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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 𝜎𝑥𝑥 , 𝜎𝑥𝑦 , 𝜎𝑥𝑧 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: (𝜎𝑥𝑥, 𝜎𝑥𝑦 , 𝜎𝑥𝑧), (𝜎𝑦𝑦 , 𝜎𝑦𝑥, 𝜎𝑦𝑧),
(𝜎𝑧𝑧, 𝜎𝑧𝑥, 𝜎𝑧𝑦)
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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, 𝜎𝑋𝑋 = 𝜎1, 𝜎𝑌𝑌 =𝜎2, 𝜎𝑍𝑍 = 𝜎3, 𝜎𝑋𝑌 = 𝜎𝑌𝑍 = 𝜎𝑋𝑍 = 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
𝜎𝑜𝑐𝑡 = 𝜎𝑃𝑁 =1
3(𝜎1 + 𝜎2 + 𝜎3)
𝝈𝑷 = 1
3(𝜎1𝑖 + 𝜎2𝑗 + 𝜎3𝑘)
𝜏𝑜𝑐𝑡 = 𝜎𝑃𝑆 =1
32𝜎1
2 + 2𝜎22 + 2𝜎3
2 − (𝜎1 + 𝜎2+𝜎3)2
=1
3(𝜎1−𝜎2)2 + (𝜎2−𝜎3)2 + (𝜎3−𝜎1)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 Mohr’s 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.
𝜎𝑁𝑆 𝑎𝑛𝑑 𝜎𝑁𝑁 are coordinate axes
to construct Mohr’s 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.