Deflection Calculation For Cantilever Beam

Cantilever Beam End Deflection Formula:

\[ \delta_{end} = \frac{P L^3}{3 E I} \]

Point Load (P):

Length (L):

Modulus of Elasticity (E):

Moment of Inertia (I):

m⁴

End Deflection (δ_end):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Cantilever Beam End Deflection?

Cantilever beam end deflection refers to the maximum displacement at the free end of a beam that is fixed at one end and free at the other, when subjected to a point load at the free end. This is a fundamental concept in structural engineering and mechanics of materials.

2. How Does the Calculator Work?

The calculator uses the standard formula for cantilever beam deflection:

\[ \delta_{end} = \frac{P L^3}{3 E I} \]

Where:

\( \delta_{end} \) — End deflection (m)
\( P \) — 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 beam cross-section (m⁴)

Explanation: The formula calculates the maximum deflection at the free end of a cantilever beam subjected to a point load. The deflection is proportional to the cube of the beam length and inversely proportional to both the modulus of elasticity and moment of inertia.

3. Importance of Deflection Calculation

Details: Calculating beam deflection is crucial in structural design to ensure that beams don't deflect beyond acceptable limits, which could lead to structural failure, serviceability issues, or discomfort for occupants. Deflection limits are often specified in building codes and design standards.

4. Using the Calculator

Tips: Enter the point load in newtons (N), beam length in meters (m), modulus of elasticity in pascals (Pa), and moment of inertia in meters to the fourth power (m⁴). 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 construction for elements like balconies, cantilever bridges, and aircraft wings.

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 direction).

Q3: How do I calculate moment of inertia?
A: Moment of inertia depends on the cross-sectional shape. For common shapes like rectangles (I = bh³/12) and circles (I = πd⁴/64), there are standard formulas available.

Q4: Does this formula work for distributed loads?
A: No, this formula is specifically for a point load at the free end. Different formulas exist for distributed loads or loads applied at different positions.

Q5: What are typical deflection limits in building design?
A: Deflection limits vary by application but are often expressed as a fraction of the span length (e.g., L/360 for floors, L/240 for roofs under live load).