Modulus of Rigidity Formula: G = (F/A)/θ, Shear Modulus

The modulus of rigidity formula is $G = \frac{\text{shear stress}}{\text{shear strain}} = \frac{F/A}{\theta}$ . For a block whose top face slides a distance Δx over a height L, the shear strain is $\theta = \frac{\Delta x}{L}$ , so $G = \frac{F L}{A\,\Delta x}$ . The modulus measures how strongly a solid resists a change of shape at constant volume.

You also see it called the shear modulus. NCERT writes G, while older Indian books use η or n; do not confuse that η with viscosity. Only solids have a modulus of rigidity, because a liquid simply flows when sheared. It is the third elastic modulus listed in the matter and fluids revision tables.

Quick facts

Formula
$G = \frac{F/A}{\theta} = \frac{F L}{A\,\Delta x}$
SI unit
pascal (Pa)
Dimensional formula
[M¹ L⁻¹ T⁻²]
Other symbols
η or n in some books
Link to Y
$Y = 2G(1 + \nu)$
Steel
about 8 × 10¹⁰ Pa

Symbols in the modulus of rigidity formula

Quantities in G = FL / (A Δx)
Symbol	Meaning	SI unit
$G$	Modulus of rigidity	Pa
$F$	Force parallel to the top face	N
$A$	Area of that face (the force lies in it)	m²
$\Delta x$	How far the top face moves sideways	m
$L$	Height of the block, at right angles to the shift	m
$\theta$	Angle of shear, $\approx \frac{\Delta x}{L}$	rad

Push a thick book's cover sideways while the back cover stays flat on the desk. The pages slide, the spine tilts and the book becomes a slanted box with the same volume. That slanted shape is shear. The force acts along the face, so A is the area of the face being pushed, not a cross section at right angles to the force, which is where shear differs from tensile stress.

Worked examples

Example 1: a rubber mounting pad

A rubber pad has a top face 0.30 m × 0.30 m and a thickness of 5.0 cm. A horizontal force of 900 N moves the top face 2.0 mm relative to the bottom. Find the modulus of rigidity.

Shear stress: $\frac{900}{0.090} = 1.0\times10^4\ \text{Pa}$ . Shear strain: $\theta = \frac{2.0\times10^{-3}}{0.050} = 0.040$ .

$G = \frac{1.0\times10^4}{0.040} = 2.5\times10^5\ \text{Pa}$

Example 2: a steel cube barely moves

A steel cube of side 10 cm (G = 8.0 × 10¹⁰ Pa) has its bottom fixed. A force of 1.0 × 10⁵ N acts along its top face. Find the angle of shear and the shift of the top face.

Area of face: $0.10 \times 0.10 = 0.010\ \text{m}^2$ , so the shear stress is $1.0\times10^7\ \text{Pa}$ .

$\theta = \frac{1.0\times10^7}{8.0\times10^{10}} = 1.25\times10^{-4}\ \text{rad}$

Shift: $\Delta x = \theta L = 1.25\times10^{-4} \times 0.10 = 1.25\times10^{-5}\ \text{m}$ , about 0.013 mm.

For shear, the height L is measured at right angles to the force, between the fixed face and the moving face.

Typical values and how G relates to Y

Approximate modulus of rigidity of some solids
Material	G in Pa
Steel	about 8 × 10¹⁰
Copper	about 4 × 10¹⁰
Aluminium	about 2.5 × 10¹⁰
Glass	about 2.6 × 10¹⁰
Rubber	about 10⁵ to 10⁶

For most solids G is roughly a third to two fifths of Young's modulus, through $Y = 2G(1+\nu)$ . Steel, with Y ≈ 2.0 × 10¹¹ Pa and ν ≈ 0.3, gives G ≈ 2.0 × 10¹¹ / 2.6 ≈ 7.7 × 10¹⁰ Pa, matching the table. For a side by side comparison of all three moduli, see the modulus of elasticity formula.

Where shear modulus matters

Twisting wires and shafts. The torque needed to twist a wire through an angle depends on G. Torsion pendulums and drive shafts are designed with it.
Earthquake waves. S waves are shear waves, and their speed is √(G/ρ). Liquids have G = 0, which is why S waves do not pass through Earth's liquid outer core.
Rubber mounts. A low G lets machinery rock slightly on its pads and absorbs vibration.

Mistakes to avoid

Using the area perpendicular to the force. In shear, the force lies along the face whose area you use.
Dividing Δx by the wrong length. Use the height between the faces, not the width of the face.
Giving θ in degrees. Shear strain is in radians.
Mixing η for rigidity with η for viscosity. Write G to stay clear.

Frequently asked questions

How do you write the modulus of rigidity formula?

G = shear stress / shear strain = (F/A) / θ. With θ = Δx / L for small shifts, G = FL / (AΔx).

What is the unit of modulus of rigidity?

The pascal (N/m²), the same as stress, because shear strain has no unit. Dimensions: [M¹ L⁻¹ T⁻²].

Is modulus of rigidity the same as shear modulus?

Yes. Both names describe the ratio of shear stress to shear strain; G is the usual symbol.

Why do liquids have no modulus of rigidity?

A liquid cannot hold a fixed shape. Any shear force, however small, makes it flow, so there is no steady shear strain to divide by.