BK50A2200
Design Methodologies and
Applications of Machine
Element Design
Lecture 2
Introduction to the textbook:
“Norton: Machine Design”
D.Sc Harri Eskelinen
Goals of this lecture
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Support the contents of the previous lectures
dealing with machine design approaches,
reliability design and wear phenomena
To get familiar with the main designing and
dimensioning criteria of the most important
machine elements (according to Norton)
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Main consecutive designing steps and aspects
Fundamental dimensioning equations
Briefly about the book
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The textbook presents an integrated approach
to the machine elements by combining the usual
set of machine element topics with a series of
case studies that illustrate the relationships
between force, stress and failure analysis in
real-world design.
The book emphasizes the design and synthesis
aspects of machine elements but it forms also a
good balance between synthesis and analysis.
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The first part of the book presents the
fundamentals of design, materials, stress,
strain, deflection, failure and fracture
theories.
The second part treats of the aspects of
machine element design, such as
designing springs, shafts, gears, bearings
etc.
PART 1. FUNDAMENTALS
Chapter 1. Introduction to Design
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This chapter partially supports the ideas of systematic design
approach (see the yellow items below) : according to Norton the
design process consists of the following ten stages:
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1 Identification of need
2 Background research
3 Goal statement
4 Task specifications
5 Synthesis
6 Analysis
7 Selection
8 Detailed design
9 Prototyping and testing
10 Production
Chapter 2. Materials and Processes
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The contents of this chapter will be discussed in details during the
university course “Introduction to Material Technology”
Basic definitions of the most common material properties are
presented briefly:
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Modulus of elasticity
Yield strength
Ultimate tensile strength
Modulus of rigidity
Fatigue strength
Toughness
Hardness
Most typical hardening and surface coating processes are presented
briefly
Basic information about some material groups is given:
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Steels
Cast iron
Aluminium
Titanium
Copper Alloys
Polymers
Ceramics
Composites
Chapter 3. Load Determination
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The content of this chapter produce the
fundamentals for the further stress, strain and
deflection analysis presented in chapter 4.
Main topics are:
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Different loading cases
Free-body diagrams
Static loading
Dynamic loading
Vibration loading
Impact loading
Beam loading
Classification of loading cases
Constant
Loads
Time-Varying
Loads
Stationary
elements
Class 1
Class 2
Moving
elements
Class 3
Class 4
Identification of different loading cases
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Identification of different loading cases in necessary to
make it possible to use proper material properties as
criteria during the material selection process
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Tension or compression  tensile or compressive stress
Bending  bending stress
Shear  shear stress
Torsion  torsion stress (shear strass)
Reverced loading  endurance limit (for reverced stress)
Pulsating loading  endurance limit (for pulsating stress)
Pulsating loading
Reverced loading
Free-Body Diagrams
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Case example: Wire connector crimping tool
Chapter 4. Stress, Strain and Deflection
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This chapter includes the basic theories of “strength of
materials”, the following topics are discussed (the most
important items are high-lighted with yellow):
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Principal stresses
Axial Tension
Bending stresses of beams
Deflection of beams
Torsion
Combined stresses
Stress concentration
Axial compression
Stresses in cylinders
Combined loading:
-Axial force
-Radial force
-Tangential force
-Torque
-  combined stresses
A
B
C
T1
Fa
Ft
Fr
D
Critical cross-sections due to
stress concentrations:
-End of the keyseat at crosssection A
-Cross-sections B, C and D of a
smaller diameter
Chapter 5. Static Failure Theories
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Chapter 5 is divided in three main sections:
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Failure of ductile materials under static loading
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Failure of brittle materials under static loading
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The main failure mode is permanent yield under static
loading  yield strength of the material is exceeded
Critical material property is yield strength
Instead on yielding brittle materials fracture
Fully hardened steels, cast iron, materials in low
temperatures can behave like brittle materials
Critical material property is toughness at certain temperature
Fracture mechanics
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This theory presumes the presence of a crack, which starts
to grow under the specific loading and finally leas to either
ductile or brittle failure
Toughness
Transition
zone
Brittle
behaviour
Temperature
Ductile
behaviour
Loading
Modes of crack displacement
Mode I
= load tends to pull the crack open in tension
Mode II
= shear crack in-plane
Mode III
= shear the crack out-of-plane
Chapter 6. Fatigue Failure Theories
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The use of typical Wöhler’s strength-lifediagrams is presented
The main principles of the use of Paris-Equation
are presented
The use of Goodman’s diagram for fatigue life
analysis is presented
Schematic fatigue-fracture surfaces of a shaft
cross-sections are presented to support further
failure mode analysis
Wöhler’s diagram
Schematic fatiguefracture surfaces
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Rotating bending
Low nominal stress
Mild stress concentration
Paris-equation
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The crack growth “speed” is presented as a function of
loading cycles:
Where
a
N
A, n
ΔK
=
=
=
=
crack width
number of cycles
material coefficients
stress intensity factor range
D
A
M
A
G
I
N
G
Region I
Crack initiation stage
Region II
Crack propagation
Region III
Unstable fracture
S
P
E
E
D
No
crack
growth
Stress intensity
Chapter 7. Surface Failure
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This chapter contains the following topics
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Mathematical theory of surface contacts
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Characteristics to describe the value of surface roughness
Spherical contact
Cylindrical contact
Dynamic contact stresses
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Designing rules to avoid surface failure
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Wear phenomena (discussed earlier during this course)
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Abrasive wear
Adhesive wear
Fatigue wear
Tribochemical wear or corrosive wear
Mathematical definition of Ra:n
l
Ra  1  y( x )dx
l0
1 n
Ra 
 yi
n i 1
Designing rules to avoid surface failure
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1 Remember the rules which were presented during the special
lesson dealing with wear phenomena
2 Choose proper materials
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3 Choose proper lubricants
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Use proper sealing constructions
Select proper material pairs (e.g. hardness pairs)
5 Avoid and minimize stress concentrations
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Take care of EHD- or HD- lubrication (avoid boundary lubrication)
Use EP-lubricants if needed(extreme pressure)
4 Take care of cleanliness
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Hardness
Surface roughness
Use of coatings
Select proper stiffness and/or geometry
6 Avoid fretting problems by taking care of possible vibration
phenomena (near joints or fits)
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Case example:
How to minimize
the stress
concentrations in a
cylindrical roller
bearing by using a
proper geometry
of the roller
elements.
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Case example:
Fretting wear
on a shaft
beneath a
press-fit hub.
PART 2. Machine Design
Chapter 8. Design Case Studies
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This brief chapter is written just to form “a
bridge” between the theories of material
science, strength of materials, failure theories
(presented in part 1) and practical dimensioning
and analysing instructions of some typical
machine elements (to be presented in part 2).
The iterative nature of designing process is
emphasized.
Chapter 9. Shafts, Keys and Couplings
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Designing of shafts step-by-step (iterative analysis):
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1 Determine the affecting loading cases
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2 Collect contacting dimensions from the construction and select
possble shaft materials
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E.g. key seats, changes of the diameters, grooves, threads etc.
6 Calculate the affecting stresses and deflections
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E.g. shaft-hub joints, diameters of bearing seats, width of gears etc.
3 Produce the free-body diagram and calculate the teaction forces
4 Draw loading (force), shear and moment diagrams
5 Find the critical cross-sections of the shaft
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E.g. gear forces, torque, forces due to belt drives etc.
E.g. tensile stress, bending stress, shear stress,
7 Calculate the critical rotating speed due to vibration and resonance
8 Calculate safety factors
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Constant and time-varying loading
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The dimensioning procedure of shafts is
based on ASME-method:
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Soderberg’s hypothesis
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(in Finland several hypothesis are used and usually
compared in university text books)
Goodman’s line
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(in Finland the use of Smith’s diagram is more
common)
Some rules of thumbs
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Estimation of shaft diameter:
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d
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Tmax
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Tsall
= required shaft diameter
= affecting torque
= allowed shear stress of the material
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Critical angular velocity:
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Bending vibration
ncr
δmax
= critical angular velocity
= maximum deflection of the shaft
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Critical angular velocity:
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Torsinal vibration
fcr
kv
J1
J2
d
G
L
m
r
=
=
=
=
=
=
=
=
=
critical angular velocity
torsional stiffness coefficient
moment of inertia (input)
moment of inertia (output)
diameter of the shaft
modulus of rigidity
length of the shaft
weight of (each) component
rotating radius of (each) component
Loading cases of shaft-hub-joints
Fa
Torque
Torque
Moment
Ft
Fr
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If the joint is able to withstand also axial loading, its torque transmission
capacity can be estimated according to the following equation:
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Where
Ttheor =
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T
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Fatheor =
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Fa
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=
=
theoretical maximum allowed torque which joint
could transmit without any axial loading
torque, which can be transmitted even though Fa
is affecting simultaneously (usually the value
which is calculated)
theroretical maximum allowed axial force, which
joint could transmit without any torque loading
axial load, which is decreasing the torque
transmission capacity (“the disturbing factor”)
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Dimensioning of parallel keys is based on SFSstandards (we skip the presentation presented
by Norton):
Main designing steps are as follows:
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Check the maximum surface stress of the hub
Check the maximum surface stress of the key
Check the maximum shear stress of the key
Ensure that the required torque transmission
capapacity is achieved
Chapter 10. Bearings and Lubrication
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This chapter includes the following
important topics:
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Lubricants and types of lubrication
Briefly about sliding bearings and their
material combinations
Rolling-element bearings
Failure of rolling-element bearings
Selection of rolling bearings
Types of lubrication
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Hydrodynamic lubrication (HD or HL)
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Elastohydrodynamic lubrication (EHD or EHL)
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HD refers to the supply of oil to the sliding interface to allow the
relative velocity of the mating surfaces to pump oil within the gap and
separate the surfaces on the dynamic film of liquid.
When the contacting surfaces are nonconforming, as with gears or cam
mechanisms, it is difficult to form a full film of oil.The affecting load
creates a contact area from the elastic deflections of the surfaces. This
area can be large and flat enough to provide full hydrodynamic film if
the relative sliding velocity is high enough. This is possible, because the
high pressure between the surfaces increase the viscosity of the fluid.
Boundary Lubrication (BL)
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Either the insufficient geometry, too high load level, low velocity or
insufficient oil quantity may prevent hydrodynamic lubrication and
cause metallic contacts between the surfaces (e.g. at the beginning or
end of the rolling)
Selection of rolling bearings
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Facilities of the selected
bearing type
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Ability to withstand axial
loads
Ability to withstand axial
bending moments or
angular assembly errors
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Allowed
dynamic load
Allowed
maximum
angular
velocity
Allowed
friction
Allowed
static load
Required
stiffness and
accuracy
Required
reliability
Some examples
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A detailed guide to select an appropriate bearing
type will be given out as a hand-out…
Tapered roller bearings
-Especially for cases in
which good axial load
standing capacity is
required
Spherical roller bearings
-Especially for cases in which bending
moment could cause additional loading
on the bearing or where possible
assembly errors may cause some
misaligning of the shaft
Basic equations of bearing design
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P=
combined dynamic equivalent load of the bearing
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p=
exponent, the value depends on the
bearing type
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L10h =
C
=
Fr = applied radial load
Fa = applied axial load
X = a radial factor
Y = an axial factor
Ball bearings p=3
Roller bearings p= 10/3
Nominal life-time (e.g. 20 000 h)
Dynamic load rating
Chapter 11. Spur Gears
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Spur gears are used to present principles of gear
dimensioning in general
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Specialized terminology is presented in details
Gear tooth theory is discussed
Equations for dimensioning gears are presented
Also gear manufacturing processes are presented briefly
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Different casting processes
Machining
Powder metallurgical processes (sintering)
Extruding and cold drawing processes
Different finishing processes
The presentation is based on standards published by the
American Gear Manufacturers Association (AGMA)
T1
Fa
At first the applying
forces on gear teeth
must be established!
Ft
Fr
Fr
Ft
Fa
T2
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Equations are based
on experimental
factors, parameters
and characteristics
describing e.g.:
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Geometric
accuracy of gears
Lubrication
conditions
Material
properties of
gears
Stress
concentration
phenomena
Surface properties
of gears
Loading conditions
of gears
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Main
dimensioning
criteria:
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Main
characteristics:
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Bending
stresses of
the teeth
Surface
stresses of
the teeth
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Gear ratio
i=Z1/Z2
Module
m=d1/Z1
Contact ratio
Functions:
To transimit
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Torque
Angular
velocity
Basic equations for spur gear design
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Power transmission capacity according to allowed
bending stresses of the teeth
Power transmission capacity according to allowed
surface stresses of the teeth
Chapter 12. Helical,Bevel and Worm Gears
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Helical gears
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Bevel gears
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Teeth are angled with respect to the axis of
rotation
Contact surface between teeth is increased
Axial load component is caused
Shafts are located usually at 90 degrees angle
Worm gears
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Shafts are located at crossing position and high
gear ratios are achieved
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The dimensioning of helical gears is based on
the equations of spur gears, so-called virtual
number of teeth should be established and then
the theory of spur gears is applied
For bevel gears either the theories of
dimensioning of spur or helical gears can be
applied:
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Straight bevel gears  spur gears
Spiral bevel gears
 helical gears
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Two main additional aspects of tooth geometry should be
considered
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Use of single- or double-enveloping tooth forms
Number of teeth in contact with worm and worm wheel
Worm gears are discussed very briefly, however the main
dimensioning aspects consist of four main steps:
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Durability against pitting (surface fatigue)
Durability against wear (abrasive wear due to different material
properties of the worm and the worm wheel)
Durability against overheat
Allowable bending deflection of the worm (shaft)
Example of dimensioning equations of worm gear design
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Safety factor SW against wear
Wlim
WP
WR
Wv
ZE
Z
= kulumislujuus
Materials wear strength
Coefficient depending on
materials
hardness
= kulumisparikerroin
(mm.
kovuus)
Coefficient depending on surface roughnesses
= pinnankarheuskerroin
= liukunopeuskerroin
Coefficient depending on the sliding velocity
= materiaalikerroin
(Eon
E2)
Coefficient depending
modulus
of elasticity
1 ja
= kosketuskerroin
(kierremuoto)
Coefficient depending
on the tooth geometry
Chapter 13. Spring Design
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Main contents of this chapter:
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Definition of the spring rate is presented
Various spring configurations and materials are
presented briefly
Dimensioning and designing criteria is presented
for the following spring configurations:
Helical compression springs
 Helical extension springs
 Helical torsion springs
 Belleville spring washers
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Example: Designing steps of helical
compression springs
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1 Decide spring configuration
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2 Establish functional properties
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1.1 Length
1.2 End details and number of active coils
1.3 Tentative material selection
2.1 Sprind index C = D/d (coil diameter/wire diameter)
2.2 Spring deflection y
2.3 Spring rate k = F/y
3 Loading cases and stress analysis
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3.1 Shear stress
3.2 Torsional shear stress
3.3 Stress concentrations
3.4 Residual stresses due to manufacturing stages (e.g coiling into the form of
helix causes tensile stress)
3.5 Buckling
3.6 Vibration and resonance phenomena
3.7 Fatigue analysis
3.8 Final material selection
Chapter 14. Screws and Fasteners
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This chapter deals with the following topics:
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Standardized thread dimensions
Power screws
Screw fasteners
Stresses in threads
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Tensile, torsion and shear
Joint stiffness
Preloaded fasteners
A bolted assembly
A bolted assembly
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The clamped construction may include two or more pieces and they may be
of different material.
Also a long bolt usually has threads over only a portion of its length having
at least two different cross-sectional areas.
These different stiffness-sections act as springs in series and their function
can be described according to the following equation:
When we know the dimensions and geometry of the bolt and pieces to be
joined it is possible to calculate the partial spring rates and combine them
according to this simple equation.
Analogical with spring analysis we can now write the relationship between
the total spring rate, deflection and applying force.
Other dimensioning aspects of bolted joints
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1 Washers
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2 Bolt’s straight length without threads
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Decrease the surface stresses at the joint
Ensure the tightness of the joint
Prevents the possible bending moment of the bolt due to
slant surface
The stress concentration at the end of the straight length before
the first thread could be critical
The straight length is planed to function “as a spring”
3 Parts to be clamped together
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In many cases the friction coefficient between
the parts to be clamped together is in key-role
There should be enough distance between the mounting holes of
the screws and edges of the parts to be clamped to avoid
fractures of the base material
Possible failure modes of bolted joints
 1 The bolt breaks under the static tensile loading
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2 The fatigue strength of the bolt is exceeded
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Tensile stress exceeds the ultimate tensile strength of
the bolt
The first thread of the bolt is cut off
The first thread of the nut is cut off
Typically the fatigue limit is only 10% of screw
material’s yield strength
3 The failures of the parts to be clamped
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The shear stress is too large near the edges
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Means to improve the fatigue strength of the
bolted joint:
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Select a taller nut (to increase the number of load
standing threads)
Select more suitable material pair for the nut and the
bolt combination (lower coefficient of elasticity for the
nut compared to that of the screw’s e.g. aluminium or
cast iron for nuts with and steel for bolts)
Use sufficient pre-loading of the bolts (equalize the
stresses applying at each thread of the bolt and nut)
Improve the surface quality of the bolt,
Select the bolt (and thread) geometry with the
smallest stress concentration coefficient
Select the advantageous manufacturing technology of
the bolts (usually cold forming is recommended)
Chapter 15. Clutches and Brakes
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This chapter forms only a brief overall picture of various types of
clutches and brakes to give the reader a sight of possible
constructions.
Classification according to the actuation
(impulse to start the function)
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Electrical (press the button)
Mechanical (push the bedal)
Pneumatic or hydraulic
Automatic
Classification according to the function
(what phenomenon the contact is based on)
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Friction between (two) surfaces
Locked geometry (e.g. toothed components)
Magnetic
Fluid coupling
SUMMARY : STRESS AND DEFLECTION ANALYSIS
Load determination and
different load cases
• utilization of free-body diagrams
• static or dynamic loading
• vibration loading
- bending vibration
- torsional vibration
• impact loading
• tension, compression, bending
shear or torsion
• reversed or pulsating loading
Pulsating loading
Failure theories
• Static failure theories
- ductile materials
- brittle materials
• Fracture mechanics theory
• Fatigue failure theories
- Wöhler
- Paris –Eguation
- Soderberg
- Goodman
• Surface failure theories
- wear
- surface contact theories
Reverced loading
Stress, strain and deflection
• allowed stress and deflection
• due to affecting load cased
• combined stresses
• stress concentrations
Material properties
• metallic materials
- steels
- aluminium
- cast iron
• polymers
• ceramics
• composites
• nanomaterials
Exercise 2
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Exercise 2A

Present typical applications of power transmission or guiding shafts under
different loading cases. Explain the reasons for the affecting combined loading.
Use illustrative figures. Present at least the following loading cases:
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Tension or compression + bending
Tension or compression + bending + torsion
Bending + torsion
Shear + any other loading case
Reversed loading
Pulsating loading
Exercise 2B
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Compare different fatigue failure theories and approaches by presenting typical
applications of different types of mechanical components or constructions.
Compare at least the following approaches:
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Wöhler’s strength-life-theory
Goodman’s theory
Paris-equation
Utilization of schematic fatigue-fracture surfaces
Fracture mechanics