YIELD CRITERIA FOR DUCTILE MATERIALS UNDER PLANE STRESS (STATIC LOADING)
Structural elements and components made from ductile materials are generally designed such that the material doesn't
yield under the expected loading condition in service. When the element is in a state of plane stress,
it's convenient to first calculate
principal stresses for the point that is going to be analyzed.
Then following calculation tool can be used to analyze the element against yielding using Maximum Shear Stress Theory and DistortionEnergy Theory
failure criteria. Calculation tool is only valid for ductile material under
static plane stress loading.
The formulas used for the calculations are given in the List of Equations
section.
Calculator:
Note: Use dot "." as decimal seperator.
RESULTS

Parameter 
Condition to be met for safe design 
Status 
MSS theory 






DE theory 
(σmax^2σmax*σmin+σmin^2)^0.5< Sy/n





Note: Dot "." in the drawing shall remain inside the polygon for safe design
according to MSS theory and shall remain inside ellipse according to DE theory.
Use design factor included graphs for evaluation.
Definitions:
Design factor (nd):The ratio of failure stress to allowable stress. The design factor is what the item is required
to withstand .The design factor is defined for an application (generally provided in advance and often set by regulatory code or policy) and
is not an actual calculation.
Ductility: A measure of the
degree of plastic deformation that has been sustained at fracture. Ductile
materials are approximately considered to be those having a fracture strain of
more than 5 %.
Maximum DistortionEnergy Theory (DE): Yielding occurs when the distortion strain
energy per unit volume reaches or exceeds the distortion strain energy per unit
volume for yield in simple tension or compression of the same material.
Maximum Shear Stress Theory (MSS):
According to the maximumshearstress theory, yielding in a material begins when the maximum shear stress in any element
of the material gets larger than the shear stress that occurs when
a specimen produced from the same material begins to yield during a tension test.
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 
Principal Stress: Maximum and minimum normal stress possible
for a specific point on a structural element. Shear stress is 0 at the orientation where principal stresses occur.
Stress: Average force per unit area which results strain of material.
Yield strength: The stress at which a material exhibits a specified permanent deformation or set.
Example: Al6061T6: 145 Mpa
Supplements:
Link 
Usage 
Principal Stress Calculator

To be able to use yield criteria of ductile material calculation tool, principal
stress shall be first calculated. Refer principal stress calculator to find
principal stresses from the plane stresses. 
List of Equations:
Parameter 
Symbol 
Formula 
Maximum
distortion energy

 
(σ_{max}^{2}σ_{max}*σ_{min}+σ_{min}^{2})^{0.5}< Sy/n 
Maximum
shear stress 
 
if σmax and σmin has same sign:
σmax<Sy/n , σmin<Sy/n
if σmax and σmin has opposite sign:
σmaxσmin<Sy/n 
Examples:
Link 
Usage 
Pressure Vessel

An example about the calculation of stresses on a pressure vessel,
evaluation of yield criteria of material and stress transformation to find shear
and perpendicular stresses on welding of the cylindrical body of the pressure
vessel. 
Torsion Of Solid Shaft

An example about the calculation of torsional stress on stepped shaft. After
calculation of torsional stress, principal stresses are calculated and
evaluation of yield criteria of material is done with these stresses. 
Reference: