How Do You Calculate Deflection In A Cantilever Beam

Maximum Deflection Formula:

\[ \delta_{max} = \frac{q L^4}{8 E I} \]

Uniform Load (q):

N/m

Length (L):

Modulus of Elasticity (E):

Moment of Inertia (I):

m⁴

Maximum Deflection (δ_max):

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1. What is Cantilever Beam Deflection?

Cantilever beam deflection refers to the displacement of a beam when it's subjected to a load. For a uniform load along the entire length of a cantilever beam, the maximum deflection occurs at the free end of the beam.

2. How Does the Calculator Work?

The calculator uses the maximum deflection formula:

\[ \delta_{max} = \frac{q L^4}{8 E I} \]

Where:

\( \delta_{max} \) — Maximum deflection (m)
\( q \) — Uniform load (N/m)
\( L \) — Length of the beam (m)
\( E \) — Modulus of elasticity (Pa)
\( I \) — Moment of inertia (m⁴)

Explanation: This formula calculates the maximum vertical displacement at the free end of a cantilever beam subjected to a uniformly distributed load along its entire length.

3. Importance of Deflection Calculation

Details: Calculating beam deflection is crucial in 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 the uniform load in N/m, length in meters, modulus of elasticity in Pascals, and moment of inertia in m⁴. All values must be positive numbers.

5. Frequently Asked Questions (FAQ)

Q1: What is a cantilever beam?
A: A cantilever beam is a rigid structural element that is fixed at one end and free at the other end, projecting horizontally into space.

Q2: What are typical deflection limits for cantilever beams?
A: Deflection limits vary by application but are typically L/180 to L/240 for live loads and L/120 to L/180 for total loads, where L is the span length.

Q3: Does this formula work for point loads?
A: No, this specific formula is for uniformly distributed loads. Point loads have a different deflection formula: δ_max = (P L³)/(3 E I).

Q4: What affects beam deflection the most?
A: Length has the greatest effect as deflection is proportional to L⁴. Material stiffness (E) and cross-sectional properties (I) also significantly influence deflection.

Q5: When is this deflection formula not applicable?
A: This formula assumes linear elastic material behavior, small deflections, and uniform cross-section along the beam length. It may not be accurate for large deflections or non-uniform beams.