The formulas for calculating the deflection of a hollow rectangular beam depend on the load type:

Where:

- δ: Maximum deflection (m)
- P: Point load (N)
- w: Uniform distributed load (N/m)
- L: Beam length (m)
- E: Elastic modulus (Pa)
- I: Second moment of area (m⁴)

The second moment of area (I) for a hollow rectangular section is calculated as:

\[I = \frac{b h^3 - b_1 h_1^3}{12}\]Where:

- b: Outer width of the beam
- h: Outer height of the beam
- b₁: Inner width of the hollow section
- h₁: Inner height of the hollow section

- Calculate the second moment of area (I) for the hollow rectangular section
- Determine the appropriate formula based on the load type (point load or uniform distributed load)
- Input all known values into the formula
- Calculate the maximum deflection

Let's calculate the deflection for a hollow rectangular beam with the following properties:

- Beam Length (L) = 5 m
- Outer Width (b) = 200 mm = 0.2 m
- Outer Height (h) = 300 mm = 0.3 m
- Inner Width (b₁) = 180 mm = 0.18 m
- Inner Height (h₁) = 280 mm = 0.28 m
- Point Load (P) = 10 kN = 10,000 N
- Elastic Modulus (E) = 200 GPa = 200 × 10⁹ Pa

- Calculate I:

I = (0.2 × 0.3³ - 0.18 × 0.28³) / 12 = 1.5573 × 10⁻⁴ m⁴ - Use the formula for point load:

δ = (P × L³) / (48 × E × I)

δ = (10,000 × 5³) / (48 × 200 × 10⁹ × 1.5573 × 10⁻⁴)

δ = 0.004178 m = 4.178 mm

Therefore, the maximum deflection of the hollow rectangular beam under the given conditions is approximately 4.178 mm.

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