Related Resources: beam bending

### Double Integration Method Example 3 Proof Cantilevered Beam

Double Integration Method Example 3 Proof Cantilevered Beam of Length L with Variable Increasing Load to ωo 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 ωo L

M = ½ ωo L [ (2/3) L]

M = (1/3) ωo L2

By ratio and proportion

z / x = ωo / L

z = x ( ωo / L )

F = ½ x z

F = ½ x ( x ( ωo / L ) )

F = ( ωo / ( 2 L ) ) x2

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

E I y'' = -( 1 / 3 ) ωo L2 + 0.5 ωo L x - (1/3) x ( ωo / ( 2 L ) x2 )

E I y'' = - ωo L2 / 3 + ( ( ωo L ) / 2 ) x- ωo / ( 6 L ) x3

E I y' = - ( ωo L2 / 3 ) x + ( ωo L / 4 ) x2 - ωo / ( 24 L ) x4 + C1

E I y = - ( ωo L2 / 6 ) x2 + ( ωo L / 12 ) x3 - ωo / ( 120 L ) x5 +C1 x + C2

At x = 0, y' = 0, therefore C1 = 0
At x = 0, y = 0, therefore C2 = 0

Therefore, the equation of the elastic curve is

E I y = - ( ωo L2 / 6 ) x2 + ( ωo L / 12 ) x3 - ωo / ( 120 L ) x5

Related:

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