Slope at Free Ends of Simply Supported Beam carrying UDL Calculator

Slope Formula:

\[ \theta = \frac{w' \times l^3}{24 \times E \times I} \]

Load per Unit Length:

N/m

Length of Beam:

Elasticity Modulus of Concrete:

Area Moment of Inertia:

m⁴

Tolerance:

Slope of Beam (radians):

Slope Range (±5%):

1. What is Slope at Free Ends of Simply Supported Beam?

Definition: The slope at the free ends of a simply supported beam carrying uniformly distributed load (UDL) is the angle of rotation at the supports.

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

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \theta = \frac{w' \times l^3}{24 \times E \times I} \]

Where:

\( \theta \) — Slope at free ends (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)

3. Importance of Slope Calculation

Details: Proper slope calculation ensures structural integrity and serviceability of beams under load.

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 (default ±5%). All values must be > 0.

5. Frequently Asked Questions (FAQ)

Q1: What is a typical modulus of elasticity for concrete?
A: Typically 25-30 GPa for normal strength concrete.

Q2: How do I find the moment of inertia?
A: It depends on the beam's cross-section. For rectangular beams: \( I = \frac{b \times h^3}{12} \).

Q3: Why include a tolerance value?
A: To account for material variations and construction tolerances in real-world applications.

Q4: Can this be used for other materials?
A: Yes, just change the modulus of elasticity to match your material (e.g., 200 GPa for steel).

Q5: How does slope relate to deflection?
A: Slope is the first derivative of the deflection curve with respect to beam length.