Slope at Free End of Cantilever Beam carrying UDL Calculator

Slope Formula:

\[ \theta = \frac{w' \times l^3}{6 \times E \times I} \]

Load per Unit Length:

N/m

Length of Beam:

Elasticity Modulus of Concrete:

Area Moment of Inertia:

m⁴

Tolerance:

Slope at Free End (rad):

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

Definition: This calculator determines the angular displacement (slope) at the free end of a cantilever beam subjected to a uniformly distributed load (UDL).

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

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \theta = \frac{w' \times l^3}{6 \times E \times I} \]

Where:

\( \theta \) — Slope at free end (radians)
\( w' \) — Load per unit length (N/m)
\( l \) — Length of beam (m)
\( E \) — Elasticity modulus of concrete (Pa)
\( I \) — Area moment of inertia (m⁴)

Explanation: The formula calculates the angle of deflection considering the beam's material properties and loading conditions.

3. Importance of Slope Calculation

Details: Proper slope estimation ensures structural integrity, prevents excessive deflection, and meets design code requirements.

4. Using the Calculator

Tips: Enter the load per unit length, beam length, elasticity modulus (default 30 GPa for concrete), moment of inertia (default 0.0016 m⁴), and tolerance percentage (default ±5%).

5. Frequently Asked Questions (FAQ)

Q1: What is a typical modulus of elasticity for concrete?
A: Normal-weight concrete typically has E between 25-35 GPa (25,000-35,000 MPa).

Q2: How does beam length affect the slope?
A: Slope increases with the cube of the length, making long beams much more susceptible to deflection.

Q3: What's the purpose of the tolerance value?
A: The tolerance accounts for material variations and construction tolerances in real-world applications.

Q4: How do I determine the moment of inertia?
A: It depends on the cross-section shape. For rectangular beams, I = (b × h³)/12.

Q5: What if my beam isn't concrete?
A: Simply input the appropriate modulus of elasticity for your material (e.g., 200 GPa for steel).