Beam Bending & Deflection Calculator

About the Beam Bending & Deflection Calculator

The Beam Bending & Deflection Calculator helps engineers, designers, and students determine the bending stress (σ), maximum deflection (δ_max), and slope (θ_max) of beams under various loading and support conditions. By inputting beam dimensions, material properties, cross-sectional moment of inertia, applied load, and support type, the calculator provides accurate results for structural analysis and design.

What You Can Calculate

• Bending Stress (σ): Maximum stress experienced by the beam’s fibers due to bending.
• Maximum Deflection (δ_max): Vertical displacement of the beam under load.
• Maximum Slope (θ_max): Angular change of the beam at points of maximum curvature.

Formulas Used in the Calculator

Results are computed using standard beam theory formulas for common support conditions:

• Simply Supported Beam (center point load):
  • Maximum Bending Stress: σ = (M * c) / I
  • Maximum Deflection: δ_max = (P * L³) / (48 * E * I)
  • Maximum Slope: θ_max = (P * L²) / (16 * E * I)
• Cantilever Beam (end point load):
  • Maximum Bending Stress: σ = (M * c) / I
  • Maximum Deflection: δ_max = (P * L³) / (3 * E * I)
  • Maximum Slope: θ_max = (P * L²) / (2 * E * I)

Where:
σ = Bending stress (Pa),
δ_max = Maximum deflection (m),
θ_max = Maximum slope (rad),
P = Applied load (N),
L = Beam length (m),
E = Modulus of elasticity (Pa),
I = Moment of inertia of the cross-section (m⁴),
c = Distance from neutral axis to extreme fiber (m).

How to Use the Calculator

1. Enter the beam length, modulus of elasticity (E), and moment of inertia (I) of the beam.
2. Input the applied load (P) acting on the beam.
3. Select the appropriate support type (simply supported or cantilever).
4. Choose the correct units for each input.
5. The calculator will instantly display the maximum bending stress, deflection, and slope.
6. Adjust inputs as needed to analyze different beam conditions or materials.

Applications of Beam Bending Analysis

• Structural Design: Determine safe dimensions for beams, girders, and joists in buildings and bridges.
• Mechanical Components: Analyze shafts, levers, and frames under bending loads.
• Deflection Checks: Ensure serviceability by verifying that beam deflection remains within acceptable limits.
• Educational Use: Learn fundamental beam theory, bending stress, and deflection concepts for engineering studies.
• Simulation Validation: Compare analytical calculations with finite element analysis (FEA) results.