Experiment Setup

An aluminum bar with mounted strain gauges will be used for measuring an

applied torque $(T)$ and transverse force $(P)$

Location A: Two gauges $($top and bottom$)$ oriented in x$-$direction, and wired in a

$\frac{1}{2}$ bridge circuit.

Location B: Two gauges mounted on the sides $($of circular section$),$ oriented at $45,$

and wired in a $\frac{1}{2}$ bridge circuit. Overview:

Predict two sets of calibration constants $(k_1,k_2,k_3,k_4)$ for the device:

Problem $1$: One set of four $k_i$ for the full length $(L)$ of the apparatus

Problem $2$: One set of four $k_i$ for a test setup with the load placed at two$-$thirds of the

full length $(\frac{2}{3}L).$ Strains at A and $B$ are function of loads $P$ and $T$

The strains at A and $B,{}_A$ and ${}_B$ respectively, are linearly related to the loads $P$ and $T$ by:

${}_A=k_{1}P+k_{2}T$

${}_B=k_{3}P+k_{4}T$

Where $k_i$ are constants found through calibration of the device.

$($What are the units for each $k_i?)$

$P=\frac{{}_A{k}_4-{}_B{k}_2}{k_1{k}_4-k_2{k}_3},T=\frac{{}_B{k}_1-{}_A{k}_3}{k_1{k}_4-k_2{k}_3}$

Once you have calibrated the device

$($determined the $k_i$ values$),$ these

equations could be used to

determine $P$ and $T.$ How is the strain $({}_A)$ at the top strain gauge related to the force $(P)?$

State of stress at top

Bending Stress at surface: $,=\frac{Mc}{\text{I}},$ where $,c=\frac{h}{2}$

strain gauge at $A?$

Moment: $,M=PL$

Moment of Inertia: $,I=\frac{1}{12}b{h}^3$

Modulus of Elasticity: $,E=\frac{}{}=10,000$ ksi

Shear Modulus: G$=3770$ksi

Poisson's ratio: V$=\mathrm{.}33$

Can determine the ${}_A$ as a function of $P.$

Then use this relationship to calculate theoretical $k_1,({}_A=k_{1}P)$

$k_1$ will be a function of $L,b,h,$EUse theory to predict $k_1$

How to find theoretical value of $k_1?$

Can we easily relate ${}_A$ to $P?$

${}_A=k_{1}P+k_{2}T$

${}_A=k_{1}P$

What if we set $T$ to zero? $($don$\text{'}$t apply a torque$)$

Only apply a force $P$

Can we determine $k_1$ based on theoretical relationship between ${}_A$ and $P?$

What is the theroetical value of K$1?$