'08 DMcSLecture Notes - Chapter 3
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Transcript of '08 DMcSLecture Notes - Chapter 3
![Page 1: '08 DMcSLecture Notes - Chapter 3](https://reader031.fdocuments.in/reader031/viewer/2022020122/549de184b4795974208b45ca/html5/thumbnails/1.jpg)
MECH 401 Mechanical Design ApplicationsDr. M. O’Malley – Master Notes
Spring 2008Dr. D. M. McStravickRice University
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Updates
HW 1 due Thursday (1-17-08)Last time
IntroductionUnitsReliability engineeringMaterials
This weekLoad and stress analysis
Quiz #1 is Jan 29 (in class)Covers material through Chapter 3 (first 2 weeks of class)
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Equilibrium
Basic equations of equilibrium enable determination of unknown loads on a body
For a body at rest, (recalling from statics)ΣF = 0 ΣM = 0
For a body in motion, (recalling from dynamics)
ΣF = ma ΣM = Iα
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Determining loads
Machine and structural components are load-carrying membersWe need to be able to analyze these loads in order to design components for the proper conditionsDetermining loads
Engines/compressors operate at known torques and speeds (easy!)Airplane structure loads depend on air turbulence, pilot decisions (not so easy)Experimental methods / past performance
Often we can determine loads by using a free body diagram (FBD) Gives a concise view of all the forces acting on a rigid body
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Steps for drawing FBD’s
1. Choose your body and detach it from all other bodies and the ground – sketch the contour
2. Show all external forcesFrom groundFrom other bodiesInclude the weight of the body acting at the center of gravity (CG)
3. Be sure to properly indicate magnitude and directionForces acting on the body, not by the body
4. Draw unknown external forcesTypically reaction forces at ground contactsRecall that reaction forces constrain the body and occur at supports and connections
5. Include important dimensions
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Example – Drawing FBD’s
2m 4mB
1.5 m
2400 kgA
CG
A - Pin - 2 Rx forces
B - Rocker - 1 Rx force
Fixed crane has mass of 1000 kgUsed to lift a 2400 kg crateFind: Determine the reaction forces at A and B
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Crane example (FBD’s) cont.
1. Choose your body and detach it from all other bodies and the ground –sketch the contour
2. Show all external forces (from ground, from other bodies). Include weight of the body acting at the center of gravity (CG)
3. Be sure to properly indicate magnitude and direction (acting ON the body, not BY the body)
2m 4mB
1.5 m
2400 kgA
CG
A - Pin - 2 Rx forces
B - Rocker - 1 Rx force
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Crane example (FBD’s) cont.
4. Draw unknown external forces (typically reaction forces at ground contacts). Recall that reaction forces constrain the body and occur at supports and connections
5. Include important dimensions
Ay
Ax
B
1000 kg
2400 kg
1.5 m
2m 4m
Ay
Ax
BW = (1000 kg)(9.81 m/s2) = 9.81 kN
(2400 kg)(9.81 m/s2) = 23.5 kN
2m 4mB
1.5 m
2400 kgA
CG
A - Pin - 2 Rx forces
B - Rocker - 1 Rx force
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Find Ax, Ay, and BΣFx = 0 ΣFy = 0 ΣM = 0
Find B: ΣMA = 0B(1.5) – (9.81)(2) – (23.5)(6) = 0B = 107.1 kN
Find Ax: ΣFx = 0Ax + B = 0Ax = -107.1 kNAx = 107.1 kN
Find Ay: ΣFy = 0Ay – 9.81 – 23.5 = 0Ay = 33.3 kN
1.5 m
2m 4m
Ay
Ax
BW = (1000 kg)(9.81 m/s2) = 9.81 kN
(2400 kg)(9.81 m/s2) = 23.5 kN
Which forces contribute to ΣMA?
B, 9.81, 23.5
Which forces contribute to ΣFX?Ax, B
Which forces contribute to ΣFy?Ay, 9.81, 23.5
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3-D Equilibrium example
2 transmission belts pass over sheaves welded to an axle supported by bearings at B and DA radius = 2.5”C radius = 2”Rotates at constant speedFind T and the reaction forces at B, DAssumptions:
Bearing at D exerts no axial thrustNeglect weights of sheaves and axle
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Draw FBD
Detach the body from ground (bearings at B and D)Insert appropriate reaction forces
24 lb
Bz
By Dz
DyT
18 lb
30 lb
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Solve for reaction forces for each axis
24 lb
Bz
By Dz
DyT
18 lb
30 lb
If we sum moments about x (alongthe shaft), what forces are involved? 24lb, 18lb, 30 lb, T
If we sum moments about y,what forces are involved? 24lb, 18lb, Dz
If we sum moments about z,what forces are involved? 30lb, T, Dy
If we sum forces in y, what forceswill we need to consider? By, 30lb, T, Dy
If we sum forces in Z, what forcesdo we need to consider? 24lb, 18lb, Bz, Dz
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Solve for reaction forces for each axisΣMx = 0 = (24)(2.5) – 18(2.5) + 30(2) – T(2)T = 37.5 lb
ΣMy = 0 = (24)(8) + (18)(8) – Dz(12)Dz = 28 lb
ΣMz = 0 = -(30)(6) – (37.5)(6) + Dy(12)Dy = 33.75 lb
ΣFy = 0 = By – 30 – 37.5 + DyBy + Dy = 67.5 By = 33.75 lb
ΣFz = 0 = 24 + 18 – Bz + Dz42 + Dz = BzBz = 70 lb
24 lb
Bz
By Dz
DyT
18 lb
30 lb
B = (33.75 lb)j – (70 lb)k D = (33.75 lb)j + (28 lb)k
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Linearly Elastic Material Behavior
Some linearly elastic materials:MetalsWoodConcretePlasticCeramic and glass
Linearly elastic materials obey Hooke’s Laws!
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Uniaxial Stress State
Hooke’s Law in uniaxialtension-compression:
σx = Eεx
Also, for isotropic and homogenous material
Poisson’s Ratioν = -εy / εx
0 (cork) < ν < 0.5 (rubber)
xx
x
y
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Thermal Stresses
Expansion of Parts due to temperaturewithout constraint – no stresseswith constraint – stress buildup
Expansion of a rod vs. a holeDifferential Thermal Expansion
Two material with differential thermal expansion rates that are bound togetherBrass and steelMetals vs. plastic
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Hooke’s Law for Shear
τ = GγG ?
Shear ModulusModulus of Rigidity
Note: No equivalent to Poisson for shear (no coupling between axes)
yx
xy
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Relating E and G
For linearly elastic, homogenous, isotropic material characterized by TWO independent parameters
)1(2 ν+=
EG
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Hooke’s Law for Biaxial Stress State
xx
yx
xy
y
y
EEyx
x
σνσε −=
EExy
yσν
σε −=
Gxy
xy
τγ =
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Hooke’s Law for triaxial state of stress
x
xz
xy
zx
y
z
zy
yz yx
Most general case of static loadingCoupled:
Decoupled:
Note:
( )zxy
y EEσσνσ
ε +−=
Gxy
xy
τγ =
Gyz
yz
τγ =
Gxz
xzτγ =
( )zyx
x EEσσνσε +−=
( )yxz
z EEσσνσε +−=
)1(2 ν+=
EG
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More on Hooke’s Law
Used to relate loading to stress state through geometryAnalytical solutions for classical forms of loading
Axial cases:Column in tension (trivial)Column in compression (non-trivial)
We’ll do this later
Other cases:Beam in pure bendingBeam in bending and shearShaft in torsion
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Column in tension
Uniaxial tension
Hooke’s Law
AF
=σ
EAF
Ey
y ==σ
ε
EAF
zxνεε −==
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Beam in pure bending
What does PURE mean?Moment is constant along the beam
Other assumptionsSymmetric cross-sectionUniform along the length of the beamLinearly elasticHomogenous
Constant properties throughoutIsotropic
Equal physical properties along each axisEnable geometric argumentsEnable the use of Hooke’s Law to relate geometry (ε) to stress (σ)
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Beam in pure bendingResult –
I is the area moment of inertia:M is the applied bending momentc is the point of interest for stress analysis, a distance (usually ymax) from the neutral axis (at y = 0)
If homogenous (E = constant), neutral axis passes through the centroid
Uniaxial tension:
modulus)(section cIZ
ZM
IMc
zx
=
==σ
EIMy
EIMy
zy
x
νεε
ε
−==
=
∫=A
z dAyI 2
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Example
Beam with rectangular cross-section
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Beam in pure bending example, cont.
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Example
Find the maximum tensile and compressive stresses in the I-beamBeam in pure bending --
IMc
=σ
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Review of centroids
Recall Parallel Axis Theorem
I is the moment of inertia about any pointIc is moment of inertia about the centroidA is area of sectiond is distance from section centroid to axis of I
2AdII c +=
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Example, cont.
Recall:c and I defined with respect to the neutral axis (NA)
First, we’ll need to find I
So…We need to find the neutral axisAssume the I-beam is homogenousNA passes through the centroid of the cross-section
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Finding the centroid
Neutral axis passes through the centroidof the cross-sectional area
Divide into simple-shaped sections to find the centroid of a composite area
∑∑=
i
ii
AAy
y
( )( ) ( )( ) hhbhb
hbhhbyhh
43
2323 24 =
+++
=
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Finding I – Moment of Inertia
Red dot shows centroid of the composite area that we just foundSimilarly, we can find the moment of inertia of a composite area
Use parallel axis theorem to find I for the blue and yellow areas
Moment of inertia about a different point is the moment of inertia of the section about its centroid plus the area of the section times square of the distance to the point
blueyellow
iz
III
IIz
+=
= ∑
2AdII c +=
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Finding I – Moment of Inertia, cont.
( )( ) ( )( ) 322
32
16133
126
bhhbb
I hh
yellow =+=
( )( ) ( )( ) 3243
3
24432
122 bhhhbhbIblue =+=
3
48125 bhIII yellowblue =+=
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Maximum tensile stress occurs at the base
Maximum compressive stress occurs at the top
Note, y is the distance from the point of interest (top or base) to the neutral axis
Total height of cross-section = 2h + h/2 = 10h/4Neutral axis is at 3h/4
Finding the bending stresses
( )⎟⎠⎞
⎜⎝⎛=== 23
48125
43
12536
bhM
bhhM
IMc
baseσ
( )⎟⎠⎞
⎜⎝⎛−
=−
== 2348
12547
12584
bhM
bhhM
IMc
topσ
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Beams in bending and shear
Assumptions for the analytical solution:
σx = Mc/I holds even when moment is not constant along the length of the beamτxy is constant across the width
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Calculating the shear stress for beams in bending
V(x) = shear forceI = Iz = area moment of inertia about NA (neutral axis)b(y) = width of beam
Where A’ is the area between y=y’ and the top (or bottom) of the beam cross-section General observations about Q:
Q is 0 at the top and bottom of the beamQ is maximum at the neutral axisτ = 0 at top and bottom of cross-section τ = max at neutral axis
Note, V and b can be functions of y
IbVQ
xy =τ
∫′
′=A
AydyQ )(
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Usually we have common cross-sections
τmax for common shapes on page 136
Example:Rectangular cross-section
Shear and normal stress distributions across the cross-section
⎥⎥⎦
⎤
⎢⎢⎣
⎡−⎟
⎠⎞
⎜⎝⎛= 2
2
22yhbQ
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Relative magnitudes of normal and shear stresses
⎟⎠⎞
⎜⎝⎛== 2max 2
3bhPL
IMyσ
bhP
bhAV P
83
43
43 2
max =⎟⎠⎞
⎜⎝⎛==τ
⎟⎠⎞
⎜⎝⎛=
lh
41
max
max
στ
For THIS loading, if h << L, then τmax << σmax and τ can be neglected
Rectangularcross-section
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Shafts in torsion
AssumptionsConstant moment along lengthNo lengthening or shortening of shaftLinearly elasticHomogenous
Where J is the polar moment of inertia
Note:Circular shaft
Hollow shaft
JTr
z =θτ
∫=A
dArJ 2
[ ]44
4
32
32
io ddJ
dJ
−=
=
π
π
T
T
T
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Recap: Primary forms of loadingAxial
Pure bending
Bending and shear
Torsion
AF
=σ
IMc
=σ
IbVQ
IMc
=
=
τ
σ
JTr
=τ
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Questions
So, when I load a beam in pure bending, is there any shear stress in the material? What about uniaxial tension?Yes, there is!The equations on the previous slide don’t tell the whole storyRecall:
When we derived the equations above, we always sliced the beam (or shaft) perpendicular to the long axis
If we make some other cut, we will in general get a different stress state
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General case of planar stress
Infinitesimal piece of material:A general state of planar stress is called a biaxial stress stateThree components of stress are necessary to specify the stress at any point
σx
σy
τxy
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Changing orientation
Now let’s slice this element at any arbitrary angle to look at how the stress components vary with orientationWe can define a normal stress (σ) and shear stress (τ)Adding in some dimensions, we can now solve a static equilibrium problem…
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Static equilibrium equations
φφ
sincos
lxly
==
( ) ( )( ) ( ) dydxdldl
dxdydldl
xyy
xyx
τσφτφσ
τσφτφσ
−−=+
−−=−
cossinsincos
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From equilibrium…
We can find the stresses at any arbitrary orientation (σx’, σy’, τxy’)
)2cos()2sin(2
)2sin()2cos(22
)2sin()2cos(22
φτφσσ
τ
φτφσσσσ
σ
φτφσσσσ
σ
xyyx
xy
xyyxyx
y
xyyxyx
x
+⎟⎟⎠
⎞⎜⎜⎝
⎛ −−=′
−⎟⎟⎠
⎞⎜⎜⎝
⎛ −−
+=′
+⎟⎟⎠
⎞⎜⎜⎝
⎛ −+
+=′
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Mohr’s Circle
These equations can be represented geometrically by Mohr’s CircleStress state in a known orientation:Draw Mohr’s circle for stress state:φ is our orientation angle, which can be found by measuring FROM the line XY to the orientation axis we are interested in
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Question from before…
Is a beam in pure bending subjected to any shear stress?Take an element…Draw Mohr’s Circleτmax occurs at the orientation 2φ = 90º
φ = 45º IMy
=σ
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Special points on Mohr’s Circle
σ1,2 – Principal stressesAt this orientation, one normal stress is maximum and shear is zeroNote, σ1 > σ2
τmax – Maximum shear stress (in plane)
At this orientation, normal stresses are equal and shear is at a maximum
Why are we interested in Mohr’s Circle?
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Mohr’s Circle, cont.
A shaft in torsion has a shear stress distribution:
Why does chalk break like this…?
Look at an element and its stress state:
JTr
=τ
JTr
=τ
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Mohr’s circle for our element:
σ1 and σ2 are at 2φ = 90ºTherefore φ = 45 ºThis is the angle of maximum shear!
The angle of maximum shear indicates how the chalk will fail in torsion
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Example #1
( )0,5.610,2
81420,2
−=⎟⎠⎞
⎜⎝⎛ −−
=⎟⎟⎠
⎞⎜⎜⎝
⎛ + yx σσ
σx = -42σy = -81τxy = 30 cwx at (σx, τxy )
x at ( -42, 30)y at (σy, τyx )
y at ( -81, -30)Center
Radius
( ) 8.352
8142302
22
22 =⎟
⎠⎞
⎜⎝⎛ −−−
+=⎟⎟⎠
⎞⎜⎜⎝
⎛ −+= yx
xyRσσ
τ
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Example #1, cont.Now we have:
x at ( -42, 30)y at ( -81, -30)C at (-61.5, 0)R = 35.8
Find principal stresses:σ1 = Cx + R = -25.7σ2 = Cx – R = -97.3τmax = R = 35.8Orientation:
cwo
yx
xy 9.563960tan
2tan2 11 =⎟
⎠⎞
⎜⎝⎛=⎟
⎟⎠
⎞⎜⎜⎝
⎛
−= −−
σστ
φ
Recall, 2φ is measured from the line XY to the principal axis. This is the same rotation direction you use to draw the PRINCIPAL ORIENTATION ELEMENT
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Example #1, cont.
Orientation of maximum shearAt what orientation is our element when we have the case of max shear?From before, we have:
σ1 = Cx + R = -25.7σ2 = Cx – R = -97.3τmax = R = 35.8φ = 28.5 º CW
φmax = φ1,2 + 45º CCW φmax = 28.5 º CW + 45º CCW
16.5 º CCW
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Example #2
( )0,400,2
401200,2
=⎟⎠⎞
⎜⎝⎛ −
=⎟⎟⎠
⎞⎜⎜⎝
⎛ + yx σσ
σx = 120σy = -40τxy = 50 ccwx at (σx, τxy )
x at ( 120, -50)y at (σy, τyx )
y at ( -40, 50)Center
Radius
( ) 3.942
40120502
22
22 =⎟
⎠⎞
⎜⎝⎛ −−
+=⎟⎟⎠
⎞⎜⎜⎝
⎛ −+= yx
xyRσσ
τ
![Page 54: '08 DMcSLecture Notes - Chapter 3](https://reader031.fdocuments.in/reader031/viewer/2022020122/549de184b4795974208b45ca/html5/thumbnails/54.jpg)
Example #2, cont.Now we have:
x at ( 120, -50)y at ( -40, 50)C at (40, 0)R = 94.3
Find principal stresses:σ1 = Cx + R = 134.3σ2 = Cx – R = -54.3τmax = R = 94.3Orientation:
( )( ) ccwo
yx
xy 0.3240120
502tan2
tan2 11 =⎟⎟⎠
⎞⎜⎜⎝
⎛−−
−=⎟
⎟⎠
⎞⎜⎜⎝
⎛
−= −−
σστ
φ
Recall, 2φ is measured from the line XY to the principal axis. This is the same rotation direction you use to draw the PRINCIPAL ORIENTATION ELEMENT
![Page 55: '08 DMcSLecture Notes - Chapter 3](https://reader031.fdocuments.in/reader031/viewer/2022020122/549de184b4795974208b45ca/html5/thumbnails/55.jpg)
Example #2, cont.
Orientation of maximum shearAt what orientation is our element when we have the case of max shear?From before, we have:
σ1 = Cx + R = 134.3σ2 = Cx – R = -54.3τmax = R = 94.3φ = 16.0 º CCW
φmax = φ1,2 + 45º CCW φmax = 16.0 º CCW + 45º CCW
61.0 º CCW = 90.0 - 61.0 º CW = 29.0 º CW
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3-D Mohr’s Circle and Max ShearMax shear in a plane vs. Absolute Max shear
Biaxial Stateof Stress
Still biaxial, but considerthe 3-D element
![Page 57: '08 DMcSLecture Notes - Chapter 3](https://reader031.fdocuments.in/reader031/viewer/2022020122/549de184b4795974208b45ca/html5/thumbnails/57.jpg)
3-D Mohr’s Circle
τmax is oriented in a plane 45º from the x-y plane (2φ = 90º)
When using “max shear”, you must consider τmax(Not τx-y max)
![Page 58: '08 DMcSLecture Notes - Chapter 3](https://reader031.fdocuments.in/reader031/viewer/2022020122/549de184b4795974208b45ca/html5/thumbnails/58.jpg)
Out of Plane Maximum Shear for Biaxial State of Stress
Case 1σ1,2 > 0σ3 = 0
21
maxστ =
231
maxσσ
τ−
=23
maxσ
τ =
Case 2σ2,3 < 0σ1 = 0
Case 3σ1 > 0, σ3 < 0σ2 = 0
![Page 59: '08 DMcSLecture Notes - Chapter 3](https://reader031.fdocuments.in/reader031/viewer/2022020122/549de184b4795974208b45ca/html5/thumbnails/59.jpg)
Additional topics we will cover
3-13 stress concentration3-14 pressurized cylinders3-18 curved beams in bending3-19 contact stresses
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Stress concentrations
We had assumed no geometric irregularitiesShoulders, holes, etc are called discontinuities
Will cause stress raisersRegion where they occur – stress concentration
Usually ignore them for ductile materials in static loadingPlastic strain in the region of the stress is localizedUsually has a strengthening effect
Must consider them for brittle materials in static loadingMultiply nominal stress (theoretical stress without SC) by Kt, the stress concentration factor.Find them for variety of geometries in Tables A-15 and A-16
We will revisit SC’s…
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Stresses in pressurized cylinders
Pressure vessels, hydraulic cylinders, gun barrels, pipesDevelop radial and tangential stresses
Dependent on radius
0for
1
1
2
2
22
2
2
2
22
2
=
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−=
o
o
io
iir
o
io
iit
p
rr
rrpr
rr
rrpr
σ
σ
![Page 62: '08 DMcSLecture Notes - Chapter 3](https://reader031.fdocuments.in/reader031/viewer/2022020122/549de184b4795974208b45ca/html5/thumbnails/62.jpg)
Stresses in pressurized cylinders, cont.
Longtudincal stresses exist when the end reactions to the internal pressure are taken by the pressure vessel itself
These equations only apply to sections taken a significant distance from the ends and away from any SCs
22
2
io
iil rr
pr−
=σ
![Page 63: '08 DMcSLecture Notes - Chapter 3](https://reader031.fdocuments.in/reader031/viewer/2022020122/549de184b4795974208b45ca/html5/thumbnails/63.jpg)
Thin-walled vessels
If wall thickness is 1/20th or less of its radius, the radial stress is quite small compared to tangential stress
( )
( ) ( )
tpd
ttdp
tpd
il
it
iavgt
4
2
2
max
=
+=
=
σ
σ
σ
![Page 64: '08 DMcSLecture Notes - Chapter 3](https://reader031.fdocuments.in/reader031/viewer/2022020122/549de184b4795974208b45ca/html5/thumbnails/64.jpg)
Curved-surface contact stresses
Theoretically, contact between curved surfaces is a point or a lineWhen curved elastic bodies are pressed together, finite contact areas arise
Due to deflectionsAreas tend to be smallCorresponding compressive stresses tend to be very highApplied cyclically
Ball bearingsRoller bearingsGearsCams and followersResult – fatigue failures caused by minute cracks
“surface fatigue”
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Contact stresses
Contact between spheresArea is circular
Contact between cylinders (parallel)Area is rectangular
Define maximum contact pressure (p0) Exists on the load axis
Define area of contacta for spheresb and L for cylinders
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Contact stresses - equations
First, introduce quantity Δ, a function of Young’s modulus (E) and Poisson’s ratio (ν) for the contacting bodies
Then, for two spheres,
For two parallel cylinders,
⎟⎟⎠
⎞⎜⎜⎝
⎛
+Δ
= 321 /1/1
908.0RR
Fa
2
22
1
21 11
EEνν −
+−
=Δ
( )⎟⎟⎠
⎞⎜⎜⎝
⎛
Δ+
= 32
210
/1/1578.0 RRFp
( )⎟⎟⎠
⎞⎜⎜⎝
⎛
Δ+
=L
RRFp 210
/1/1564.0 ( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛
+Δ
=21 /1/1
13.1RRL
Fb
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Contact stresses
Contact pressure (p0) is also the value of the surface compressive stress (σz) at the load axisOriginal analysis of elastic contact
1881Heinrich Hertz of Germany
Stresses at the mating surfaces of curved bodies in compression:
Hertz contact stresses
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Contact stresses
Assumptions for those equationsContact is frictionlessContacting bodies are
ElasticIsotropicHomogenousSmooth
Radii of curvature R1 and R2 are very large in comparison with the dimensions of the boundary of the contact surface
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Elastic stresses below the surface along load axis (Figures4-43 and 4-45 in JMB)
Surface
Below surfaceSpheres
Cylinders
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Mohr’s Circle for Spherical Contact Stress
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Mohr’s Circle for Roller Contact Stress
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Bearing Failure Below Surface
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Contact stresses
Most rolling members also tend to slideMating gear teethCam and followerBall and roller bearings
Resulting friction forces cause other stressesTangential normal and shear stressesSuperimposed on stresses caused by normal loading
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Curved beams in bending
Must use following assumptionsCross section has axis of symmetry in a plane along the length of the beamPlane cross sections remain plane after bendingModulus of elasticity is same in tension and compression
![Page 75: '08 DMcSLecture Notes - Chapter 3](https://reader031.fdocuments.in/reader031/viewer/2022020122/549de184b4795974208b45ca/html5/thumbnails/75.jpg)
Curved beams in bending, cont.
( )
o
oo
i
ii
n
n
AerMc
AerMc
yrAeMy
rdAAr
−=
=
−=
=
∫
σ
σ
σ