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

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?

∑ M_R2 = 0

4 R₁ = 300 ( 2 ) ( 3 )

R₁ = 450 N

∑ M_R1 = 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 y_midspan = 75 ( 2³ ) - 12.5 ( 2⁴) + 12.5 ( 2 - 2 )⁴ - 450 (2)

E I y_midspan = -500 N·m³

E I δ_midspan = 500 N·m³

Maximum midspan deflection

δ_midspan = L/360 = 4 / 360 = M / 90

δ_midspan = ( 100 / 9 )

Thus,

10,000 I (100 / 9) = 500 ( 1000³ )

I = 4,500,000 mm⁴ or

I = 4.5 x 10⁶ mm⁴

Related:

• Double Integration Method for Beam Deflections
• Double Integration Method Example 1 Simply Supported Beam of Length L with Concentrated Load at Mid Span
• Double Integration Method Example 3 Proof Cantilevered Beam
• 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.
• Curved Beam Stress and Deflection Design Spreadsheet Calculator
• Curved Circular (Cylinder) Beam Stress Formulas and Calculator

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