Slope at Free End of Cantilever Beam Carrying Concentrated Load at Free End Calculator

Slope Formula:

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

Point Load (P):

Length of Beam (l):

Elasticity Modulus (E):

Area Moment of Inertia (I):

m⁴

Tolerance:

Slope (radians):

Slope (degrees):

With Tolerance (±5%):

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

Definition: The slope (θ) at the free end of a cantilever beam carrying a concentrated load is the angle of rotation due to bending under the applied load.

Purpose: This calculation is essential in structural engineering to ensure beam deflection remains within acceptable limits.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( \theta \) — Slope at free end (radians)
\( P \) — Point load at free end (N)
\( l \) — Length of beam (m)
\( E \) — Modulus of elasticity (Pa)
\( I \) — Area moment of inertia (m⁴)

Explanation: The formula calculates the angular displacement at the free end of a cantilever beam under a concentrated load.

3. Importance of Slope Calculation

Details: Proper slope calculation ensures structural integrity and serviceability of cantilever beams in construction projects.

4. Using the Calculator

Tips: Enter the point load, beam length, modulus of elasticity (default 30 GPa for concrete), moment of inertia (default 0.0016 m⁴), and tolerance percentage (default ±5%). All values must be > 0 except tolerance which can be 0.

5. Frequently Asked Questions (FAQ)

Q1: What is a typical modulus of elasticity for concrete?
A: For normal weight concrete, E typically ranges from 20-30 GPa (20,000-30,000 MPa).

Q2: How do I find the moment of inertia for my beam?
A: It depends on the cross-section shape. For rectangular beams: \( I = \frac{b \times h^3}{12} \), where b is width and h is height.

Q3: Why include a tolerance percentage?
A: Tolerance accounts for material variations, construction tolerances, and safety factors in real-world applications.

Q4: Can this calculator be used for other beam types?
A: No, this formula is specific for cantilever beams with concentrated load at the free end.

Q5: What if my beam has distributed loads?
A: You would need a different formula that accounts for distributed load conditions.