# CALCULATION OF PLANE STRESS STATE WITH STRAIN GAGE ROSETTE RESULTS

Strain measurements on machine components or structural elements are necessary in real life problems to understand actual stress states on them. Strain measurements are conveniently and accurately done with strain gages. Strain gages are consisted of thin electrical wire and are glued parallel to the direction where measurements are desired. As the material elongates (or shorten), the electrical resistance of strain gage changes. By measuring the current passing through the gage, strain can be determined accurately for the loading condition.

By measuring 3 normal strain for a point on a surface of machine component (ε1, ε2, ε3), two normal and one shear strain can be calculated for the point on xy plane (εx, εy, and γxy). The arrangement of three strain gages is shown in the figure and this configuration is called strain rosette.

For plane stress situation such as stress on the free surface of a machine component, if x-y plane is assumed to be the plane which plane stress state occurs, then following situation holds for homogenous isotropic material which obeys Hooke’s law. (σz=0, τxz=0, τyz=0, γxy=0, γxy=0). By using strain rosette measurement results and plane stress assumption, principal stresses can be calculated.

This calculator is compromise of the stress-strain calculators to calculate principal stresses of plane stress situation with the usage of strain gage rosette measurement results. The formulas used for the calculations are given in the "List of Equations" section. INPUT PARAMETERS Parameter Value Unit Elastic modulus (E) GPa psi*10^6 Poisson's ratio (v) --- Measured strain-1 (ε1) μm/m (μin/in) Measured strain-2 (ε2) Measured strain-3 (ε3) Line-1 angle wrt x-axis (θ1) deg Line-2 angle wrt x-axis (θ2) Line-3 angle wrt x-axis (θ3)

Note: Use dot "." as decimal separator.

 RESULTS Parameter Value Unit STRAINS ON REFERENCE AXIS (X-Y) Normal strain in x-direction (εx) --- μm/m (μin/in) Normal strain in y-direction (εy) --- Shear strain in xy-direction (γxy) --- PLANE STRESSES Normal stress in x-direction (σx) --- MPa psi Normal stress in y-direction (σy) --- Shear stress in xy plane (τxy) --- PRINCIPAL STRESSES Maximum principal stress (σmax) --- MPa Minimum principal stress (σmin) --- Maximum shear stress (τmax) --- Average principal stress (σavg) --- Von Mises stress (σmises) ---

### Definitions:

Normal Strain: The ratio of length change to original length of the material. ε=σ/E

Normal Stress: Stress acts perpendicular to the surface (cross section).

Plane Stress: A loading situation on a cubic element where two faces the element is free of any stress. Such a situation occurs on free surface of a structural element or machine component, at any point of the surface of that element which is not subjected to an external force. Another example for plane stress is structures which are built from sheet metals where stresses across the thickness are negligible. Plane stress example - Free surface of structural element

Strain Gage: An electrical measurement device to measure strain.

Strain Rosette: Strain gauge arrangement to measure three normal strains (ε1, ε2, ε3).

Shear Strain: The angular distortion on element caused by shear stress. γ=τ/G.

Shear stress: A form of a stress acts parallel to the surface (cross section) which has a cutting nature.

Stress: Average force per unit area which results strain of material.

### List of Equations:

 Parameter Formula Measured strain-1 (ε1) εx(cosθ1)2+εy(sinθ1)2+γxysinθ1cosθ1 Measured strain-2 (ε2) εx(cosθ2)2+εy(sinθ2)2+γxysinθ2cosθ2 Measured strain-3 (ε3) εx(cosθ3)2+εy(sinθ3)2+γxysinθ3cosθ3 Normal strain (εx) (σx/E-vσy/E-vσz/E) Normal strain (εy) (σy/E-vσz/E-vσx/E) Normal strain (εz) (σz/E-vσx/E-vσy/E) Shear strain (γxy) τxy/G Shear strain (γyz) τyz/G Shear strain (γzx) τzx/G Modulus of rigidity (G) E/(2(1+v))

### Examples:

 Link Usage Principal Stress Calculation Example with Strain Gauge Rosette Measurement An example about the calculation of normal and shear strain and principal stresses on a machine component with the usage of strain results which have been measured with 45° Rosette.

### Reference:

• Beer.F.P. , Johnston.E.R. (1992). Mechanics of Materials , 2nd edition. McGraw-Hill