Beam Cantilever Calculator

Cantilever Beam End Deflection Formula:

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

Uniform 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 the Cantilever Beam End Deflection Formula?

The cantilever beam end deflection formula calculates the maximum deflection at the free end of a cantilever beam subjected to a uniformly distributed load. This is important for structural engineering applications where deflection limits must be maintained.

2. How Does the Calculator Work?

The calculator uses the cantilever beam deflection formula:

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

Where:

\( w \) — Uniformly distributed load (N/m)
\( L \) — 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: The formula shows that deflection is proportional to the load and the fourth power of the length, and inversely proportional to both the modulus of elasticity and moment of inertia.

3. Importance of End Deflection Calculation

Details: Calculating end deflection is crucial for ensuring structural integrity, preventing excessive deformation, and meeting building code requirements for cantilever beam designs.

4. Using the Calculator

Tips: Enter the uniform load in N/m, length in meters, modulus of elasticity in Pa, and moment of inertia in m⁴. All values must be positive and non-zero.

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 engineering applications.

Q2: Why does length have such a significant effect on deflection?
A: Deflection is proportional to the fourth power of length (L⁴), which means even small increases in length result in dramatically larger deflections.

Q3: What are typical values for modulus of elasticity?
A: Steel has E ≈ 200 GPa, aluminum ≈ 70 GPa, wood varies from 8-14 GPa depending on species and grain direction.

Q4: How do I calculate moment of inertia for different cross-sections?
A: Moment of inertia formulas vary by shape. For rectangular sections: I = bh³/12, for circular sections: I = πd⁴/64.

Q5: Are there deflection limits for cantilever beams?
A: Yes, building codes typically limit deflection to L/180 to L/480 depending on the application and material to prevent serviceability issues.