If the percentage change in length of the rod due to loading is 3% and the corresponding change in resistivity of the strain gauge material is 0.3%, estimate the percentage of change in the resistance of the strain gauge and its gage factor, Possion’s ratio = 0.3. If the strain gage is connected to a measurement device capable of determining change in resistances with an accuracy of ±0.02Ω, what is the uncertainty in stress that would result in using this resistance measurement device?

€ = $\frac{(△l)}{l}$=0.03

$\frac{dρ}{ρ}$=0.003

E = 200 GPa

R = 120Ω

V = 0.3

Gage Factor

$\frac{dR}{R}$= $\frac{dρ}{ρ}$+ $\frac{dl}{l}$ (1+2v) = 0.003 + 0.03 (1 + 2 × 0.3)

$\frac{dR}{R}$=0.051

Sg = $\frac{(\frac{dR}{R})}{(\frac{dl}{l})}$= $\frac{0.051}{0.03}$=1.7

Stress uncertainty

µ△R = ±0.02Ω

Sg = 1.7

R =120Ω

µσ =± $\frac{∂σ}{(∂△R)}$= µ△R

σ = €E = $\frac{∆R}{R}$ $\frac{E}{Sg}$

$\frac{∂σ}{∂∆R}$= $\frac{E}{RSg}$

µσ = ±$\frac{E}{RSgµσR}$

µσ = ± $\frac{(200 ×10^9×0.02)}{(120 ×1.7)}$

µσ = ±19.6 MPa

Strain Uncertainty

µ€ = ±$\frac{1}{RSgµ△R}$

µ€ = $\frac{0.02}{(120 ×1.7)}$= ±98μϵ

µ€ = ±$\frac{(µσ)}{E}$

µ€= $\frac{(19.6 ×10^6)}{(200×10^9 )}$

µ€ = ±98μϵ