**Related Resources: beam bending**

### Double Integration Method Example 3 Proof Cantilevered Beam

**Beams Deflection and Stress Formulas and Calculators
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**Engineering Mathematics**

**Double Integration Method Example 3 Proof Cantilevered Beam of Length L with Variable Increasing Load to ω _{o} at free end. **

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

Equations of elastic curve

V = ½ ω_{o} L

M = ½ ω_{o} L [ (2/3) L]

M = (1/3) ω_{o} L^{2}

By ratio and proportion

z / x = ω_{o} / L

z = x ( ω_{o} / L )

F = ½ x z

F = ½ x ( x ( ω_{o} / L ) )

F = ( ω_{o} / ( 2 L ) ) x^{2}

E I y'' = -M + V x - F ( x / 3 )

E I y'' = -( 1 / 3 ) ω_{o} L^{2} + 0.5 ω_{o} L x - (1/3) x ( ω_{o} / ( 2 L ) x^{2} )

E I y'' = - ω_{o} L^{2} / 3 + ( ( ω_{o} L ) / 2 ) x- ω_{o} / ( 6 L ) x^{3}

E I y' = - ( ω_{o} L^{2} / 3 ) x + ( ω_{o} L / 4 ) x^{2} - ω_{o} / ( 24 L ) x^{4} + C_{1}

E I y = - ( ω_{o} L^{2} / 6 ) x^{2} + ( ω_{o} L / 12 ) x^{3} - ω_{o} / ( 120 L ) x^{5} +C_{1} x + C_{2}

At x = 0, y' = 0, therefore C_{1} = 0

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

Therefore, the equation of the elastic curve is

E I y = - ( ω_{o} L^{2} / 6 ) x^{2} + ( ω_{o} L / 12 ) x^{3} - ω_{o} / ( 120 L ) x^{5}

Related:

- Double Integration Method for Beam Deflections
- Double Integration Method Example 1 Simply Supported Beam of Length L with Concentrated Load at Mid Span

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