Deflection Calculation Formula

General Beam Deflection Formula:

\[ \delta = \iint \frac{M}{E I} dx dx \]

Moment (M):

Modulus of Elasticity (E):

Moment of Inertia (I):

m⁴

Beam Length (x):

Deflection (δ):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Beam Deflection Formula?

The double integration method for beam deflection calculates the vertical displacement of a beam under load using the moment equation integrated twice with respect to position along the beam. The general formula represents the fundamental approach to determining deflection in structural engineering.

2. How Does the Calculator Work?

The calculator uses a simplified approach based on the general formula:

\[ \delta = \iint \frac{M}{E I} dx dx \]

Where:

\( \delta \) — Deflection (m)
\( M \) — Bending moment (Nm)
\( E \) — Modulus of elasticity (Pa)
\( I \) — Moment of inertia (m⁴)
\( x \) — Position along the beam (m)

Explanation: The actual double integration requires specific boundary conditions and moment equations for different loading scenarios. This calculator provides a simplified estimation.

3. Importance of Deflection Calculation

Details: Accurate deflection calculation is crucial for structural design, ensuring beams and structures meet serviceability requirements and don't experience excessive deformation that could affect functionality or appearance.

4. Using the Calculator

Tips: Enter bending moment in Nm, modulus of elasticity in Pa, moment of inertia in m⁴, and beam length in meters. All values must be positive and valid for meaningful results.

5. Frequently Asked Questions (FAQ)

Q1: What are typical modulus values for common materials?
A: Steel: ~200 GPa, Aluminum: ~69 GPa, Concrete: ~20-30 GPa, Wood: ~8-12 GPa (varies by species and grade).

Q2: How does moment of inertia affect deflection?
A: Deflection is inversely proportional to moment of inertia. Larger I-values (e.g., I-beams) significantly reduce deflection for the same loading.

Q3: What are acceptable deflection limits?
A: Typically L/360 for floors, L/240 for roofs under live load, and L/180 to L/120 for total load, where L is span length.

Q4: When is double integration method most appropriate?
A: For statically determinate beams with relatively simple loading conditions where the moment equation can be easily expressed.

Q5: What are alternatives to double integration?
A: Moment-area method, conjugate beam method, virtual work method, and finite element analysis for complex structures.