Failures Theories and Design

Abstract

Failure theories play an important role in designing machine components. This chapter first gives an outline of yielding and fracture as well as a new theory of failure. The commonly applied theories of failure are explained; these theories include: the maximum principal normal stress theory (or Rankine theory), the maximum shear stress theory (or Tresca theory), and von Mises theory. In particular, Tresca theory and von-Mises theories are discussed with reference to their applications in shaft design. Relevant mathematical models for each theory are presented and useful formulae have been derived. This chapter contains 9 worked examples, 26 formulae, 6 exercise problems, and 5 MCQs with their answers given at the end of the book.

Questions and Problems

• 11.1. What is the technological importance of theories of failure?

• 11.2. Explain the maximum principal normal stress theory with the aid of a diagram.

• 11.3. Explain Tresca theory giving its mathematical and graphical representations.

• 11.4. State Von-Mises failure theory and prove that: $\left(\sqrt{1-\alpha +{\alpha}^2}\right)\ {\sigma}_1={\sigma}_{ys}$

• 11.5. A machine component is made of titanium with a yield strength of 450 MPa. At a point of interest on the free surface of the component, the stresses, in the state of plane stress, with respect to a convenient coordinate system in the plane of the surface are σ_x = 200 MPa, σ_y = 230 MPa, and τ_xy = 30 MPa. Apply MSS theory to predict failure at the specified point.

• 11.6. A machine component is made of cast iron with a tensile strength of 500 MPa. At a point of interest on the free surface of the component, the stresses with respect to a convenient coordinate system in the plane of the surface are σ_x = 330 MPa, σ_y = 250 MPa, and τ_xy = 140 MPa. Will the failure occur at the specified point?

• 11.7. By using the data in 11.5, predict the failure by using the Von-Mises failure theory.

• 11.8. A machine component is made of aluminum with a yield strength of 35 MPa. At a point of interest on the free surface of the component, the stresses with respect to a convenient coordinate system in the plane of the surface are σ_x = 100 MPa, σ_y =  − 80 MPa, and τ_xy = 40 MPa. Predict failure by applying MSS theory.

• 11.9. A shaft is made of carbon steel with a yield strength of 470 MPa. There are loads on pulleys mounted on the shaft to keep it in equilibrium. The greatest bending moment at pulley A on the shaft is 620 × 10³ N ‐ mm, and the torque is 560 × 10³ N ‐ mm. Design the shaft by calculating its minimum diameter by using Tresca theory. Take the factor of safety as 3.

• 11.10. A 45-mm-diameter solid shaft is made of carbon steel with a yield strength of 430 MPa. There are forces on pulleys mounted on the shaft to keep it in equilibrium. The greatest bending moment at a pulley on the shaft is 670 × 10³ N ‐ mm, and the torque is 565 × 10³ N ‐ mm. The shaft showed satisfactory service in the machine. Calculate the factor of safety by using Von-Mises Theory.

• 11.11. (MCQs). Encircle the most appropriate answers for the following questions.

  • (a) Which failure theory relates to the root mean square (RMS) value of the maximum shear stresses to a critical value?

    • (i) Tresca theory, (ii) Rankine theory, (iii) Von-Mises theory, (iv) Umantsev theory

  • (b) Which failure theory is used for predicting failure of brittle materials?

    • (i) Tresca theory, (ii) Rankine theory, (iii) Von-Mises theory.

  • (c) Which failure theory relates the greatest maximum shear stress to σ_ys/2?

    • (i) Tresca theory, (ii) Rankine theory, (iii) Von-Mises theory, (Iv) Umantsev theory.

  • (d) The excessive plastic deformation rendering the machine part unsuitable to perform refers to:

    • (i) Necking, (ii) yielding, (iii) plastic instability, (iv) fracture.

  • (e) Which failure theory includes all major components of the mechanical behavior of ductile materials (e.g. strain/work-hardening, Bauschinger effect, etc.) and covers all regimes of viscoplastic tensile/compressive loading/unloading?

    • (i) Umantsev theory, (ii) Tresca theory, (iii) Rankine theory, (iv) Von-Mises theory.