How To Calculate Deflection In Cantilever Beam

Cantilever Beam Slope at End Formula:

\[ \theta_{end} = \frac{w L^3}{6 E I} \]

Distributed Load (w):

N/m

Length (L):

Modulus of Elasticity (E):

Moment of Inertia (I):

m⁴

Slope at End (θ_end):

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1. What is Cantilever Beam Slope Calculation?

The cantilever beam slope calculation determines the angular displacement at the free end of a cantilever beam under uniform load. This is important for structural analysis and ensuring beams meet design requirements for deflection limits.

2. How Does the Calculator Work?

The calculator uses the cantilever beam slope formula:

\[ \theta_{end} = \frac{w L^3}{6 E I} \]

Where:

\( \theta_{end} \) — Slope at the free end (radians)
\( w \) — 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 beam cross-section (m⁴)

Explanation: The formula calculates the angular displacement at the free end of a cantilever beam subjected to a uniformly distributed load along its length.

3. Importance of Slope Calculation

Details: Calculating beam slope is essential for structural design to ensure that deflections and rotations remain within acceptable limits for safety, functionality, and serviceability of structures.

4. Using the Calculator

Tips: Enter the distributed load in N/m, length in meters, modulus of elasticity in Pascals, and moment of inertia in m⁴. All values must be positive numbers.

5. Frequently Asked Questions (FAQ)

Q1: What is a cantilever beam?
A: A cantilever beam is a rigid structural element that extends horizontally and is supported at only one end.

Q2: 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 grade).

Q3: How do I calculate moment of inertia?
A: Moment of inertia depends on the cross-sectional shape. For common shapes like rectangles or circles, standard formulas are available in engineering handbooks.

Q4: What are acceptable slope limits?
A: Acceptable limits vary by application and building codes, but typically range from L/180 to L/360 for most structures.

Q5: Does this formula work for point loads?
A: No, this specific formula is for uniformly distributed loads. Different formulas exist for point loads and other loading conditions.