Principal Stresses from Principal Strains

As previously noted, a three-element strain gage rosette must be employed to determine the principal strains in a general biaxial stress state when the directions of the principal axes are unknown. The usual goal of experimental stress analysis, however, is to arrive at the principal stresses, for comparison with some criterion of failure. With the strain state fully established as described previously , the complete state of stress (in the surface of the test part) can also be obtained quite easily when the test material meets certain requirements on its mechanical properties. Since some types of strain gage instrumentation, such as the Vishay Measurements Group's System 5000, calculate both the principal strains and the principal stresses, the following material is provided as background information.

If the test material is homogeneous in composition, and is isotropic in its mechanical properties (i.e., the properties are the same in every direction), and if the stress/strain relationship is linear, with stress proportional to strain, then the biaxial form of Hooke's law can be used to convert the principal strains into principal stresses. This procedure requires, of course, that the elastic modulus ( ) and Poisson's ratio ( ) of the material be known. Hooke's law for the biaxial stress state can be expressed as follows:

   (515.10a)

   (515.10b)

The numerical values of the principal strains calculated from Eq. ( 515.3 ) or Eq. ( 515.6 ) can be substituted into Eq. (515.10), along with the elastic properties, to obtain the principal stresses. As an alternative, Eq. (515.3) or Eq. (515.6) (depending on the rosette type) can be substituted algebraically into Eq. (515.10) to express the principal stresses directly in terms of the three measured strains and the material properties. The results are as follows:

Rectangular :
 

   (515.11)

  (515.12)

When the test material is isotropic and linear-elastic in its mechanical properties (as required for the validity of the preceding strain-to-stress conversion), the principal stress axes coincide in direction with the principal strains. Because of this, the angle from Grid 1 of the rosette to the principal stress direction is given by Eq. ( 515.5 ) for rectangular rosettes, and by Eq. ( 515.8 ) for delta rosettes.

Page 13 of 19