Deflection Calculation Steel Beam

Fourier Series Approximation For Deflection:

\[ \delta = \frac{M L^2}{8 E I} \times \left(1 - \cos\left(\frac{\pi x}{L}\right)\right) \]

Moment (M):

Length (L):

Modulus of Elasticity (E):

Moment of Inertia (I):

m⁴

Position (x):

Deflection (δ):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Fourier Series Deflection Equation?

The Fourier series approximation for beam deflection provides an analytical solution for calculating the deflection of a simply supported beam under various loading conditions. This particular equation calculates deflection at any point along the beam length.

2. How Does the Calculator Work?

The calculator uses the Fourier series deflection equation:

\[ \delta = \frac{M L^2}{8 E I} \times \left(1 - \cos\left(\frac{\pi x}{L}\right)\right) \]

Where:

\( \delta \) — Deflection (m)
\( M \) — Applied moment (Nm)
\( L \) — Beam length (m)
\( E \) — Modulus of elasticity (Pa)
\( I \) — Moment of inertia (m⁴)
\( x \) — Position along beam (m)
\( \pi \) — Mathematical constant (3.1416)

Explanation: The equation calculates beam deflection at any point x along the beam length using Fourier series approximation, accounting for material properties and loading conditions.

3. Importance of Deflection Calculation

Details: Accurate deflection calculation is crucial for structural design, ensuring beams meet serviceability requirements and don't deflect excessively under load, which could cause damage or discomfort.

4. Using the Calculator

Tips: Enter all values in consistent units. Moment of inertia depends on beam cross-section. Position x must be between 0 and L. All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What types of beams does this equation apply to?
A: This equation applies to simply supported beams with specific loading conditions that can be represented by Fourier series.

Q2: How accurate is the Fourier series approximation?
A: The approximation becomes more accurate with more terms in the series, though this single-term solution provides reasonable estimates for many engineering applications.

Q3: What are typical deflection limits for beams?
A: Deflection limits vary by application but are typically L/360 for floors and L/240 for roofs under live load conditions.

Q4: How does material affect deflection?
A: Higher modulus of elasticity (stiffer materials) results in less deflection. Steel has higher E than wood or concrete.

Q5: Can this be used for composite beams?
A: For composite beams, equivalent section properties must be calculated considering different materials' moduli of elasticity.