3. Theory
3.1 Stress, strain and Hooke's law
Within the elastic range, the normal stress and strain in a bar under uniaxial stress are proportional:
\[ \sigma = E\,\varepsilon \]
where E is Young's modulus. Strain is dimensionless and small, so it is quoted in microstrain: 1 µε = 10−6. The same stress also makes the bar contract sideways. Poisson's ratio ν links the transverse strain to the longitudinal strain:
\[ \varepsilon_{\text{trans}} = -\nu\,\varepsilon_{\text{long}} \]
3.2 Bending of the cantilever
A load W hung at distance a from the clamp produces a hogging bending moment at any section between the clamp and the load:
\[ M(x) = W\,(a - x), \qquad x < a \]
Simple bending theory gives the stress at distance y from the neutral axis, and hence the strain at the extreme fibre, distance c from the neutral axis:
\[ \sigma = \frac{M\,y}{I}, \qquad \varepsilon_b = \frac{M\,c}{E\,I} \]
The top face is in tension (positive strain) and the bottom face in compression (negative). The section properties of the two bars on this bench are:
| Section | Second moment of area I | Extreme fibre c | Area A |
|---|
| Rectangular, width b, depth h | \(bh^3/12\) | \(h/2\) | \(bh\) |
| Solid circular, diameter d | \(\pi d^4/64\) | \(d/2\) | \(\pi d^2/4\) |
At the gauge station the strain is therefore \( \varepsilon_b = W\,(a - x_g)\,c/(EI) \). The reading depends on the moment at the gauges, so moving the hanger changes it even when the load stays the same.
3.3 Deflection
For a perfectly rigid clamp, the downward deflection v at position x under a load W at a is
\[ v(x) = \begin{cases} \dfrac{W x^2 (3a - x)}{6EI}, & x \le a \\[2ex] \dfrac{W a^2 (3x - a)}{6EI}, & x \ge a \end{cases} \]
With the load at the free end this gives the familiar \(v(L) = WL^3/(3EI)\).
A real clamp lets the root of the beam rotate slightly. If the clamp has rotational stiffness \(k_\theta\), the root rotates by \(\theta_0 = M_0/k_\theta = Wa/k_\theta\), and every point along the beam moves down by a further \(\theta_0 x\):
\[ v(x) = v_{\text{beam}}(x) + \frac{W a\, x}{k_\theta} \]
Strain readings are unaffected, because statics fixes the moment at the gauges. Deflection readings include the extra term, so a value of E calculated from deflection with the rigid-clamp formula comes out low. For a dial position \(x_d \ge a\), dividing by Wa gives a straight-line form that separates the two effects:
\[ \frac{v(x_d)}{W a} = \frac{1}{EI}\cdot\frac{a\,(3x_d - a)}{6} + \frac{x_d}{k_\theta} \]
A plot of \(v/(Wa)\) against \(a(3x_d - a)/6\) for several hanger positions has slope \(1/EI\) and intercept \(x_d/k_\theta\).
3.4 Axial load, superposition and tension stiffening
An axial tension P produces a uniform strain across the section:
\[ \varepsilon_a = \frac{P}{E A} \]
With small deflections, the strains from bending and axial load add (superposition):
\[ \varepsilon_{\text{top}} = \varepsilon_a + \varepsilon_b, \qquad \varepsilon_{\text{bottom}} = \varepsilon_a - \varepsilon_b \]
Superposition needs one correction on this bench. Once the beam has deflected, the tension at the free end acts along a line below the beam axis at every section nearer the clamp, and its moment opposes the bending. The bending moment becomes
\[ M(x) = W\,(a - x) - P\,\big(v(L) - v(x)\big), \qquad x < a \]
and the deflection obeys \( EI\,v'' - P\,v = W(a - x) - P\,v(L) \). The size of the effect is set by the dimensionless group
\[ kL = L\sqrt{\frac{P}{EI}} \]
For a load at the free end and a rigid clamp, the solution gives the moment at any section and the tip deflection:
\[ M(x) = \frac{W}{k}\,\frac{\sinh k(L - x)}{\cosh kL}, \qquad \frac{v(L)}{WL^3/3EI} = \frac{3\,(kL - \tanh kL)}{(kL)^3} \]
Dividing M(x) by its value without tension, W(L − x), gives the fraction of the bending moment that remains at section x:
\[ \frac{M(x)}{W(L - x)} = \frac{\sinh k(L - x)}{k(L - x)\cosh kL} \]
Both ratios equal 1 when P = 0 and fall as P rises. At kL = 1 the root moment is 24 % lower and the tip deflection 28 % lower than with no tension. Slender bars reach large values of kL at modest tensions.
3.5 Strain gauges
A foil strain gauge is a fine metal grid on a thin backing, bonded to the surface. When the surface strains, the grid strains with it and its resistance changes:
\[ \frac{\Delta R}{R} = GF\cdot\varepsilon \]
GF is the gauge factor, close to 2 for metal foil. A gauge responds to strain along its grid lines. A longitudinal gauge reads the strain along the beam, and a transverse gauge on the same face reads \(-\nu\) times that strain.
Temperature also changes a gauge's resistance, through the resistivity of the foil and the difference in thermal expansion between gauge and specimen. The reading this produces with no stress present is the apparent strain:
\[ \varepsilon_{\text{app}} = \beta\,\Delta T \]
The gauges on this bench are self-compensated for steel, so β is small on a steel bar and larger on materials that expand more than steel (Table A1). A single gauge cannot tell apparent strain from real strain. The bridge arrangement can.
3.6 The Wheatstone bridge
Four arms form a diamond. The excitation \(V_s\) is applied between the top and bottom corners, and the output \(V_o\) is measured between the side corners A and B. Arms are numbered clockwise from the top left. With arm resistances \(R_1\) to \(R_4\):
\[ V_o = V_s\left(\frac{R_3}{R_2 + R_3} - \frac{R_4}{R_1 + R_4}\right) \]
With four equal arms the bridge is balanced and \(V_o = 0\). For small changes \(\Delta R_i/R_i = GF\,\varepsilon_i\), the output is linear in the four strains:
\[ \frac{V_o}{V_s} \approx \frac{GF}{4}\left(\varepsilon_1 - \varepsilon_2 + \varepsilon_3 - \varepsilon_4\right) \]
Strains in adjacent arms subtract and strains in opposite arms add. Arms 1 and 3 are opposite each other, and so are arms 2 and 4. An arm filled by a completion resistor contributes no strain. To predict what a configuration measures, write each gauge's strain in terms of \(\varepsilon_a\), \(\varepsilon_b\), ν and \(\varepsilon_{\text{app}}\), then combine them with these signs.
| Preset | Arm 1 | Arm 2 | Arm 3 | Arm 4 |
|---|
| Quarter I | T1 | R | R | R |
| Quarter II | T1 | DG | R | R |
| Half I | T1 | TP | R | R |
| Half II | T1 | B1 | R | R |
| Full I | T1 | B1 | T2 | B2 |
| Full II | T1 | TP | BP | B1 |
| Full III | T1 | TP | B1 | BP |
For example, Half II has T1 in arm 1 and B1 in arm 2, so its output is proportional to \((\varepsilon_a + \varepsilon_b + \varepsilon_{\text{app}}) - (\varepsilon_a - \varepsilon_b + \varepsilon_{\text{app}}) = 2\varepsilon_b\), provided both gauges are at the same temperature. Experiments 1 and 2 ask you to predict and then measure the response of every preset.
3.7 The strain display
The display converts the bridge output into an indicated strain, assuming n equal active arms:
\[ \varepsilon_{\text{ind}} = \frac{4\,(V_o - V_{\text{zero}})}{V_s\cdot GF_{\text{set}}\cdot n} \]
\(V_{\text{zero}}\) is the output stored when you press Zero. Gauge and resistor tolerances leave every bridge slightly out of balance, so before zeroing the display reads tens to hundreds of microstrain with nothing on the beam. If a configuration's output is K times the strain at one gauge (K is its bridge factor), the indicated strain equals the strain at one gauge only when GFset is the packet value and n = K. The ARMS switch offers 1, 2 and 4, the standard quarter, half and full bridges. When K is not 1, 2 or 4, as for the presets that use the transverse gauges (K contains 1 + ν), set n = 1 and divide the reading by K in your analysis.
3.8 The dial test indicator
The large needle turns once per millimetre, over 100 divisions of 0.01 mm. The small dial counts whole millimetres. The scale is printed on the bezel, which turns by the nub at its top without moving either needle. To set zero, turn the bezel until its 0 lines up with the large needle. Tapping the nub lines it up exactly. The small dial is not reset, so always read both. You can estimate to about half a division.
A reading is the whole millimetres from the small dial plus the hundredths from the large needle. Note the unloaded reading, then subtract it from each loaded reading to obtain the deflection. If the plunger reaches the end of its travel the face shows END OF TRAVEL, and readings stop changing. Move the stand nearer the clamp or reduce the load.
3.9 Settling of readings
The loaded beam is a spring carrying a mass. When a mass is placed on the hanger, the beam vibrates at its natural frequency, between a few hertz and a few tens of hertz on this bench, and the vibration dies away over one to three seconds. The strain display samples three times a second without averaging, so a reading taken too early can be far from the settled value. Wait until the display and the dial needle stop changing.
3.10 Uncertainty and analysis
Resolution and scatter. The display resolves 1 µV and 1 µε, the dial 0.01 mm and the load cell 1 N. Electrical noise adds a scatter of about 1.5 µV RMS to the bridge output. Repeating a reading and averaging reduces the random part of the uncertainty but not the systematic part.
Fitting a straight line. Most experiments here give a straight line \(y = mx + c\). For n points the least squares slope and intercept are
\[ m = \frac{\sum (x_i - \bar x)(y_i - \bar y)}{\sum (x_i - \bar x)^2}, \qquad c = \bar y - m\,\bar x \]
and the standard error of the slope is
\[ s_m = \sqrt{\frac{\sum (y_i - m x_i - c)^2}{(n - 2)\sum (x_i - \bar x)^2}} \]
The Plot tab draws this line and reports m, c, their standard errors and R². Show the fit in your report, but carry out the calculation of E and its uncertainty yourself.
Propagating uncertainty. For a result formed from products and quotients, relative uncertainties combine in quadrature. For a rectangular bar, \(I/c = bh^2/6\), so E from the bending-strain slope \(m = (a - x_g)\,c/(EI)\) carries
\[ \frac{u(E)}{E} = \sqrt{\left(\frac{u(m)}{m}\right)^2 + \left(\frac{u(b)}{b}\right)^2 + \left(\frac{2\,u(h)}{h}\right)^2 + \left(\frac{u(a - x_g)}{a - x_g}\right)^2 + \left(\frac{u(GF)}{GF}\right)^2} \]
Deflection depends on \(h^3\), so thickness uncertainty weighs more heavily there. For a circular bar, replace the b and h terms with \(3\,u(d)/d\) for strain and \(4\,u(d)/d\) for deflection.
Comparing two estimates. Two values \(E_1 \pm u_1\) and \(E_2 \pm u_2\) are consistent if their difference is within about twice the combined uncertainty \(\sqrt{u_1^2 + u_2^2}\). A larger difference points to a systematic effect, which your report should identify.