Complex/Principal Stresses and Strains

Abstract

This chapter reinforces the concepts of stresses and strains with particular reference to complex stress and strain. Firstly, complex stress is defined and its technological significance is emphasized. Then the state of plane stress (both in 2D and 3D) is introduced with reference to principal stresses. The direct and shear stresses have been analyzed so as to derive mathematical models to calculate the stresses. Additionally, mathematical relationships for the principal normal stresses and the principal shear stresses have been derived with the aid of differential calculus tools. The Mohr’s Circle has been explained with the aid of worked examples. The principal stresses and the maximum shear stresses in 3D are also considered. Finally, complex strains are discussed and the relationships for the principal normal strains and the principal shear strains are presented along with formulas for the angles of rotations for the coordinates axes of the principal strains. This chapter contains 11 diagrams, 26 mathematical models, and 17 calculations/worked examples.

Questions and Problems

• 8.1. (a) Draw a diagram illustrating the various types of stresses acting on machine components.

  (b) Give four examples of machines/components acted upon by complex/multiple stresses.

• 8.2. (a) Define the terms complex stress, complex strain, and the state of plane stress.

  (b) Give a 2D representation of the state of plane stress and analyze the stresses to derive equations for the direct stress and the shear stress.

• 8.3. (a) Why is the knowledge of principal stress important for a design engineer?

  (b) By using differential calculus tools, derive an equation for calculating the angle of rotation to the coordinate axes for the principal normal stresses.

  (c) Explain the fracture shown in Fig. 1.4(a) with reference to the principal shear stresses.

• 8.4. The stresses at a point on the free surface of a component are σ_x = 140 MPa, σ_y = 210 MPa, and τ_xy = 150 MPa. Calculate: (a) the angle of rotation to the coordinate axes for the principal normal stresses, (b) the principal normal stresses, and (c) the principal shear stresses.

• 8.5. An element of material is acted upon by a tensile stress of 130 MPa and a compressive stress of 70 MPa on two mutually perpendicular planes. Calculate the stresses on a plane inclined 40° to the plane of the 130 MPa stress.

• 8.6. Solve Problem 8.5 by using the Mohr’s circle method.

• 8.7. An element of material is acted upon by direct stresses of 110 MPa tensile and 90 MPa compressive on two mutually perpendicular planes. A clock-wise shear stress acts on the plane with the tensile stress, and an equal and opposite shear stress acts on the other plane. The maximum principal (tensile) stress is 125 MPa. Calculate the shear stress on the planes.

• 8.8. A cubic element of material is acted upon by the following generalized plane stress:

σ_x = 130 MPa, σ_y =  − 70 MPa, σ_z = 80 MPa, τ_xy = 65 MPa, τ_yz = τ_zx = 0

Calculate the (a) principal normal stresses, and (b) the principal axes for the state of stresses.

• 8.9. There exist the following strains at a point on an unloaded surface of a machine component made of aluminum: ε_x = 0.0065, ε_y =  − 0.0007 and γ_xy = 0.004. Assume that the behavior of the material is isotropic linear-elastic. Calculate: (a) the principal normal strains, and (b) the angle of rotation to the coordinate axes for the principal normal strains.

• 8.10. By using the data in Problem 8.9, calculate: (a) the principal shear strains in the three directions, and (b) angle of rotation to the coordinate axes for the principal shear strains.

• 8.11. By using the data in Problem 8.9, calculate the normal strain accompanying the shear strains.