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 (ε_{1}, ε_{2}, ε_{3}), two normal and one shear strain can be calculated
for the point on xy plane (ε_{x}, ε_{y}, and γ_{xy}).
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 xy 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. (σ_{z}=0, τ_{xz}=0, τ_{yz}=0, γ_{xy}=0, γ_{xy}=0). By using strain rosette measurement results
and plane stress assumption, principal stresses can be calculated.
This calculator is compromise of the stressstrain 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.
Note: Use dot "." as decimal separator.
RESULTS 
Parameter 
Symbol 
Value 
Unit 
STRAINS ON REFERENCE AXIS (XY) 
Normal strain in xdirection 
ε_{x} 


μm/m (μin/in) 
Normal strain in ydirection 
ε_{y} 


Shear strain in xydirection 
γ_{xy} 


PLANE STRESSES 
Normal stress in xdirection 
σ_{x} 



Normal stress in ydirection 
σ_{y} 


Shear stress in xy plane 
τ_{xy} 


PRINCIPAL STRESSES 
Maximum principal stress 
σ_{max} 


MPa

Minimum principal stress 
σ_{min} 


Maximum shear stress 
τ_{max} 


Average principal stress 
σ_{avg} 


Von Mises stress 
σ_{mises} 


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 (ε_{1}, ε_{2}, ε_{3}).
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 
Symbol 
Formula 
Measured strain1

ε_{1} 
ε_{x}(cosθ_{1})^{2}+ε_{y}(sinθ_{1})^{2}+γ_{xy}sinθ_{1}cosθ_{1
}

Measured strain2 
ε_{2} 
ε_{x}(cosθ_{2})^{2}+ε_{y}(sinθ_{2})^{2}+γ_{xy}sinθ_{2}cosθ_{2 } 
Measured strain3

ε_{3} 
ε_{x}(cosθ_{3})^{2}+ε_{y}(sinθ_{3})^{2}+γ_{xy}sinθ_{3}cosθ_{3 } 
Normal strain 
ε_{x} 
(σ_{x}/Evσ_{y}/Evσ_{z}/E) 
Normal strain 
ε_{y} 
(σ_{y}/Evσ_{z}/Evσ_{x}/E) 
Normal strain 
ε_{z} 
(σ_{z}/Evσ_{x}/Evσ_{y}/E) 
Shear strain 
γ_{xy} 
τ_{xy}/G 
Shear strain 
γ_{yz} 
τ_{yz}/G 
Shear strain 
γ_{zx} 
τ_{zx}/G 
Modulus of rigidity 
G 
E/(2(1+v)) 
Examples:
Link 
Usage 
Principal Stress Calculation with Strain Results

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: