Transformations of Stress and Strain

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Chapter 7 Transformations of Stress and
Strain
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7.1 Introduction
Goals determine 1. Principal Stresses
2. Principle Planes 3. Max. Shearing
Stresses
3 normal stresses
-- ?x, ?y, and ?z
General State of Stress
3 shearing stresses -- ?xy, ?yz, and ?zx
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Plane Stress condition
2-D State of Stress
Plane Strain condition
A. Plane Stress State
?z 0, ?yz ?xz ?yz ?xz 0 ?z ? 0, ?xy ?
0
B. Plane Stress State
?z 0, ?yz ?xz ?yz ?xz 0 ?z ? 0, ?xy ?
0
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Examples of Plane-Stress Condition
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Thin-walled Vessels
In-plane shear stress
Shear stress
Out-of-plane shear stress
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Max. ?x ?y
(Principal stresses)
Max. ?xy
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7.2 Transformation of Plane Stress
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After rearrangement
(7.1)
(7.2)
Knowing
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Eqs. (7.1) and (7.2) can be simplified as
(7.5)
(7.6)
Can be obtained by replacing ? with (? 90o) in
Eq. (7.5)
(7.7)
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1. ?max and ?min occur at ? 0 2. ?max and ?min
are 90o apart. ?max and ?min are 90o apart. 3.
?max and ?min occur half way between ?max and
?min
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7.3 Principal Stresses Maximum Shearing Stress
Since ?max and ?min occur at ?xy 0, one can
set Eq. (7.6) 0
(7.6)
It follows,
(a)
Hence,
(b)
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Substituting Eqs. (a) and (b) into Eq. (7.5)
results in ?max and ?min
(7.14)
This is a formula of a circle with the center at
and the radius of the circle as
(7.10)
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Mohrs Circle
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The ?max can be obtained from the Mohrs circle
Since ?max is the radius of the Mohrs circle,
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Since ?max occurs at 2? 90o CCW from
?max, Hence, in the physical plane ?max is ?
45o CCW from ?max.
?? In the Mohrs circle, all angles have been
doubled.
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7.4 Mohrs Circle for Plane Stress
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Sign conventions for shear stresses
CW shear stress ? and is plotted above the
?-axis, CCW shear stress ? and is plotted
below the ?-axis
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7.5 General State of Stress 3-D cases
Definition of Direction Cosines
with
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Dividing through by ?A and solving for ?n, we have
(7.20)
We can select the coordinate axes such that the
RHS of Eq. 7.20) contains only the squares of
the ?s.
(7.21)
Since shear stress ?ij o, ?a, ?b, and ?c
are the three principal stresses.
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7.6 Application of Mohrs Circle to the 3-D
Analysis of Stress
?A gt ?B gt ?C
radius of the Mohrs circle
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7.9 Stresses in Thin-Walled Pressure Vessels
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Hoop Stress ?1
(7.30)
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Longitudinal Stress ?2
Assuming the end cap or the fluid inside takes
the pressure
Solving for ?2
(7.31)
Hence
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Using the Mohrs circle to solve for ?max
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7.8 Fracture Criteria for Brittle Materials
under Plane stress
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7.10 Transformation of Plane Strain
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7.11 Mohrs Circle for Plane Strain
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7.12 3-D Analysis of Strain
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7.13 Measurements of Strain Strain Rosette
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2
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2
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