Calculating Deflection Of A Cantilever Beam

Cantilever Beam Deflection Formula:

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

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 Cantilever Beam Deflection?

Cantilever beam deflection refers to the displacement of a beam when it's subjected to a load while being fixed at one end. The deflection formula calculates how much a beam will bend under a uniform load along its length.

2. How Does the Calculator Work?

The calculator uses the cantilever beam deflection formula:

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

Where:

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

Explanation: This equation calculates the vertical displacement at any point x along a cantilever beam subjected to a uniformly distributed load.

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 all values in the specified units. Ensure that the position x is between 0 and the beam length L. All values must be positive numbers.

5. Frequently Asked Questions (FAQ)

Q1: What is a cantilever beam?
A: A cantilever beam is a structural element that is fixed at one end and free at the other, commonly used in bridges, buildings, and aircraft structures.

Q2: What is modulus of elasticity?
A: Modulus of elasticity (E) is a material property that measures its stiffness or resistance to elastic deformation under load.

Q3: What is moment of inertia?
A: Moment of inertia (I) is a geometric property that represents how a beam's cross-sectional area is distributed relative to its neutral axis, affecting its resistance to bending.

Q4: What are typical deflection limits?
A: Deflection limits vary by application but are often limited to L/240 to L/360 of the span length for live loads, where L is the beam length.

Q5: Does this formula account for all load types?
A: No, this specific formula is for uniformly distributed loads only. Different formulas exist for point loads, varying loads, and other loading conditions.