5 -Maximum shear stress and failure theory

Question 1

Tri-axial stress

Answer 1

• The stress element has a third dimension (z)
• The element has one normal stress and two shear stress components acting on each face
• Where the stress components have the same sign the maximum in-plane shear stress value is not the absolute maximum shear stress to which the material is subjected too
• When two stress components have opposite signs (i.e. sigma(max) and sigma(min) are opposite signs), these stresses will be the maximum and minimum and the maximum in-plane shear stress equals the absolute maximum shear stress

Question 2

Safety Factor - Why use?

Answer 2

• Normal to include a safety factor to take into account imponderables (unknowns/uncertainties) which arise when forecasting likely service loads or operating conditions or to make allowance for variations in material properties or behaviour
• Sources of inaccuracy include
  
  • Changing service conditions leading to dynamic fluctuating or impact situations
  • Incorrect knowledge of the mechanical properties of the materials used
  • Method of manufacture. i.e. welding may induce residual stresses in the component
  • Complexity of design which gives rise to difficult analysis problems which even after analysis may only be a reasonable estimate of stress conditions

Question 3

Safety factor calculations

Answer 3

Question 4

The five main theories for ductile materials

Answer 4

• Maximum principal stress theory (Rankine)
• Maximum shear stress theory (Tresca)
• Maximum principal strain theory (Saint –Venant)
• Total strain energy per unit volume (Haigh)
• Shear strain energy per unit volume (Maxwell-HuberVon Mises)

Question 5

Maximum principal stress theory (Rankine)

Answer 5

• Failure could also occur in compression if the least principal stress were compressive and its value reached the value of yield stress in compression for the material concerned before the maximum principal stress reached the stress at the yield point in a tensile test
• Generally not used in practice as experimental evidence has shown that the theory should not be applied for ductile materials

Question 6

Maximum shear stress theory (Tresca)

Answer 6

• Yielding of the material will begin when the absolute maximum shear stress in the material reaches the maximum shear stress that causes the same material to yield when it is subjected only to axial tension
• Order the principal stresses as s(max) ≥ s(int) ≥ s(min)
• Trescas Hexagon - Plot two none zero principle stresses as s(x) and s(y) - join the points. Failure occurs outside of these lines.

Question 7

Energy per unit volume of material

Answer 7

Strain Energy Density

Question 8

Maximum shear strain energy per unit volume theory - Von Mises

Answer 8

• Failure occurs when the maximum shear strain energy component in the complex stress system is equal to that at the yield point in the tensile test
• Widely regarded as the most reliable basis for design particularly when dealing with ductile materials
• Yielding predicted to occur when von mises stress is larger than yield stress
  *