DEPARTMENT OF MECHANICAL AND MANUFACTURING ENGINEERING UNIVERSITY OF NAIROBI DEPARTMENT OF MECHANICAL AND MANUFACTURING ENGINEERING ENGINEERING DESIGN II FME 461 PART 9 GO NYANGASI November 2008

DISTORTION ENERGY THEORY A THEORY OF FAILURE APPLICABLE TO DUCTILE MATERIALS

STATEMENT OF THE THEORY When Yielding occurs in any material, The distortion strain energy per unit volume At the point of failure Equals or exceeds When yielding occurs in the tension test specimen.

DISTORTION ENERGY THEORY Is based on yielding Applies to ductile materials

STRAIN ENERGY AT A LOCATION OF THE ELEMENT SEGREGATED INTO THREE CATEGORIES: Total strain energy per unit volume of the stressed element, arising from the three principal stresses Strain energy per unit volume arising from the hydrostatic stress that causes change of volume only, and which is uniform in all three directions Strain energy per unit volume arising from stresses causing distortion of the element, and this can be expressed as the difference between category (1) and (2).

THREE DIMENSIONAL STRESS General Case

TRI-AXIAL STRESS SITUATION

ELASTIC STRESS-STRAIN RELATIONS UNI-AXIAL STRESS This is the case of a single principal stress Principal strains are then given in terms of principal stresses by the expressions in next slide

ELASTIC STRESS-STRAIN RELATIONS: Uni-Axial stress The variables are: Principal strain in the direction of the principal stress Poisson’s ratio for the material Modulus of elasticity for the material Principal stress

Uni-Axial Stress One dimensional (Normal/Shear)

ELASTIC STRESS-STRAIN RELATIONS: Bi-Axial stress In this case the stress situation consists of two principal stresses, The strains[1] are given by in terms of the two principal stresses as shown in next slide [1] Mechanical Engineering Design; Shigley, Joseph, pg 124, McGraw Hill, Seventh Edition, 2004

STRAINS IN BI-AXIAL STRESS Stress situation consists of two principal stresses and strains are given by the expressions

Bi-Axial Stress Two Dimensional (Plane)

ELASTIC STRESS-STRAIN RELATIONS Tri-Axial Stress This is the case of three principal stresses The most general case Three strains in the directions of the principal stresses Given by in terms of the three principal stresses as shown in next slide

STRAINS IN TRI-AXIAL STRESS Strains are given by the expressions

Tri-Axial Stress Three Dimensional stress

ENERGY PER UNIT VOLUME Tri-Axial Stress Total strain energy The total strain energy is the strain energy caused by the three principal stresses. It is given by the expressions

TOTAL STRAIN ENERGY Substituting for elastic strains

Tri-Axial Stress Three Dimensional stress

STRAIN ENERGY DUE HYDROSTATIC STRESS Hydrostatic stress is the stress that causes change of volume only Hydrostatic stress may be considered as the average of the three principal stresses and derived and expressed as

HYDROSTATIC STRAIN ENERGY Using the equation for total strain energy yields an expression for hydrostatic strain energy:

HYDROSTATIC STRAIN ENERGY Simplifying for hydrostatic strain energy

DISTORTION STRAIN ENERGY This is the difference between total strain energy and the hydrostatic strain energy

Tri-Axial Stress Three Dimensional stress

DISTORTION STRAIN ENERGY

CASE OF SIMPLE TENSION When yielding occurs in simple tension test

Uni-Axial Stress One dimensional (Normal/Shear)

DISTORTION ENERGY THEORY For the general three dimensional stress situation When Yielding occurs in any material, The distortion strain energy per unit volume At the point of failure, Equals or exceeds When yielding occurs in the tension test specimen.

THREE DIMENSIONAL STRESS WHEN YIELDING OCCURS Comparing three dimensional case with simple tension

Tri-Axial Stress Three Dimensional stress

THREE DIMENSIONAL STRESS WHEN YIELDING OCCURS Equating the two conditions

EQUIVALENT STRESS Left hand side of equation referred to as the Equivalent, or Von-Mises stress

APPLICATION OF DESIGN EQUATION Principal stresses are Determined by stress analysis. Stress analysis describes the principal stresses as a function of Load carried, Geometry and dimensions of the machine or structural element.

APPLICATION OF DESIGN EQUATION Left hand side of design equation Equivalent stress in terms of Loads and Dimensions of machine or structural element, Right hand side of design equation Indicator of strength expressed as Working, (design, allowable) stress a function of strength of the material, and a factor of safety.