How To Calculate Deflection Of Cantilever Beam

Cantilever Beam Deflection Formula:

\[ \delta = \frac{P x^2}{6 E I} (3 L - x) \]

Load (P):

Distance from Fixed End (x):

Beam Length (L):

Modulus of Elasticity (E):

Moment of Inertia (I):

m⁴

Deflection (δ):

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1. What is Cantilever Beam Deflection?

Cantilever beam deflection refers to the displacement of a beam that is fixed at one end and free at the other when subjected to external loads. Understanding deflection is crucial for structural design and ensuring safety in engineering applications.

2. How Does the Calculator Work?

The calculator uses the cantilever beam deflection formula:

\[ \delta = \frac{P x^2}{6 E I} (3 L - x) \]

Where:

\( \delta \) — Deflection at distance x (m)
\( P \) — Applied load (N)
\( x \) — Distance from fixed end (m)
\( L \) — Total beam length (m)
\( E \) — Modulus of elasticity (Pa)
\( I \) — Moment of inertia (m⁴)

Explanation: This formula calculates the deflection at any point along a cantilever beam when a concentrated load is applied at the free end.

3. Importance of Deflection Calculation

Details: Accurate deflection calculation is essential for structural integrity, preventing excessive deformation, meeting design specifications, and ensuring safety in construction and mechanical design.

4. Using the Calculator

Tips: Enter all values in the specified units. Ensure all inputs are positive values. The distance x must be less than or equal to the beam length L.

5. Frequently Asked Questions (FAQ)

Q1: What is the maximum deflection in a cantilever beam?
A: The maximum deflection occurs at the free end (x = L) and is given by \( \delta_{max} = \frac{P L^3}{3 E I} \).

Q2: What are typical values for modulus of elasticity?
A: Steel: ~200 GPa, Aluminum: ~70 GPa, Concrete: ~20-30 GPa, Wood: ~8-12 GPa (varies by species and grade).

Q3: How does moment of inertia affect deflection?
A: Deflection is inversely proportional to moment of inertia. Beams with larger I values (thicker cross-sections) deflect less under the same load.

Q4: Can this formula be used for distributed loads?
A: No, this formula is specifically for a concentrated load at the free end. Different formulas apply for distributed loads.

Q5: What are the limitations of this formula?
A: This formula assumes linear elastic material behavior, small deflections, and homogeneous, isotropic materials. It may not be accurate for large deformations or composite materials.