Calculate The Maximum Deflection And Slope Of Cantilever Beam

Cantilever Beam Formulas:

\[ \delta_{max} = \frac{P L^3}{3 E I} \] \[ \theta_{max} = \frac{P L^2}{2 E I} \]

Point Load (P):

Length (L):

Modulus of Elasticity (E):

Moment of Inertia (I):

m⁴

Maximum Deflection (δ_max):

Maximum Slope (θ_max):

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1. What is Cantilever Beam Deflection?

Cantilever beam deflection refers to the displacement of a beam when it is subjected to external loads. The maximum deflection (δ_max) occurs at the free end of the beam, while the maximum slope (θ_max) represents the angle of rotation at that point.

2. How Does the Calculator Work?

The calculator uses the standard cantilever beam formulas:

\[ \delta_{max} = \frac{P L^3}{3 E I} \] \[ \theta_{max} = \frac{P L^2}{2 E I} \]

Where:

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

Explanation: These formulas are derived from Euler-Bernoulli beam theory and assume small deflections, linear elastic material behavior, and uniform cross-section.

3. Importance of Deflection Calculation

Details: Calculating beam deflection is crucial for structural design to ensure that deformations remain within acceptable limits for both functionality and safety. Excessive deflection can lead to serviceability issues and structural failure.

4. Using the Calculator

Tips: Enter all values in consistent SI units. Ensure all inputs are positive values. The calculator assumes a point load at the free end of a cantilever beam with uniform cross-section.

5. Frequently Asked Questions (FAQ)

Q1: What are typical deflection limits for cantilever beams?
A: Deflection limits vary by application, but common limits are L/180 to L/360 for live loads and L/240 to L/480 for total loads, where L is the span length.

Q2: How does distributed load differ from point load?
A: For distributed loads, different formulas apply. The maximum deflection for a uniformly distributed load is wL⁴/(8EI), where w is the load per unit length.

Q3: What affects moment of inertia (I)?
A: Moment of inertia depends on the cross-sectional shape and dimensions. Common shapes have standard formulas (e.g., I = bh³/12 for rectangular sections).

Q4: Are these formulas valid for large deflections?
A: No, these formulas assume small deflections (less than span/10). For large deflections, more complex nonlinear analysis is required.

Q5: How does material choice affect deflection?
A: Materials with higher modulus of elasticity (E) will have less deflection under the same load. Steel (E ≈ 200 GPa) deflects less than aluminum (E ≈ 70 GPa) or wood (E ≈ 10 GPa).