Cantilever Beam Stress And Deflection Calculator

Cantilever Beam Formulas:

\[ \sigma_{\text{max}} = \frac{P L c}{I} \] \[ \delta = \frac{P L^3}{3 E I} \]

Load (P):

Length (L):

Distance to Extreme Fiber (c):

Moment of Inertia (I):

m⁴

Modulus of Elasticity (E):

Maximum Stress (σ_max):

Deflection (δ):

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1. What is Cantilever Beam Stress and Deflection?

A cantilever beam is a structural element fixed at one end and free at the other, carrying loads perpendicular to its axis. Maximum stress occurs at the fixed end, while maximum deflection occurs at the free end.

2. How Does the Calculator Work?

The calculator uses the cantilever beam formulas:

\[ \sigma_{\text{max}} = \frac{P L c}{I} \] \[ \delta = \frac{P L^3}{3 E I} \]

Where:

\( \sigma_{\text{max}} \) — Maximum stress (Pa)
\( \delta \) — Deflection (m)
\( P \) — Load applied at free end (N)
\( L \) — Length of beam (m)
\( c \) — Distance to extreme fiber (m)
\( I \) — Moment of inertia (m⁴)
\( E \) — Modulus of elasticity (Pa)

Explanation: The stress formula calculates the maximum bending stress, while the deflection formula calculates the vertical displacement at the free end.

3. Importance of Stress and Deflection Calculation

Details: Calculating stress and deflection is crucial for structural design to ensure beams can safely carry loads without excessive deformation or failure.

4. Using the Calculator

Tips: Enter all values in consistent SI units. Load (P) in Newtons, length (L) and distance (c) in meters, moment of inertia (I) in m⁴, and modulus of elasticity (E) in Pascals.

5. Frequently Asked Questions (FAQ)

Q1: What is a cantilever beam?
A: A cantilever beam is fixed at one end and free at the other, commonly used in bridges, buildings, and aircraft structures.

Q2: Where does maximum stress occur?
A: Maximum bending stress occurs at the fixed end of the cantilever beam, at the extreme fiber of the cross-section.

Q3: What affects beam deflection?
A: Deflection increases with load and length cubed, but decreases with higher modulus of elasticity and moment of inertia.

Q4: Are these formulas valid for all materials?
A: These formulas are valid for linear elastic materials that obey Hooke's law within the elastic range.

Q5: What are typical values for modulus of elasticity?
A: Steel: ~200 GPa, Aluminum: ~70 GPa, Concrete: ~20-30 GPa, Wood: ~10-15 GPa (varies by species and direction).