Deflection In Beam Calculator

Beam Deflection Formula:

\[ \theta = \int \frac{M}{E I} dx \]

Bending Moment (M):

Modulus of Elasticity (E):

Moment of Inertia (I):

m⁴

Length (x):

Slope/Rotation (θ):

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1. What is Beam Deflection?

Beam deflection refers to the displacement or rotation of a beam under load. The slope (θ) represents the angle of rotation at a specific point along the beam's length, which is crucial for structural analysis and design.

2. How Does the Calculator Work?

The calculator uses the beam deflection formula:

\[ \theta = \int \frac{M}{E I} dx \]

Where:

\( \theta \) — Slope/rotation (radians)
\( M \) — Bending moment (Nm)
\( E \) — Modulus of elasticity (Pa)
\( I \) — Moment of inertia (m⁴)
\( x \) — Length/distance (m)

Explanation: This formula calculates the angular rotation (slope) of a beam at a specific point by integrating the ratio of bending moment to the product of modulus of elasticity and moment of inertia over the beam's length.

3. Importance of Deflection Calculation

Details: Calculating beam deflection is essential for structural engineering to ensure that beams will not deform excessively under load, which could lead to structural failure or serviceability issues.

4. Using the Calculator

Tips: Enter bending moment in Nm, modulus of elasticity in Pa, moment of inertia in m⁴, and length in meters. All values must be positive and non-zero.

5. Frequently Asked Questions (FAQ)

Q1: What is the practical significance of beam slope?
A: Beam slope indicates how much a beam rotates at a specific point, which is important for understanding deformation patterns and ensuring structural integrity.

Q2: How does modulus of elasticity affect deflection?
A: Higher modulus of elasticity (stiffer material) results in less deflection, while lower modulus (more flexible material) leads to greater deflection.

Q3: What is moment of inertia and why is it important?
A: Moment of inertia measures a beam's resistance to bending. Beams with higher moment of inertia deflect less under the same load.

Q4: Are there different formulas for different beam types?
A: Yes, different support conditions and load types (point load, distributed load) require different deflection formulas. This calculator uses the general integral form.

Q5: When is deflection calculation most critical?
A: Deflection calculations are critical in designing structures where excessive deformation could affect functionality, such as in floors, bridges, and support beams.