Related Resources: beam bending

Double Integration Method Example 1 Simply Supported Beam with Concentrated Load at Mid Spa

Beams Deflection and Stress Formulas and Calculators
Engineering Mathematics

Double Integration Method Example Proof Simply Supported Beam of Length L with Concentrated Load at Mid Span

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

Determine the maximum deflection δ in a simply supported beam of length L carrying a concentrated load P at midspan.

E I y'' = 0.5 Px - P(x - 0.5 L)

E I y' = 0.25 P x² - 0.5 P( x - 0.5 L )² + C₁

E I y = ( 1/12 ) P x³ - ( 1/6 ) P ( x - 0.5 L )³ + C₁ x + C₂

At x = 0, y = 0, therefore, C₂ = 0

At x = L, y = 0

0 = ( 1/12 ) P L³ - (1/6) P ( L - 0.5 L)³ + C₁ L

0 = ( 1/12 ) P L³ - ( 1/48 ) P L³ + C₁ L

C₁ = - ( 1/16 ) P L²

Thus,

E I y = ( 1/12 ) P x³ - (1/6) P ( x - 0.5 L)³ - (1/16) P L² x

Maximum deflection will occur at x = ½ L (midspan)

E I y_max = ( 1/12 )P (½ L)³ - (1/6) P ( 0.5 L - 0.5 L )³ - (1/16) P L ² ( ½ L) )

E I y_max = ( 1/96 ) P L³ - 0 - (1/32) P L³

y_max = - ( P L³ ) / ( 48 E I )

The negative sign indicates that the deflection is below the undeformed neutral axis.

Therefore,

δ_max = - ( P L³ ) / ( 48 E I )

Related:

• Double Integration Method for Beam Deflections
• Beam Supported on Both Ends Uniform Loading Deflection and Stress Stress, Deflection Equation and Calculator using total applied load and line load.
• 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.
• Beam Distributed Load Deflection, Shear, Deflection and Stress Equations and Calculator Beam Deflection, Shear and Stress Equations and Calculator for a Beam supported One End, Pin Opposite End and Partial Distributed Load
• Calculator for a Beam supported One End, Pin Opposite End and Single Moment at End Applied Deflection and Stress Calc Beam Deflection, Shear and Stress Equations and Calculator for a Beam supported One End, Pin Opposite End and Single Moment at End Applied
• Stress & Deflection Deflection Equations and Calculator Beam Supported Uniform loading Applied on Cantilevered End Stress Deflection Equations and Calculator Beam Supported Uniform loading Applied on Cantilevered End

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