Simply Supported Beam Calculator Deflection

Deflection Equation:

\[ \delta(x) = \frac{w x}{24 E I} (L^3 - 2 L x^2 + x^3) \]

Uniform Load (w):

N/m

Position (x):

Beam Length (L):

Modulus of Elasticity (E):

Moment of Inertia (I):

m⁴

Deflection at x (δ):

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1. What Is The Simply Supported Beam Deflection Equation?

The equation δ(x) = (w x / (24 E I)) (L³ - 2 L x² + x³) calculates the deflection at any point x along a simply supported beam with uniform load w. This formula is derived from beam theory and considers the beam's material properties (E, I) and geometry (L).

2. How Does The Calculator Work?

The calculator uses the deflection equation:

\[ \delta(x) = \frac{w x}{24 E I} (L^3 - 2 L x^2 + x^3) \]

Where:

\( \delta(x) \) — Deflection at position x (m)
\( w \) — Uniform load (N/m)
\( x \) — Position along the beam (m)
\( L \) — Beam length (m)
\( E \) — Modulus of elasticity (Pa)
\( I \) — Moment of inertia (m⁴)

Explanation: The equation calculates how much a simply supported beam bends under a uniformly distributed load, considering the beam's material stiffness and cross-sectional properties.

3. Importance Of Deflection Calculation

Details: Deflection calculations are crucial in structural engineering to ensure beams don't deform excessively under load, which could compromise structural integrity or cause serviceability issues.

4. Using The Calculator

Tips: Enter all values in consistent units (meters for length, Pascals for modulus, m⁴ for moment of inertia). Ensure x ≤ L and all values are positive.

5. Frequently Asked Questions (FAQ)

Q1: What is a simply supported beam?
A: A beam supported at both ends with pinned connections that allow rotation but prevent vertical movement.

Q2: What are typical deflection limits?
A: Building codes often limit deflection to L/360 for live loads and L/240 for total loads, where L is the span length.

Q3: How does moment of inertia affect deflection?
A: Deflection is inversely proportional to moment of inertia. Beams with larger I values (deeper or wider sections) deflect less.

Q4: What materials have high modulus of elasticity?
A: Steel has high E (~200 GPa), concrete has lower E (~20-30 GPa), making steel stiffer and less prone to deflection.

Q5: Does this equation work for all beam types?
A: No, this specific equation is only for simply supported beams with uniform loads. Other support conditions and load types have different equations.