3d Stress 515y3x

Triaxial Stress State There is no Mohr’s circle solution for problems of triaxial stress state

3D Principal – Triaxial Stress

 max   int   min

 3   2  1

3D Stress – Principal Stresses The three principal stresses are obtained as the three real roots of the following equation:

 3  I1 2  I 2  I 3  0 where

I1   x   y   z I 2   x y   x z   y z   xy2   xz2   yz2 I 3   x y z  2 xy xz yz          2 x yz

2 y xz

2 z xy

I1, I2, and I3 are known as stress invariants as they do not change in value when the axes are rotated to new positions.

Stress Invariants for Principal Stress I1   1   2   3

I 2   1 2   2 3   1 3 I 3   1 2 3 Zero shear stress on principal planes

I1   x   y   z

I 2   x y   x z   y z   xy2   xz2   yz2 I 3   x y z  2 xy xz yz          2 x yz

2 y xz

2 z xy

3 D Mohr’s circle (only if three principle stresses are known).

The shaded area represent stress states that does not contain any of the three principle axes.

2D Mohr circle at Complete 3D Mohr circle at point point A A Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved.

A

B

C

When one of the principle stresses equals zero, 3D Mohr’ circle reduces to 2D Mohr’s circle. However, In case A and B, where two principle stresses have the same sign using a 2D analysis will reach to wrong conclusion.

Figure 4.34 (p. 145) Example of biaxial stress where correct determination of max requires taking σ3 into consideration. Internally pressurized cylinder illustrates biaxial stress states where correct determination of max requires taking σ3 into . Note that (1) for an element on the inside, σ3 is negative and numerically equal to the internal fluid pressure and (2) for thinwalled cylinders σ2  σ1/2.

Fundamentals of Machine Component Design, 4/E by Robert C. Juvinall and Kurt M. Marshek