Calculate Deflection Of Cantilever Beam

End Deflection Formula:

\[ \delta_{end} = \frac{w L^4}{8 E I} \]

Distributed Load (w):

N/m

Length (L):

Modulus of Elasticity (E):

Moment of Inertia (I):

m⁴

End Deflection (δ_end):

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1. What is End Deflection of a Cantilever Beam?

End deflection (δ_end) is the maximum displacement at the free end of a cantilever beam under a uniformly distributed load. It's a critical parameter in structural engineering for assessing beam performance and ensuring designs meet safety and serviceability requirements.

2. How Does the Calculator Work?

The calculator uses the end deflection formula:

\[ \delta_{end} = \frac{w L^4}{8 E I} \]

Where:

\( w \) — Uniformly distributed load (N/m)
\( L \) — Length of the beam (m)
\( E \) — Modulus of elasticity of the material (Pa)
\( I \) — Moment of inertia of the cross-section (m⁴)

Explanation: The formula shows that deflection is proportional to the load and the fourth power of length, and inversely proportional to both the modulus of elasticity and moment of inertia.

3. Importance of Deflection Calculation

Details: Calculating end deflection is essential for ensuring structural integrity, preventing excessive deformation that could affect functionality, and meeting building code requirements for serviceability limits.

4. Using the Calculator

Tips: Enter all values in consistent SI units. Ensure all inputs are positive values. The calculator provides deflection in meters, which can be converted to other units if needed.

5. Frequently Asked Questions (FAQ)

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

Q2: Why does length have such a strong effect (L⁴) on deflection?
A: The fourth power relationship comes from the integration of the bending moment equation twice to obtain deflection, making length the most influential factor.

Q3: What are typical values for modulus of elasticity?
A: Steel: ~200 GPa, Aluminum: ~69 GPa, Concrete: ~20-30 GPa, Wood: ~8-14 GPa (varies by species and grain direction).

Q4: How do I calculate moment of inertia for different cross-sections?
A: Different formulas exist for different shapes: rectangle (bh³/12), circle (πd⁴/64), I-beam (complex calculation based on dimensions).

Q5: Are there limitations to this formula?
A: This formula assumes linear elastic material behavior, small deflections, uniform cross-section, and perfectly fixed support conditions.