Slope at Free End of Cantilever Beam Carrying Couple at Free End Calculator

Slope Formula:

\[ \theta = \frac{M_c \times l}{E \times I} \]

Moment of Couple (M_c):

N·m

Length of Beam (l):

Elasticity Modulus (E):

Area Moment of Inertia (I):

m⁴

Tolerance:

Slope (θ) in radians:

Slope (θ) in degrees:

Acceptable Range (±5%):

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 when a couple moment is applied at that end.

Purpose: It helps structural engineers and designers analyze beam deflection under applied moments.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \theta = \frac{M_c \times l}{E \times I} \]

Where:

\( \theta \) — Slope at free end (radians)
\( M_c \) — Moment of couple applied at free end (N·m)
\( l \) — Length of beam (m)
\( E \) — Modulus of elasticity of material (Pa)
\( I \) — Area moment of inertia of beam cross-section (m⁴)

Explanation: The slope is directly proportional to the applied moment and beam length, and inversely proportional to the material stiffness and cross-section moment of inertia.

3. Importance of Slope Calculation

Details: Calculating beam slope is crucial for ensuring structural integrity, serviceability, and compliance with design codes that limit deflections.

4. Using the Calculator

Tips: Enter the moment, 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.

5. Frequently Asked Questions (FAQ)

Q1: What is a cantilever beam?
A: A beam fixed at one end and free at the other, with no support at the free end.

Q2: What's a typical modulus for concrete?
A: Concrete typically has E = 25-30 GPa, but varies with strength and composition.

Q3: How do I find the moment of inertia?
A: It depends on cross-section shape. For rectangular beams, I = (b × h³)/12.

Q4: Why include tolerance?
A: To account for material variations, construction tolerances, and safety factors.

Q5: Can this be used for other materials?
A: Yes, just input the correct E value (e.g., 200 GPa for steel).