**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 x ^{2} - 0.5 P( x - 0.5 L )^{2} + C_{1} *

E I y = ( 1/12 ) P x^{3} - ( 1/6 ) P ( x - 0.5 L )^{3} + C_{1} x + C_{2}

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

At x = L, y = 0

0 = ( 1/12 ) P L^{3} - (1/6) P ( L - 0.5 L)^{3} + C_{1} L

0 = ( 1/12 ) P L^{3} - ( 1/48 ) P L^{3} + C_{1} L

C_{1} = - ( 1/16 ) P L^{2}

Thus,

E I y = ( 1/12 ) P x^{3} - (1/6) P ( x - 0.5 L)^{3} - (1/16) P L^{2} 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 L^{3} - 0 - (1/32) P L^{3}

y_{max} = - ( P L^{3} ) / ( 48 E I )

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

Therefore,

δ_{max} = - ( P L^{3} ) / ( 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