ME TUMechanical Engineering DepartmentFaculty of Engineering, Thammasat University
ME311MECHANICAL DESIGN
Module 2Review of Solid Mechanics
Dulyachot CholaseukMechanical Engineering Department
Thammasat University
ME TUMechanical Engineering DepartmentFaculty of Engineering, Thammasat University
ME311 Module 2: Review of Solid Mechanics 2
Contents
1. Stress Analysis2. Theory of Failure3. Deformation analysis
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ME311 Module 2: Review of Solid Mechanics 3
Stress Analysis
Normal Stress from Axial Load
Normal Stress from Bending
Transverse Shear Stress
Shear Stress from Torsion
Contact Stress
Basic Types of Loads and Stresses
1Stress is intensity of internal force.
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ME311 Module 2: Review of Solid Mechanics 4
Normal Stress from Axial Load
σx
σx
Tensile (+)
Compressive (-)
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Normal stress from Bending
-max
xy
+max
σxσx
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ME311 Module 2: Review of Solid Mechanics 6
Transverse shear stress
max
τxy
xy
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ME311 Module 2: Review of Solid Mechanics 7
Shear stress from torsion
max
τxy
xy
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Contact stress
Called Hertzian Stress
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ME311 Module 2: Review of Solid Mechanics 9
Stress concentrations
σavg = F/A
σmax > σavg
F
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Application of stress concentration
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Stress concentration factors
σmax > σavg
Let σmax = Kσavg
K = σmax / σavg
F
Value of K …•depends on geometry.•can be obtained by photoelastic experiment
or by computer simulation.
Stress concentration factor,
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ME311 Module 2: Review of Solid Mechanics 12
Photoelastic experiment
Computer simulation (FEA)
Photoelastic experiment
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ME311 Module 2: Review of Solid Mechanics 13
Stress concentration factor chart
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ME311 Module 2: Review of Solid Mechanics 14
Stress concentration factor chart
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ME311 Module 2: Review of Solid Mechanics 15
Design to avoid stress concentration factor
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ME311 Module 2: Review of Solid Mechanics 16
Exercise
Is point A or B the critical point of the beam?
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Exercise
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Combined Stress
Machine elements
are subjected to different
kind of stress simultaneously
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Curved Beam
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ME311 Module 2: Review of Solid Mechanics 20
Example: Curved Beam
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ME311 Module 2: Review of Solid Mechanics 21
Example: Curved Beam
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ME311 Module 2: Review of Solid Mechanics 22
Example: Curved Beam
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ME311 Module 2: Review of Solid Mechanics 23
Example: Curved Beam
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ME311 Module 2: Review of Solid Mechanics 24
Example: Curved Beam
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Drill bit
T
F
+ =
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ME311 Module 2: Review of Solid Mechanics 26
Shaft
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State of Stress
Tri-axial Plane Stress
σx=100 MPa, σy= 80 MPa and τxy= 30 MPa,while Sy = 120 MPa. Is it safe?
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Stress transformation
Values of stresses depend on directions
σy
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Stress transformation
Values of stresses depend on directions
( ) ( ) ( ) ( ) 0coscossincossinsincossin;0 =−−−−=∑ ′ θθσθθτθθσθθτσθ AAAAAF xxyyxyx
( ) ( ) ( ) ( ) 0sincoscoscoscossinsinsin;0 =−−−+=∑ ′ θθσθθτθθσθθττθ AAAAAF xxyyxyy
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Stress transformation
θτθσσσσ
σθ 2sin2cos22 xy
yxyx +
−+
+=
θτθσσ
τθ 2cos2sin2 xy
yx +
−−=
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Principal stresses
02cos22sin2
2 =+
−−= θτθ
σσθσθ
xyyx
dd
yx
xy
σστ
φσ −=
22tan
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ME311 Module 2: Review of Solid Mechanics 32
Principal stresses
yx
xy
σστ
φσ −=
22tan
22
21 22, xy
yxyx τσσσσ
σσ +
−±
+=
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Maximum shear stress
°±= 45στ φφ
22
max 2 xyyx τ
σστ +
−±=
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Mohr circle
θτθσσσσ
σθ 2sin2cos22 xy
yxyx +
−+
+=
θτθσσ
τθ 2cos2sin2 xy
yx +
−−=
22
21 22, xy
yxyx τσσσσ
σσ +
−±
+=
22
max 2 xyyx τ
σστ +
−±=
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ME311 Module 2: Review of Solid Mechanics 35
Absolute maximum shear stress
03 =σ 1σ2σ σ
τ
maxτ
max,absτ
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Exercise
Consider a 1-meter-long solid shaft of 15 mm diameter. Answer the
following questions:
(a) Locate the critical point.
(b) Find the state of stress at the critical point.
(c) Find principal stresses and the principal direction at the critical
point.
(d) Draw the corresponding Mohr’s circle.
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Principal stress trajectories
φσ= 0°
φσ= 20°
φσ= 45°
φσ= 60°
φσ= 90°
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Principal stress (σ1) field and trajectories
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Principal stress trajectories in a cantilever beam
σ1
σ2
Principal stress trajectories derived from beam theory(consider both bending stress and tranverse shear)
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Sketch of principal stresses trajectories in human femur(based on cantilever curved beam model)
Principal stress trajectories in human femur
Anterior-to-posterior roentgenogram of a thin-sectioned human proximal femur
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Trabecular Microstructure
“Trabecular Microstructure Differs Greatly between Trabecular Groups in
Proximal Femurs of Postmenopausal Women”
Wang, J; Zhou, B; Guo, X Columbia University, New York, NY, USA
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Stress distribution
High stress
High stress
Low stress
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Maximum shear stress distribution
SHEAR"
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F
−
=2
,2
,2
max 2121max
σσσστ
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ME311 Module 2: Review of Solid Mechanics 44
Examples of stress distribution
High
Low
Stress magnitude
Principal stress (beam theory)
Von Mises stress (FEA)
Von Mises stress (FEA)
Von Mises stress (FEA)
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Stress distribution and stress trajectories
SHEAR"
6587866133745679625225504771384317263863153409032954912500792046671592561138446843223020.1
F
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ME311 Module 2: Review of Solid Mechanics 46
Theory of Failure2STATIC LOAD
BRITTLE – MNST
DUCTILE – MSST
– DET
FATIGUE
SODERBERG
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Tensile Testing
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Brittle Failure
Brittle material fail when bonding
between molecules breaks.
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Maximum Normal Stress Theory
utS=1σ
1σ
2σ
ucS
ucS
utS
utS
Failure occurs when
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ME311 Module 2: Review of Solid Mechanics 50
Failure of concrete beam
Principal stress trajectories
Maximum shear
stress trajectories
Insufficient bottom
reinforcement
Sufficient bottom
reinforcement
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ME311 Module 2: Review of Solid Mechanics 51
Ductile Failure
Ludwig line
Ductile material fail when the molecule
deforms permanently.
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Maximum Shear Stress Theory
Ductile material fail when the
molecule deforms permanently. 2max,y
abs
S=τ
1σ
2σ
yS
yS
yS
yS
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ME311 Module 2: Review of Solid Mechanics 53
Distortion Energy Theory
Failure of ductile material occurs when
strain energy per unit volume of the
material exceeds the strain energy per
unit volume at the yield point of the same
material under tensile test. ye S=σ
1σ
2σ
yS
yS
yS
yS
xyyyxx
e
τσσσσ
σσσσσ
322
2221
21
++−=
+−=
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ME311 Module 2: Review of Solid Mechanics 54
Comparison
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Chalk twisting experiment
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Exercise
Which one is brittle?
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Exercise
Consider a 1-meter-long solid shaft of 15 mm diameter. Answer the
following questions:
(e) If the material is gray cast iron (Su=125MPa), will it fail?
(f) If the material is medium carbon steel (Sy=300MPa), find Ns
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Fatigue
Cumulative damage caused by alternated load.
Stress lower than the static threshold.
Source of more than 80% of mechanical part failure.
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Alternated Load
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Alternated load in machines
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Rotating beam experiment
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Universal Testing Machine
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S-N Diagram
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Endurance Limit
Endurance limit is maximum the value of fatigue stress that
will not cause failure of the parts within 107 cycles.
Aluminum does not have endurance limit
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Low cycle fatigue (N<1000)
ul SS 9.0=′
ul SS 75.0=′
ul SS 72.0=′
Bending
Axial load
Torsion
Steel parts will last for about 1000 cycles
if fatigue stress is less than lS ′
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High cycle fatigue (103<N<107)
( ) sbt
cf NS ′=′ 10
Steel parts will last for about N cycles
if fatigue stress is fS ′
′′
−=e
ls S
Sb log31
( )e
l
SSc′′
=2
log
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Modified Endurance Limit
Actual conditions differ from the experiments,
adjustment factors are needed:
Surface finishing factor
Size factor
Temperature factor
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Soderberg theory
1=+e
a
y
m
SSσσ
For non-zero mean stress
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Exercise
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Deformation Analysis3Other than stress compliance, a design has to comply with
deformation criteria.
Deformation = function of geometry,
load and Young's modulus
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Strain Energy
The external work done on an elastic member in deforming
it is transformed into strain energy (like spring). If the
member is deformed a distance y, this energy is equal to
the product of the average force and the deflection,
for spring kFyFU
2
2==
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Strain Energy in Various Load Types
Rod under tension or compression: AElFU
2
2
=
Rod under torsion: AG
lTU2
2
=
Element under shear:AG
lFU2
2
=
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Strain Energy in Various Load Types
Beam under pure bending moment: ∫= dxEI
MU2
2
Beam under transverse shear: ∫= dxAG
CVU2
2
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ME311 Module 2: Review of Solid Mechanics 74
Strain Energy Density
Tension and compression Eu
2
2σ=
Direct shear Gu
2
2τ=
TorsionG
u4
2maxτ
=
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ME311 Module 2: Review of Solid Mechanics 75
Castigliano’s Theorem
When forces act on elastic systems subject to small
displacements, the displacement corresponding to any force,
collinear with the force, is equal to the partial derivative of the
total strain energy with respect to that force.
ii F
U∂∂
=δ
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Statically Indeterminate Problems
Use compatibility conditions
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Buckling
Failure of long members under compression occurs
before yield point. [More detail in MODULE 6:
POWER SCREWS]
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ME311 Module 2: Review of Solid Mechanics 78
Buckling
Lateral deformation due to axial load