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Structural Analysis Fundamentals / Stress and strain

Shear strain and the Poisson effect

The second kind of stress, and why stretching one way shrinks the others.

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Why this matters

Shear strain describes angular distortion, and Poisson’s ratio relates lateral to axial strain in a uniaxial test. These quantities are used in torsion, stress transformation and restrained-deformation calculations.

By the end of this lesson you should be able to

  • Define shear stress and shear strain, and say how they differ from the direct quantities
  • Use Hooke's law in shear, τ = Gγ
  • Define Poisson's ratio and calculate lateral strain
  • Explain why a restrained member develops stress in more than one direction

What you should already know

  • Direct stress and strain (this module, lesson 1)
  • Hooke's law and Young's modulus (this module, lesson 2)

Direct stress acts perpendicular to a surface: it pulls the material apart or pushes it together. Shear stress acts along the surface — it tries to slide one part of the material past the neighbouring part.

The definition looks the same as for direct stress, but the area means something different:

Average shear stress
τ=VA\tau = \frac{V}{A}

What it calculates: the mean shear stress on a surface

τ
shear stress (N/mm²)
V
force acting along the surface (N)
A
area of the surface being sheared (mm²)

This assumes

  • The shear is spread evenly over the area, which is an approximation
  • Elastic behaviour

In plain terms: For direct stress the area is perpendicular to the force. For shear stress it is parallel to it. Getting the two mixed up is the commonest error in a bolt or weld check.

Where direct strain measures a change of length, shear strain measures a change of angle.

Take a small square of material and shear it. It becomes a parallelogram. The corners were right angles and are no longer. The shear strain γ is the amount by which a right angle has changed, measured in radians:

Because it is an angle change, shear strain is dimensionless — like direct strain, and for the same reason: it is a ratio of a sideways movement to the distance over which that movement occurs.

Hooke's law in shear
τ=Gγ\tau = G\gamma

What it calculates: shear stress from shear strain

τ
shear stress (N/mm²)
G
shear modulus, sometimes called the modulus of rigidity (N/mm²)
γ
shear strain, the change in a right angle (radians)

This assumes

  • Linearly elastic material
  • Small strains

In plain terms: Exactly parallel to σ = Eε. G plays the part for shape change that E plays for length change. For structural steel G is about 79 000 N/mm², a little under two-fifths of E.

Predict first

A 20 mm bolt connects two overlapping plates in single shear. The shear plane crosses its unthreaded shank. Which area is used for average bolt shear stress?

Under uniaxial tension, common structural materials with positive Poisson’s ratio contract laterally. Poisson’s ratio quantifies this response. Some materials have negative Poisson’s ratio, so lateral contraction is not universal.

Poisson's ratio
ν=−εlateralεaxial\nu = -\frac{\varepsilon_{lateral}}{\varepsilon_{axial}}

What it calculates: how much a material contracts sideways when stretched

ν
Poisson's ratio (dimensionless)
εlateral
strain across the member (dimensionless)
εaxial
strain along the member (dimensionless)

This assumes

  • Isotropic, linearly elastic material
  • Uniaxial stress

In plain terms: The minus sign makes ν positive for ordinary materials, since the two strains have opposite signs. Steel is about 0.30, aluminium 0.33, concrete 0.20.

Try it

Stretch a bar

Stress depends on the load and the area. Extension also depends on the length and the material.

kN (tension)

mm

mm²

Material

Stress σ = P/A
100.0 N/mm²
Strain ε = σ/E
4.878e-4
Extension δ = PL/AE
1.220 mm
Member stiffness AE/L
98.4 kN/mm
strain εσ100 N/mm²

A stiffer material gives a steeper line: the same strain needs more stress.

Worked example

A bolt in shear, and a bar that gets thinner

Given

  • A 20 mm diameter bolt carries 45 kN across one unthreaded-shank shear plane
  • Separately: a steel bar 60 mm wide and 20 mm thick carries 240 kN in tension
  • E = 205 000 N/mm², ν = 0.30

Find

The shear stress in the bolt, and the change in width of the bar.

    Practice

    A 20 mm bolt carries 45.0 kN across one unthreaded-shank shear plane. What is its average shear stress, in N/mm²?

    Practice

    A steel bar 60.0 mm wide carries a tensile stress of 200 N/mm². With E = 205 000 N/mm² and ν = 0.30, what is the magnitude of the lateral strain?

    Practice

    A block of material is sheared so that a right angle changes by 0.0012 radians. With G = 79 000 N/mm², what shear stress is acting, in N/mm²?

    Summary

    • Direct stress acts perpendicular to a surface; shear stress acts along it
    • Shear strain is a change of angle, not of length, and is measured in radians
    • Hooke's law in shear: τ = Gγ, with G ≈ 79 000 N/mm² for steel
    • Poisson's ratio ν = −εlateral/εaxial, about 0.30 for steel
    • Lateral strain produces lateral stress only where the movement is restrained