Free Beam Calculator

Understanding the Free Beam Calculator

A free beam, also known as a cantilever beam, is a structural element that is fixed at one end and free at the other. This type of beam is commonly found in applications such as balconies, diving boards, and aircraft wings. Calculating the deflection (how much it bends) and stress (internal forces) in a free beam under load is crucial for ensuring structural integrity and safety.

This Free Beam Calculator helps engineers, architects, and students quickly estimate these critical parameters for a simply supported beam subjected to a concentrated load at its free end.

The Physics Behind the Calculations

The calculations are based on fundamental principles of structural mechanics and beam theory.

• Modulus of Elasticity (E): This material property represents a material's stiffness. A higher modulus means the material is stiffer and will deflect less. It's typically measured in Pascals (Pa) or pounds per square inch (psi).
• Area Moment of Inertia (I): This geometric property describes how the cross-sectional area of the beam is distributed with respect to its neutral axis. A larger moment of inertia indicates a stronger resistance to bending. It's measured in meters to the fourth power (m⁴) or inches to the fourth power (in⁴).
• Beam Length (L): The total length of the free beam from the fixed support to the free end.
• Magnitude of Load (P): The concentrated force applied at the free end of the beam.

Formulas Used:

For a free beam with a concentrated load P at the free end:

• Maximum Deflection ($\delta_{max}$) occurs at the free end and is calculated as:
  $\delta_{max} = \frac{PL^3}{3EI}$
• Maximum Bending Moment ($M_{max}$) occurs at the fixed support and is calculated as:
  $M_{max} = PL$
• Maximum Bending Stress ($\sigma_{max}$) occurs at the outermost fibers at the fixed support and is calculated as:
  $\sigma_{max} = \frac{M_{max} \cdot c}{I} = \frac{PL \cdot c}{I}$ where 'c' is the distance from the neutral axis to the outermost fiber of the beam's cross-section. For simplicity in this calculator, we assume a standard shape or require 'c' to be implicitly handled within the Area Moment of Inertia calculation or by user input if they have specific cross-section data. For common shapes, 'c' is often related to the depth of the beam (e.g., for a rectangular beam of height 'h', c = h/2). If you have specific geometric data, you can often find 'I' and 'c' for common shapes online. This calculator simplifies this by directly using 'I' and assuming 'c' is implicitly accounted for in the context of typical stress calculations or by providing a standard value if 'c' is not directly user-definable.

How to Use the Calculator:

1. Magnitude of Load (P): Enter the total force applied at the free end of the beam.
2. Beam Length (L): Enter the total length of the beam from the fixed support to the free end.
3. Length Units: Select the units for your Load and Length inputs.
4. Modulus of Elasticity (E): Enter the material's modulus of elasticity. Use scientific notation (e.g., 200e9 for 200 GPa, or 29e6 for 29 million psi).
5. Area Moment of Inertia (I): Enter the beam's area moment of inertia. Use scientific notation (e.g., 1.5e-5 for 0.000015 m⁴).
6. Results Units: Choose whether you want the output (deflection and stress) in SI or Imperial units.
7. Click 'Calculate' to see the maximum deflection and maximum bending stress.

Example Calculation:

Consider a steel cantilever beam with:

• Load (P): 500 N
• Length (L): 2 meters
• Units: SI (meters)
• Modulus of Elasticity (E): 200 GPa (which is 200 x 10⁹ Pa, or 200e9)
• Area Moment of Inertia (I): 8 x 10^-6 m⁴ (or 8e-6)
• Results Units: SI Units

Using the calculator:

• Maximum Deflection = (500 N * (2 m)³) / (3 * 200e9 Pa * 8e-6 m⁴) = 0.002083 meters (or 2.083 mm)
• Maximum Bending Moment = 500 N * 2 m = 1000 Nm
• Assuming a relevant 'c' value for stress calculation (e.g., if the beam is 10cm high and square, c might be ~0.05m): Maximum Stress $\approx$ (1000 Nm * 0.05 m) / 8e-6 m⁴ = 6,250,000 Pa (or 6.25 MPa)

The calculator will provide the deflection and stress values directly based on your inputs.