Design Contents |

Preliminary Matters

Design Methodology

..brain storming

..evaluation matrix

..QFD

Statistical Considerations

..variability in materials

..variability in dimensions

..variability in loading

..preferred sizes

Tolerances

Design Factor

Introduction to Failure

Failure Theories

Application of von Mises

..criterion in 2 D

Stress Concentration

..and notch sensitivity

Failure Under Combined Loading

..combined bending and torsion

Failure Under Cyclic Loading

..fatigue

..fracture mechanics

Instability - Buckling

Concentrically Loaded Strut

..slender columns

..Euler formula

..effective length

..short and intermediate columns

Eccentrically Loaded Strut

.. theory

Shock Loading

..deflection

..stress

Failure Theories |

**Introduction**

A number of failure theories have been proposed to explain the ways components
manufactured from different types of materials fail that covers all types of loading. One complication is that some materials have
a greater strength in compression than in tension. Although the main criteria are summarised below,
only the following are widely used:

- von Mises failure criterion - applicabe to ductile materials (most Finite element stress analysis packages calculate von Mises stresses - manual calculation can be tedious).
- Maximum Shear stress criterion - also applicable to ductile materials, but slightly conservative compared to the von Mises criterion, but usually easier to calculate manually.
- Maximum principal stress criterion - applies to brittle materials.

**The Maximum - Normal - Stress Theory (Rankine)**

Failure occurs when one of the three principal stresses equals the strength.

Assuming then failure occurs when or

where S_{t} is the tensile strength and S_{c} is the compressive strength.

This criteria is often used when designing with brittle materials such as concrete or some cast irons.

**The Maximum Normal - Strain - Theory** (Also called St Venant's theory)

Applies only in the elastic range. States yielding occurs when the largest of the 3 principal strains becomes equal to the strain corresponding to the yield strength. If it is assumed that the yield strength in tension and compression are equal, conditions for yielding are:

**The Maximum - Shear - Stress Theory (Tresca)**

States that yielding begins when the maximum shear stress becomes equal to the maximum shear stress in a tension test specimen of the same material when that specimen begins to yield.

or

This theory also predicts that the yield strength in shear is

For principal stresses:

etc

Decompose the three normal principal stresses into the components -

etc so these equal stresses are called hydrostatic components.

If then the three shear stresses would all be zero and there could be no yielding - regardless of the hydrostatic stress. The magnitude of the hydrostatic stress has no effect on the size of the Mohr circle, but move it along the normal stress axis.

This criteria may be used when designing with ductile materials (most commonly used metals) but gives somewhat conservative designs for some combinations of loadings.

**Maximum Strain Energy Theorem**

Suggests failure by yielding occurs when the total strain energy per unit volume reaches or exceeds the strain energy in the same volume corresponding to the yield strength in tension or compression.

se / unit vol. when stressed uniaxially to yield: u_{s} = S_{y}^{2}/2E, Energy in a unit vol. subject to combined stresses:

This is not much used presently.

**The von Mises Theory** (also known as the maximum distortion energy theory. This gives the same result as the von Mises-Hencky theory or the octahedral shear stress theory).

This states that yielding will occur when the distortion energy in a unit vol. equals the distortion energy in a unit vol. when uniaxially stressed to the yield strength. This was derived from the observation that yielding is not affected by a volume change caused by compression, so may be related to the angular distortion of a stressed element. With some algebra, the effective or von Mises stress is defined by:

vms =

and yielding occurs when vms*S _{y}*

This criterion is commonly used when designing with ductile metals - it gives a better fit to experimental data than the Tresca criterion.

2-D Stress investigation - Mohr's cicle and von Mises failure criterion - interactive plot.

**Coulomb Mohr or Internal friction theory**

This theory and variants try to cover materials whose yield stress in compression is not the same as their yield strength in tension.

David J Grieve, updated: 25th January 2014, 24th January 2010, 13th August 2004, original: 1st November 1999.

**Contact the Author:**

Please contact me for comments and / or corrections or to purchase the book, at: davejgrieve@aol.com