Slope at Free End of Cantilever Beam Carrying UVL with Maximum Intensity at Fixed End Calculator

Slope of Beam Formula:

\[ \theta = \frac{q \cdot l^3}{24 \cdot E \cdot I} \]

Uniformly Varying Load (q):

N/m

Length of Beam (l):

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 slope (angle of deflection) at the free end of a cantilever beam subjected to a uniformly varying load with maximum intensity at the fixed end.

Purpose: It helps structural engineers analyze beam deflection and ensure designs meet required specifications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \theta = \frac{q \cdot l^3}{24 \cdot E \cdot I} \]

Where:

\( \theta \) — Slope at free end (radians)
\( q \) — Uniformly varying load (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 based on beam properties and loading conditions.

3. Importance of Slope Calculation

Details: Accurate slope calculation is crucial for ensuring structural integrity, serviceability, and compliance with building codes.

4. Using the Calculator

Tips: Enter the load in N/m, length in meters, elasticity modulus in Pa (default 30 GPa for concrete), moment of inertia in m⁴ (default 0.0016), and tolerance percentage (default ±5%). All values must be > 0.

5. Frequently Asked Questions (FAQ)

Q1: What is a uniformly varying load?
A: A load that varies linearly along the beam length, from zero at the free end to maximum at the fixed end.

Q2: Why is modulus of elasticity important?
A: It measures the material's stiffness and directly affects how much the beam will deflect under load.

Q3: What affects the moment of inertia?
A: The beam's cross-sectional shape and dimensions - it resists bending deformation.

Q4: What does the tolerance percentage mean?
A: It indicates the acceptable variation (±%) in the calculated slope value.

Q5: Can this be used for other beam types?
A: No, this formula is specific for cantilever beams with UVL and maximum intensity at the fixed end.