What is Beam Slope and Deflection?

In structural engineering, beam deflection refers to the vertical displacement of a structural element under a specific load. Understanding the slope (the angle of rotation) and the deflection is critical for ensuring the safety and serviceability of structures like bridges, buildings, and mechanical shafts. This calculator uses standard Euler-Bernoulli beam theory formulas to provide instantaneous results.

How to Use the Slope and Deflection Calculator

To calculate the deflection and slope of a beam, follow these simple steps:

1. Select Beam Configuration: Choose between a Cantilever beam (fixed at one end) or a Simply Supported beam (supported at both ends).
2. Enter the Load (P): Input the concentrated force applied to the beam in Newtons.
3. Enter Beam Length (L): Input the total span of the beam in millimeters.
4. Material Properties: Enter the Young's Modulus (E) in GPa and the Moment of Inertia (I) in mm⁴. These values depend on the material (like steel or wood) and the cross-sectional shape of the beam.
5. Click Calculate: The tool will generate the step-by-step breakdown of the formula and the final numerical result.

Key Formulas

For a cantilever beam with a point load at the free end, the maximum deflection is calculated using δ = PL³ / 3EI. For a simply supported beam with a center point load, the maximum deflection is δ = PL³ / 48EI. Our tool handles the conversion of units (like GPa to N/mm²) automatically to ensure accuracy.

Frequently Asked Questions

Q: Why is my deflection value so high?
A: Ensure your units are consistent. For example, if E is in GPa, it must be multiplied by 10³ to convert to N/mm² if your length and inertia are in mm.

Q: What is Moment of Inertia (I)?
A: It is a property of a cross-section that predicts its resistance to bending and deflection. For a rectangular beam, it is calculated as bh³/12.