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 Beam
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 ymax = ( 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 ymax = ( 1/96 ) P L3 - 0 - (1/32) P L3

ymax = - ( P L3 ) / ( 48 E I )

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

Therefore,

δmax = - ( P L3 ) / ( 48 E I )

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
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