Bolts
•General
•Sizes and strength
•Stiffness and preload
•Fatigue
General
•Purpose: to clamp 2 or more parts together.
•Use washer under head to prevent drill-burrs from scratching fillet,
causing stress concentration.
•Tighten nut until bolt elongation approaches elastic limit. (Theory later.)
•Do not reuse nuts. (Properly tightened nuts yield to distribute load among threads.)
•Bolted joints can be dangerous if not
•designed by a trained engineer
•assembled by a trained mechanic.
•Common threaded connections:
•a) Bolt and nut
•b) Machine screw
•c) Stud
Machine screws
Hexagonal Nuts: a)general
end view b)washer-faced c)both
sides chamfered d)jam nut, washer
faced e)jam, both faced chamfered
Cap-screw heads.
Machine screw heads
(Cap screws use no nuts. One
member has threaded hole)
Thread geometry
•Major diameter d = largest diameter of thread
•Pitch p = distance between adjacent threads (M), or 1/#of
threads per inch (US)
•Minor diameter dr = smallest diameter of screw thread
•Pitch diameter dp = diameter where thread width=gap width.
• β = 60o.
•UN = Unified National thread standard
•UNR = UN with specified root radius. Root is rounded to
reduce stress concentration factor
higher fatigue strength.
Thread Dimensions
(UN)
Table 15.6
Experimental
result:
Ατ = π dt2/4,
where dt = (minor
diameter + pitch
diameter)/2.
Or, if you want to
know,
At =
π
0.9743 
 dc −

4
n 
2
(15.27)
Cres t
diameter,
d c, in.
0.0600
0.0730
0.0860
0.0990
0.1120
0.1250
0.1380
0.1640
0.1900
0.2160
0.3500
0.3125
0.3750
0.4735
0.5000
0.5625
0.6250
0.7500
0.8750
1.000
1.125
1.250
1.375
1.500
1.750
2.000
Coars e Threads (U NC)
N umber of
Tens ile
threads per
s tres s area,
inch, n
A t, in. 2
64
0.00263
56
0.00370
48
0.00487
40
0.00604
40
0.00796
32
0.00909
32
0.0140
24
0.0175
24
0.0242
20
0.0318
18
0.0524
16
0.0775
14
0.1063
13
0.1419
12
0.182
11
0.226
10
0.334
9
0.462
8
0.606
7
0.763
7
0.969
6
1.155
6
1.405
5
1.90
4 1/2
2.50
Fine Threads (U N F)
N umber of
Tens ile
threads per
s tres s area,
inch, n
A t, in. 2
80
0.00180
72
0.00278
64
0.00394
56
0.00523
48
0.00661
44
0.00830
40
0.01015
36
0.01474
32
0.0200
28
0.0258
28
0.0364
24
0.0580
24
0.0878
20
0.1187
20
0.1599
18
0.203
18
0.256
16
0.373
14
0.509
12
0.663
12
0.856
12
1.073
12
1.315
12
1.581
-
Thread Dimensions - Metric
Table 15.7
At =
π
(d c − 0.9382 p )2
4
(15.28)
Crest
diameter,
dc, mm
1
1.6
2
2.5
3
4
5
6
8
10
12
16
20
24
30
36
42
48
Coarse Threads (MC)
Tensile
Pitch, p,
stress area,
mm
At, mm2
0.25
0.460
0.35
1.27
0.4
2.07
0.45
3.39
0.5
5.03
0.7
8.78
0.8
14.2
1
20.1
1.25
36.6
1.5
58.0
1.75
84.3
2
157
2.5
245
3
353
3.5
561
4
817
4.5
1121
5
1473
Fine Threads (MF)
Tensile
Pitch, p,
stress area,
mm
At, mm2
0.20
1.57
.25
2.45
.35
3.70
.35
5.61
.5
9.79
.5
16.1
.75
22
1
39.2
1.25
61.2
1.25
92.1
1.5
167
1.5
272
2
384
2
621
3
865
-
Bolt
strengths
(SAE, US)
• Proof load =
maximum force a
bolt can withstand
without yielding.
• Proof strength =
proof load/tensile
stress area (At)
Add to Table 15.4 Strength of steel bolts for various sizes in inches.
Bolt
strengths
(ASTM, US)
• ASTM bolts have
shorter thread and
longer shanks for
structural (mainly
shear) loads.
Table not in
Hamrock
Bolt
strengths
(Metric)
Add to Table 15.5
Strength of steel
bolts for various
sizes in millimeters.
Nut
heights
• Needed for specifying bolt length
when buying.
bolt
joint
joint
thickness
nut
height
• Thread lengths are not always
standardized as
L t’ =
(15.22)
L t’ =
(15.23)
Preload
•For bolted connection to work properly, bolt must be tightened.
•Amount of preload force Pi recommended by Shigley is
Pi = 0.75 Pp for reused connection
Pi = 0.90 Pp for permanent connection.
(15.33)
•Pp = proof load
• Pp= SpAt (Equation below 15.33)
•Sp = proof strength, see Table 15.4, 15.5, etc.
•At = tensile strength area, Eq. (5.27) or (5.28)
•Alternate method (for automobile wheel nuts etc): tighten nut
180o beyond snug-tight (finger-tight).
•Proper preload stresses the bolt to almost 0.85 yield strength!
However, Shigley says this is good. (More on this soon).
Pretorque
•Preload is good and necessary, however, measuring it is very
difficult. (E.g. in lab with a special bolt instrumented with strain gages)
•What a mechanic asks from the engineer is a specified bolt
tightening torque, which s(he) can realize using a torque wrench or a
pneumatic impact wrench.
•The engineer must calculate this tightening torque Ti:
Ti = KPid
( Not in Hamrock)
Shigley Table 8-10 Torque factors
Pi
d
K
= preload (Eq 15.33)
= major diameter
= torque factor
(depends on friction etc.)
Bolt Sizing for Shear Loads
3 in
12 in
500 lbf
Assume that the clamping force of each bolt is
concentrated at the center of the bolt. This force creates
the necessary friction force to prevent slipping. The
coefficient of friction between the bolted members is
0.25. Use factor of safety against slipping = 2. For the
bigger bolt, specify:
a) The required preload to prevent slipping.
b) The required pre-torque, for a non-plated, black-finished bolt, reused
connection, and c) The minimum size of a grade 5 bolt.
(b and c may require iteration).
Bolt Sizing fo Shear
Purposes of Preload, (in a nutshell. More on these later.)
• Preload prevents slipping of the bolted members,
which causes shear loading of the bolt. Always design
bolted joints to prevent slipping!
• Preload prevents joint separation, which causes leaks,
increases possibility of loosening, and other bad stuff.
•Preload increases the rigidity of the bolted connection.
•Preload increases the effective fatigue strength of the bolt.
• Preload achieves all of the above by effectively
distributing the external load P into
• increasing the bolt stress (~10% of P)
• decreasing the joint (pre)stress (~90% of P)
Safety Factor against Joint Separation
If a force Pj pulls on the joint, then the safety factor against
separation is
nsj =
Pi
Pj (1 − Ck )
( 15.32)
Pi = preload (Eq 15.33)
Pj = external load
Ck = joint constant (dimensionless stiffness parameter)
Ck =
kb
kb + k j
kb = bolt stiffness
kj = joint stiffness
Figure 15.15, page 690
( 15.17)
Bolt stiffness, kb
Calculate bolt stiffness from
(Derived from k =
AE
L
1
4  Ls + 0.4d c Lt + 0.4d r 
=
+

 ( 15.21)
kb πE 
d c2
d r2

with the following lengths and cross-section areas:)
c = crest
r = root
s = shank
t = threaded
part
Text Reference: Figure 15.12, page 682
Bolt dimensions for
stiffness
To calculate bolt stiffness,
1
4  Ls + 0.4d c Lt + 0.4d r 
=
+


k b πE 
d c2
d r2

Ls = shank length
Lt = tensioned thread length
Use the above Lt to compute stiffness.
This Lt’ is only for specifying bolt length when buying.
Calculate root diameter dr from
3
3
p = pitch
ht =
p=
2
2n
n = # of threads/inch
d r = d c − 1.25 ht
Joint stiffness kj , Shigley’s formula
k ji =
Stiffness of ith layer is
washer
frustum
πEi d c tan α f
 (L tan α f + d i − d c )(d i + d c ) 
2 ln  i

 (Li tan α f + d i + d c )(d i − d c ) 
( 15.24)
E = Young’s modulus
αf = frustum angle = 30o
Li = thickness of “layer”i
di = frustum diameter
For frustum close to a washer, di = 1.5 dc. So
k ji =
0.577πEd c
 0.577 Li + 0.5d c
2 ln 5
 0.577 Li + 2.5d c



( 15.24a)
Total stiffness of all layers is calculated
from
1
1
1
1
=
+
+
+L
k j k j1 k j 2 k j 3
( 15.25)
Joint Stiffness kj, Wileman’s formula
To calculate joint stiffness, Wileman formula can also be used.
Correct
Stiffness of ith layer is
k ji = Ei d c Ai e Bi d c / Li
( 15.26)
Total stiffness of all layers is calculated from
1
1
1
1
=
+
+
+L
k j k j1 k j 2 k j 3
( 15.25)
Table 15.3 Constants used in joint stiffness formula [Eq. (15.26)] [From Wileman et al
(1991)]
Material
Steel
Aluminum
Copper
Gray cast iron
Poiss on’s
ratio, ν
0. 291
0. 334
0. 326
0. 211
Modulus of
Elasticity, E,
GPa
206. 8
71.0
118. 6
100. 0
Text Reference: Table 15.3, page 684
Numerical Constants
Ai
Bi
0. 78715
0.62873
0. 79670
0.63816
0. 79568
0.63553
0. 77871
0.61616
Summary of Joint Constant Ck calculation
Bolt dimensions for stiffness
To calculate bolt stiffness,
1
4  Ls + 0.4dc Lt + 0.4d r 
=
+


kb πE 
d c2
d r2

Ls = shank length
Lt = tensioned thread length
Use the above Lt to compute stiffness.
This Lt’ is only for specifying bolt length when buying.
Ck =
Calculate root diameter dr from
3
3
p = pitch
ht =
p=
2
2n
n = # of threads/inch
d r = d c − 1.25 ht
kb
kb + k j
Joint stiffness kj , Shigley’s formula
Stiffness of ith layer is
washer
frustum
k ji =
πEi d c tan α f
 (L tan α f + d i − d c )(d i + d c )
2 ln  i

 (Li tan α f + d i + d c )(d i − d c )
To calculate joint stiffness, Wileman formula can also be used.
E = Young’s modulus
αf = frustum angle = 30o
Li = thickness of “layer”i
di = frustum diameter
k ji = E i d c Ai e Bi dc / Li
Stiffness of ith layer is
or
For frustum close to a washer, di = 1.5 dc. So
k ji =
Joint Stiffness kj, Wileman’s formula
( 15.24)
Correct
( 15.26)
Total stiffness of all layers is calculated from
1
1
1
1
=
+
+
+L
k j k j1 k j 2 k j 3
( 15.25)
0.577πEd c
 0.577 Li + 0.5d c 

2 ln  5
 0.577 Li + 2.5d c 
( 15.24a)
Total stiffness of all layers is calculated
from
1
1
1
1
=
+
+
+L
k j k j1 k j 2 k j 3
( 15.25)
Table 15.3 Constants used in joint stiffness formula [Eq. (15.26)] [From Wileman et al
(1991)]
Material
Steel
Aluminum
Copper
Gray cast iron
Pois s on’s
ratio , ν
0. 291
0. 334
0. 326
0. 211
Modulus o f
Elasticity , E,
GPa
206. 8
71.0
118. 6
100. 0
Numerical Co ns tants
Ai
Bi
0. 78715
0.62873
0. 79670
0.63816
0. 79568
0.63553
0. 77871
0.61616
Text Reference: Table 15.3, page 684
Example 15.6
Material
Steel
Aluminum
Copper
Gray cast iron
Poisson’s
ratio, ν
0.291
0.334
0.326
0.211
Text Reference: Figure 15.14, page 685
Modulus of
Elasticity, E,
GPa
206.8
71.0
118.6
100.0
Numerical Constants
Ai
Bi
0.78715
0.62873
0.79670
0.63816
0.79568
0.63553
0.77871
0.61616
Example 15.6
k ji =
πEi d c tan α f
 (L tan α f + d i − d c )(d i + d c ) 
2 ln  i

 (Li tan α f + d i + d c )(d i − d c ) 
Smallest of
frustrum
diameter


tana := tan  30⋅
d 2 :=
3
2
π


tana = 0.577
180 
−3
⋅ d c + 2⋅ 25⋅ 10 ⋅ tana
−2
d 2 = 4.987 × 10
Example 15.6
k ji = E i d c Ai e Bi dc / Li
Material
Steel
Aluminum
Copper
Gray cast iron
Poisson’s
ratio, ν
0.291
0.334
0.326
0.211
Modulus of
Elasticity, E,
GPa
206.8
71.0
118.6
100.0
Numerical Constants
Ai
Bi
0.62873
0.78715
0.79670
0.63816
0.79568
0.63553
0.77871
0.61616