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)

11/24/2014

2

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

11/24/2014

3

Stresses on Arbitrary Planes Stresses on Arbitrary Planes

Stresses on Arbitrary Planes Normal Stress and Shear Stress on an Oblique Plane

11/24/2014

4

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

11/24/2014

5

Transformation of Stress in 3D Transformation of Stress in 3D

Principal Stresses in 3D Principal Stresses in 3D

11/24/2014

6

Example 2 Example 2

Example 2 Example 2

11/24/2014

7

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.

11/24/2014

8

Mohr’s Circle in 3D Example 4

Example 4

The normal and shear stresses acting on the planes with normal vectors N1

and N2 are :