener in Tension
Primary application of bolts and nuts is to clamp parts together, such
that applied loads put bolt(s) in tension. See Fig. 14-21
Joints are preloaded by tightening bolts with sufficient torques to create
tensile loads that approach their proof strengths.
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Fasteners Cont’d
For statically loaded assemblies, preload that generates bolt stress as
high as 90% of proof strength is sometimes used.
For dynamically loaded assemblies (fatigue loading) preload that
generates bolt stress as high as 75% or more of proof strengths is
commonly used.
If bolts are suitably sized for applied loads, these high preloads increase
reliability of the bolts.
Reasons for this are subtle and require an understanding of how
elasticities of bolts and clamped members interact when bolt is tightened
and when external load is later applied.
Clamped members have spring constant .
Bolt, being elastic, will stretch when tightened.
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Spring Constants of Bolt
Fig. 14.23 shows bolt clamping cylinder of known cross section and length.
We want to examine loads, deflections, and stresses in both bolt and
cylinder under preload and after an external load is applied.
To examine above parameters, we must determine spring constants of
bolt and members.
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Spring Constants of Bolt
Bolt:
For bolt of diameter d and axially loaded thread length lt within its clamped
zone of length l as shown in Fig. 14-21, spring constant is
lt
l  lt
lt
ls
1




Kb
At Eb
Ab Eb
At Eb
Ab Eb
14.11a
where:
Ab is total cross-sectional area and At is tensile stressed area of bolt, and
ls is length of unthreaded shank.
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Spring Constants of Bolt Cont’d
1

(2d  4 ) in
LT  lth  
1
(2d  ) in
2

 (2d  6) m m

LT  lth   (2d  12) m m
(2d  25) m m

L  6 in
Table 8.7 (Shigley)
L  6 in
L  125, d  48 m m
125  L  200 m m
L  200 m m
Table 8.7 (Shigley)
Bolts shorter than standard thread lengths are threaded as close to head as
possible
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Determining Joint Stiffness Factor Cont’d
Fig. 14-30 shows results of finite element analysis (FEA) study of stress
distribution in two-part joint-sandwich clamped together with single,
preloaded bolt.
Stress distribution around bolt resembles truncated-cone (or cone-frusta)
barrel shape, as shown in Fig. 14-30a.
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FIGURE 17.19
Lines of equal compressive stress in joint. Bolt loaded
to 100 kip. (Reprinted from [20], courtesy Marcel Dekker
Inc.)
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Spring Constants of Members
(Cylindrical Model)
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Members:
For cylindrical material geometry shown in Fig. 14-23 (ignoring flanges),
material spring constant becomes:
l1
l2
4l1
4l2
1
14.11b




Km
Am1E1
Am 2 E2
 D2eff 1E1  D2eff 2 E2
where:
Am are effective areas of clamped materials and Deff are effective diameter
of those areas
Deff2  D2  d 2
14.11c
If both clamped materials are same
Am Em
14.11d
Km 
l
If Am can be defined as solid cylinder with effective diameter Deff equation
14.11d becomes
2
 Deff
Em
Am Em
Km 
or K m 
l
4l
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ME3180
External Load on Bolted Connection
Let us consider what happens when external tensile load P is applied to bolted
connection in Fig. 14-23.
Assume clamping force which we call preload Fi, has been correctly applied
by tightening nut before P is applied.
Fi = preload
P = external tensile load
Pb = portion of P taken by bolt
Pm = portion of P taken by members
Fb = Pb + Fi = resultant bolt load
Fm = Fi - Pm = resultant load on members
Fm < 0
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External Load on Bolted Connection Cont’d
These results are valid as long as some clamping load remains in the members.
P = P b + Pm
The load P causes connection to stretch, or elongate.
Pb = CP, where
14.12a
K
Pb  b Pm
Km
14.13b
Kb
Pb 
P
Km  Kb
14.13c
Kb
Km  Kb
C is called joint’s stiffness constant or joint constant.
C is typically less than 1, and if Kb is small compared to Km, C will be small
fraction. This confirms that bolt will see only portion of the applied load P.
C
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External Load on Bolted Connection Cont’d
Pm  0.8P. Also members can take even greater percentage of P, if grip is
longer. See Table 8.12 (Shigley)
Km
14.13d
Pm 
P  (1  C)P
Kb  Km
Fm  Fi  (1  C)P
14.14a
Fb Fi  CP
14.14b
Load Po required to separate the joint can be found from equation 14.14a by
setting Fm = 0.
F
Po  i
14.14c
1 C
Safety factor (or load factor) guarding against joint separation is
Po
Fi
N

P P(1  C)
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14.14d
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External Load on Bolted Connection Cont’d
Tensile stress in bolt can be found by dividing Fb = CP + Fi by At.
b 
CP Fi

At At
section 8.9 in Shigley
Limiting value of b is the proof strength Sp. Thus with introduction of load
factor, above equation becomes:
CnP Fi
  Sp
At At
n
Sp A t  Fi
CP
Any value of load factor (factor of safety), n  1, ensures that
 b  Sp
This implies that bolt will not fail.
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External Load on Bolted Connection Cont’d
From Shigley

0.75 Fp for reused connections also used for dynam icloading
Fi  

0.90 Fp for perm anentconnections also used for static loading
F p = A tS p
For other materials not in Tables 14.6 and 14.7, use Sp = 0.85Sy
Where Sy is yield strength of that material.
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FIGURE 14-19
Head Marks for SAE Bolts
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FIGURE 14-20
Head Marks-Metric Bolts
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Joints-MemberStiffness
(Cone Frusta Model)
• Both stiffness of members and fasteners must be known in order to
learn what happens when assembled connection is subjected to external
tensile loading.
• More than two members could be included in grip of fastener. All
together these act like compressive springs in series, and hence total
spring rate of members is
1
1 1 1
1
     
km k1 k2 k3
ki
(Equ.8-18 Shigley)
• If one of members is soft gasket, its stiffness relative to other
members is usually so small that for all practical purposes others can be
neglected and only gasket stiffness is used.
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Joints-Member Stiffness (cont.)
• Figure 8-15 illustrates general cone geometry using half-apex angle α.
Angle α = 45° has been used, but Little reports that this overestimates
3
clamping stiffness. In this book we shall use α = 30° except in cases in
which material is insufficient to allow frusta to exit.
• Thus spring rate or stiffness of this frustum is
 Ed tan 
k 
 ln (2t tan   D  d )( D  d )
(2t tan   D  d )( D  d )
P
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(8-19)
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Joints-Member Stiffness (cont.)
With α = 30°, this becomes
0.5774 Ed
k
(1.155t  D  d )( D  d )
ln
(1.155t  D  d )( D  d )
(8-20)
• Equations (8-20), or (8-19), must be solved separately for each
frustum in the joint. Then individual stiffnesses are assembled to obtain
κ m using Eq.(8-18).
• If members of joint have same Young’s modulus E with symmetrical
frusta back to back, then they act as two identical springs in series.
From Eq.(8-18) we learn that κ m =κ/2. Using grip as l=2t and d w as
diameter of washer face, we find spring rate of members to be
 Ed tan 
km 
(l tan   d w  d )(d w  d )
2 ln
(l tan   d w  d )(d w  d )
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(8-21)
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Joints-Member Stiffness (cont.)
• Diameter of washer face is about 50 percent greater than fastener
diameter for standard hexagon-head bolts and cap screws. Thus we can
simplify Eq.(8-21) by letting d W =1.5d . If we also use α = 30°, the
Eq.(8-21) can be written as
km 
0.5774 Ed
  0.5774l  0.5d  
2 ln  5

0.5774
l

2.5
d

 
(8-22)
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