**Related Resources: beam bending**

### Double Integration Method Example 4 Proof Simply Supported Beam of Length L with Partial Distributed Load

**Beams Deflection and Stress Formulas and Calculators**

**Engineering Mathematics**

**Double Integration Method Example 4 Proof Simply Supported Beam of Length L with Partial Distributed Load. **

The Double Integration Method, also known as Macaulay’s Method is a powerful tool in solving deflection and slope of a beam at any point because we will be able to get the equation of the elastic curve.

Elastic Curve

Compute the value of EI δ at midspan for the beam loaded as shown in the figure above. If E = 10 GPa, what value of I is required to limit the midspan deflection to 1/360 of the span?

∑ M_{R2} = 0

4 R_{1} = 300 ( 2 ) ( 3 )

R_{1} = 450 N

∑ M_{R1} = 0

4 R_{2} = 300 ( 2 ) ( 1 )

R_{2} = 150 N

E I y'' = 450 x - 0.5 (300) x^{2} + 0.5 (300) < x - 2 >^{2}

E I y'' = 450 x - 150 x^{2} + 150 < x - 2 >^{2}

E I y' = 225 x^{2} - 50 x^{3} + 50 < x - 2 >^{3} + C_{1}

E I y = 75 x^{3} - 12.5 x^{4} + 12.5 < x - 2 >^{4} + C_{1} x + C_{2}

At x = 0, y = 0, therefore C_{2} = 0

At x = 4 m, y = 0

0 = 75 (4^{3}) - 12.5 (4^{4}) + 12.5 ( 4 -2 )^{4} +4 C_{1}

C_{1} = -450 N·m^{2}

Therefore,

E I y = 75 x^{3} - 12.5 x^{4} + 12.5 < x - 2 >^{4} - 450 x

At x = 2 m (midspan)

E I y_{midspan} = 75 ( 2^{3} ) - 12.5 ( 2^{4}) + 12.5 ( 2 - 2 )^{4} - 450 (2)

E I y_{midspan} = -500 N·m^{3}

E I δ_{midspan} = 500 N·m^{3}

Maximum midspan deflection

δ_{midspan} = L/360 = 4 / 360 = M / 90

δ_{midspan} = ( 100 / 9 )

Thus,

10,000 I (100 / 9) = 500 ( 1000^{3} )

I = 4,500,000 mm^{4} or

I = 4.5 x 10^{6} mm^{4}

Related:

- Double Integration Method for Beam Deflections
- Double Integration Method Example 1 Simply Supported Beam of Length L with Concentrated Load at Mid Span
- Double Integration Method Example 3 Proof Cantilevered Beam
- Beam Calculator Cantilevered Beam with One Load Applied at End Deflection and Stress Structural Beam Deflection, Stress, Bending Equations and calculator for a Cantilevered Beam with One Load Applied at End.
- Curved Beam Stress and Deflection Design Spreadsheet Calculator
- Curved Circular (Cylinder) Beam Stress Formulas and Calculator

Reference:

- Dr. ZM Nizam Lecture Notes
- Shingley Machine Design, 4-3 "Deflection Due to Bending"
- Beam Deflection by Integration Lecture Presentation Paul Palazolo, University of Memphis,
- Beam Deflections Using Double integration, Steven Vukazich, San Jose University