# THEORIES OF FAILURE 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 theories of failure calculator for ductile materials can be used to analyze the element against yielding using Maximum Shear Stress Theory and Distortion-Energy Theory failure criteria. Calculation tool is only valid for ductile material under static plane stress loading.

In themaximum distortion-energy theory (DE), also known as the von Mises Criterion, 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.

According to themaximum-shear-stress (MSS) theory, also known as the Tresca criterion, 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.

The formulas used for the calculations are given in the List of Equations section.

### Theories of Failure Calculator for Ductile Materials:

 INPUT PARAMETERS Parameter Value Unit Max. principal stress (σmax) MPa psi Min principal stress (σmin) Yield strength (Sy) Design factor (nd)

Note: Use dot "." as decimal seperator.

 RESULTS Parameter Condition to be met for safe design Status MSS theory --- --- --- DE theory (σmax^2 - σmax*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 %.

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: Al6061-T6: 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 Formula Maximum distortion energy (σmax2-σmax*σmin+σmin2)0.5< Sy/n Maximum shear stress if σmax and σmin has same sign: |σmax|

### Examples:

 Link Usage Thin Walled Pressure Vessel Stress Calculation Example 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:

• Callister.W.D,JR. (2007). Materials Science and Engineering: An Introduction . 7th edition. John Wiley & Sons, Inc.
• Beer.F.P. , Johnston.E.R. (1992). Mechanics of Materials , 2nd edition. McGraw-Hill