Cantilever Beam Deflection Calculation

Cantilever Beam Deflection Formula:

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

Load (P):

Length (L):

Position (x):

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 a load. This calculation is essential in structural engineering to ensure beams can withstand expected loads without excessive bending.

2. How Does the Calculator Work?

The calculator uses the cantilever beam deflection formula:

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

Where:

\( \delta(x) \) — Deflection at position x (m)
\( P \) — Applied load (N)
\( L \) — Length of the beam (m)
\( x \) — Position along the beam (m)
\( E \) — Modulus of elasticity (Pa)
\( I \) — Moment of inertia (m⁴)

Explanation: This formula calculates the deflection at any point x along a cantilever beam with a point load applied at the free end.

3. Importance of Deflection Calculation

Details: Calculating beam deflection is crucial for structural design to ensure that beams will not deform excessively under load, which could lead to structural failure or serviceability issues.

4. Using the Calculator

Tips: Enter all values in the specified units. Ensure position x is between 0 and L (0 ≤ x ≤ L). 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 that is fixed at one end and free at the other, commonly used in bridges, buildings, and aircraft wings.

Q2: What is modulus of elasticity?
A: Modulus of elasticity (E) is a material property that measures its stiffness or resistance to elastic deformation under load.

Q3: What is moment of inertia?
A: Moment of inertia (I) is a geometric property that represents how a beam's cross-sectional area is distributed relative to its neutral axis, affecting its resistance to bending.

Q4: Where is maximum deflection in a cantilever beam?
A: Maximum deflection occurs at the free end of the beam (x = L) when a point load is applied at the free end.

Q5: What are typical values for modulus of elasticity?
A: Steel has E ≈ 200 GPa, aluminum ≈ 70 GPa, wood varies from 8-14 GPa depending on species and grain direction.