Free Beam Calculator

Calculate the maximum deflection and bending stress for a simply supported beam under a uniformly distributed load.

Understanding the Free Beam Calculator

A "free beam" in structural engineering typically refers to a beam that is supported at both ends, allowing it to bend freely under load without significant restraint at the supports. This type of beam is commonly known as a simply supported beam. This calculator helps determine key structural parameters for such beams under a uniformly distributed load (UDL).

What it Calculates:

• Maximum Deflection: The maximum vertical displacement of the beam from its original position. Excessive deflection can lead to aesthetic issues, damage to non-structural elements, and affect the serviceability of the structure.
• Maximum Bending Stress: The peak stress experienced within the beam's cross-section due to bending. This is critical for ensuring the beam does not yield or fracture under load.

The Physics & Formulas:

For a simply supported beam subjected to a uniformly distributed load (w) over its entire length (L), the following formulas are standard in beam theory:

• Maximum Deflection (δ_max): occurs at the mid-span of the beam.
  Formula: δ_max = (5 * w * L⁴) / (384 * E * I) Where:
  • w = Uniformly Distributed Load (Force per unit length, e.g., N/m)
  • L = Length of the beam (m)
  • E = Young's Modulus of the beam material (Pascals, Pa)
  • I = Moment of Inertia of the beam's cross-section (m⁴)
  The result is typically in meters (m).
• Maximum Bending Stress (σ_max): also occurs at the mid-span and at the top and bottom fibers of the beam's cross-section.
  Formula: σ_max = (M_max * y) / I First, the maximum bending moment (M_max) for this load case is: M_max = (w * L²) / 8 Then, assuming 'y' represents the distance from the neutral axis to the outermost fiber of the cross-section (this calculator simplifies by assuming a standard calculation using the total depth or similar property, but in practice, 'y' is derived from the cross-section geometry). For a general indication, we often relate stress to the section modulus (S = I/y) where σ_max = M_max / S. This calculator directly uses the maximum moment and provided Moment of Inertia (I) to indicate the bending intensity.
  For a simpler direct output related to the input moment of inertia: σ_max ≈ (M_max) / (I / y_effective) Note: For simplicity in this calculator, we'll output the maximum bending moment and relate it conceptually to stress, as the specific 'y' value (distance to extreme fiber) is not an input. A professional calculation would require cross-section dimensions. The result is typically in Pascals (Pa) or Megapascals (MPa).

How to Use This Calculator:

1. Uniformly Distributed Load (w): Enter the total load spread evenly across the beam's length, in Newtons per meter (N/m).
2. Beam Length (L): Input the total span of the beam in meters (m).
3. Young's Modulus (E): Provide the material's stiffness. Common values include ~200 GPa (200 x 10⁹ Pa) for steel and ~10 GPa (10 x 10⁹ Pa) for wood.
4. Moment of Inertia (I): Enter the beam's cross-sectional resistance to bending, in meters to the fourth power (m⁴). This value depends on the shape and dimensions of the beam's cross-section (e.g., I-beam, rectangular).
5. Click "Calculate" to see the maximum deflection and bending moment.

Use Cases:

• Civil Engineering: Designing floor joists, beams in bridges, and other structural elements.
• Mechanical Engineering: Analyzing components like shafts, levers, and frames that experience bending loads.
• Architecture: Ensuring aesthetic requirements are met by controlling excessive sagging in visible beams.
• DIY Projects: Estimating the behavior of custom-built wooden or metal beams.

Disclaimer: This calculator provides an estimate based on simplified physics. For critical applications, always consult a qualified structural engineer and refer to detailed design codes and standards. Material properties and boundary conditions can significantly affect actual performance.

Please enter valid positive numbers for all fields.

Results:

' + deflectionDisplay + '

' + momentDisplay + '

(Note: Bending Stress requires cross-section geometry, typically represented by 'y', the distance from the neutral axis to the extreme fiber. Consult engineering resources for stress calculation.)