3.2 Shear Stress and Strain
Shear Strain | Shear Deformation | General Shear Strain | Shear Stress | Complementary Shear Stress | Shear Modulus | Elastic Shear Strain Energy Density

» Shear Strain
A hollow shaft, of length, L, radius R, and thickness, t (t<<R), is subjected to a torque, T, acting about the axis passing through the centroid of the cross-section. If the torque, cross-section and material properties are constant over the length, L, then the amount of rotation of a cross-section at any distance, x, from the left hand side (point A) is linear: q(x) = (x/L)q.

The deflection (of Point B) per unit length of the shaft is termed SHEAR STRAIN. The symbol used for shear strain is g (gamma). The deflection at the right hand side is Rq, and so the deflection per unit length of the shaft or shear strain is:


» Shear Deformation
Consider a square element on a thin-walled shaft with both ends free to rotate. As a torque is applied to the tube, the square becomes a rhombus.

» General Shear Strain
Consider a 2-dimensional square element has width, dx, and height, dy. Shear deformations cause the square to change shape into a rhombus as shown at right. SHEAR STRAIN, g, is equal to the change in right angle of a square element, a (radians). Since a is generally small, tan(a) ~ a, therefore:

  • On an element, Shear Strain is defined as positive if it causes the right angle of the 1st quadrant (between the +x and +y-axes) to decrease;
  • Shear Strain is negative if it causes the right angle in the 1st quadrant to increase.
  • The angle is measured in Radians, which is a non-unit (shear strain is dimensionless).

» Shear Stress
Consider the thin-walled shaft (t<<R) subjected to a torque as above. If a cut is taken perpendicular to the axis, the torque is distributed over the cross-section of area, A=2pRt. The shear force per unit area on the face of this cut is termed SHEAR STRESS. The symbol used for shear stress in most engineering texts is t (tau). Therefore, shear stress in a thin-walled shaft is:

  • Shear Stress has units of force per unit area (ksi, MPa, etc.).

The cross-sectional area of a
thin-walled shaft (t<<R) is:

SHEAR YIELD STRENGTH: the value of shear stress when the shear stress-shear strain relationship is no longer linear.

» Complementary Shear Stress
Consider an element from the cylinder above with stresses acting on the top and bottom face. To prevent angular acceleration in the element, there necessarily must be an equal and opposite resisting force. This resisting force is distributed across the adjacent faces and is called the COMPLEMENTARY SHEAR STRESS.

  • Shear Stresses come in pairs, txy and tyx;
  • Shear Stresses on mutually perpendicular planes are equal; txy = tyx;
  • Their arrowheads (or tails) meet.

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Complementary Shear Stress

» Shear Modulus
As with axial stress and strain, a relationship exists between Shear Stress and Shear Strain. Instead of Young's Modulus, E, being the proportional constant, it is the SHEAR MODULUS, G, that relates t and g. Hooke's Law for Shear Stress and Shear Strain is:

For isotropic, homogeneous materials only, i.e., steel, aluminum, etc.. NOT true for composites and other non-isotropic or non-homogeneous materials such as fiber-glass, steel-reinforced concrete, etc..

For for most metals, n ~ 1/3, which means the G can be approximated:

» Elastic Shear Strain Energy Density
The elastic SHEAR strain energy density - the elastic strain energy per unit volume - stored in an axial member is:

The maximum value of the elastic strain energy is the |SHEAR RESILIENCE. It occurs when the stress in the stress reaches the shear yield strength: