Lab Report: Verification of Maxwell’s Law of Reciprocal Deflection

The primary objectives of this experiment are:

• To experimentally verify Maxwell’s law of reciprocal deflections.
• To compare the deflections observed in the experiment with their theoretical values.
• To calculate and compare the mutual deflections at various points on the beam.

Maxwell’s law of reciprocal deflections states that for a linearly elastic structure, the deflection at a point A in a specific direction when a force is applied at a point B is equal to the deflection at point B in the direction of the same force when it is applied at point A.

Mathematically, this can be expressed as:

Where:

• \(\delta_{AB}\) is the deflection at point A due to a load at point B.
• \(\delta_{BA}\) is the deflection at point B due to the same load at point A.

The experiment utilizes a beam setup with specific loading points and measurement locations. Key components include:

• A simply supported beam.
• Weights to apply loads at designated points (B, C, and D).
• Dial gauges to measure deflections at these points.

The experiment was conducted as follows:

1. A stabilizing load of 5N was initially applied at point B, and the initial readings on the dial gauges at points C and D were recorded.
2. Additional loads of 5N, 10N, 15N, and 20N were sequentially applied at point B, and the corresponding dial gauge readings at C and D were noted for each load increment.
3. The process was repeated by moving the load to point C and then to point D, with the dial gauges repositioned to measure the corresponding reciprocal deflections.
4. All observations were systematically recorded in tables.

The following data was recorded during the experiment:

Table 1: Load Applied at Point B

Table 2: Load Applied at Point C

Table 3: Load Applied at Point D

Theoretical Deflection

Castigliano’s theorem was used to determine the theoretical deflection. The deflection is given by the formula:

For the beam with the specified properties (b=25mm, h=5mm, E=2×10¹¹N/m², I=260.417×10^-12m⁴), the theoretical deflection was calculated as follows:

• Theoretical \(\delta_{bc}\): \(\frac{55}{192EI}\) = 5.5 mm
• Theoretical \(\delta_{cb}\): \(\frac{55}{192EI}\) = 5.5 mm

The theoretical calculations confirm Maxwell’s theorem, showing that \(\delta_{bc} = \delta_{cb}\).

Experimental Deflection

From the experimental data for a 20N load:

• The deflection at C due to the load at B (\(\delta_{cb}\)) was 6.04 mm.
• The deflection at B due to the load at C (\(\delta_{bc}\)) was 6.05 mm.

The experimental results show a close agreement, with \(\delta_{cb} \approx \delta_{bc}\).

Percentage Error

The percentage error between the average experimental value (6.045 mm, approximated to 6.04 mm in the document) and the theoretical value (5.5 mm) is calculated as:

The experiment successfully verified Maxwell’s Law of Reciprocal Deflection. The theoretical deflection was calculated to be 5.5 mm, while the average measured practical deflection was 6.04 mm under a 20N load.

The observed difference of 9.81% between the theoretical and experimental values can be attributed to potential sources of error, including:

• Imperfections in the material of the beam.
• Human error during the measurement of deflections.
• The assumptions of ideal material behavior and perfect support conditions in the theoretical calculations.