Kon-41.2010 Machine design basics B (4 cr)
Strength calculation................................................................................................................ 1
Symbols and units ..........................................................................................................................................1
Linear motion .................................................................................................................................................1
Stresses...........................................................................................................................................................2
Failure theories...............................................................................................................................................4
Static load.......................................................................................................................................................4
Fatigue loads ..................................................................................................................................................4
Stress concentration factors............................................................................................................................5
Reversed stress (mean stress zero) .................................................................................................................6
Smith diagram ................................................................................................................................................7
Smith diagrams (non-alloy structural steels) ..................................................................................................9
Engineering materials........................................................................................................... 10
Steels ............................................................................................................................................................10
Cast irons......................................................................................................................................................12
Aluminium ...................................................................................................................................................13
Copper alloys ...............................................................................................................................................13
Physical properties of steels and cast irons ..................................................................................................14
Physical properties of materials ...................................................................................................................15
Bolted joint........................................................................................................................... 16
1 Stresses of a bolt during tightening ...........................................................................................................16
2 Torque required to tighten the bolt............................................................................................................17
Welded connections ............................................................................................................. 19
Stresses in fillet weld....................................................................................................................................19
Simple calculation method ...........................................................................................................................19
Parallel keys ......................................................................................................................... 20
Interference fits .................................................................................................................... 21
Spring design........................................................................................................................ 22
1 Helical extension and compression springs...............................................................................................22
2 Belleville springs.......................................................................................................................................23
3 Rubber springs...........................................................................................................................................24
Gears..................................................................................................................................... 25
Helical gears (external gears) .......................................................................................................................27
Forces on gear teeth......................................................................................................................................28
Couplings .....................................................................................................................................................29
Narrow V-belt drives (SFS 3527) ........................................................................................ 30
Datum lengths of narrow V-belts and datum diameters of pulleys .............................................................31
Rolling bearings ................................................................................................................... 33
Lubrication and lubricant classification ............................................................................... 38
1 Lubrication mechanisms............................................................................................................................38
2 Oil classification........................................................................................................................................39
Design of pressure vessels.................................................................................................... 41
1 Pressure equipment directive.....................................................................................................................41
2 Nominal design stress................................................................................................................................41
3 Cylindrical and spherical shells.................................................................................................................41
4 Dished ends ...............................................................................................................................................43

Machine Elements/SK 2009
Greek alphabet (upright and sloping types) alfa beeta gamma delta epsilon zeeta eeta theeta ioota kappa lambda myy nyy ksii omikron pii rhoo sigma tau ypsilon fii khii psii omega
α
β
λ
μ
ν
ξ
ο
π , ϖ
ρ
σ
γ
δ
ε
ζ
η
θ , ϑ
ι
κ
τ
υ
φ , ϕ
χ
ψ
ω
Α
Β
Γ
Δ
Ε
Ζ
Η
Θ
Φ
Χ
Ψ
Ω
Ρ
Σ
Τ
Υ
Ν
Ξ
Ο
Π
Ι
Κ
Λ
Μ
α
β
γ
δ
ε
ζ
η
θ, ϑ
ι
κ
λ
μ
ν
ξ
ο
π, ϖ
ρ
σ
τ
υ
φ, ϕ
χ
ψ
ω
Υ
SI prefixes
10
10 -1
10 -2
10 -3
10 -6
10 -9
10 -12
10 -15
Factor Name Symbol
10 15
10 12
10 9
10 6
10 3
10 2 peta tera giga mega kilo hehto
P
T
G
M k h deka desi sentti milli mikro nano piko femto da d c m
μ n p f

Machine Elements/SK 2009
1
Acceleration
Modulus of elasticity
Force
Gravity
Moment of inertia
Torque
Mass
Rotation speed
Power
Work
Radius
Diameter
Length a
E
F
G
J
M v
, T m n
P
W r d l m/s 2
N/mm 2 , MPa
N
N kgm 2
Nm kg r/min, r/s
W
Nm, J m, mm m, mm m, mm
Area
Pressure
Density
Stress
(tensile, compression, bending)
Shear stress
Extension
Strain
Time
Velocity
Angular velocity
Angular acceleration
Efficiency
Friction coefficient v
ω
α
η
μ
A p
ρ
σ
τ
∆ l
(
)
ε t
Distance
Velocity
Acceleration
Force
Moment
Power
Work
s
= vt Rotational angle ϕ ω t
= 2 π nt
v
a
=
= s/t v/t
Tangential
Angular velocity
Angular
ω
ω ϕ
& r
=
2
&&
π
π dn n
= v r
/ t
F
= ma
M
=
Fl
Torque
M v
=
J
P
=
Fv
W
=
Fs
Power
Work
P
W
=
=
M
M v v
ω ϕ
Kinetic energy
W
= 1
2 mv
2 Kinetic energy
A part may fail if it yields and distorts sufficiently to not function properly. A part may also fail by fracturing and separating. Only ductile materials may yield significantly before fracturing (fig. 1). Brittle materials proceed to fracture without significant shape change.
σ
W
= 1
2
J ω
2
-
- m 2
Pa, N/m 2 , bar kg/m 3
N/mm 2 , MPa
N/mm 2 , MPa m, mm
- s m/s rad/s rad/s 2
R m
R eH
R eL
= F / A tensile stress
A cross-section area
length change (extension)
=
/
L strain
Modulus of elasticity E = tan β
β
ε
Fig. 1. S tress-strain–diagram (low carbon steel).
R eH upper yield strength
R eL lower yield strength
R m tensile strength.

Machine Elements/SK 2009
Tensile stress
♦ Hooke’s law
Shear stress
σ =
F
A
σ =
E ε
=
τ =
F
A
2
Surface pressure p
=
F
A
F
B projected area
D
Bending stress
Torsion stress
σ =
M
W
τ =
M v
W v
W
W z
=
W y
=
≈ , d
3
π d
3
32
W v
π d
3
16
≈ 0 2 d
3
W z
=
W y
=
π ( D
4 − d
4
)
32 D
π (
D
4 − d
4
)
≈
16
(
D
D
4 − d
4
)
D
Cross-section area
A
A
=
π d
2
4
A
=
π (
D
2 − d
2
)
4

Machine Elements/SK 2009
Table 1 . End reactions, moment, deflection and slope formulas.
Load Deflection and slope
1.
End reactions and moment
B
=
F
M
= −
Fx
M max
= −
Fl y
=
Fl
6
3
EI
⎝
2 3 l x
+
β
A
Fl
3 y max
=
= −
3
EI
Fl
2
2
EI x
3 l
3 ⎠
2. B
=
Q
= ql
Qx
2
M
= −
2 l
M max
= −
Ql
2 y
=
Ql
24
3
EI
⎝
3 4 l x
+
β
A
Ql
3 y max
=
= −
8
EI
Ql
2
6 EI x
4 l
4 ⎠
3.
A B
M max
F
=
=
2
Fx
Fl
2
4
=
Fl
48
3
EI
⎝
3 x
4 l
−
β
A
Fl
3 y max
=
48
EI
Fl
2
= − β
B
=
16
EI x
3 l
3 ⎠
4.
=
Fbl
6
EI
2 ⎡
⎜ 1
⎣
⎝
− b
2 l
2
⎞
⎠ l
A
=
Fb l
B
=
Fa l
=
Fbx l
: =
Fa
⎛
⎝⎜
1 − l x ⎞
⎠⎟
M max
=
Fab l y
C
= y max
=
3 EIl
Fbl
27
2
EI
3 1
⎝
− b
2 l
2 ⎠
3 kun
β
A
= x
= ( l
2 − b
2
) / 3
Fbl
6
EI
⎝
1 − b
2 l
2 ⎠
β
B
= −
Fal
6 EI
⎝
1 − a
2 l
2 ⎠ ja a
≥ b l x
3
3
⎤
⎥
3
5.
A B Q
/ 2
(
Q
= ql
)
M
=
Qx
2
⎛
⎝⎜
1 − l x ⎞
⎠⎟
M max
=
Ql
8 y
=
Ql
24
3
EI
⎝ l x
− 2
5
Ql
3 y max
β
A
=
= −
384 EI
β
B
=
Ql
2
24
EI x
3 l
3
+ x
4 l
4 ⎠

Machine Elements/SK 2009
4
Distortion energy theory, effective stress
σ vert
= σ
2 + 3 τ
2
Maximum shear stress theory, effective stress
σ
2 + 4 τ
2
(1)
(2) σ vert
=
A. Ductile (tough) material
Effective stress
σ vert
≤ σ sall
=
R eL (3) n where R eL
is a yield strength and n safety factor. Normally n = 1,2...2.
B. Brittle material
Effective stress
σ vert
≤
R m (4) n where R m
is a tensile strength and safety factor n = 2...4.
a) c) Fluctuating
Fig. 2.
Fatigue

Machine Elements/SK 2009
Bending
5
Torsion
Fig. 3.
Stress concentration factor for a shaft shoulder.
The maximum stress (bending)
σ max
= K ft
σ nim
(5)
σ nim
is a nominal stress, K ft
is a stress concentration factor
K ft
= 1 + q
(
K tt
- 1) concentration factor (fig. 3). where q
is a notch sensitivity of the material (steel S355: q
≈ 0,9) and
K tt
(6)
geometric stress

Machine Elements/SK 2009
1 k
1
0,9
0,8
0,6
0,8
1,6
3,2
0,7
6,3
0,6
Rolled, forged or casted
25
0,5
300 400 500 600 700 800 900 1000 1100 1200 1300 1400
Tensile strength R m
(N/mm )
Fig. 4.
Surface quality factor k
1
.
1 k
2
0,9
0,8
0,7
0,6
10 20 30 40 50 60 70 80 90 100 110 120 d (mm)
Fig. 5.
Size factor k
2
.
Bending or tensile-compression load (mean stress σ n
= k
1 k
K
2 ft
w nim m
= 0)
Torsion load (mean stress n
= k
1 k
2
K fv
w nim
m
= 0)
In other cases the safety factor is calculated using Smith diagram.
Table 2.
Physical properties of structural steels.
Tensile
(N/mm 2 )
Bending
(N/mm 2 )
Torsion
(N/mm 2 )
Steel
R e
σ w
R te
σ tw
τ vs
τ vw
(7)
(8)
S235 (Fe 37) E295 (Fe 50) S355 (Fe 52)
235 295 355
175 230 245
335
195
170
135
410
250
205
175
490
260
240
215
6

Machine Elements/SK 2009
R e
σ w
P
2
P
1
σ w
’
σ
Q v,a
OP' = PP' = σ v,m
PQ = σ v,a
O
45°
σ
P v,m
σ v,m
P ’
P
1
’
σ m
σ w
' = k
1 k
2
σ w
σ w
is fatigue limit k k
R
1
2 e
surface quality factor
size factor
yield strength
Fig.
6.
Smith diagram.
Safety factor n
1. mean stress σ v,m
and amplitude σ n
=
OP
1
OQ
=
OP
1
'
OP ' v,a
will grow in the same ratio
2. σ v,m
is constant, σ v,a
increases n
=
PP
2
PQ
Torsional and bending load, effective stress:
v, m
= ( K ft
tm
)
2 + 3 ( K fv
vm
)
2
v, a
= (
K ft
ta
)
2 + 3 (
K fv
va where K ft
stress concentration factor for bending
K fv
stress concentration factor for torsion
σ tm
mean bending stress
τ vm
mean torsional stress
σ ta
bending stress amplitude
τ va
torsional stress amplitude.
)
2
(9)
(10)
(11)
(12)
7

Machine Elements/SK 2009
Notched specimen Shape Stress concentration factor
K f
Bending
K fv
Groove 1,5...2 1,3...1,8
Retaining ring groove
Shoulder fillet
Transverse hole
End-milled keyway *
Sled-runner keyway *
Shaft-hub connection: interference fit
Shaft-hub connection: key
2,5...3,5
≈ 1,5 r/d d/D
= 0,1 and d/D
1,4...1,8
= 0,7
= 0,14
2,6...3
2...2,5
1,7...1,9
2...2,4
* Stress concentration factor depends on corner radius and material.
Fig. 7 .
Preliminary design values for stress concentration factors.
2,5...3,5
≈
≈
1,25 r/d d/D
= 0,1 and d/D
1,4...1,8
= 0,7
= 0,14
2,3
2...2,5
1,3...1,4
1,5...1,6
8

Machine Elements/SK 2009
9
Raaka-ainekäsikirja 1. Muokatut teräkset. 3. uudistettu painos. Metalliteollisuuden Kustannus
Oy 2001. 361 s. ISBN 951-817-751-1.
N/mm 2
400
S355
R eH
=
355
E295
300
245
230
200
175
S235
295
235
σ w
100
0
σ w
-100
-175
-200
-245
-300
N/mm 2
500
-230
100 200 300 400
σ m
Tensile - compression a)
400
E295
S235
S355
490
410
335 tw
300
260
200
195
250
100
0
100 200 300 400 500
- tw
-100
-195
-200
-300
-250
-265 m
Bending b)
Fig. 8.
τ vw
N/mm 2
300
215
200
175
135
100
E295
S235
S355
τ vs
=
240
205
170
0
τ vw
-100
-135
-175
-200
100
τ m
200 300
-215
Torsion c)

Machine Elements/SK 2009
10
According to SFS-EN 10027-1
1 Steels designated according to their application and mechanical or physical properties
Principal symbols:
• S structural steel
• P steels for pressure purposes
• L steels for pipelines
• E engineering
ƒ followed by a number being the specified minimum yield strength (N/mm 2 ), e.g. S235, E295
ƒ for steel casting the name shall be preceded by the letter G
ƒ additional symbols for impact strength etc, e.g. S355J2
Table 1.
Structural steels.
SFS 200 SFS-EN
10025 v. 2004
S235JR
S235J0
S235J2
S275JR
S275J0
S275J2
S355JR
S355J0
S355J2
S355K2
Yield strength 1)
Tensile strength 2)
R eH
(N/mm 2 ) R m
(N/mm 2
235
235
235
360...510
360...510
360...510
275
275
275
355
355
355
355
430...580
430...580
430...580
510...680
510...680
510...680
510...680
Impact strength
27 / 20
27 / 0
27 / -20
27 / 20
27 / 0
27 / -20
27 / 20
27 / 0
27 / -20
40 / -20
SFS-EN
10025 v. 1991
Fe 360 B FN
Fe 360 C
Fe 360 D2
Fe 430 B
Fe 430 C
Fe 430 D2
Fe 510 B
Fe 510 C
Fe 510 D2
Fe 510 DD2
S185
E295
E335
E360
3)
3)
3)
185
295
335
360
310...540
490...660
590...770
690...900
- / -
- / -
Fe 310-0
Fe 490-2
Fe 590-2
Fe 690-2
1)
Nominal thickness ≤ 16 mm.
2)
Nominal thickness < 3 mm.
3)
Engineering steels.
Classification by impact strength
(SFS-EN 10027-1)
Test temperature Impact strength (J)
°C 27 J 40 J 60 J
20
0
JR
JO
KR
KO
LR
LO v. 1986
Fe 37 B
Fe 44 B
Fe 52 C
Fe 33
Fe 50
Fe 60
Fe 70

Machine Elements/SK 2009
11
2 Steels designated according to chemical composition
(Examples in tables 2…4)
Non-alloy steels
• letter C and the carbon content % multiplied by 100
Non-alloy steels (with Mn ≥ 1 %), non-alloy free-cutting steels and alloy steels (except high speed steels) where the content, by weight, of every alloying element is < 5 %
• carbon content % multiplied by 100
• chemical symbols indicating the alloy elements (in decreasing order)
• numbers indicating the values of contents of alloy elements
Alloy steels (except high speed steels)
• letter X
• carbon content % multiplied by 100
• chemical symbols indicating the alloy elements (in decreasing order)
• numbers indicating the values of contents of alloy elements
Table 2.
Quenched and tempered steels (SFS-EN 10083).
Material
R e
(N/mm 2 )
R m
(N/mm 2 )
2 C 45
25 CrMo 4
42 CrMo 4
34 CrNiMo 6
370
450
650
800
630...780
700...850
900...1100
1000...1200
(40 mm < d < 100 mm)
ƒ heat treatment including hardening and annealing in relative high temperature (500…700 ° C)
ƒ shafts, couplings, gears, bolts and nuts.
Table 3.
Case hardening steels.
SFS-EN 10084
20NiCrMo2-2
R e
(N/mm
490
2
)
R m
(N/mm
740...1030
2
) Hardness HB
265
16MnCr5 590
20NiCrMo5 690
18CrNiMo7-6 780
790...1080
1030...1370
1080...1330
285
345
370
ƒ higher carbon content in thin surface layer
ƒ high wear resistance and fatigue strength and bending strength
ƒ gears and shafts.
Table 4.
Stainless steels.
10088-2
R p0,2 strength Modulus of elasticity
(N/mm 2
X2CrNi19-11 200
)
R m
(N/mm 2 )
E
2 )
000
X2CrNi18-9 200 500...650 200
X5CrNi18-10 210
X2CrNiMo17-12-2 220
X3CrNiMo17-13-3 220
520...670
530...730
ƒ corrosion resistant
ƒ ductile at low temperatures
ƒ pipes, vessels, valves, machinery in process industry, containers and tanks.

Machine Elements/SK 2009
Table 5.
Grey cast irons.
SFS-EN 1561
R m
(N/mm 2 )
R p0,1
(N/mm 2 ) Elongation
12
ƒ low cost, good for casting and easy machining, absorption of vibration
ƒ machine beds, valves, pipes, cylinders and lining, brake drums and disks.
Table 6.
Spheroidal graphite cast irons (ductile irons).
SFS-EN 1563
R m
(N/mm 2 )
R p0,2
(N/mm 2
EN-GJS-350-22 350 220
EN-GJS-400-18 400 240
EN-GJS-400-15 400 250
EN-GJS-450-10 450 310
EN-GJS-500-7 500 320
EN-GJS-600-3 600 370
EN-GJS-700-2 700 420
EN-GJS-800-2 800 480
EN-GJS-900-2 900 600
22
18
15
10
7
3
2
2
2
ƒ high strength compared to grey cast iron, heat treating possible
ƒ gears, bodies and frames, power transmission, combustion engine and paper machine components.
Table 7.
Austempered Ductile Irons (ADI).
EN 1564
Yield strength
(N/mm 2 )
800-8 500
1000-5 700
1200-2 850
1400-1 1100
Tensile strength
(N/mm 2 )
800
1000
1200
1400
Elongation
(%)
8
5
2
1
Hardness
(HB)
260...320
300...360
340...440
380...480

Machine Elements/SK 2009
ƒ low weight
ƒ corrosion resistant
ƒ good heat and electricity conductivity
ƒ special alloys with high strength
ƒ aluminium profiles
• economical manufacturing
• material extruded trough profile tool
ƒ aluminium casting
• low weight
• ductile
• easy to machine
Table 8.
Aluminium profile alloys.
Alloy
Al 99,5
AlMg2,5
E-AlMgSi
AlSi1Mg
AlSi1MgPb
AlZn5Mg1
Yield strength
(N/mm 2 )
20
80
180
260
180
280
Tensile strength
(N/mm 2 )
70
180
220
300
280
330
Modulus of elasticity
E
≈ 70 000 N/mm 2
ƒ journal bearings are most important applications
Table 9.
Common copper alloys.
Alloy
CuZn39Pb3
Lead brass
CuZn35Mn2AlFe
Special brass
CuSn6
Tin bronze
GK-CuZn40Pb
Lead brass
GS-CuSn12
Tin bronze
GS-CuPb10Sn10
Lead tin bronze
GS-CuAl10Fe3
Aluminium bronze
Products
Bolts, nuts, valves, connectors
Shafts, piston rods, gears, bolts, nuts, valves
Springs, valve and pump components
Components of devices, locks, decorative parts
Gears and worm wheels, sliding surfaces, journal bearings
Heavily loaded journal bearings (edge contact)
Crane wheels, bushings, gears, journal bearings
Yield strength
(N/mm 2 )
250...
430
270...
440
390...
490
120
160
80
180
Elongation
A
5
23
(%)
14
10
8
8
10
Tensile strength
(N/mm 2 )
430...
520
470...
590
470...
550
280
280
180
500
Hardness
(HB)
18...25
35...45
65...75
95...115
85...95
115...125
Elongation
A
5
(%)
Hardness
(HB)
15
12
7
13
70
95
65
115
13
15...30 115...
155
15...30 135...
170
15...40 -

Machine Elements/SK 2009
14
Material
Structural steels
Quenched and tempered steels
Case hardening steels
Stainless steels:
X4CrNi 18 9
X4CrNiMo 17 12 3
E
(GN/m 2 )
206
206
206
200
200
Poisson's ratio
ν
0,3
0,3
0,3
0,3
0,3
Density ρ
(kg/m 3 )
7850
7850
7850
7900
8000
Linear expansion coefficient
α (1/K)
12 ⋅ 10 -6
12 ⋅ 10 -6
12 ⋅ 10 -6
17 ⋅ 10 -6
16,5 ⋅ 10 -6
Thermal conductivity
λ (W/(m K))
52…63
42…59
42…59
15
13,5
Specific heat capacity c
(kJ/(kg K))
0,50
0,50
0,50
0,44
0,44
Grey cast irons
GJL-150 (GRS 150)
GJL-200 (GRS 200)
GJL-250 (GRS 250)
GJL-300 (GRS 300)
GJL-350 (GRS 350)
Spheroidal graphite cast irons
GJS-350
GJS-400 (GRP 400)
GJS-450
GJS-500 (GRP 500)
GJS-600 (GRP 600)
GJS-700 (GRP 700)
GJS-800 (GRP 800)
GJS-900
78…103
88…113
103…118
108…137
123…143
169
169
169
169
174
176
176
176
0,26
0,26
0,26
0,26
0,26
0,275
0,275
0,275
0,275
0,275
0,275
0,275
0,275
7100
7150
7200
7250
7300
7100
7100
7100
7100
7200
7200
7200
7200
11,7 ⋅ 10
-6
11,7 ⋅ 10
-6
11,7 ⋅ 10
-6
11,7 ⋅ 10
-6
11,7 ⋅ 10
-6
12,5 ⋅ 10
-6
12,5 ⋅ 10
-6
12,5 ⋅ 10
-6
12,5 ⋅ 10
-6
12,5 ⋅ 10
-6
12,5 ⋅ 10
-6
12,5 ⋅ 10
-6
12,5 ⋅ 10
-6
52,5
50,0
48,5
47,5
45,5
36,2
36,2
36,2
35,2
32,5
31,1
31,1
31,1
1)
2)
ADI - Austempered ductile cast irons
GJS-800-8
GJS-1000-5
GJS-1200-2
GJS-1400-1
1)
2) t = 100 °C t = 300 °C
References
170
168
167
165
0,27
0,27
0,27
0,27
7100
7100
7100
7100
14,6 ⋅ 10
-6
14,3 ⋅ 10
-6
14,0 ⋅ 10
-6
13,8 ⋅ 10
-6
22,1
21,8
21,5
21,2
Raaka-ainekäsikirja 1. Muokatut teräkset. 3. uudistettu painos. Metalliteollisuuden Kustannus Oy
2001. 361 s. ISBN 951-817-751-1.
Raaka-ainekäsikirja 2. Valuraudat ja valuteräkset. 2. uudistettu painos. Metalliteollisuuden Kustannus
Oy 2001. 196 s. ISBN 951-817-757-0.
Raaka-ainekäsikirja 1. Muokatut teräkset. 2. tarkistettu ja uudistettu painos. Metalliteollisuuden Kus-
0,46
0,46
0,46
0,46
0,46
0,515
0,515
0,515
0,515
0,515
0,515
0,515
0,515 tannus Oy 1993. 353 s. ISBN 951-817-564-0.
SFS-Käsikirja 138. Valurauta. Yleis- ja ainestandardit. Suomen Standardisoimisliitto 1999. 176 s.
ISBN 952-5143-38-4.

Machine Elements/SK 2009
15
Material
Aluminium alloy
Copper
Brass
Aluminium bronze
Lead bronze
Magnesium alloy
Babbitt (lead)
Babbitt (tin)
Zinc alloy
Nickel alloy
Steel
Stainless steel
Titanium
Grey cast iron 1)
Diamond (natural)
Synthetic diamond
2)
2)
Aluminium oxide
(polycrystal) 3)
Silicon carbide 3)
Silicon nitride 3)
Titanium carbide
Tungsten carbide
Graphite
Nylon 3)
3)
3)
3)
Reinforced nylon 3)
Polyimide 3)
Teflon 3)
Silicon oxide (glass)
E
(GN/m
70
124
100
117
97
41
29
52
50
207
200
193
110
76...176
965
1000
345
400
310
393
655
14
2,6...3,3
9,6...14
3,2...5,2
0,26...0,45
68
2 )
Poisson's ratio
ν
0,33
0,33
0,33
0,33
0,33
0,33
0,27
0,30
0,30
0,30
0,33
0,2...0,3
0,20
0,20
0,23
0,15
0,27
0,21
0,24
0,30
0,32...0,36
0,32...0,36
0,41
0,45
0,16
Density
(kg/m 3
2700
8900
8600
7500
8900
1800
10100
7400
6700
7800
7800
ρ
)
7100...7300
3515
3515
3900
3200
3200
6000
15100
1900
1140
1420
1430
2200
2200
Linear expansion coefficient
α
(1/K)
24 ⋅ 10 -6
18 ⋅ 10 -6
19 ⋅ 10 -6
18 ⋅ 10 -6
18 ⋅ 10 -6
27 ⋅ 10 -6
20 ⋅ 10 -6
23 ⋅ 10 -6
27 ⋅ 10 -6
11,9 ⋅ 10 -6
11 ⋅ 10 -6
17 ⋅ 10 -6
10,3 ⋅ 10 -6
10…13 ⋅ 10 -6
1,34 ⋅ 10 -6
1,34 ⋅ 10 -6
7 ⋅ 10 -6
4 ⋅ 10 -6
1,26 ⋅ 10 -6
9 ⋅ 10 -6
5 ⋅ 10 -6
3 ⋅ 10 -6
81 ⋅ 10 -6
(25…38) ⋅ 10 -6
(45…50) ⋅ 10 -6
(135…151) ⋅ 10-6
0,6 ⋅ 10 -6
Thermal conductivity
λ (W/(m K))
146 (cast) 4)
170
120
50
4)
4)
47
110
24
56
110
35
15
31...53
800
2000
30
50
30,7
55
102
178
0,25
0,22…0,48
0,36…0,98
0,24
1,25
Specific heat capacity c
(kJ/(kg K))
0,900
0,380
0,390
0,380
0,380
1,000
0,150
0,210
0,400
0,450
0,450
0,46...0,54
0,510
0,510
0,752
0,670
0,710
0,543
0,205
0,710
1,670
-
1,13…1,30
1,050
0,800
4)
1) Values are representative. Exact values vary with composition and processing.
2) Materials are anisotropic. Values vary with crystallographic orientation.
3) Typical properties of bearing quality materials. Ceramics are hot pressed or equivalent sintered. These properties are representative and depend on detailed composition and processing. t = 100 °C
References
Hamrock B. J.
Fundamentals of Fluid Film Lubrication. McGraw-Hill, New York 1994. 690 s. ISBN
0-07-025956-9.
Wear Control Handbook (Ed. Peterson & Winer). New York 1980. 1358 p.

Machine Elements/SK 2009
A flange joint is a typical bolted joint (fig. 1-1).
16
Fig.
1-1. Flange joint.
When the bolt is tightened, a tensile stress and torsional stress is developed in the bolt. For
ISO metric threads (thread angle 60 ° ) the friction torque in threads is /1/
M
G
= 1
2 d
2
F
M
⎛
⎜⎜
1 , 155
G
+
P d
⎞
⎟⎟
(1-1)
2 where F d
2
M
is the preload (from tightening)
the pitch diameter (table 1-1)
μ
G
the friction coefficient in threads
P
the pitch.
The torsional stress in a round section (diameter d
S
) is
=
M
W v
G =
8 d
2 d
F
3
S
M
⎛
⎜⎜
1 , 155
G
+
P d
2
⎞
⎟⎟
(1-2)
The equation for the diameter d S
of the thread is /1, 2/ d
S
= d
2
+ d
2
3 (1-3) where d
3
is the root diameter of the thread. If the bolt has a reduced diameter (< d S minimum diameter d
T
), use the
. The tensile stress in the cross-section due to the preload force is
S
=
4
F d
M
2
S
The effective stress is (theory of constant energy of distortion)
(1-4)
σ vert
= σ
2
S
+ 3 τ
2
(1-5)

Machine Elements/SK 2009
17
The effective stress should not be more than 90 % of the yield stress (0,9 R p0,2 maximum tensile stress during tightening is /1, 3/
or 0,9 R eL
). The
S
=
1 + 3
⎛
⎝
2 d d
2
S
0 , 9
R p 0 , 2
( 1 , 155
G
+
P d
2
)
⎞
⎟⎟
2
(1-6)
The friction coefficient in threads depends on the material, surface treatment and lubrication.
(table 1-2). For bolts M6...M16 σ
S
≈ 0,7
R eL
, when the friction coefficient in threads is μ
0,15. The maximum axial force (in assembled state) is
G
=
F
SP
= σ
S
A
S
(1-7) where A
S
is the tensile stress area of the bolt (table 1-1). Property classes of bolts are in the table 1-3 (SFS-ISO 898-1).
Table 1-1.
Selected dimensions of ISO metric threads.
Thread
Nominal diameter d
/mm
Pitch
P
/mm
Pitch diameter d
2
/mm
Root diameter d
3
/mm
Tensile stress area
A
S
/mm 2
Width across
flats s
/mm
SFS-ISO 272
M 6
M 8
M 10
M 12
M 16
M 20
6
8
10
12
16
20
1,0
1,25
1,5
1,75
2,0
2,5
5,350
7,188
9,026
10,863
14,701
18,376
Table 1-2.
Friction coefficient μ
G
in threads /4/.
4,773
6,466
8,160
9,853
13,546
16,933
20,1
36,6
58,0
84,3
157
245
10
13
16
18
24
30
Surface treatment Dry
Untreated
Phosphated
Phosphated black
Zinc electroplated
Cadmium electropl.
0,20...0,35
0,28...0,40
0,26...0,37
0,14...0,20
0,10...0,19
Oiled
0,16...0,23
0,16...0,33
0,24...0,27
0,14...0,19
0,10...0,17
MoS
2
0,13...0,19
0,13...0,19
0,14...0,21
0,10...0,17
0,13...0,19
Table 1-3.
Property classes (strength grades) of bolts.
R m
/ N/mm 2
R eL
or R p0,2
/ N/mm
Property class
(nominal)
2
(nominal)
5.6 6.8 8.8 10.9 12.9
500 600 800 1000 1200
300 480 640 900 1080
R m
tensile strength,
R eL
or
R p0,2 yield strength.
The total torque required to tighten the bolt is a sum of the friction torque in threads and torque between the head or nut and the surface (fig. 2-1). The friction torque
M
K
between the nut and the surface is
M
K
= 1
2
μ
K
D km
F
M
(2-1)

Machine Elements/SK 2009
18 where μ
D
K
is the friction coefficient between the nut (or head) and the surface d
K km
= ( d
K
+
D
K
)/2 the mean diameter (location of friction force)
the outside diameter of the nut (or head) ≈ width across flats s
(wrench opening)
D
K
the diameter of the hole.
The friction coefficient between the nut (or head) and the surface is μ
K
≈ 0,08...0,22 depending on the material, surface treatment and lubrication. The friction coefficient of stainless steels (between the nut (or head) and the surface or in threads) can be even 0,5.
The total torque required to tighten the bolt is
M
A
=
M
G
+
M
K
=
1
2
F
M
1 , 155 μ
G d
2
+ μ
K
D km
+
P
π
(2-2)
Fig. 2-1.
Bolt tightening using wrench.
The preload F
M
depends on friction coefficients and torque. With hand tools only bolts M10
(10.9) and M12 (8.8) are tightened properly (preload of small bolts is usually too high and preload of big bolts is too small) /1/.
1. Verho Ruuviliitokset ja liikeruuvit. Julkaisussa: Airila M. et al. Koneenosien suunnittelu, 2. painos.
Porvoo: WSOY 1997. S. 161...243. ISBN 951-0-20172-3.
Verlag 1995. 677 s. ISBN 3-446-17966-6.
3.
VDI Richtlinie 2230 Blatt 1 . Systematische Berechnung hochbeanspruchter Schraubenverbindungen.
Düsseldorf: VDI-Verlag 1986. (Systematic calculation of high duty bolted joints)
4. Haberhauer H. & Bodenstein F.
Maschinenelemente. Gestaltung, Berechnung, Anwendung. 10. Auflage.
Berlin: Springer-Verlag 1996. 626 s. ISBN 3-540-60619-X.

Machine Elements/SK 2009
19
The stresses of the fillet weld are calculated for the minimum cross section A = al ( a is the throat thickness (height of the cross section area) and l
is the length of the weld). The minimum cross section area is located at 45 ° to the legs. The stresses of the area are divided into three components (fig. 1). a
⊥
⊥
Fig. 1.
Stresses on the throat section of a fillet weld.
In the simple calculation method the equation for the stress of the weld σ w
is regardless of the direction of the load
σ w
=
F d (1) al
F d
is the design load = γ
F
F , γ
F
is a partial safety factor for the load. The resistance of the weld is sufficient if (SFS-EN 1993-1-8)
w
≤ f vw, d
=
3
f u w
M 2
(2) f u
is the tensile strength (table 1) and
M2
= 1 , 25 partial safety factor of weld. The calculation method is valid when a
≥ 3 mm (SFS-EN 1993-1-8). The length of the weld has also limitations. Mechanical properties of structural steels are in the table 1.
Table 1.
Mechanical properties of structural steels. f y
is yield strength .
Steel Thickness (mm) f y
(N/mm 2 ) f u
(N/mm 2 ) f vw,d
(N/mm 2 ) Factor
w
S 235
S 275
S 355
≤ 40
40 < t
≤ 80
≤ 40
40 < t
≤ 80
≤ 40
40 < t
≤ 80
235
215
275
255
355
345
360
360
430
410
510
470
208
208
234
223
262
241
0,8
0,85
0,9

Machine Elements/SK 2009
20
The torque that can be transmitted (the bearing action between the side of the key and the hub material) (fig. 1) where p t
M vn l
is the length of the key
2 n
= p n l t
2
( d
+ t
2
)/2
is the compressive stress of the hub
the depth of the keyway in the hub d the diameter of the shaft.
(1)
The torque that can be transmitted (the bearing action between the side of the key and the shaft material)
M va
= p a l t
1
( d
- t
1
)/2 where p a
is the compressive stress of the hub and t
(2)
1
the depth of the keyway in the shaft. p n p a
The compressive stress
◊
◊
◊ the
p is:
150 grey cast iron o spheroidal graphite cast iron
90 N/mm
2
110 N/mm 2 .
The load factor is in the table 2.
M v
Fig. 1.
Parallel key (SFS 2636).
Table 1.
Dimensions of keys (SFS 2636).
Key length is in the standard.
Width b h9) 2 3 4 5 6 8 10 12 14 16 18 20 22 25 28 32 36 40 45 50
Height h
2 3 4 5 6 7 8 8 9 10 11 12 14 14 16 18 20 22 25 28
Diameter of shaft d
> 6 8 10 12 17 22 30 38 44 50 58 65 75 85 95 110 130 150 170 200
≤ 8 10 12 17 22 30 38 44 50 58 65 75 85 95 110 130 150 170 200 230
Depth of keyway (shaft) t
1
1,2 1,8 2,5 3 3,5 4 5 5 5,5 6 7 7,5 9 9 10 11 12 13 15 17
Depth (hub) t
2
1 1,4 1,8 2,3 2,8 3,3 3,3 3,3 3,8 4,3 4,4 4,9 5,4 5,4 6,4 7,4 8,4 9,4 10,4 11,4
Table 2.
The design compressive stress p sall
=
Cp o
.
One-way load, static
0,8 p o
One-way load, light shocks
0,7 p o
One-way load, heavy shocks
0,6 p o
Reverse load, light shocks
0,45 p o
Reverse load, heavy shocks
0,25 p o

Machine Elements/SK 2009
21
A press fit is obtained by machining the hole in the hub to a slightly smaller diameter than that of the shaft. Only relative small parts can be press-fitted. For large parts a shrink fit can be made by heating the hub to expand its inside diameter. a) b) c)
σ t
σ r
σ vn
σ tn d a d i
D i
D a
D
F p u a u n
σ ra p
σ rn
σ va
σ ta
Fig. 1.
An interference fit and stresses in interference fits.
Table 1.
Interference fits (sizes mm).
Nominal sizes
>
3
≤
3
6
H7
Deviations
+0,010
0
+0,012
0 compression s6
Deviations
+0,020
+0,014
+0,027
+0,019 t6
Deviations u6
Deviations
+0,024
+0,018
+0,031
+0,023 v6
Deviations
6
180
10
200
+0,015
0
+0,032
+0,023
+0,037
+0,028
10 14 +0,018 +0,039
0 +0,028
+0,044
+0,033 +0,050
+0,039 14
18
24
30
40
50
24 +0,021 +0,048
30
0 +0,035 +0,054
+0,041
+0,054 +0,060
+0,041 +0,047
+0,061
+0,048
+0,068
+0,055
50
+0,064
40 +0,025 +0,059 +0,048
0 +0,043 +0,070
+0,054
+0,076
+0,060
+0,086
+0,070
+0,084
+0,068
+0,097
+0,081
+0,072
65 +0,030
80
+0,053
0 +0,078
+0,059
+0,085
+0,066
+0,094
+0,075
+0,106
+0,087
+0,121
+0,102
+0,121
+0,102
+0,139
+0,120 65
+0,093
80 100 +0,035
100 120
+0,071
0 +0,101
+0,079
120 140
+0,117
+0,092
+0,113
+0,091
+0,126
+0,104
+0,147
+0,122
+0,146
+0,124
+0,166
+0,144
+0,195
+0,170
+0,168
+0,146
+0,194
+0,172
+0,227
+0,202
140
18
160
+0,040
0
+0,125
+0,100
+0,159
+0,134
+0,215
+0,190
+0,253
+0,228
160 180
+0,133
+0,108
+0,171
+0,146
+0,235
+0,210
+0,277
+0,252
+0,151
+0,122
+0,195
+0,166
+0,265
+0,236
+0,313
+0,284

Machine Elements/SK 2009
22
Common forms of helical springs are in fig. 1. For springs with end treatments the total number of coils n t
is bigger than the number of active coils n
. Other forms are possible such as conical helical compression springs. If the place for a spring is small it is possible to put several helical springs within each other.
Fig. 1.
Helical compression springs (a) and extension spring (b).
The force of a helical spring is
F
=
Gd
4 f
8 where
G
is the shear modulus of elasticity f d
the wire diameter
D the mean coil diameter n
the number of active coils
the deflection.
The spring rate (spring constant) for a helical spring is (
F
= kf
)
Gd
4 k
=
8
The nominal shear stress of the wire’s cross-section is
τ =
8 DF
π d
3
=
Gd
π nD
2 f
The maximum shear stress is
τ tod
= k w
τ
(1)
(2)
(3)
(4)

Machine Elements/SK 2009
23 where k w tor k w
is the stress concentration factor for the dynamic load. The stress concentration fac-
(the Wahl factor) is as a function of the spring index
C
=
D/d
in fig. 2.
The stress concentration factor for the static load is k s
= 1 +
1
2
C
(5) k w
=
4
C
4
C
− 1
− 4
+
0 , 615
C
Fig. 2.
Stress concentration factor or Wahl factor.
Groups 1 and 2
φ
D e t l
0
Group 3
φ
D i
φ
D e h
0
Class
A
B
C
D e
/ t
18
28
40 h
0
/ t
0,4
0,75
1,3
OM I l
0
IV
II h
0 t'
III
φ
D i
Fig. 3.
Forms of Belleville springs, the top and bottom of springs in group 3 are chamfered. Belleville springs have three dimension classes A, B and C (DIN 2093).
The force-deflection relationship is nonlinear. The allowed deflection f ≤ 0,75 h
0
.

Machine Elements/SK 2009
Fig. 4.
Deflection of Belleville spring.
24
The modulus of elasticity
E
and
G
(in shear) for rubber depends on the durometer hardness number (e.g. IRHD). Dynamically loaded rubber springs have higher stiffness than statically loaded. A cylindrical rubber spring is frequently used as a compression spring (fig. 5).
Fig. 5. Cylindrical rubber spring with compression loading.
Fig. 6.
Simple rubber shear spring.
Fig. 7.
Cylindrical rubber spring
(torsion loading).

Machine Elements/SK 2009
25
Gears are used to transmit torque and angular velocity in many applications. There is a wide variety of gear types to choose from.
Spur gears Helical gears Spur gears, internal set
Rack and pinion
Crossed helical gears Bevel gears Worm and worm gear
Fig. 1.
A variety of gear types.

Machine Elements/SK 2009
Motor
P
1 n
1
Coupling
Gear
P n
2
2
Coupling
Fig. 2.
Mechanical power transmission.
Gear ratio =
1
/
2
Driven machine
26
Fig. 3.
Shaft mounted speed reducer (Benzler-Sala).
Debarking drum X
Girth ring ( z
2
) AC-motor
(assembled from segments)
X
Fig. 4.
Driving unit for a debarking drum (Moventas).
Pinion z
1
Gear

Machine Elements/SK 2009
27
Normal module m n
, pressure angle α n
= 20 ° , helix angle β , number of teeth z
, facewidth b
and addendum modification coefficient x
(SFS 3390).
Equation
Tranverse module
(1)
Transverse pressure angle
Tranverse pitch m t
= m n cos β
α t
= arctan tan cos
α
β n p t
= m t
π
(2)
(3)
Tranverse base pitch p bt
= p t cos α t
(4)
Reference diameter (5)
Base diameter
Addendum of gear tooth d
= m t z d b
= d cos α t h a
= m n
( 1 + x
) − Δ h a
(6)
(7)
Correction of addendum
Dedendum
Tip diameter (outside diameter)
Root diameter
Base centre distance
(no profile- shift)
Centre distance
Working pressure angle
Involute function
Transverse contact ratio
Δ h a
= m n
⎛
⎜⎜
2 z
1
+ cos z
2
β
+ x
1
+ x
2
⎞
⎟⎟
− a w
, then ∆ h a
= 0 h f
= m n
( 1 , 25 − x ) d a
= d
+ 2 h a d f
= d
− 2 h f a
= a w
= m t a
( z
1
+ z
2
)
2 cos α t cos α wt cos inv α wt
= a cos α t a w inv α t
+
2 ( x
1
+ z
1 x
2
+
) tan α n z
2 inv α = tan α − α
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
ε
α
=
1 p bt
⋅ d
2 a 1
2
− d
2 b 1 + d
2 a 2
2
− d
2 b 2 − a w sin α wt
(16)
Overlap ratio
Total contact ratio
ε
β
ε
γ
= b tan β p t
= ε
α
+ ε
β
(17)
(18)
Fig. 5.
Involute gear (a), bottom clearance c and backlash j (b).

Machine Elements/SK 2009
28
Transmitted load (tangential load)
F t
=
M r
1 v1 =
M r
2 v 2 =
π
P d n
=
π
P
(19)
M v1,2 gear).
is a torque on a gear, n
1,2
rotational speed,
P
power and d
1,2
pitch diameter (1 pinion, 2
Radial force
F r
=
F t tan α t
=
Axial force
F a
=
F t tan β
F t tan α n
/ cos β (20)
(21)
On spur gears the teeth are straight and aligned with the axis of the gear, the helix angle β = 0.
F n
α
F
N
F r
β
F t
β F a b
F n
Fig. 6.
Forces on gear teeth: F t
tangential force, F r
radial force and F a
axial force.
Gear ratio i
= n
1 n
2
=
ω
ω
1
2
= d
2 d
1
= z
2 z
1 where index 1 is for the driving gear (pinion) and index 2 for the driven gear.
(22)
Driving Driven r
1 r
2 n
1 pitch point n
2 a
Fig. 7.
Two gears in mesh.

Machine Elements/SK 2009
Fig. 8.
Gear coupling. a) b) flexible part
Fig. 9.
Flexible couplings (KUMERA).
29

Machine Elements/SK 2009
30
1. If the diameter of the small pulley d p
D p
using the speed ratio i
is known, calculate the diameter of the large pulley i
= n
1 (1) n
2
D p
= id p
π (
D p
+ d p
) +
(
D p
− d p
4
E
(2)
2. If the required centre distance of a V-belt system E and diameters D p length of the V-belt is
and d p
are known, the
L
≈ 2
E
+ 1
2
)
2
(3)
If the length L differs from the standard datum length L p distance of a V-belt system is
(SFS-ISO 4184), the new centre
E p
= +
L p
−
L
(4)
2
The recommended centre distance of a V-belt system is
E
= 0,75...1,0( d p
+
D p
).
3. Initial tension
The initial tension of the belt is critical because it ensures that the belt will not slip under the design load. The too high tension can damage the belts and bearings. The proper belttensioning can be calculated according to the standard.
4. Adjustment for the centre distance
The adjustable length for the mounting is y
= 20...30 mm depending on the belt profile.
The adjustable length for the tension is x = 0,03 L p
. (SFS-ISO 155)
L j n
1
D p
β v d p
E x y
Fig. 1 .
Adjustable lengths of the centre distance between the pulley shafts.

Machine Elements/SK 2009
31
Standard datum lengths
L d
of narrow V-belts are in the table 1. Datum diameters
d d are in the table 2. Grooves of pulleys are in the figure 2. The datum width w d the groove profile. The groove angle α of the pulley is 34 or 38° (SFS-ISO 4183).
of pulleys
is characterizing
Table 1.
Standard datum lengths of narrow V-belts and distribution according to the sections, dimensions in millimeters (SFS-ISO 4184).
630
710
800
900
1000
1120
1250
1400
1600
1800
2000
2240
2500
2800
3150
3550
4000
4500
5000
5600
6300
7100
8000
9000
10000
11200
12500
Nominal Section datum length L d
(= L p
)
SPZ SPA SPB SPC
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ + +
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
± 25
± 25
± 32
± 32
± 40
± 40
± 50
± 50
± 6
± 8
± 8
± 10
± 10
± 13
± 13
± 16
± 16
± 20
± 20
± 63
± 63
± 80
± 80
± 100
± 100
± 125
± 125 between the lengths of the belts of the same set
2
4
6
10
16
Fig. 2.
Grooves of the pulleys (mm) (SFS-ISO 4183). w d b (min.) h (min.) e f (min.)
α = 34°, d d
:
α = 38°, d d
:
8,5
2
9
12
7
≤ 80
>80
11
2,75
11
15
9
≤ 118
>118
14
3,5
14
19
11,5
≤ 190
>190
19
4,8
19
25,5
16
≤ 315
>315

Machine Elements/SK 2009
Table 2.
Datum diameter d d
(SFS-ISO 4183).
Datum diameter d d
(= d p
)
Recommendation 1) Radial and axial runout
Nominal Tol. Z A B C D E mm diam. mm
63
67
71
75
50
53
56
60
80
85
90
95
100
425
450
475
500
530
560
600
630
265
280
300
315
335
355
375
400
170
180
190
200
212
224
236
250
106
112
118
125
132
140
150
160
SPZ
± 0,8 % +
+
∗
∗
∗
∗
∗
∗
± 0,8 %
∗
± 0,8 %
∗
∗
∗
∗
∗
∗
∗
± 0,8 %
∗
∗
∗
± 0,8 %
∗
∗
∗
∗
∗
SPA
∗
∗
∗
∗
∗
∗
∗
∗
+
∗
∗
∗
+
+
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
1) + only classical V-belts (Z, A...E)
∗ narrow and classical V-belts
SPB SPC
0,2
∗
∗
∗
∗
∗
∗
∗
+
+
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
0,3
∗
∗
∗
∗
+
∗
∗
∗
∗
∗
+
∗
∗
∗
∗
∗
∗
0,4
+
+
+
+
+
+
+
+
+
+
0,5
+
+
+
+
+
0,6
32

Machine Elements/SK 2009
33
Fig. 1. a) deep groove ball bearing, b) self-aligning ball bearing, c) angular contact ball bearing, d) cylindrical roller bearing, e) needle roller bearing, f) spherical roller bearing, g) taper roller bearing, h) thrust ball bearing, i) cylindrical roller thrust bearing, j) spherical roller thrust bearing (SKF).
Fig. 2.
Bearing housing (SKF), rolling bearing and adapter sleeve with nut and locking device.
Locating bearing Non-locating
Fig. 3.
Bearing arrangement.
Locating bearing Non-locating

Machine Elements/SK 2009
Basic rating life equation
L
10
= ⎛⎝⎜
C
P
⎞
⎠⎟ p where L
10
is the basic rating life (millions of revolutions)
C
the basic dynamic load (N)
P p
the equivalent dynamic bearing load (N)
= 3 for ball bearings and 10/3 for roller bearings.
For bearings operating at constant speed the basic rating life (operating hours) is
(1)
34
L
10 h
=
10
6
60 n
C
P p
(2) where n is the rotational speed (r/min).
Adjusted rating life (million of revolutions)
L nm
= a
1 a
ISO
L
10
(3) where a
1
is a life adjustment factor for reliability
is a life modification factor, which considers the influence of lubrication, fatigue a
ISO stress limit and contamination on bearing life (fig. 5 and 6)
The n
represents the difference between the requisite reliability and 100 %.
Table 2.
Values for life adjustment factor a
1
(ISO 281, 2007).
Reliability % 90 95 96 97 98 99 99,2 99,4 99,6 99,8 99,9 99,92 99,94 99,95 a
1
The condition of the lubricant separation is described as the ratio of the actual viscosity the reference kinematic viscosity
ν
κ =
ν
1
1
. The viscosity ratio is
(4)
to
The reference kinematic viscosity ν
ν
ν
1
1
=
=
45000
4500 ⋅
⋅ n
− 0 , 83 n
− 0 , 5 ⋅
⋅
D
− 0 pw
, 5
D
− 0 pw
, 5
for
1
can be calculated with following equations (ISO 281)
for n n
≥
< 1000 r/min
1000 r/min
(5)
D pw
is the pitch diameter, the mean bearing diameter d bore diameter and
D
outside diameter (ISO 281, 2007). m
= 0 , 5 ( d
+
D
) can also be used. d
is
Guide values for the contamination factor can be taken from table 3, which shows typical levels of contamination for well lubricated bearings.

Machine Elements/SK 2009
35
1000
2
1 500 5
10
200
20
100 n
(r
/m in
)
200
100
50
50
20
10
5
500
10
500
5
100
00
100
300
150
0
0
0
00
200
500
0
0
200
00
3
10 20 50 100 200 500 1000 2000
Fig. 4.
The
D pw
ν
1
required at the operating temperature to ensure adequate lubrication.
is the pitch diameter, the mean bearing diameter used (ISO 281, 2007).
D pw
(mm) d m
= 0 , 5 ( d
+
D
) can also be
Table 3.
Contamination factor η c
(ISO 281 2007).
Level of contamination η c
D pw
< 100 D pw
≥ 100
Extreme cleanliness
Particle size of the order of lubricant film thickness, laboratory conditions
1 1
0,8…0,6 0,9…0,8 High cleanliness
Oil filtered through extreme fine filter, conditions typical of bearing greased for life and sealed
Normal cleanliness
Oil filtered through fine filter, conditions typical of bearing greased for life and shielded
Slight contamination
Slight contamination in lubricant
Typical contamination
Conditions typical of bearings without integral seals, course filtering, wear particles and ingress from surroundings
Severe contamination
Bearing environment heavily contaminated and bearing arrangement with inadequate sealing
Very severe contamination
D pw
is pitch diameter, the mean diameter 0,5( d + D ) can also be used (SKF, 2005).
0,6…0,5 0,8…0,6
0,5…0,3 0,6…0,4
0,3...0,1 0,4...0,2
0,1...0 0,1...0
0 0

Machine Elements/SK 2009
50
36 a
ISO
20
10
5
2
1
0,5
κ =
4
2
1
0
,8
0
,6
0,
5
0,
4
0,
3
0,2
0,1
5
0,2
0,1
0,1
0,05
0,005 0,01 0,02 0,05 0,1 0,2 0,5 1 2 5
η c
P u
P
Fig. 5. Life modification factor a
ISO tio
= ν / ν
for radial ball bearings as a function of the viscosity ra-
1
.
P u
is the fatigue load limit of bearing.
Some EP additives in the lubricant can extend bearing service life where lubrication might otherwise be poor. When the viscosity ratio
≥ 0,2, a value of
< 1 and the factor for the contamination level
c
= 1 can be used in the calculation, if a lubricant with proven effective EP additives is used. In this case the life modification factor a
ISO value if this a
ISO
value is above 3. This motivation for increasing the able smoothening effect of the contacting surfaces can expected when an effective EP additive is used. In the case of severe contamination (
c be proven under actual lubricant contamination.
shall be limited to a
ISO
≤ 3, respectively to the life modification factor calculated for normal lubricants with the actual
value is that a favour-
< 0,2), the efficiency of the EP additives shall

Machine Elements/SK 2009
50
37 a
ISO
20
10
5
2
1
0,5
κ =
4
2
1
0
,8
0
,6
0,
5
0,
4
0,
3
0,2
0,15
0,1
0,2
0,1
0,05
0,005 0,01 0,02 0,05 0,1 0,2 0,5 1 2 5
Fig. 6.
Life modification factor a
ISO ratio
= ν / ν
η c
P u
P
for radial roller bearings as a function of the viscosity
1
. P u
is the fatigue load limit of bearing.
P = XF r
+ YF a
(6) where
F r
is the radial bearing load (N),
F a
the axial bearing load (N)
X
the radial load factor for the bearing,
Y the axial load factor for the bearing.
Table 4.
Load factors for deep groove ball bearings (normal clearance). f
0
F a
/ C
0 e X
F a
/ F r
≤ e
Y X
F a
/
F r
> e
Y
0,172 0,19 1 0 0,56 2,30
0,345 0,22 1 0 0,56 1,99
0,689 0,26 1 0 0,56 1,71 f
0
is a calculation factor (bearing tables). C
0
is a basic static load rating (a total permanent deformation of rolling element and raceway is approximately 0,0001 of the rolling element diameter).

Machine Elements/SK 2009
38
In fluid film lubrication the rubbing surfaces are completely separated by a thick film of lubricant. Fluid films are formed in three ways: hydrodynamic, elastohydrodynamic or hydrostatic film (fig. 1).
In elastohydrodynamic lubrication (EHD) the viscosity of lubricant increases, as the pressure on an oil increases and elastic deformation of two surfaces occurs due to the pressure of lubricant.
1. Film and pressure is formed by motion of lubricated surfaces
Hydrodynamic lubrication EHD-lubrication p p
EHD u
1 h c h min u
2. Film is formed by pumping fluid under pressure
Hydrostatic lubrication
F pressure p
T h u velocity h film p pressure
F load u
2 p
P
Fig. 1.
Fluid film lubrication.
Boundary Mixed Fluid film
lubrication lubrication lubrication
Hydrodynamic bearing
Hydrostatic bearing
Fig. 2.
Effect of speed on bearing friction.
Speed

Machine Elements/SK 2009
39
The relationship between the roughnesses of the surfaces and the film thickness is important.
The film thickness increases as the speed is increased, the lubricant viscosity is increased, the load is decreased, or the geometric conformity of the mating surfaces is improved. Boundary lubrication occurs when speeds are low or applied loads are very high. For this type of lubrication EP-additives are required to prevent welding of the contact and adhesive wear.
Role of lubricant
‰
‰
‰ reduce friction and power loss reduce wear cooling
‰
‰ prevent corrosion eliminate harmful particles
• wear particles
• deposits
Fig 3.
Internal combustion engine
(Neste Oil).
Additives of lubricants
‰ pour point depressants
‰
• lower the temperature at which a mineral oil is immobilized by wax viscosity index improvers
• reduce the effect of temperature on viscosity
‰
‰
‰
‰
‰ foam inhibitors oxidation inhibitors rust inhibitors detergents and dispersants
• reduce deposits of sludge in internal combustion engines antiwear and extreme pressure (EP) agents.
ACEA classification is in table 1 and SAE viscosity classification for engine and automotive gear oils is given in tables 2 and 4. ISO viscosity classification for industrial oils is given in table 6.
Performance classification for engine and automotive gear oils is given in tables 3 and 5.
Table 1. ACEA classification (Association of European automotive manufacturers).
Petrol (gasoline) and diesel engine oils
Catalyst compatible oils, petrol and diesel engines
Heavy duty diesel engine oils
A1/B1, A3/B3, A3/B4 ja A5/B5
C1, C2, ja C3
E2, E4, E6 ja E7

Machine Elements/SK 2009
Table 2.
SAE viscosity grades for engine oils.
SAE grade
Visc. cP max.
Pumping temp. max.
0W
5W
10W
15W
20W
25W
20
30
40
50
60
6200/-35 °C
6600/-30 °C
7000/-25 °C
7000/-20 °C
9500/-15 °C
13000/-10 °C
-
-
-
-
-
Table 3.
API engine oil classification.
-40 °C
-35 °C
-30 °C
-25 °C
-20 °C
-15 °C
-
-
-
-
- mm
Viscosity
2 /s (100 °C)
min. max.
3,8
3,8
4,1
5,6
5,6
9,3
5,6
9,3
12,5
16,3
21,9
-
-
-
-
-
-
< 9,3
< 12,5
< 16,3
< 21,9
< 26,1
Gasoline engine oil categories Diesel engine oil categories
(SA …SH), SJ, SL, SM
→
(CA…CE), CF, CG, CH, CI, CJ-4
-better performance →
The performance requirements for each classification are defined in terms of performance in engine tests (protection against wear, oxidation, deposits and corrosion).
Table 4.
SAE viscosity grades for axle and manual transmission oils.
SAE grade
70W
75W
80W
85W
90
140
250
Max. temperature for a viscosity
150000 cP
-55 °C
-40 °C
-26 °C
-12 °C
Viscosity mm 2 /s (100 °C)
min. max.
4,1
4,1
7,0
11,0
13,5
24,0
41,0
<24,0
<41,0
Table 5.
API gear oil classification.
GL-1
GL-2
GL-3
GL-4
GL-5
Gear oils without EP additives
Mildly fortified gear oils for worm wheels
Lubricant with light EP for non-hypoid gears and bevel wheels
Medium EP effect lubricant for moderate load hypoid gears
High EP effect lubricant for hypoid gear drives
GL-1, GL-4 and GL-5 are in common use.
Table 6.
ISO viscosity classes.
ISO VG (ISO 3448)
2
22
220
3
32
320
5
46
460
7
68
680
10
100
1000
15
150
1500
ISO viscosity class (ISO VG) is a kinematic viscosity (mm 2 /s) at temperature +40 ° C, allowed variation ± 10 %.
40

Machine Elements/SK 2009
41
Directive 97/23/EC applies to the design, manufacture and conformity assessment of pressure equipment and assemblies with a maximum allowable pressure PS greater than 0,5 bar.
Pressure equipment means vessels, piping, safety accessories and pressure accessories. Where applicable, pressure equipment includes elements attached to pressurized parts, such as flanges, nozzles, couplings, supports, lifting lugs, etc.
Vessel means a housing designed and built to contain fluids under pressure including its direct attachments up to the coupling point connecting it to other equipment.
Piping means piping components intended for the transport of fluids, when connected together for integration into a pressure system. Piping includes in particular a pipe or system of pipes, tubing, fittings, expansion joints, hoses, or other pressure-bearing components as appropriate.
The pressure equipment must satisfy the essential requirements. Pressure equipment must be designed, manufactured and checked, and if applicable equipped and installed, in such a way as to ensure its safety when put into service in accordance with the manufacturer's instructions, or in reasonably foreseeable conditions.
The maximum allowed value of the nominal design stress is (other than austenitic steels,
A
<
30 %) f d
= min
⎛
⎜⎜
R p 0 , 2
1,5
/ t ;
R m / 20
2,4
⎞
⎟⎟
(1) where
R p0,2
/t
is the 0,2 % proof strength at temperature t
(yield strength
R lieu of R p0,2
) and R m/20 eH
may be used in
is the tensile strength at temperature 20 °C. For testing category 4 the nominal stress shall be multiplied by 0,9. Numbers 1,5 and 2,4 are safety factors. Equations for austenitic steels are in standard SFS-EN 13445-3.
Mechanical properties of steels for pressure purposes at elevated temperatures is given in table 2.
The required thickness of cylindrical shells shall be calculated from the equation (SFS-EN
13445-3 /3/) e
=
2 pD fz
− i p
(2) where p
is the calculation pressure,
D stress f
≤ f d group ( z
= 1; 0,85 or 0,7). i
the inside diameter of the pressure vessel, the design
and z
the weld joint coefficient. The weld joint coefficient is related to the testing

Machine Elements/SK 2009
Table 1.
Mechanical properties of steels for pressure purposes.
42
R m
Steel Standard (N/mm 2 ) t
≤ 100 mm 1)
P235GH
P265GH
P295GH
P355GH
SFS-EN 10028-2
SFS-EN 10028-2
SFS-EN 10028-2
SFS-EN 10028-2
360...480
410...530
460...580
510...650
2) t
≤ 16 mm
235
265
295
355
R eH
(N/mm 2 )
> 16
≤ 40 mm
225
255
290
345
> 40
≤ 60 mm
215
245
285
335
Fine grain steels
P275N
P355N
P460N
1)
2)
Standard
SFS-EN 10028-3
SFS-EN 10028-3
SFS-EN 10028-3
(N/mm 2 ) t
≤ 70 mm 1)
390...510
490...630
570...720 t
≤ 16 mm
275
355
460
Table 2.
0,2 % proof stress at elevated temperatures.
> 16
≤ 35 mm
275
355
450
In standards mechanical properties for product thickness up to t = 150 mm.
Product thickness ≤ 60 mm.
> 35
≤ 50 mm
265
345
440
> 60
≤ 100 mm 1)
200
215
260
315
> 50
≤ 70 mm 1)
255
325
420
Steel
P235GH
P265GH
P295GH
P355GH
P275NH
P355NH
P460NH
Temperature ° C
20 50 100 150 200 250 300 350 400 Standard t / mm
≤ 60
≤ 60
≤ 60
≤ 60
≤ 35
≤ 35
≤ 35
234
272
318
336
-
215
250
290
304
402
0,2 % proof stress / N/mm 2
180
205
235
270
226
284
373
170
195
225
255
196
245
333
150
175
205
235
177
226
314
130
155
185
215
147
216
294
120
140
170
200
127
196
265
110
130
155
180
108
167
235
SFS-EN 10028-2
SFS-EN 10028-2
SFS-EN 10028-2
SFS-EN 10028-2
SFS-EN 10028-3
SFS-EN 10028-3
SFS-EN 10028-3
If the outside diameter D e tion
is known, the required thickness shall be calculated from the equae
=
2 pD fz
+ e p
(3)
The equations are valid for e
/
D e
not greater than 0,16. Tolerances and fabrication allowances shall be additional (fig. 1). c c c
2
1
0 c
2 c c
1
0 e the minimum required thickness without allowances the corrosion allowance c
0 c
1
the absolute value of possible negative tolerance on nominal thickness (from material standards) c
2 the allowance for possible thinning during manufacturing process e e r
ε e ord e n e a e r
the required thickness with allowances
ε the additional thickness resulting from the selection of the ordered thickness e ord e n e a
the ordered thickness
the nominal thickness (on drawings)
the analysis thickness, used for the check of the strength
Fig. 1.
Wall thickness.

Machine Elements/SK 2009
43
The required thickness of spherical shells shall be calculated from one of he following two equations e
=
4 pD fz
− i p e
=
4 pD fz
+ e p
(4)
(5)
The following requirements are limited in application to ends for which all following conditions are met (see fig. 2):
• r
≤ 0,2
• r
≥ 2 e
D i
• r
≥ 0,06
D i
• e
≤ 0,08 D
• e a e
≥ 0,001
D e
•
R
≤
D e a) H r b) H r
D e
D e
R D e
R D e e e r D e r D e
H D e
-0,445 e H D e
-0,635 e
V D e e 3
Fig. 2.
Dished ends: a) Klöpper-end, b) korbbogen-end.
The required thickness e
= max( e s
, e y
, e e b
shall be greatest of
) e s
, e y
and e b
V D e e 3 e s e y
=
= pR
2 fz
p
(
− 0 , 5
0 , 75
R p
+ 0 , 2
D i
) f where e b
= ( 0 , 75
R
+ 0 , 2
D i
)
⎡
⎢
⎢ p
⎣
111 f b
D i r
0 , 825
⎥
⎦
⎤
⎝
⎜
⎛ 1
1 , 5 ⎠
⎟
⎞ f b
=
R p 0 , 2 / t
1,5 f b
=
1 , 6 R p 0 , 2 / t
1,5
(for cold spun seamless austenitic stainless steel)
(6)
(7)
(8)
(9)
(10)
(11)

Machine Elements/SK 2009
Formulae for calculation of factor
Y
= min e
R
; 0 , 04
Z
= log
10
1
Y
X
= r
D i
N
= 1 , 006 −
1
[
6 , 2 + ( 90
Y
)
4
]
♦
X
= 0,06: β
0 , 06
=
N
( − 0 , 3635
Z
3 + 2 , 2124
Z
2 − 3 , 2937
Z
+ 1 , 8873 )
♦ 0,06 <
X
< 0,1: β = 25
[
( 0 , 1 −
X
) β
0 , 06
+ (
X
− 0 , 06 ) β
0 , 1
]
♦
X
= 0,1: β
0 , 1
=
N
( − 0 , 1833
Z
3 + 1 , 0383
Z
2 − 1 , 2943
Z
+ 0 , 837 )
♦ 0,1 <
X
< 0,2: β = 10
[
( 0 , 2 −
X
) β
0 , 1
+ (
X
− 0 , 1 ) β
0 , 2
]
♦
X = 0,2: β
0 , 2
= max
[
0 , 95 ( 0 , 56 − 1 , 94 Y
− 82 , 5 Y
2
); 0 , 5
]
The calculation method for
is iterative. Computer procedure is recommended.
44
1. Painelaitteet. Turvatekniikan keskus (TUKES). http://www.tukes.fi/painelaitteet/esitteet_ ja_oppaat/ painelaiteopas.pdf. 2.12.2004. 16 s.
2.
Heikkilä E. & Huhdankoski E.
Rautaruukin paineastiakäsikirja 1999, 4. painos. Raahe:
Rautaruukki Oy 1999. 176 s. ISBN 952-5010-27-9.
3. SFS-EN Lämmittämättömät painesäiliöt. Osa 3: Suunnittelu. Unfired pressure vessels. Part 3. Design. Suomen Standardisoimisliitto 2002. 708 s.
4.
Teollisuusputkistot ja painelaitesäädäntö. Kunnossapitokoulu n:o 71. Kunnossapito 10
2001. 9 s.
5.
Hovi K . Paineastiat, putkistot ja niiden koneenosat. Julkaisussa: Airila M. et al. (toim.)
Koneenosien suunnittelu 4, WSOY 1985. S. 13...165. ISBN 951-0-13223-3.
6.
SFS-EN 13445-2.
Lämmittämättömät painesäiliöt. Osa 2: Materiaalit. Unfired pressure vessels. Part 2. Materials. Suomen Standardisoimisliitto 2002. 101 s.
Other standards:
SFS-EN 13480 Parts 1…5 Metalliset teollisuusputkistot. Metallic industrial piping
SFS-EN 12952 Osat 1…8 Vesiputkikattilat. Water-tube boilers
SFS-EN 12953 Osat 1…8 Tulitorvikattilat. Shell boilers