8.8 Strain Gauges, Rosettes & Experimental Stress Analysis

Why Measure Strain Rather Than Stress

Stress cannot be measured directly. What an instrument can detect is deformation, and stress is then inferred through the constitutive law. Every experimental stress analysis technique therefore measures strain and computes stress from it.

This matters practically in a coal operation: dragline booms, shovel dipper handles, conveyor gantries and haul-truck chassis are all validated by strain gauging under service load, because their real loading is far more complex than any design idealisation.

The Electrical Resistance Strain Gauge

A thin metal foil grid is bonded to the surface so that it deforms with it. Stretching the foil makes it longer and thinner, both of which raise its electrical resistance. For a conductor of resistivity $\rho$, length $L$ and area $A$:

$R = \frac{\rho L}{A}$

Differentiating and collecting terms gives

$\frac{\Delta R}{R} = (1 + 2\mu)\epsilon + \frac{\Delta\rho}{\rho}$

The bracketed term comes from the geometry change and the last from the piezoresistive effect. Collecting everything into a single constant defines the gauge factor:

$\boxed{G_f = \frac{\Delta R/R}{\epsilon}}$

The number to remember is that a common foil gauge has $G_f \approx 2$.

The magnitude problem

Typical elastic strains are of the order of $1000,\mu\epsilon$, that is $10^{-3}$. With $G_f = 2$, the fractional resistance change is only $2\times10^{-3}$ — for a 120 ohm gauge, a change of 0.24 ohm. Detecting that reliably requires a bridge circuit, not a plain ohmmeter.

The Wheatstone Bridge

The gauge forms one arm of a Wheatstone bridge. At balance the output is zero; a small resistance change unbalances it and produces a measurable voltage.

Temperature compensation

Temperature affects a gauge in two ways: the foil's resistivity changes, and differential thermal expansion between gauge and specimen produces apparent strain. Both are indistinguishable from real strain in a single gauge.

The standard remedy is a dummy gauge: an identical gauge bonded to an unstressed piece of the same material, kept at the same temperature, and placed in the adjacent bridge arm. Both gauges see the same thermal effect, which therefore cancels in the bridge output, while only the active gauge sees mechanical strain.

In a bending application, mounting one gauge on the tension face and one on the compression face in adjacent arms doubles the output and compensates temperature simultaneously — the reason half-bridge arrangements are standard in load cells and torque transducers.

Strain Rosettes

A single gauge measures normal strain in one direction only. A general plane strain state has three independent components — $\epsilon_x$, $\epsilon_y$ and $\gamma_{xy}$ — so three gauge readings at known angles are required. That cluster is a rosette.

The general transformation for a gauge at angle $\theta$ from the reference axis is

$\epsilon_\theta = \frac{\epsilon_x + \epsilon_y}{2} + \frac{\epsilon_x - \epsilon_y}{2}\cos2\theta + \frac{\gamma_{xy}}{2}\sin2\theta$

which is the strain analogue of the stress transformation used in Mohr's circle.

Rectangular (45-degree) rosette

Gauges at 0, 45 and 90 degrees, giving readings $\epsilon_a$, $\epsilon_b$ and $\epsilon_c$:

$\epsilon_x = \epsilon_a, \qquad \epsilon_y = \epsilon_c, \qquad \gamma_{xy} = 2\epsilon_b - (\epsilon_a + \epsilon_c)$

The principal strains follow:

$\epsilon_{1,2} = \frac{\epsilon_a + \epsilon_c}{2} \pm \frac{1}{\sqrt2}\sqrt{(\epsilon_a - \epsilon_b)^2 + (\epsilon_b - \epsilon_c)^2}$

and the principal direction from

$\tan2\theta_p = \frac{2\epsilon_b - \epsilon_a - \epsilon_c}{\epsilon_a - \epsilon_c}$

Delta (60-degree) rosette

Gauges at 0, 60 and 120 degrees:

$\epsilon_{1,2} = \frac{\epsilon_a+\epsilon_b+\epsilon_c}{3} \pm \frac{\sqrt2}{3}\sqrt{(\epsilon_a-\epsilon_b)^2 + (\epsilon_b-\epsilon_c)^2 + (\epsilon_c-\epsilon_a)^2}$

The delta rosette distributes the gauges more evenly and is slightly less sensitive to any single reading error, but the rectangular rosette is easier to align and far more common.

From strain to stress

Once principal strains are known, principal stresses follow from the two-dimensional Hooke's law:

$\sigma_1 = \frac{E}{1-\mu^2}\left(\epsilon_1 + \mu\epsilon_2\right), \qquad \sigma_2 = \frac{E}{1-\mu^2}\left(\epsilon_2 + \mu\epsilon_1\right)$

The factor $1/(1-\mu^2)$ is essential. Computing $\sigma = E\epsilon$ from a rosette reading is a frequent and serious error, because it ignores the transverse stress entirely.

Worked example

A rectangular rosette on a dragline boom reads $\epsilon_a = 600,\mu\epsilon$, $\epsilon_b = 200,\mu\epsilon$, $\epsilon_c = -200,\mu\epsilon$. Find the principal strains.

$\epsilon_{1,2} = \frac{600 - 200}{2} \pm \frac{1}{\sqrt2}\sqrt{(600-200)^2 + (200+200)^2}$

$= 200 \pm \frac{1}{\sqrt2}\sqrt{160000 + 160000} = 200 \pm \frac{565.7}{1.414} = 200 \pm 400$

So $\epsilon_1 = 600,\mu\epsilon$ and $\epsilon_2 = -200,\mu\epsilon$. With $E = 200$ GPa and $\mu = 0.3$:

$\sigma_1 = \frac{200000}{1-0.09}\left(600 + 0.3(-200)\right)\times10^{-6} = \frac{200000}{0.91}\times540\times10^{-6} = 118.7 \text{ MPa}$

Other Experimental Methods

The practical workflow in industry combines them: a brittle coating or photoelastic model first identifies where the peak stress is and in what direction, and strain gauges or rosettes are then bonded there to measure it accurately.