Tapered Cantilever Beam Deflection Calculator

Tapered Cantilever Beam Deflection Equation:

\[ \delta = \frac{w L^4}{30 E I_{tip}} \times k \]

Distributed Load (w):

N/m

Length (L):

Modulus of Elasticity (E):

Moment of Inertia at Tip (I_tip):

m⁴

Taper Coefficient (k):

dimensionless

Deflection (δ):

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1. What is the Tapered Cantilever Beam Deflection Equation?

The tapered cantilever beam deflection equation calculates the maximum deflection at the free end of a tapered cantilever beam under uniformly distributed load. It accounts for the varying cross-section along the beam's length through the taper coefficient.

2. How Does the Calculator Work?

The calculator uses the tapered cantilever beam deflection equation:

\[ \delta = \frac{w L^4}{30 E I_{tip}} \times k \]

Where:

\( \delta \) — Deflection at free end (m)
\( w \) — Uniformly distributed load (N/m)
\( L \) — Length of the beam (m)
\( E \) — Modulus of elasticity (Pa)
\( I_{tip} \) — Moment of inertia at the tip (m⁴)
\( k \) — Taper coefficient (dimensionless)

Explanation: The equation accounts for the tapered geometry of the beam through the taper coefficient k, which depends on the specific taper ratio and profile of the beam.

3. Importance of Deflection Calculation

Details: Accurate deflection calculation is crucial for structural design to ensure beams meet serviceability requirements, prevent excessive deformations, and maintain structural integrity under applied loads.

4. Using the Calculator

Tips: Enter all values in consistent SI units. The distributed load should be in N/m, length in meters, modulus of elasticity in Pascals, moment of inertia in m⁴, and taper coefficient as a dimensionless value.

5. Frequently Asked Questions (FAQ)

Q1: What is the taper coefficient k?
A: The taper coefficient accounts for the variation in cross-section along the beam length. Its value depends on the specific taper ratio and profile of the beam.

Q2: How is moment of inertia at the tip determined?
A: The moment of inertia at the tip is calculated based on the cross-sectional dimensions at the free end of the tapered beam.

Q3: What are typical values for modulus of elasticity?
A: Steel: ~200 GPa, Aluminum: ~69 GPa, Wood: ~10 GPa (varies by species and grade).

Q4: When is this equation applicable?
A: This equation applies to linearly tapered cantilever beams with small deflections and within the elastic range of the material.

Q5: How does taper affect beam deflection?
A: Tapered beams generally have different deflection characteristics compared to uniform beams, with the deflection pattern depending on the taper ratio and profile.