A light rectangular beam is clamped horizontally at one end and a heavy mass is attached at the other. Find the depression at the loaded end.

To determine the depression at the loaded end of a light rectangular beam clamped at one end with a heavy mass attached at the other, we can use principles from mechanics and material science. The problem involves understanding how beams behave under load, particularly focusing on bending and deflection.

Understanding Beam Deflection

When a beam is subjected to a load, it bends, and this bending is quantified as deflection. The amount of deflection depends on several factors, including the material properties of the beam, its dimensions, the type of load applied, and the boundary conditions (in this case, one end is clamped).

Key Variables

• Length of the beam (L): The distance from the clamped end to the loaded end.
• Load (P): The weight of the mass attached at the free end.
• Modulus of Elasticity (E): A measure of the material's stiffness.
• Moment of Inertia (I): A geometric property that reflects how the beam's cross-section resists bending.

Deflection Formula

For a cantilever beam (which is what we have here, since one end is fixed), the deflection (\( \delta \)) at the free end due to a point load can be calculated using the following formula:

\( \delta = \frac{P L^3}{3 E I} \)

Breaking Down the Formula

Let’s break down the components of the formula:

• P: This is the force applied at the end of the beam, which is equal to the weight of the mass (P = mg, where m is the mass and g is the acceleration due to gravity).
• L: This is the length of the beam from the clamped end to the point where the load is applied.
• E: The modulus of elasticity is a property of the material. For example, steel has a high modulus of elasticity, meaning it will deform less under the same load compared to a material like wood.
• I: The moment of inertia depends on the beam's cross-sectional shape. For a rectangular beam, it can be calculated using the formula \( I = \frac{b h^3}{12} \), where \( b \) is the width and \( h \) is the height of the beam's cross-section.

Example Calculation

Let’s say we have a beam that is 2 meters long (L = 2 m), a mass of 10 kg is attached (P = 10 kg × 9.81 m/s² = 98.1 N), the beam is made of a material with a modulus of elasticity of 200 GPa (E = 200 × 10⁹ Pa), and the beam has a rectangular cross-section of 0.1 m by 0.2 m.

First, calculate the moment of inertia:

I = \frac{0.1 \times (0.2)^3}{12} = \frac{0.1 \times 0.008}{12} = 6.67 \times 10^{-4} m^4

Now, substitute the values into the deflection formula:

\( \delta = \frac{98.1 \times (2)^3}{3 \times (200 \times 10^9) \times (6.67 \times 10^{-4})} \)

Calculating this gives:

\( \delta \approx 0.00049 m \) or \( 0.49 mm \)

Final Thoughts

This example illustrates how to calculate the deflection of a cantilever beam under a point load. Understanding these principles is crucial in fields like civil engineering and architecture, where ensuring structural integrity is paramount. If you have any further questions or need clarification on any step, feel free to ask!