8

1
Thread Standards and Definitions
8

2
The Mechanics of Power Screws
8

3
Strength Constraints
8

4
Joints

Fasteners Stiffness
8

5
Joints

Member Stiffness
8

6
Bolt Strength
8

7
Tension Joints

The External Load
8

8
Relating Bolt Torque to Bolt Tension
8

9
Statically Loaded Tension Joint with Preload
8

10
Gasketed
Joints
8

11
Fatigue Loading of Tension Joints
8

12
Shear Joints
8

13
Setscrews
8

14
Keys and Pins
8

15
Stochastic Considerations
8

3
Strength Constraints
8

4
Joints

Fasteners Stiffness
8

5
Joints

Member Stiffness
8

6
Bolt Strength
8

4
Joints: Fastener Stiffness
When
a
connection
is
desired
that
can
be
disassembled
without
destructive
methods
and
that
is
strong
enough
to
resist
external
tensile
loads,
moment
loads,
and
shear
loads,
or
a
combination
of
these,
then
the
simple
bolted
joint
using
hardened
steel
washers
is
a
good
solution
.
Twisting
the
nut
stretches
the
bolt
to
produce
the
clamping
force
.
This
clamping
force
is
called
the
pretention
or
bolt
preload
.
This
force
exists
in
the
connection
after
the
nut
has
been
properly
tightened
.
Figure
8

13
A
bolted
connection
loaded
in
tension
by
the
forces
P
.
Note
the
use
of
two
washers
.
Note
how
the
threads
extend
into
the
body
of
the
connection
.
This
is
usual
and
is
desired
.
L
G
is
the
grip
of
the
connection
.
Figure
8

14
Section
of
cylindrical
pressure
vessel
.
Hexagon

head
caps
crews
are
used
to
fasten
the
cylinder
head
to
the
body
.
Note
the
use
of
an
O

ring
seal
.
LG’
is
the
effective
length
of
the
connection
(See
Table
8

7
)
.
Joints: Fastener Stiffness
Figure
8

14
shows
another
tension

loaded
connection
.
This
joint
uses
cap
screws
threaded
into
one
of
the
members
.
Joints: Fastener Stiffness
An
alternative
approach
to
this
problem
(of
not
using
a
nut)
would
be
to
use
studs
.
A
stud
is
a
rod
threaded
on
both
ends
.
The
stud
is
screwed
into
the
lower
member
first
;
then
the
top
member
is
positioned
and
fastened
down
with
hardened
washers
and
nuts
Studs
are
regarded
as
permanent
,
and
so
the
joint
can
be
disassembled
merely
by
removing
the
nut
and
washer
.
Joints: Fastener Stiffness
Spring
Rate
:
The
ratio
between
the
force
applied
to
the
member
and
the
deflection
produced
by
that
force
.
The
grip
L
G
of
a
connection
is
the
total
thickness
of
the
clamped
material
.
Total
distance
between
the
underside
of
the
nut
to
the
bearing
face
of
the
bolt
head
;
includes
washer,
gasket
thickness
etc
.
The
grip
L
G
here
is
the
sum
of
the
thicknesses
of
both
members
and
both
washers
.
Table
8

7
Suggested
Procedure
for
Finding
fastener
Stiffness
To find different parameters use table 8

7
1 1 1
,
b d t
d t
b
d t
d t
d t
d t
d t
b
d t t d
k k k
k k
k
k k
A E A E
k k
l l
A A E
k
A l Al
A
t
: tensile stress area (Tables 8

1, 8

2),
l
t
: length of threaded portion of the grip;
A
d
:
major diameter area of fastener;
l
d
: length of unthreaded portion in grip.
k
b
: is the
estimated effective stiffness of the bolt or cap screw in the clamped zone.
For short bolts
k
b
=
k
t
From Chapter 5
(Springs in series)
In
joint
under
tension
the
members
are
under
compression
and
the
bolt
under
tension
:
k
b
=
equivalent
spring
constant
of
bolt
composed
of
threaded
k
t
and
unthreaded
k
d
parts
acting
as
springs
in
series
.
d
t
There may be more than two members included in the grip of the
fastener.
All together these act like compressive springs in series.
Equivalent spring constant
k
m
If one of the members is a soft gasket, its stiffness relative
to the other members is usually so small that for all
practical purposes the others can be neglected and only
gasket stiffness used.
With no gasket, the stiffness of the members is difficult to
obtain, except by experimentation.
Compression spreads out between the bolt head and the
nut and area is not uniform.
1 2 3
1 1 1 1 1
....
m i
k k k k k
8

5
Joints

Member Stiffness
8

5
Joints Member Stiffness
Joint pressure distribution theoretical models
Ito
used
ultrasonic
techniques
to
determine
pressure
distribution
at
the
member
interface
.
Results
show
that
pressure
stays
high
out
to
about
1
.
5
bolt
radii
.
Ito
suggested
the
use
of
Rotscher’s
pressure
cone
method
for
stiffness
calculations
with
a
variable
cone
angle
.
This
method
is
quite
complicated
.
8

5
Joints Member Stiffness
Figure 8

15
We choose a simpler approach using a fixed cone
angle.
The contraction of an element of the cone of
thickness
dx
is subjected to a compressive force
P
is, from Eq. (5

3),
The area of the element is
Pdx
d
EA
8

5
Joints Member Stiffness
Figure 8

15b general cone
geometry using a half

apex
angle
.
2 2
2 2
0
tan
2 2
tan tan
2 2
i
D d
A r r x
D d D d
x x
Substituting this into the previous equation and integrating the resulting equation from
0
to
t
gives.
For Members made of Aluminum, hardened steel and cast iron
25
o
<
<33
o
With
=30
o
, this becomes
8

5
Joints Member Stiffness
(2 tan )( )
ln
tan (2 tan )( )
tan
(2 tan )( )
ln
(2 tan )( )
P t D d D d
Ed t D d D d
P Ed
k
t D d D d
t D d D d
(8

19)
0.574
(1.55 )( )
ln
(1.55 )( )
E d
k
t D d D d
t D d D d
(8

20)
Substituting this into the previous equation and integrating the resulting equation from
0
to
t
gives.
With
=30
o
, this becomes
8

5
Joints Member Stiffness
(2 tan )( )
ln
tan (2 tan )( )
tan
(2 tan )( )
ln
(2 tan )( )
P t D d D d
Ed t D d D d
P Ed
k
t D d D d
t D d D d
(8

19)
0.574
(1.55 )( )
ln
(1.55 )( )
E d
k
t D d D d
t D d D d
(8

20)
8

5
Joints Member Stiffness
CH

8 LEC 36 Slide
16
If members of the joint have the
same E with symmetrical frusta (l=2t)
,
then they act as two identical springs in series
k
m
=
k/2
.
For
=
30
°
and
D =
d
w
= 1.5 d,
this can be written as
,
Finite element analysis agree with
= 30
o
recommendation coinciding
exactly at the
aspect ratio
d/l
= 0.4
.
Additionally, FEM offered an exponential curve

fit of the form
where A and B are given in Table 8

8.
0.5774
0.5774 0.5
2ln 5
0.5774 2.5
m
E d
k
l d
l d
(8

22)
(/)
Bd l
m
k
Ae
Ed
(8

23)
8

5
Joints Member Stiffness
Figure 8

16
The dimensionless plot
of stiffness versus aspect
ratio of the members of
a bolted joint, showing
the relative accuracy of
method of
Rotsher
,
Michke
and
Motosh
,
compared to Finite
Element Analysis (FEA)
conducted by
Wileman
,
Choudury
, and Green.
8

5
Joints Member Stiffness
Bolt strength is specified by:
1.
minimum proof strength S
p
2.
or minimum proof load
F
p
,
3.
and minimum tensile strength,
S
ut
The
proof load
is the maximum load (force) that a
bolt can withstand without acquiring a
permanent set.
The
proof strength
is the
quotient
of the
proof
load
and the
tensile

stress area
.
The
proof strength
is about
90%
of the
0.2%
offset yield strength.
8

6
Bolt Strength
The SAE specifications are given in
Table 8

9
bolt grades are numbered
according to
minimum tensile strength.
The ASTM Specs for steel bolts (structural)
are in
Table 8

10
.
Metric Specs are in
Table 8

11.
If
S
p
not available use:
S
p
=0.85
S
y
F
p
=
A
t
S
p
8

6
Bolt Strength
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