1. What is the Beam Deflection Equation?

The beam deflection equation \( EI y'' = M(x) \) is a fundamental differential equation in structural engineering that relates the bending moment \( M(x) \) to the second derivative of the beam's deflection \( y'' \), where \( E \) is the elastic modulus and \( I \) is the moment of inertia.

2. How Does the Calculator Work?

The calculator uses the beam deflection equation:

Where:

• \( E \) — Elastic modulus (Pa)
• \( I \) — Moment of inertia (m⁴)
• \( M(x) \) — Bending moment (Nm)
• \( y'' \) — Second derivative of deflection (m⁻¹)

Explanation: The equation shows that the curvature of the beam (y'') is directly proportional to the bending moment and inversely proportional to the flexural rigidity (EI).

3. Importance of Beam Deflection Calculation

Details: Accurate deflection calculation is crucial for structural design to ensure that beams and other structural elements meet serviceability requirements and don't deflect excessively under load.

4. Using the Calculator

Tips: Enter elastic modulus in Pascals, moment of inertia in meters to the fourth power, and bending moment in Newton-meters. All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What is the significance of the second derivative of deflection?
A: The second derivative of deflection (y'') represents the curvature of the beam, which is directly related to the bending moment and stress distribution.

Q2: How do I obtain the actual deflection from this calculation?
A: You need to integrate the second derivative twice with appropriate boundary conditions to obtain the deflection function y(x).

Q3: What are typical values for elastic modulus?
A: For steel: ~200 GPa, for aluminum: ~70 GPa, for concrete: ~20-30 GPa, depending on the specific material and grade.

Q4: When is this equation applicable?
A: This equation is valid for small deflections and linear elastic material behavior (Hooke's law applies).

Q5: What are the limitations of this approach?
A: The equation assumes linear elastic behavior, small deflections, and doesn't account for shear deformation, which may be significant for short, deep beams.