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DEPARTMENT OF MECHANICAL AND MANUFACTURING ENGINEERING UNIVERSITY OF NAIROBI DEPARTMENT OF MECHANICAL AND MANUFACTURING ENGINEERING ENGINEERING DESIGN II FME 461 PART 6 GO NYANGASI November 2008

BI-AXIAL STRESS AND MSS THEORY FAILURE BOUNDARIES

MAXIMUM SHEAR STRESS Statement of the theory When Yielding occurs in any material The maximum shear stress at the point of failure Equals or Exceeds The maximum shear stress when yielding occurs In the tension test specimen.

PLANE STRESS SITUATION Three stresses exists on one plane and stresses on the third dimension are zero

PRINCIPAL STRESSES IN BIAXIAL STRESS SITUATION In plane stress situation, the principal stresses are given by the expression

PRINCIPAL STRESSES IN BIAXIAL STRESS SITUATION ORDER OF MAGNITUDE The principal stresses shown in the previous slide can represent two alternatives, which determine the order of magnitude of principal stresses Principal stresses in the plane are of opposite sign Principal stresses in the plane are of same sign

TRI-AXIAL STRESS SITUATION Maximum shear stress at a location Extreme values of shear stresses in each of the three principal planes are then given by the expressions below

BIAXIAL STRESS SITUATION For plane (bi-axial) stress situation Third principal stress is zero The extreme values of shear stress are then expressed as:

UNI-AXIAL STRESS SITUATION Simple tension is an example When yielding occurs in tension test specimen

MOHR’S CIRCLE DIAGRAMS

BI-AXIAL STRESS PRINCIPAL STRESSES: of OPPOSITE sign

MAXIMUM SHEAR STRESS THEORY IN BI-AXIAL STRESS Principal stresses: of OPPOSITE sign

PRINCIPAL STRESSES: of OPPOSITE sign

BI-AXIAL STRESS PRINCIPAL STRESSES: of SAME sign Expression for Maximum shear stress shown below- Illustrated in the next Slide Expression for Failure boundaries

MAXIMUM SHEAR STRESS THEORY IN BI-AXIAL STRESS Principal stresses: of SAME sign

PRINCIPAL STRESSES: of SAME sign Expression for Failure boundaries

MAXIMUM SHEAR STRESS THEORY IN BI-AXIAL STRESS Failure boundaries in bi-axial stress