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Title: Determination of Young's Modulus of a Material using a Simply Supported Beam

Aim: To determine the Young's Modulus ($E$) of the material of a given beam by measuring its central deflection under varying loads when simply supported at its ends.

Apparatus:

• Test beam (e.g., steel, aluminum, wood)
• Two knife-edge supports
• Loading arrangement (e.g., hanger with slotted weights)
• Dial gauge or micrometer screw gauge for measuring deflection
• Measuring tape or ruler
• Vernier caliper or micrometer screw gauge for measuring beam dimensions (width and depth)

Theory:
When a beam is simply supported at its ends and subjected to a central point load, it undergoes deflection. For a simply supported beam of length $L$ with a central point load $W$, the maximum deflection ($\delta$) at the center is given by the formula:

$$\delta = \frac{WL^3}{48EI}$$

Where:

• $W$ is the applied central load (in Newtons, N)
• $L$ is the effective length of the beam between the supports (in meters, m)
• $E$ is the Young's Modulus of the beam material (in Pascals, Pa or $N/m^2$)
• $I$ is the moment of inertia of the beam's cross-section (in $m^4$)

For a rectangular cross-section beam with width $b$ and depth $d$:

$$I = \frac{bd^3}{12}$$

Substituting $I$ into the deflection formula, we get:

$$\delta = \frac{WL^3}{48E \left(\frac{bd^3}{12}\right)} = \frac{WL^3}{4Ebd^3}$$

From this, Young's Modulus $E$ can be calculated as:

$$E = \frac{WL^3}{4\delta bd^3}$$

By plotting a graph of load ($W$) versus deflection ($\delta$), a straight line should be obtained. The slope of this $W-\delta$ graph is $\frac{W}{\delta}$. Therefore, $E$ can also be determined from the slope ($m$) as:

$$E = \frac{L^3}{4bd^3} \times m$$

where $m = \frac{\Delta W}{\Delta \delta}$.

Procedure:

1. Measure the width ($b$) and depth ($d$) of the beam's cross-section using a Vernier caliper or micrometer screw gauge at several points and record the average values.
2. Place the beam on the two knife-edge supports, ensuring they are at the same level.
3. Measure the effective length ($L$) between the knife-edge supports.
4. Position the loading hanger exactly at the center of the beam.
5. Place the dial gauge (or micrometer screw gauge) directly under the center of the beam, ensuring its stem touches the beam. Adjust the dial gauge to read zero or a convenient initial reading.
6. Apply a small initial load (e.g., $0.5 \, kg$) to ensure the beam is in contact with the supports and the dial gauge. Record this initial load and the corresponding deflection reading.
7. Incrementally add weights to the hanger (e.g., $0.5 \, kg$ or $1 \, kg$ at a time). For each added load, record the total load ($W$) and the corresponding deflection ($\delta$) from the dial gauge.
8. Continue adding weights until the maximum safe load for the beam is reached or a significant deflection is observed.
9. After reaching the maximum load, incrementally remove the weights and record the deflection readings for decreasing loads. This helps check for hysteresis and permanent deformation.
10. Plot a graph of total load ($W$) versus central deflection ($\delta$).

Calculation:

1. Calculate the average width ($b$) and depth ($d$) of the beam.
2. Calculate the moment of inertia ($I = \frac{bd^3}{12}$) for the beam's cross-section.
3. From the $W-\delta$ graph, determine the slope ($m = \frac{\Delta W}{\Delta \delta}$) of the linear portion.
4. Calculate Young's Modulus ($E$) using the formula: $$E = \frac{L^3}{4bd^3} \times m$$ Alternatively, for each load-deflection pair, calculate $E = \frac{WL^3}{4\delta bd^3}$ and then find the average $E$.

Table:

S.No.	Load (W) (N)	Dial Gauge Reading (mm)	Deflection ($\delta$) (mm)	Deflection ($\delta$) (m)
1	Initial Load
2	$W_1$		$\delta_1$	$\delta_1/1000$
3	$W_2$		$\delta_2$	$\delta_2/1000$
...	...	...	...	...
n	$W_n$		$\delta_n$	$\delta_n/1000$

Beam Dimensions:

• Length ($L$) = $\boxed{... \, m}$
• Width ($b$) = $\boxed{... \, m}$
• Depth ($d$) = $\boxed{... \, m}$

Precautions:

• Ensure the knife-edge supports are perfectly level and parallel to each other to prevent twisting of the beam.
• The load should be applied gradually and centrally to avoid eccentric loading.
• The dial gauge should be placed directly under the point of load application to measure maximum deflection accurately.
• Avoid exceeding the elastic limit of the beam material to prevent permanent deformation.
• Take readings carefully, ensuring the dial gauge stem is perpendicular to the beam surface.
• Measure the beam dimensions ($b$, $d$, $L$) accurately using appropriate measuring instruments.
• Ensure there is no slack in the loading arrangement before taking initial readings.