# Determination of Material Stresses from Strain Measurements

In the elastic deformation range of materials the methods of calculating the material stresses from the measured strains are based on **Hooke's Law**. In its simplest form Hooke's Law is:

*σ= ε ⋅ Ε *

σ= material stress [N/mm^{2}]

ε= strain [m/m]

Ε= modulus of elasticity, i.e. Young’s modulus [N/mm^{2}]

This version of **Hooke's Law** only applies to the uniaxial stress state. Biaxial and multiaxial stress states require extended versions.

*Note: In strain measurements, only, the difference between an initial output condition and a condition which occurs later can be determined. The initial condition may be a load-free condition, but it may also be a condition with significant pre-loading, for example due to the object's own weight as with a bridge.*

Preload or also residual stress conditions can only be measured if interference with the object is allowed e.g., making a small drill hole.

The material stress σ should be calculated from the measured strain ε only, according to Hooke’s Law for the **uniaxial stress** state as in the equation stated above, if the strain ε is measured in the active direction of the force (0° direction).

In the transverse direction (90° direction), there is no material stress present despite the measurable strain (transverse contraction, transverse dilation).

Therefore, for reliable results the active direction of the force must be known and the strain must be measured in this direction. If this direction is unknown or only known approximately, then the measurements and their evaluation should be carried out with the biaxial stress state in unknown principal directions.

In problems in experimental stress analysis, the uniaxial stress state is more the exception rather than the rule. The biaxial stress state is met far more often, and its determination should not be undertaken with the simple method used for the uniaxial stress state; this would lead to significant errors.

For a plane stress condition, the extreme normal stresses σ_{1} and σ_{2} occur in the perpendicular directions 1 and 2. The stresses σ_{1} and σ_{2} are termed the principal stresses and similarly the directions 1 and 2 are the principal directions of the stress state. If the principal normal stresses and their active directions are known, the biaxial stress condition is unambiguously defined.

Known principal directions of stress are found, for example, on the surface of a round cylindrical vessel under internal pressure, on a shaft loaded with pure torsion, and in an area away from the edges, on a bent plate.

With other objects and with the simultaneous action of different variables, such as normal force and bending or torsion and bending, and so forth. the principal directions must be assumed to be unknown.

The principal normal stresses σ_{1} and σ_{2 }of the biaxial stress state are calculated according to the extended version of **Hooke's Law** from the measured principal strains ε_{1} and ε_{2}, the material's modulus of elasticity E and Poisson's ratio v for the material:

It is assumed that the stress σ_{3} in the principal direction 3 (perpendicular to the surface) is equal to zero.

To simplify mounting procedures, X rosettes are suitable for measurements in the biaxial stress field with known principal directions. The axes of the two measuring grids must be mounted in alignment with the axes of the principal normal stresses (principal strain directions).

For objects of complex shape, with the superimposition of different types of loading (normal, bending or torsional loadings) or for points of inhomogeneity (e.g. changes in cross-sectional area), prediction of the principal directions of the stress state is generally not possible.

In each case where the directions of the principal stresses are not clearly defined, the stress analysis must be carried out according to the methods described below.