Related Resources: beam bending

Double Integration Method Example 4 Proof Simply Supported Beam of Length L with Partial Distributed Load

Beams Deflection and Stress Formulas and Calculators
Engineering Mathematics

Double Integration Method Example 4 Proof Simply Supported Beam of Length L with Partial Distributed Load.

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

Elastic Curve

Compute the value of EI δ at midspan for the beam loaded as shown in the figure above. If E = 10 GPa, what value of I is required to limit the midspan deflection to 1/360 of the span?

∑ MR2 = 0

4 R1 = 300 ( 2 ) ( 3 )

R1 = 450 N

∑ MR1 = 0

4 R2 = 300 ( 2 ) ( 1 )

R2 = 150 N

E I y'' = 450 x - 0.5 (300) x2 + 0.5 (300) < x - 2 >2

E I y'' = 450 x - 150 x2 + 150 < x - 2 >2

E I y' = 225 x2 - 50 x3 + 50 < x - 2 >3 + C1

E I y = 75 x3 - 12.5 x4 + 12.5 < x - 2 >4 + C1 x + C2

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

At x = 4 m, y = 0

0 = 75 (43) - 12.5 (44) + 12.5 ( 4 -2 )4 +4 C1

C1 = -450 N·m2

Therefore,

E I y = 75 x3 - 12.5 x4 + 12.5 < x - 2 >4 - 450 x

At x = 2 m (midspan)

E I ymidspan = 75 ( 23 ) - 12.5 ( 24) + 12.5 ( 2 - 2 )4 - 450 (2)

E I ymidspan = -500 N·m3

E I δmidspan = 500 N·m3

Maximum midspan deflection

δmidspan = L/360 = 4 / 360 = M / 90

δmidspan = ( 100 / 9 )

Thus,

10,000 I (100 / 9) = 500 ( 10003 )

I = 4,500,000 mm4 or

I = 4.5 x 106 mm4

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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