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3d stress explained
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11/24/2014
1
EE3280
Lecture 11
3D Stress
3D Stress
3D Stress Notation
The stress components ππ₯π₯ , ππ₯π¦ , ππ₯π§ acting on the
positive face are taken to be positive when they are
directed in the positive x, y and z directions.
The state of stress at a point consists of 9
components of stress: (ππ₯π₯, ππ₯π¦ , ππ₯π§), (ππ¦π¦ , ππ¦π₯, ππ¦π§),
(ππ§π§, ππ§π₯, ππ§π¦)
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The state of stress at a point is not a scalar or a vector. It
is a more complicated object, called a second order tensor.
Scalars: defined by magnitude, e.g. temperature, density.
Vectors: defined by magnitude and direction, e.g. force,
displacement, velocity.
Second-order tensors: defined by magnitude and two
directions, e.g. stress, strain, electromagnetic field
strength.
Tensor Stress Tensor
Stresses on Arbitrary Planes
Stresses on Arbitrary Planes
N
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Stresses on Arbitrary Planes Stresses on Arbitrary Planes
Stresses on Arbitrary Planes Normal Stress and Shear Stress on an Oblique Plane
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Example 1
Determine the stresses acting on a plane of particular importance in
failure theory, represented by face ABC in the figure with QA=QB=QC.
Solution
X, Y and Z axis are principal axes, πππ = π1, πππ =π2, πππ = π3, πππ = πππ = πππ = 0. Plane ABC is one of
the eight faces of a regular octahedron.
The normal stress on this plane is octahedral normal
stress,oct, and the shear stress on it is the octahedral
shearing stress, oct.
Solution
ππππ‘ = πππ =1
3(π1 + π2 + π3)
ππ· = 1
3(π1π + π2π + π3π)
ππππ‘ = πππ =1
32π1
2 + 2π22 + 2π3
2 β (π1 + π2+π3)2
=1
3(π1βπ2)2 + (π2βπ3)2 + (π3βπ1)2
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Transformation of Stress in 3D Transformation of Stress in 3D
Principal Stresses in 3D Principal Stresses in 3D
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Example 3 Example 3
Example 3 Mohrβs Circle in 3D
For any plane through the point,
let N axis be normal to the plane
and S axis coincide with the
shear component of the stress
for the plane.
πππ πππ πππ are coordinate axes
to construct Mohrβs circle.
The stress components for any
plane passing through the point
locates a point either on one of
the three circles or in one of the
two shaded areas.