3b. Shear Stress and Strain  (SSES Ch. 3.2)
Shear StrainShear DeformationGeneral Shear StrainShear Stress
Complementary Shear StressShear Modulus
Elastic Shear Strain Energy Density
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Shear Strain
A thin-walled circular shaft (cylinder) of length L, average radius R, and thickness t (t<<R), is subjected to torque T, acting about the axis passing through the centroid of the shaft's cross-section. Here, the left end (A) of the shaft is taken as fixed, and the right end (B) rotates by angle q (in radians). Angle q is the angle of twist over length L. If the torque, cross-section and material properties are all constant over length L, then the rotation of a cross-section at any distance x from the left hand side (point A) is linear: q(x) = (x/L)q.

The displacement (of Point B) per unit length of the shaft is the shear strain g (gamma). The displacement of Point B is Rq - perpendicular to the shaft's axis - and so the shear strain is:

Shear Deformation
Consider a square element on a thin-walled shaft. As torque is applied to the shaft, the square becomes a rhombus.

General Shear Strain
Consider a square element of width dx and height dy ~ h. Shear deformations cause the square to change into a rhombus. 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:

Shear strain g 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 the cut is the shear stress. The symbol used for shear stress is t (tau). In a thin-walled shaft the effective force F acting around the area divided by the cross-sectional area 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
Now consider an element with shear stresses acting on the left and right faces (these faces are on the cross-sections of the last cylinder).
To prevent rotation of element, there must be an equal and opposite resisting couple caused by shear stresses at the top and bottom faces of the stress element. These shear stresses are the complementary shear stresses.
  • Shear stresses come in pairs, txy and tyx;
  • At any material point, shear stresses on mutually perpendicular planes (in 2D the sides of the square) are equal; txy = tyx;
  • The heads and tails of the shear stresses meet.

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Complementary Shear Stress

Shear Modulus
As with normal stress and strain, a relationship exists between shear stress and shear strain. In the linear-elastic region, shear modulus G relates t and g. Hooke's Law for Shear Stress and Shear Strain is:

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

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 shear strain energy (Tq/2) 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:

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Updated: 05/10/2009 DJD