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Lecture 1Dr Muhammad Adil Khattak
Plane Stress
Stress Transformation in Plane Stress
Principal Stresses & Maximum Shear Stress
Stress Transformation
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Stress Analysis - Review1. Determine critical point
2. Solve for internal forces at that point (orreduce to cantilever)
3. Solve for stresses at that point
4. Add like stresses, i.e.:
Wtotal = W1 +W2 +W3 + ...
Xtotal = X1 +X2 +X3 + 5. Summarize stresses at that point on a stress
element.
6. May be necessary to use Stress
Transformation or Mohrs circle to get max
Chapter 8
Chapter 9
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Drill Bit Isolator (
Thrust Load = 8,000 to
10,000 lb
Bending Load= 125 lbs)=K*H
Torsion Load= 300 lb-ft
Drill Rod
Chuck
Isolator
Drill Bit
HOT
SPOT!
Recall from Chapter 8, already did steps 1 5:
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Now What????
Solve for stress at a
point using standard
Equations.
Summarize these
stresses on an initial
stress element or
alignedstress element.
Must find MAXIMUM
stresses at that point,may be different then
the applied stresses can
occur at some other
orientation plane or
angle.
Compare
max stresses
to material
allowablesto
determine:
Is it safe,
will it fail???
step 6
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Introduction
We have learned Axially
In Torsion
In bending
These stresses act on cross sections of
the members. Larger stresses can occur on inclined
sections.
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Concrete Beam subject to Bending and Shear
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Ductile material failed in Axial loading
Brittle Material failed in Axial Loading
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Ductile Material failed in Torsion
Brittle Material failed in Torsion
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State of stresses varies in the body
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6Stress Components
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Plane Stress Plane Stress The state of stress
when we analyzed bars in tension
and compression, shafts in torsion,and beams in bending.
Consider a 3 dimensional stresselement
Material is in plane stress in the xy
plane Only the x and y faces of the element
are subjected to stresses
All stresses act parallel to the x and yaxis
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Plane Stress
Normal stress
subscript identifies the face on which thestress acts
Sign Convention
Tension positive
compression negative
xW
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Plane Stress
Shear Stress - Two subscripts
First denotes the face on which the stress acts
Second gives the direction on that face
xyX
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Plane Stress
A2-dimensionalview can depict therelevant stressinformation, fig. 9.1c
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A. Given Plane Stress State: B. What are new stresses at
element rotation ofU??:
Note, positivestress
directionsshown.
Note, positive angle (ccw)
shown.
Sign convention
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Stresses onInclined Planes
First we know Wx, Wy, andXxy,
Consider a new stresselement Located at the same point in
the material as the original
element, but is rotatedabout the z axis
x and y axis rotatedthrough an angle U
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Stresses onInclined Planes
The normal and shear stresses acting
on they new element are:
Using the same subscript designations
and sign conventions described.
Remembering equilibrium, we knowthat:
'''' ,,
yxyxXWW
''''xyyx
XX !
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Stresses onInclined Planes
The stresses in the xyplane can be expressed interms of the stresses onthe xy element by usingequilibrium.
Consider a wedge shapedelement
Inclined face same as the xface of inclined element.
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Stresses onInclined Planes
Construct a FBD showing all theforces acting on the faces The sectioned face is (A.
Then the normal and shearforces can be represented onthe FBD.
Summing forces in the x and ydirections and rememberingtrig identities, we get:
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UX
U
WWWWW
2sin2cos22' xyyxyx
x
!
UXUWW
X 2cos2sin2
''xy
yx
yx
!
UXUWWWWW 2sin2cos22
' xy
yxyx
y
!
(9.1)
(9.2)
(9.3)
1.990 equationinforSubstitute UUU !
0'
!Y
F
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Stresses onInclined Planes
These are called the transformation equationsfor plane stress.
They transfer the stress component form one setof axes to another.
The state of stress remains the same.
Based only on equilibrium, do not depend onmaterial properties or geometry
There are Strain Transformation equations thatare based solely on the geometry of deformation.
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Mohrs Circle for Plane Stress
The transformation equations for plane
stress can be represented in a graphicalformat known as Mohrs circle.
This representation is useful invisualizing the relationships betweennormal and shear stresses acting onvarious inclined planes at a point in astressed body.
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Mohr s CircleRules
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Construction of Mohrs circle 1.Determine the point on the body in which the principal stresses are to be determined. 2. Treating the load cases independently and calculated the stresses for the point
chosen. 3. Choose a set of x-y reference axes and draw a square element centered on the axes. 4.Identify the stresses x, y, and xy = yx and list them with the proper sign. 5.Draw a set of - coordinate axes with being positive to the right and being
positive in the upward direction. Choose an appropriate scale for the each axis. 6.Using the rules on the previous page, plot the stresses on the x face of the element in this coordinate system (point V).Repeat the process for the y face
(point H). 7.Draw a line between the two point V and H. The point where this line crosses the axis establishes the center of the circle.
8.Draw the complete circle. 9. The line from the center of the circle to point V identifies the x axis or reference axis for angle measurements (i.e. = 0). Note: The angle between the reference axis and the axis is equal to 2p.
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