Machine elements:
Those elements that a machine consists of.
1. Some elements are bought readymade. Selection is made
according to guidelines given by the supplier.
Typical examples are: rolling element bearings, electro
motors, different standard gearboxes, springs.
2. Other are bought ready made but are used under so
different conditions that the supplier is not able to give a
sufficient set of equations/guidelines for the
dimensioning.
Typical examples: bolts and screws, gear wheels, springs.
3. Other elements you design and manufacture yourself
because the element is designed for a specific application.
Typical examples: shafts and hubs, shaft-hub
connections, bolts, springs, sliding bearings,
transmissions of different kinds for example gear wheels.
In group 2 and 3 the general methods from the mechanics
of materials are used – maybe with some modifications.
Some elements for example springs and gear wheels
appears in more groups, but this is natural and depends on
the application.
Alexander Kielland platform 1980
GEOMETRY
Well defined information
about the real geometry
of the structure
LOAD
PHYSICS
Well defined information
about the real loads on
the structure
Well defined information
about the physical
properties of the
components in the
structure
Remember the influence
from the environment
Well defined use of the
data
Well defined evaluation
of the reliability of the
structure
The most important types of failure
Fracture (brute force, low number of load cycles –
often called static loading)
Yielding (permanent deformation)
Unacceptable large deformations
Creep (Load over long time at relative high
temperature)
Instability (buckling of columns)
Mechanical wear
Chemical degradation (ex. Corrosion)
Fatigue failure (high number of load cycles – often
called dynamic loading)
Example of creep: lead pipes
Machine Design: An Integrated Approach, Fifth Edition
Robert L. Norton
Copyright ©2014 by Pearson Education, Inc.
All rights reserved.
Machine Design: An Integrated Approach, Fifth Edition
Robert L. Norton
Copyright ©2014 by Pearson Education, Inc.
All rights reserved.
Machine Design: An Integrated Approach, Fifth Edition
Robert L. Norton
Copyright ©2014 by Pearson Education, Inc.
All rights reserved.
Machine Design: An Integrated Approach, Fifth Edition
Robert L. Norton
Copyright ©2014 by Pearson Education, Inc.
All rights reserved.
Machine Design: An Integrated Approach, Fifth Edition
Robert L. Norton
Copyright ©2014 by Pearson Education, Inc.
All rights reserved.
Measures for checking the ability of a material to deform:
A = ((Lu – L0)/L0)100 [%], fracture elongation. No index
on A means the measuring length L0 = 5d. A10 means
L0 = 10d. A80 means L0 = 80mm independent of
diameter of the test specimen.
Z = ((S0 – Su)/S0)100 [%], area reduction (degree of
deformation)
Both the elongation and area of the fracture are measured on
the test specimen after fracture.
Machine Design: An Integrated Approach, Fifth Edition
Robert L. Norton
Copyright ©2014 by Pearson Education, Inc.
All rights reserved.
Machine Design: An Integrated Approach, Fifth Edition
Robert L. Norton
Copyright ©2014 by Pearson Education, Inc.
All rights reserved.
Machine Design: An Integrated Approach, Fifth Edition
Robert L. Norton
Copyright ©2014 by Pearson Education, Inc.
All rights reserved.
Safety factors
Ductile materials
Brittle materials
𝑁𝑁𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ≅ 𝑀𝑀𝑀𝑀𝑀𝑀(𝐹𝐹1, 𝐹𝐹2, 𝐹𝐹3)
𝑁𝑁𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 ≅ 2 ∙ 𝑀𝑀𝑀𝑀𝑀𝑀(𝐹𝐹1, 𝐹𝐹2, 𝐹𝐹3)
For preliminary design computations for ductile materials it is
often useful to start with a safety factor of 𝑁𝑁 ≈ 3
y
σy
τyx
τyz
τxz
σx
τxy
σz
τzx
τzy
σz
τyx
τzy
τzx
τyz
z
τyx
τxy
τxz
τxy
c
y
σx
τxy
x
τyx
z
σy
a
x
b
y
y
τyx
a
τyx
τxy
c
τxy
τxy
c
τxy
τyx
z
x
x
b
τyx
a
0 = τxy · ✘a✘·✘b✘·✘c − τyx · ✘a✘·✘b✘·✘c
✘
τxy = τyx
✘
⇓
y
y
τyx
γ
2
a
τxy
τyx
π
2 +γ
c
τxy
τxy
τxy
π
2 −γ
π
2 +γ
x
τyx
τyx
γ er skjærtøyningen (shear strain)
Hookes lov for forskyvning
τ = G·γ
Skjærmodulen
E
G =
2 · (1 + ν)
γ
2
x
Stress computation
VON MISES
EFFECTIVESTRESS It is often convenient in situations involving combined tensile and shear stresses acting on the same point to define an e f f d v e stress that
can be used to represent the stress combination. The distortion-energy approach provides a good means to do this for ductile materials. The von Mises effective stress a'
is defined as the uniaxial tensile stress that would create the same distortion energy as
ir created by the actual combination of applied stresses. This approach allows us to treat
cases of combined, multiaxial tension and shear stresses as if they were due to pure
tensile loading.
The von'Mises effectivestress d for the three-dimensional case is found from equation 5.6b:
This can also be expressed in terms of the applied stresses:
and for the two-dimensional case from equation 5 . 6 ~(with 02 = 0):
and if expressed in terms of the applied stresses:
Use these effective stresses for any combined stress situation. (See Example 5-1.)
This von Mises effective stress will be revisited later when examples of combined
stresses are encountered.
SAFETY
FACTOR Equations 5.6b and 5 . 6 ~define the conditions at failure. For
design purposes it is convenient to include a chosen safety factor N in the calculation
so that the stress state will be safely inside the failure-stress ellipse of Figure 5-3.
Stress concentrations
·
A1
P
A2
P
Stress concentrations
·
A1
P
For
D/d = 1.35
r/d = 0.125
A2
P
Kt = 1.68
0.125
Nominal stress is
computed on the
thin part
Stress concentrations
• Linear interpolation
•
1.35
•
0.125
.
•
0.99682 + 1.35-1.3
• &
−0.25751 + 1.35 − 1.3
(
•
·
! . "#
. ! .$
! . #
! ! .
. ! .$
0.997508 · 0.125 ! . "$"
0.997508
−0.26369
1.72602
• The approach by computing Kt is recommended for the quiz.
10
σ∞
9
8
7
a
6
5
4
c
3
2
1
0
0 1 2 3 4 5 6 7 8 9 10
a/c
σ∞
=
=1+2∙
∙ 1+2∙
Note the reciprocal term c/a
222
MACHINE DESIGN
An lntegrated Approach
Chapter 4
This stress-concentration function is really a three-dimensional surface with the axes,
rid, D/d, and K1• In Figure 4-36, we are looking at lines on that 3-D surface computed
at different values of D/d and projected forward to the r/d-K1 plane. The geometry of
the part and its stress equation are defined in the figure, as is the function that defines
each stress-concentration curve. In Figure 4-36 it is an exponential function of the form
Kt =Ax b
(4.32b)
where x represents the independent variable or rId in this case. The values of the coef­
ficient A and exponent b for any one value of DId are determined by nonlinear regression
on several data points taken from the experimental data. The resulting values of A and
b for various magnitudes of the second independent variable DId are given in the table
within the figure. A and b for other values of D/d can be interpolated. The file name
that evaluates these functions and interpolates between them is also noted in this figure
and in Appendix C for each of the 14 cases shown there.
The stress-con7entration plots and functions provided in Appendix C along with
their corresponding files will prove useful in the design of machine parts throughout this
text and in your practice of engineering. For loading and geometry cases not covered in
Appendix C of this text, see references 3 and 4.
Designing to Avoid Stress Concentrations
,Complicated geometry is often necessary for the proper function of machine parts. For
example, a crankshaft must have particular contours for its purpose. The designer is
always faced with the problem of stress concentrations at sections having abrupt changes
- f)-n-�:�-�-r�r.-1.-��-r::-:r:?- -::�.-:-TT- ,- - - , :TT:-:'.}j::}- ----?-, �-�
4.0
--:----:---:
3.5
Dld
Line
-+---:- ' . '---r---�
l�
3.00--2.00 ---1.30 ---1.20 - - -
3.0
' '
··(:::
:
h
Mc
M
Cinom = - = 6--2
I
hd
Cimax = K1Cinom
and:
-'
'
'
where:
2.0
1.5
1.0
0
FIGURE 4-36
0.1
0.2
0.3
ri
0.4
0.5
0.6
D/d
A
b
3.00
2.00
1.30
1.20
1.10
1.05
1.01
0.907 20
0.932 32
0.958 80
0.995 90
1.016 50
1.022 60
0.966 89
-0.333 33
-0.303 04
-0.272 69
-0.238 29
-0.215 48
-0.191 56
-0.154 17
el
Geometric Stress-Concentration Factors and Functions for a Stepped Flat Bar in Bending - Also see the File APP C-10
Source: Fig. 73, p. 98, R. E. Peterson, Stress Concentration Factors, John Wiley & Sans, 1975, with the publish;r's permission.
STRESS, STRAIN, AND DEFLECTION
of shape. The best that can be done is to minimize their effects. A study of the stress­
concentration curves for various geometries in Appendix C will show that, in general,
the sharper the corner and the !arger in magnitude the change in contour, the worse will
be the stress concentration. For the stepped bar in Figure 4-36, !arger DId ratios and
smaller rId ratios give worse stress concentration. From these observations, we can state
some general guidelines for designing to minimize stress concentrations.
Avoid abrupt and/or large-magnitude changes in cross section if possible.
2 Avoid sharp corners completely and provide the !argest possible transition radii
between surfaces of different contours.
These guidelines are fine to state and better to observe, but practical design con­
straints often intervene to preclude strict adherence to them. Some examples of good and
bad designs for stress concentration are shown in Figures 4-37 through 4-39 along with
some common tricks used by experienced designers to improve the situation.
FORCE-FLOW ANALOGY Figure 4-37a shows a shaft with an abrupt step and a
sharp corner, while Figure 4-37b shows the same step in a shaft with a large transition
radius. A useful way to visualize the difference in the stress states in contoured parts
such as these is to use a "force-flow" analogy, which considers the forces (and thus the
stresses) to flow around contours in a way similar to the flow of an ideal incompress­
ible fluid inside a pipe or duct of changing contour. (See also Figure 4-34.) A sudden
narrowing of the pipe or duct causes an increase in fluid velocity at the neckdown to
maintain constant flow. The velocity profile is then "concentrated" into a smaller region.
Streamlined shapes are used in pipes and ducts (and on objects that are pushed through
a fluid medium, such as aircraft and boats) to reduce turbulence and resistance to flow.
"Streamlining" our part contours (at least internally) can have similar beneficia! effects
in reducing stress concentrations. The force-flow contours at the abrupt step transition
in Figure 4-37a are more concentrated than in the design of Figure 4-37b.
The example in Figure 4-38 is a stepped shaft to which a ball bearing is to be fitted.
A step is needed to locate the bearing axially as well as radially on the shaft diameter.
Commercial ball- and roller bearings have quite small radii on their corners, which
forces the designer to create a fairly sharp corner at the shaft step. To reduce the stress
concentration at the step (a), a larger radius is needed than the bearing will allow. Three
possible design modifications to create a better force flow around the step are shown
in the figure. The first design (b) removes additional material at the corner in order to
increase the radius and then "returns" the contour to provide the needed axial locating
surface for the bearing. The second approach (c) removes material behind the step to
improve the force streamlines. The third approach (d) provides a suitably large corner
radius and adds a special washer that bridges the radius to provide the bearing seat. The
stress concentration is reduced in each case versus the original sharp-cornered design.
A similar approach of removing material to improve the force flow is seen in Figure
4-39a, which shows a snap-ring groove in a shaft with additional relief grooves provided
on each side to smooth the effective transition of the cross-sectional dimension. The
effect on the force flow lines is similar to that shown in Figure 4-38c. Another common
source of stress concentration is a key needed to torque-couple gears, pulleys, flywheels,
etc., to a shaft. The keyway groove creates sharp corners at locations of maximum bend­
ing and torsional stresses. Different key styles are available, the most common being the
square key and the circular-segment Woodruff key as shown in Figures 4-38b and 4-38c.
See Section 10.10 for more information on keys and keyways.
(a) Force flow
around a
sharp corner
(b) Force flow
around a
radiused corm
FIGURE 4-37
The Force-Flow Analogy
for Contoured Parts
The original diagram is ok but the students in
MAS236 do not have sufficient background to use
it. Therefore it is modified to be used in MAS236.
Morten Kjeld Ebbesen, August 2024
Static Loading Failure Analysis
Materials are assumed homogenous and isotropic
Find all applied forces, moments, torques, etc. and
draw free-body diagrams to show them applied to
the part’s geometry.
Based on the load distributions over the part’s
geometry, determine what cross sections of the
part are most heavily loaded
Draw a stress element for each of the selected
points of interest within the section and identify the
stresses acting on it.
Calculate the applied nominal stresses on each
element and then calculate the principal stresses
and maximum shear stress resulting therefrom.
Ductile material
Brittle material
If the material is ductile, then calculate the von
Mises effective stress at each selected stress
element based on the calculated principal
stresses. Local stress concentrations are not
taken into account.
If the material is brittle, calculate the CoulombMohr effective stress at each selected stress
element based on its principal stresses. use the
largest resulting tensile stresses or shear
stresses. Local stress concentrations are taken
into account.
Choose a trial material and compute a safety
factor based on tensile yield strength of that
material.
Choose a trial material and compute a safety
factor based on the ultimate tensile strength of
that material.
If a known or suspected crack is present, calculate
the stress intensity factor from equation 5.14 and
compare it to the fracture toughness of the
material to determine if there is any danger of a
crack propagation failure.
Modified figure 5.26 from Norton to be used in MAS236