Introduction
To determine the performance of a bolted joint it is necessary to calculate
the joint stiffness. That is the deflection of the joint under a bolt loading condition
When geometry of the bolted joint is an annulus with an OD less than 2,5 x the bolt diameter the joint
stiffness can be conveniently calculated using k = EA/l . When the joint includes a nonmetallic gasket
or material with a low modulus of elasticity then this item may have such a low stiffness value that the stiffness
of the metal parts of the connection will have very low impact on the overall stiffness.
It is very difficult to obtain theoretical values of joint stiffness because a
number of factors affect the result including.
 The effect of the angle of the screw thread
 Screw hole size
 Surface finish of contacting faces
 Flatness of contacting faces
 Use of Washer
 Selected size of contacting surface (d and D)
 Condition of surfaces between clamped plates

Any of these factors can have a significant effect on the resulting joint stiffness
The notes below identify methods which approximate the loading regime
which has been found to occur by various studies using different methods.
Overall stiffness evaluation for multicomponent joints
A bolted joint can include a number of separate parts. The relationship
between the individual stiffness values of the parts and the total stiffness of the clamped
joint is as shown below..
The sketch below shows a bolted surface. It has been shown using using
ultrasonics and FEA that the pressure in a bolted surface is greatest under the bolt
head and reduces as the distance from the bolt interface increases. A method of
be approximating the force distribution is based on use of the cone geometry as
shown. Various cone halfapex angles are used but for systems using a
washerface annulus and with hardened steel, cast iron or aluminium an angle of
25^{o} to 35^{o} considered reasonable.
These notes relate to half apex angle of 30^{o}
Referring to the contraction of a small element dx of cone subject to a load P as shown.
The area is determined as follows
Substituting this into the equation for contraction of an element..
To obtain the total deflection the equation above is integrated between 0 and t.
Using integration manipulation techniques this results in
The spring rate of the truncated cone is therefore ..
The bolt hole diameter is d, the diameter of the applied pressure e.g.the washer
diameter, is and the plate thickness is t.
For a half apex angle of 30^{o} which provides a reasonable value
for many engineering metals, the equation simplifies to..
Important Note:
Widely different stiffness values result from different studies. It has also
been proved that different loading conditions and surface conditions also affect
the resulting stiffness.
The variation in half apex angle α can be notice from the different
studies completed. Differences result from various reason including
D/d ratios differences and surface finish difference
Originator  Technique  Surface Type  α 
Link 3 below  Ultrasonics  Ground  41^{o} 
Link 3 below  Ultrasonics  Turned /Ground  68^{o} 
Shigley and Mischke  Analysis  Any  30^{o} 
Ito  Ultrasonics  Ground  70^{o} 
Gould and Mikic  Analysis  Smooth  38^{o} 
Table A
Showing stiffness values for steel plate E = 207 kN/mm^{2} and a half apex
angle α of 30 ^{o} calculated using equation X above..
d= bolt dia and D = 1,5 d = approximately as would result from a
typical bolt or cap screw .
 

Plate Thickness (mm) 


6 
12 
20 
40 
50 
60 
80 
100 
d 
D 
k = Stiffness (MN/mm ) based on...E = 207 kN/mm^{2 } and α = 30^{o} 
6 
9 
2.757 
2.102 
1.83 
1.619 
1.576 
1.547 
1.511 
1.489 
8 
12 
4.245 
3.098 
2.622 
2.254 
2.178 
2.127 
2.063 
2.024 
10 
15 
6.009 
4.236 
3.503 
2.934 
2.817 
2.739 
2.639 
2.579 
12 
18 
8.047 
5.514 
4.47 
3.66 
3.493 
3.381 
3.239 
3.153 
16 
24 
12.945 
8.49 
6.661 
5.245 
4.953 
4.757 
4.508 
4.356 
20 
30 
18.932 
12.017 
9.191 
7.006 
6.556 
6.253 
5.868 
5.634 
24 
36 
26.008 
16.095 
12.055 
8.94 
8.3 
7.867 
7.32 
6.986 
30 
45 
38.658 
23.236 
16.974 
12.163 
11.175 
10.509 
9.664 
9.149 
36 
54 
53.75 
31.605 
22.638 
15.768 
14.36 
13.41 
12.207 
11.474 
Table B
Showing stiffness values for steel plate E = 207 kN/mm^{2} and a half apex
angle α of 60 ^{o} calculated using Equation X above. This table is provided
to identify the variation is stiffness using two different, but still relevant, values of α
d= bolt dia and D = 1,5 d = approx as would result from a
typical bolt or cap screw .
 

Plate Thickness (mm) 


6 
12 
20 
40 
50 
60 
80 
100 
d 
D 
k = Stiffness (MN/mm ) based on...E = 207 kN/mm^{2 } and α = 60^{o} 
6 
9 
5.627 
4.929 
4.642 
4.423 
4.378 
4.349 
4.312 
4.289 
8 
12 
8.108 
6.886 
6.382 
5.995 
5.917 
5.864 
5.798 
5.759 
10 
15 
10.88 
8.996 
8.216 
7.616 
7.494 
7.412 
7.31 
7.248 
12 
18 
13.939 
11.255 
10.142 
9.284 
9.11 
8.993 
8.845 
8.757 
16 
24 
20.909 
16.216 
14.27 
12.764 
12.456 
12.25 
11.99 
11.833 
20 
30 
29 
21.76 
18.758 
16.431 
15.955 
15.635 
15.231 
14.988 
24 
36 
38.203 
27.878 
23.602 
20.284 
19.604 
19.146 
18.569 
18.219 
30 
45 
54.079 
38.121 
31.526 
26.407 
25.355 
24.647 
23.753 
23.211 
36 
54 
72.426 
49.629 
40.231 
32.938 
31.437 
30.426 
29.148 
28.373 
Bolted joint comprising two equal plates
The force distribution on a typical bolted joint comprising two plates is shown below:
It is necessary to obtain a value for the stiffness for each part of the joint. In the case
above for each of the two plates.
If the plates are of the same material and similar thickness then the overall stiffness of the joint
is..
FEA work has been completed (Wileman ,Choudury & Green)for similar 2 part bolted joint with typical
washers (D_{w} = 1,5d) and joint members of the same material and an approximate
curve fit curve for the resulting relationship has been derived..
Values for A and B are provided in the following table...

Poisson Ratio 
Youngs Modulus 
A 
B 
Material 

GPa 


Steel 
0,291 
207 
0,78715 
0,62873 
Aluminium 
0,334 
71 
0,79670 
0,63816 
Copper 
0,326 
119 
0,79568 
0,63553 
Gray Cast Iron 
0,211 
100 
0,77871 
0,61616 
General Case 


0,78952 
0,62914 
Example comparing stiffness value using two methods.
Consider two plates 20mm thick clamped with a 10m bolt using the table with a half apex
angle α of 30 ^{o}
k for each plate from Table A above = 3 503 kN/mm .... 1/ k _{t} = 1/k + 1/k .
Therefore k _{t} = k/2 = 1 7513 kN/mm.
Using the formula above with A = 0,78715,B = 0,62873,d = 10, l=40 and E = 207 OOO
k _{t} = E.d.A exp (Bd/l)
= 1 906 kN/mm
