Cantilever Beam Deflection Formula Calculator

Cantilever Beam Deflection Formula:

\[ \delta_C = \frac{F a^2 (3 L - a)}{6 E I} \]

Force (F):

Distance (a):

Length (L):

Modulus of Elasticity (E):

Moment of Inertia (I):

m⁴

Deflection at End (δ_C):

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1. What is the Cantilever Beam Deflection Formula?

The cantilever beam deflection formula calculates the deflection at the end of a cantilever beam with an intermediate load. This formula is essential in structural engineering for determining how much a beam will bend under a specific load.

2. How Does the Calculator Work?

The calculator uses the cantilever beam deflection formula:

\[ \delta_C = \frac{F a^2 (3 L - a)}{6 E I} \]

Where:

\( \delta_C \) — Deflection at end (m)
\( F \) — Applied force (N)
\( a \) — Distance from fixed end to load point (m)
\( L \) — Total length of beam (m)
\( E \) — Modulus of elasticity (Pa)
\( I \) — Moment of inertia (m⁴)

Explanation: The formula calculates the maximum deflection at the free end of a cantilever beam when a load is applied at an intermediate point along its length.

3. Importance of Deflection Calculation

Details: Accurate deflection calculation is crucial for structural design, ensuring beams and structures can withstand applied loads without excessive bending that could lead to failure or serviceability issues.

4. Using the Calculator

Tips: Enter force in newtons, distances in meters, modulus of elasticity in pascals, and moment of inertia in meters to the fourth power. All values must be positive numbers.

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 mechanical applications.

Q2: When is this formula applicable?
A: This formula applies to cantilever beams with a point load applied at an intermediate position, assuming linear elastic material behavior and small deflections.

Q3: 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 direction).

Q4: 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.

Q5: What are deflection limits in structural design?
A: Deflection limits vary by application but are typically L/360 for floors, L/240 for roofs, and L/180 for ceilings under live loads, where L is the span length.