Slope at Free End of Cantilever Beam Carrying Concentrated Load at Any Point from Fixed End Calculator

Slope Formula:

\[ \theta = \frac{P \times x^2}{2 \times E \times I} \]

Point Load (P):

Distance from Support (x):

Elasticity Modulus (E):

Area Moment of Inertia (I):

m⁴

Tolerance:

Slope at Free End (θ):

1. What is Slope at Free End of Cantilever Beam?

Definition: This calculator determines the angular displacement (slope) at the free end of a cantilever beam when a concentrated load is applied at any point along its length.

Purpose: It helps structural engineers and designers analyze beam deflection characteristics for proper design and safety assessment.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \theta = \frac{P \times x^2}{2 \times E \times I} \]

Where:

\( \theta \) — Slope at free end (radians)
\( P \) — Point load (Newtons)
\( x \) — Distance from fixed support to load point (meters)
\( E \) — Modulus of elasticity of beam material (Pascals)
\( I \) — Area moment of inertia of beam cross-section (m⁴)

Explanation: The formula calculates the slope based on beam material properties, geometry, and applied load.

3. Importance of Slope Calculation

Details: Slope calculation is crucial for ensuring structural integrity, serviceability, and compliance with building codes that limit deflection.

4. Using the Calculator

Tips: Enter the point load, distance from support, modulus of elasticity (default 30 GPa for concrete), moment of inertia (default 0.0016 m⁴), and tolerance percentage (default ±5%). All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What is a typical modulus of elasticity for concrete?
A: Normal-weight concrete typically has E between 20-30 GPa (20,000-30,000 MPa).

Q2: How do I find the moment of inertia for my beam?
A: Use standard formulas based on cross-section shape (e.g., I = b×h³/12 for rectangular sections).

Q3: Why include a tolerance percentage?
A: It accounts for material property variations, construction tolerances, and safety factors.

Q4: Can this be used for other beam types?
A: No, this formula is specific for cantilever beams with concentrated loads.

Q5: What if my load is distributed, not concentrated?
A: Use a different formula specifically for distributed loads on cantilever beams.