Beam deflection

Modulus of elasticity

Modulus of elasticity is a measure of a material's ability to deform when subjected to an external force. It is a measure of the stiffness of a material. Given in GPa:

1. Mild steel: $E = 200 \, \text{GPa}$
2. Aluminum: $E = 70 \times 10^9 \, \text{Pa}$

Moment of inertia

1. Round tube

   The moment of inertia of a round tube is given by the following equation:

   $I = \frac{\pi}{64} \cdot (D_o^4 - D_i^4)$

   where $D_o$ is the outer diameter and $D_i$ is the inner diameter.

Deflection

• Variables
  • $F$ is the force applied to the beam
  • $L$ is the length of the beam
  • $E$ is the modulus of elasticity
  • $I$ is the moment of inertia

1. Cantilever beam, end load

   The deflection of a cantilever beam is given by the following equation:

   $\delta_{max} = \frac{F \cdot L^3}{3 \cdot E \cdot I}$

2. Cantilever beam, uniformly distributed load

   The deflection of a cantilever beam with a uniformly distributed load is given by the following equation:

   $\delta_{max} = \frac{5 \cdot F \cdot L^4}{384 \cdot E \cdot I}$

3. Cantilever beam, load at $x$

   The deflection of a cantilever beam with a load at $x$ is given by the following equation:

   $\delta_{max} = \frac{F \cdot x \cdot (L - x)^2}{3 \cdot E \cdot I \cdot L}$

Appendix

• Modulus of elasticity for wood
• Structural properties of wood
• Modulus of elasticity for steel https://www.engineersedge.com/manufacturing_spec/properties_of_metals_strength.htm https://en.wikipedia.org/wiki/Young%27s_modulus