Experiment: Bending of a Beam

Study of Bending of a Beam and Determination of Young's Modulus

Method: Non-Uniform Bending (Central Loading Method)

1. Aim

To study the non-uniform bending of a rectangular beam supported on two knife edges and to determine Young's modulus of the material by applying a load at the centre of the beam.

2. Apparatus Required

Rectangular beam (metallic or wooden), two knife-edge supports, weight hanger, slotted weights, travelling microscope, metre scale, Vernier calipers and screw gauge.

3. Theory

When a rectangular beam is supported horizontally on two knife edges and a load is applied at its midpoint, the beam bends downward. The bending moment varies along the length of the beam; therefore, this type of bending is called non-uniform bending.

For a simply supported beam carrying a central load, the depression at the midpoint is given by

\[ y=\frac{Wl^3}{48YI} \]

where:

• \(y\) = vertical depression at the centre (m)
• \(W=mg\) = applied load (N)
• \(m\) = total applied mass, including the hanger if applicable (kg)
• \(g\) = acceleration due to gravity (\(\mathrm{m\,s^{-2}}\))
• \(l\) = distance between the knife edges (m)
• \(Y\) = Young's modulus of the beam material (\(\mathrm{N\,m^{-2}}\))
• \(I\) = second moment of area of the cross-section (\(\mathrm{m^4}\))

For a rectangular cross-section,

\[ I=\frac{bd^3}{12} \]

where \(b\) is the breadth and \(d\) is the thickness of the beam.

Substituting this expression into the bending equation,

\[ y=\frac{mgl^3}{48Y\left(\frac{bd^3}{12}\right)} \] \[ y=\frac{mgl^3}{4Ybd^3} \]

\[ \boxed{Y=\frac{mgl^3}{4bd^3y}} \]

Physical significance: Within the elastic limit, the central depression is directly proportional to the applied load. The thickness appears as \(d^3\) in the denominator of the expression for Young's modulus, making accurate thickness measurement particularly important.

4. Experimental Arrangement

Arrangement of the experiment:

1. Place the rectangular beam horizontally on two knife-edge supports.
2. Keep the supports at equal distances on either side of the centre.
3. Suspend the weight hanger at the midpoint of the beam.
4. Position the travelling microscope to measure the vertical displacement of a reference pin at the centre.

The distance between the knife edges is \(l\), the breadth of the beam is \(b\), and its vertical thickness is \(d\).

5. Procedure

1. Place the rectangular beam horizontally on two knife-edge supports.
2. Measure the distance \(l\) between the knife edges using a metre scale.
3. Measure the breadth \(b\) using Vernier calipers.
4. Measure the thickness \(d\) using a screw gauge at several positions and calculate its mean value.
5. Suspend the weight hanger exactly at the midpoint of the beam.
6. Focus the travelling microscope on the reference pin fixed at the centre of the beam.
7. Record the initial microscope reading for the chosen reference load.
8. Add masses gradually and record the corresponding microscope readings.
9. Remove the masses gradually and record the unloading readings.
10. Calculate the mean reading for each load and determine the central depression relative to the reference reading.
11. Convert the mass, dimensions and depression into consistent SI units.
12. Calculate Young's modulus using the theoretical expression. If required, determine the slope of the load–depression graph.

Keep the reference condition consistent. If the hanger is already attached during the initial reading, either include its mass in the total load or calculate the incremental depression relative to that initial condition.

6. Observations

A. Dimensions of the Beam

Quantity	Symbol	Illustrative Reading
Distance between supports	\(l\)	0.60 m
Breadth of beam	\(b\)	0.025 m
Thickness of beam	\(d\)	0.005 m
Acceleration due to gravity	\(g\)	9.81 m s−2

B. Observation Table

The following readings reproduce the supplied illustrative dataset. Editable cells are provided so that actual laboratory readings can be entered directly in the table.

Mass (kg)	Set I			Set II			Mean Reading (cm)	Depression (cm)
	MSR (cm)	VSR (div.)	Total (cm)	MSR (cm)	VSR (div.)	Total (cm)
0	1	0.021	1.021	1	0.029	1.029	1.025
0.5	1.1	0.025	1.125	1.1	0.035	1.135	1.130	0.105
1	1.21	0.061	1.271	1.21	0.052	1.262	1.2665	0.1365
1.5	1.32	0.038	1.358	1.32	0.078	1.398	1.378	0.1115
2	1.42	0.042	1.462	1.42	0.017	1.437	1.4495	0.0715
2.5	1.53	0.036	1.566	1.53	0.095	1.625	1.5955	0.146

The central depression is calculated from the mean microscope reading:

\[ \overline{y}=0.114cm \]

Here, \(\overline{R}_i\) is the mean reading for a particular load, and \(\overline{R}_0\) is the mean reference reading. If the reading is in centimetres, convert the depression into metres before using the formula for Young's modulus.

Data verification: The supplied observation table and the separate calculation below do not form one internally consistent dataset. In particular, the listed total readings should be checked against the instrument's least count and vernier convention. The depression values above are calculated from the displayed mean readings and are not presented as verified experimental results.

7. Calculations

Formula:

\[ Y=\frac{mgl^3}{4bd^3y} \]

The following calculation uses the illustrative values supplied separately for a mass of 200 g.

Given:

• Mass, \(m=500\,\mathrm{g}=0.500\,\mathrm{kg}\)
• Depression, \(y=0.1141\,\mathrm{cm}=1.14\times10^{-3}\,\mathrm{m}\)
• Distance between supports, \(l=0.60\,\mathrm{m}\)
• Breadth, \(b=0.025\,\mathrm{m}\)
• Thickness, \(d=0.005\,\mathrm{m}\)
• Acceleration due to gravity, \(g=9.81\,\mathrm{m\,s^{-2}}\)

Substituting the values,

\[ Y= \frac{0.500\times9.81\times(0.60)^3} {4\times0.025\times(0.005)^3 \times(1.14\times10^{-3})} \]

\[ \boxed{Y=7.435\times10^{10}\ \mathrm{N\,m^{-2}}} \]

Important: This numerical result follows from the separately supplied values of mass and depression. It should not be treated as the result calculated from the observation table unless the measurements are reconciled. The mass must represent the appropriate total load, or the depression must be defined relative to a fixed preload.

8. Graph

Plot the applied mass \(m\) along the X-axis and the central depression \(y\) along the Y-axis. The following graph is generated from the illustrative depression values in the observation table.

Graph of applied mass against central depression.

Interpretation:

Within the elastic limit, the central depression should be directly proportional to the applied load. Therefore, the load–depression relationship should be linear, ideally passing through the origin when both quantities are measured relative to the same zero-load reference.

The plotted points are illustrative, not verified measurements. Use the actual observation data for the final practical graph.

9. Result

From the separately supplied illustrative values, Young's modulus of the beam material is calculated as:

\[ \boxed{Y=7.43\times10^{10}\ \mathrm{N\,m^{-2}}} \]

The final experimental value should be recalculated from the verified dimensions, load and central depression obtained in the laboratory.

10. Precautions

1. The beam must be horizontal and supported symmetrically on the knife edges.
2. The load must be applied exactly at the midpoint of the beam.
3. The applied load must not exceed the elastic limit of the material.
4. The travelling microscope must be focused carefully to avoid parallax error.
5. The thickness should be measured at several positions, and the mean value should be used.
6. Readings should be recorded only after the beam becomes stationary.
7. Loads should be added and removed gradually to avoid oscillations.
8. The same reference condition should be maintained throughout the experiment.
9. All quantities must be converted into SI units before calculation.

11. Viva-Voce Questions

Q1. Why is this bending called non-uniform bending?

Because the bending moment varies along the length of the beam.

Q2. Where is the maximum bending moment produced?

At the midpoint of the simply supported beam when a concentrated load is applied at its centre.

Q3. What is the maximum bending moment for a central load?

\[ M_{\max}=\frac{Wl}{4} \]

Here, \(W\) is the applied load and \(l\) is the distance between the supports.

Q4. Where is the maximum depression of the beam?

At the midpoint of the beam, directly beneath the applied central load.

Q5. What is the SI unit of Young's modulus?

The SI unit is pascal (Pa), equivalent to \(\mathrm{N\,m^{-2}}\).

Q6. Why is the thickness of the beam important?

The second moment of area of a rectangular beam is proportional to the cube of its thickness. Consequently, the calculated Young's modulus is highly sensitive to the thickness measurement:

\[ Y\propto\frac{1}{d^3} \]

Q7. What is the relation between load and depression?

Within the elastic limit, the depression is directly proportional to the applied load:

\[ y\propto W \]

Q8. What is Young's modulus?

Young's modulus is the ratio of longitudinal stress to longitudinal strain within the elastic limit.

\[ Y=\frac{\text{Longitudinal stress}} {\text{Longitudinal strain}} \]

Q9. Why is a travelling microscope used?

It measures small vertical displacements of the reference pin with greater precision than an ordinary scale.

Q10. What assumptions are used in the bending equation?

The beam is initially straight, has a uniform rectangular cross-section, obeys linear elastic behaviour, is simply supported, and undergoes small deflections. The load is assumed to act at the midpoint.

12. Theoretical Note

The central-loading formula assumes a simply supported beam with a concentrated load at its midpoint. The second moment of area must be calculated using the dimension perpendicular to the direction of bending as the thickness.

\[ I=\frac{bd^3}{12}, \qquad Y=\frac{mgl^3}{4bd^3y} \]

The hanger mass and the reference preload must be handled consistently. If an initial preload is used, the change in load and the corresponding change in depression should be considered together.

Physics Laboratory Practical | Non-Uniform Bending of a Beam