Maximum Beam Deflection Calculator

Cantilever Beam Deflection Formula:

\[ \delta_{max} = \frac{P L^3}{3 E I} \]

Point Load (P):

Length (L):

Modulus of Elasticity (E):

Moment of Inertia (I):

m⁴

Maximum Deflection (δ_max):

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1. What is Maximum Beam Deflection?

Maximum beam deflection refers to the greatest displacement of a structural element under load. For cantilever beams with a point load at the free end, the maximum deflection occurs at the end of the beam and is calculated using the formula δ_max = PL³/(3EI).

2. How Does the Calculator Work?

The calculator uses the cantilever beam deflection formula:

\[ \delta_{max} = \frac{P L^3}{3 E I} \]

Where:

\( P \) — Point load applied at the free end (N)
\( L \) — Length of the cantilever beam (m)
\( E \) — Modulus of elasticity of the material (Pa)
\( I \) — Moment of inertia of the beam cross-section (m⁴)

Explanation: This formula calculates the maximum vertical displacement at the free end of a cantilever beam subjected to a point load at its tip.

3. Importance of Deflection Calculation

Details: Calculating beam deflection is crucial in structural engineering to ensure that beams don't deflect excessively under load, which could lead to serviceability issues or structural failure. Deflection limits are often specified in building codes.

4. Using the Calculator

Tips: Enter the point load in newtons, length in meters, modulus of elasticity in pascals, and moment of inertia in meters to the fourth power. All values must be positive numbers.

5. Frequently Asked Questions (FAQ)

Q1: What is a cantilever beam?
A: A cantilever beam is fixed at one end and free at the other, commonly used in structures like balconies, diving boards, and aircraft wings.

Q2: What are typical deflection limits?
A: Deflection limits vary by application but are often L/360 for live loads and L/240 for total loads in building design, where L is the span length.

Q3: How does material affect deflection?
A: Materials with higher modulus of elasticity (like steel) deflect less than those with lower modulus (like wood) under the same load.

Q4: What if the load is distributed instead of point load?
A: Different formulas apply for distributed loads. The maximum deflection for a uniformly distributed load on a cantilever is wL⁴/(8EI).

Q5: How does cross-section shape affect deflection?
A: The moment of inertia (I) depends on the cross-sectional shape and size. Beams with larger I values (like I-beams) deflect less than those with smaller I values.