Copyright© 2001 1 Content ïµ Stress Transformation ïµ A Mini Quiz ïµ Strain Transformation Click here Click here Click here Click here Click here Click here

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PowerPoint PresentationPlane Stress Loading
~ where all elements of the body are subjected to normal and shear stresses acting along a plane (x-y); none perpendicular to the plane (z-direction)
z = 0; xz = 0; zy = 0
x
y
Plane Stress Loading
Therefore, the state of stress at a point can be defined by the three independent stresses:
x; y; and xy
If x, y, and xy are known, …
A
x
y
x
y
xy
A
…what would be ’x, ’y, and ’xy?
A
x
y
’x
’y
’xy
A
x’
y’
x
y
x’
y’
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Principal Planes
~ are the two planes where the normal stress () is the maximum or minimum
~ the orientations of the planes (p) are given by:
~ there are no shear stresses on principal planes
~ these two planes are mutually perpendicular
gives two values (p1 and p2)
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max = R
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Maximum Shear
x
s1
s2
90
p = s ± 45
Equation of a circle, with variables being x’ and xy’
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Mohr Circles
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Mohr Circles
A point on the Mohr circle represents the x’ and xy’ values on a specific plane.
is measured counterclockwise from the original x-axis.
Same sign convention for stresses as before. i.e., on positive planes, pointing positive directions positive, and ….
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Mohr Circles
When we rotate the plane by 180°, we go a full round (i.e., 360°, on the Mohr circle. Therefore….
x’
xy’
= 90
= 0
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Mohr Circles
…..when we rotate the plane by °, we go 2° on the Mohr circle.
x’
xy’
Mohr circle is a simple but powerful technique
Mohr circle represents the state of stress at a point; thus different Mohr circles for different points in the body
Quit
Continue
The stresses at a point A are shown on right.
A Mohr Circle Problem
orientations of principal planes,
maximum shear stress, and
A
on Mohr Circle
(kPa)
(kPa)
A
Oh, NO!
The state of stress at a point A is shown.
What would be the maximum shear stress at this point?
Answer 1: 50 kPa
Press RETURN to continue
Answer 2:
Press RETURN to continue
Answer 3:
Press RETURN for the answer
Press RETURN to continue
The state of stress at a point B is shown.
What would be the maximum shear stress at this point?
Answer 4:
0
This is hydrostatic state of stress (same in all directions). No shear stresses.
Press RETURN for the answer
Press RETURN to continue
Plane Strain Loading
~ where all elements of the body are subjected to normal and shear strains acting along a plane (x-y); none perpendicular to the plane (z-direction)
z = 0; xz = 0; zy = 0
x
y
by , and
by /2
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e.g.,
x positive
y negative
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Principal Strains
~ maximum (1) and minimum (2) principal strains
~ occur along two mutually perpendicular directions, given by:
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~ measures normal strain (), from the change in electrical resistance during deformation
electrical resistance strain gauge
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Strain Rosettes
~ measure normal strain () in three directions; use these to find x, y, and xy
e.g., 45° Strain Rosette
x
45°
45°
0
90
45
measured