Slope at Free Ends of Simply Supported Beam carrying Concentrated Load at Center Calculator

Slope Formula:

\[ \theta = \frac{P \times l^2}{16 \times E \times I} \]

Point Load (P):

Length of Beam (l):

Elasticity Modulus (E):

Area Moment of Inertia (I):

m⁴

Tolerance:

Slope (θ):

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

Definition: This calculator determines the angular displacement (slope) at the free ends of a simply supported beam carrying a concentrated load at its center.

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

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \theta = \frac{P \times l^2}{16 \times E \times I} \]

Where:

\( \theta \) — Slope at free ends (radians)
\( P \) — Point load at center (Newtons)
\( l \) — Length of beam (meters)
\( E \) — Elasticity modulus of concrete (Pascals)
\( I \) — Area moment of inertia (m⁴)

Explanation: The formula calculates the angle of rotation at the beam supports due to the applied central load.

3. Importance of Slope Calculation

Details: Proper slope calculation ensures structural integrity, prevents excessive deflection, and helps meet design specifications.

4. Using the Calculator

Tips: Enter the point load, beam length, elasticity modulus (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 elasticity modulus for concrete?
A: Normal concrete typically has an E value around 30 GPa (30,000,000,000 Pa), but this varies with concrete strength.

Q2: How do I find the moment of inertia for my beam?
A: It depends on the cross-section shape. For rectangular beams, I = (width × height³)/12.

Q3: Why include a tolerance percentage?
A: The 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 specific formula applies only to simply supported beams with a central point load.

Q5: How do I convert radians to degrees?
A: Multiply radians by 180/π (≈57.2958). Our result shows radians as it's the standard unit in structural calculations.