How To Calculate Cantilever Beam Deflection

Cantilever Beam Deflection Equation:

\[ \delta(x) = \frac{w}{24 E I} (6 L^2 x^2 - 4 L x^3 + x^4) \]

Uniform Load (w):

N/m

Beam Length (L):

Distance from Fixed End (x):

Modulus of Elasticity (E):

Moment of Inertia (I):

m⁴

Deflection (δ):

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1. What is Cantilever Beam Deflection?

Cantilever beam deflection refers to the displacement of a beam that is fixed at one end and free at the other when subjected to loads. The deflection equation calculates how much the beam bends under a uniformly distributed load.

2. How Does the Calculator Work?

The calculator uses the cantilever beam deflection equation:

\[ \delta(x) = \frac{w}{24 E I} (6 L^2 x^2 - 4 L x^3 + x^4) \]

Where:

\( \delta(x) \) — Deflection at distance x from fixed end (m)
\( w \) — Uniformly distributed load (N/m)
\( L \) — Length of the beam (m)
\( x \) — Distance from fixed end (m)
\( E \) — Modulus of elasticity (Pa)
\( I \) — Moment of inertia (m⁴)

Explanation: This equation calculates the vertical displacement of a cantilever beam at any point along its length when subjected to a uniform load.

3. Importance of Deflection Calculation

Details: Calculating beam deflection is crucial for structural engineering to ensure that beams don't deflect beyond acceptable limits, which could lead to structural failure or serviceability issues.

4. Using the Calculator

Tips: Enter all values in consistent units (meters for length, Newtons per meter for load, Pascals for modulus, and meters to the fourth for moment of inertia). All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What is the maximum deflection of a cantilever beam?
A: The maximum deflection occurs at the free end (x = L) and is given by: \( \delta_{max} = \frac{wL^4}{8EI} \)

Q2: What are typical values for modulus of elasticity?
A: Steel: ~200 GPa, Aluminum: ~69 GPa, Wood: ~10 GPa (varies by species and grade)

Q3: How do I calculate moment of inertia?
A: Moment of inertia depends on the cross-sectional shape. For rectangular sections: \( I = \frac{bh^3}{12} \), where b is width and h is height.

Q4: Does this equation work for point loads?
A: No, this equation is specifically for uniformly distributed loads. Different equations are used for point loads.

Q5: What are acceptable deflection limits?
A: Deflection limits vary by application but are typically L/180 to L/360 for live loads and L/240 to L/480 for total loads, where L is the span length.