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Stress Transformation
9.1-9.3
Plane Stress
Stress Transformation in Plane Stress
Principal Stresses & Maximum Shear Stress
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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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Introduction
We will look at stress elements toanalyze the state of stress produce by asingle type of load or by a combinationof loads.
From the stress element, we will derive
the Transformation Equations
Give the stresses acting on the sides of anelement oriented in a different direction.
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Introduction
Stress elements: only one intrinsic state of stress exists at a point in a stressed body,
regardless of the orientation of the elementfor that state of stress.
Two elements with different orientations at
the same point in a body, the stress acting onthe faces of the two elements are different,but represent the same state of stress
The stress at the point under consideration.
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Introduction
Remember, stresses are not vectors.
Are represented like a vector withmagnitude and direction
Do not combine with vector algebra
Stresses are much more complex
quantities than vectors Are called Tensors (like strain and I)
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Plane Stress
Plane Stress – The state of stresswhen we analyzed bars in tensionand 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
x
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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
Sign convention Positive when acts on a positive face of an
element in the positive direction of an axis (++)or (--)
Negative when acts on a positive face of anelement in the negative direction of an axis (+-)
or (-+)
xy
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Plane Stress
A 2-dimensionalview can depict the
relevant stressinformation, fig. 9.1c
Special cases
Uniaxial Stress Pure shear
Biaxial stress
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Stresses on Inclined Planes
First we know x, y, andxy,
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
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Stresses on Inclined Planes
The normal and shear stresses actingon they new element are:
Using the same subscript designationsand sign conventions described.
Remembering equilibrium, we knowthat:
'''' ,, y x y x
'''' x y y x
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Stresses on Inclined Planes
The stresses in the x’y’ plane can be expressed in
terms of the stresses onthe xy element by usingequilibrium.
Consider a wedge shapedelement
Inclined face same as the x’ face of inclined element.
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Stresses on Inclined Planes
Construct a FBD showing all theforces acting on the faces
The sectioned face is A. Then the normal and shear
forces can be represented onthe FBD.
Summing forces in the x and ydirections and rememberingtrig identities, we get:
2cos2sin2
2sin2cos22
xy
y x
y x
xy
y x y x
x
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Stresses on Inclined 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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Stresses on Inclined Planes
Special case simplifications
Uniaxial stress- y
& Txy
= 0
Pure Shear - x & y = 0
Biaxial stress - Txy = 0
Transformation equations are simplifiedaccordingly.
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Principal & Maximum Shear
Stresses Since a structural member can fail due
to excessive normal or shear stress, weneed to know what the maximumnormal and stresses are at a point.
We will determine the maximum and
minimum stress planes for whichmaximum and minimum normal andshear stresses act.
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Principal & Maximum Shear
Stresses Principal stresses – maximum and minimum
normal stresses.
Occurs on planes where:
Applying to eq 9.1 we get:
p=the orientation of the principal planes
The planes on which the principal stresses act.
0'
d
d x
y x
xy
p
2
2tan
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Principal & Maximum Shear
Stresses Two values of the angle 2p are obtained
from the equation.
One value 0-180, other 180-360 Therefore p has two values 0-90 & 90-180
Values are called Principal Angles.
For one angle x is maximum, the other x isminimum.
Therefore: Principal stresses occur onmutually perpendicular planes.
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Principal & Maximum Shear
Stresses We could find the principal stress by
substituting this angle into thetransformation equation and solving
Or we could derive general formulas forthe principal stresses.
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Principal Stresses
Consider the right triangle
Using the trig from the triangle
and substituting into thetransformation equation fornormal stress, we get
Formula for principal stresses.
2
2
2,122
xy
y x y x
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Shear Stresses on the
Principal Planes If we set the shear stress x’y’ equal to
zero in the transformation equation and
solve for 2, we get equation 9-4. The angles to the planes of zero shear
stress are the same as the angles to the
principal planesTherefore:The shear stresses are zero on
the principal planes
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The Third Principal Stress
We looked only at the xy plane rotating aboutthe z-axis.
Equations derived are in-plane principalstresses
BUT, stress element is 3D and has 3 principal
stresses. By Eigenvalue analysis it can be shown thatz=0 when oriented on the principal plane.
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Maximum In-Plane Shear
Stress Consider the maximum shear stress and
the plane on which they act.
The shear stresses are given by thetransformation equations.
Taking the derivative of x’y’
withrespect to and setting it equal to zerowe can derive equation 9-7
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Maximum Shear Stress
The maximum negative shear stress min has the same magnitude but opposite
sign.
The planes of maximum shear stressoccur at 45 to the principal planes
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Maximum Shear Stress
If we use equation 9-5, subtract 2 from 1, and compare with equation 9-
7, we see that:
Maximum shear stress is equal to ½ thedifference of the principal shear stress.
2
21
max
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Average Normal Stress
The planes of maximum shear stressalso contain normal stresses.
Normal stresses acting on the planes of maximum positive shear stress can bedetermined by substituting the
expressions for the angle
s into theequations for x’ .
Result is Equation 9-8.
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Important Points
The principal stresses are the max and min normalstress at a point
When the state of stress is represented by theprincipal stresses, no shear stress acts on theelement
The state of stress at the point can also berepresented in terms of max in-plane shear stress .
In this case an average normal stress also acts onthe element
The element in max in-plane shear stress is oriented45° from the element in principal stresses.
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