Hollow Rectangular Beams Deflection Calculator

The Formulas

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

1. For Point Load at the Center:

\[\delta = \frac{P L^3}{48 E I}\]

2. For Uniform Distributed Load:

\[\delta = \frac{5 w L^4}{384 E I}\]

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)

Second Moment of Area for Hollow Rectangular Section

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

Calculation Steps

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

Example Calculation

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

1. Calculate I:
   I = (0.2 × 0.3³ - 0.18 × 0.28³) / 12 = 1.5573 × 10⁻⁴ m⁴
2. 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.