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A simply supported beam with a point load in the middle is presented

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A simply supported beam with a point load in the middle is presented. Using the Bernoulli beam equation drive the deflection of the beam at any location x. Show that the maximum deflection of the beam given by:

                                                       d = 1 FL³/48 EI

The beam has a Young’s modulus E = 210 Gpa, Poisson’s ratio, v = 0.3 and density, p = 7850 kg/m³. If the beam cross-section is square with one of its side w = 20 mm, and a length L =1.5 m. Evaluate the maximum deflection for a point load placed in the middle. Validate your calculation using COMSOL structural mechanics beam module.

F

L

Hints:

1. Strat with the beam equation:

  Eid²y/dx² = M

2. Since the beam is simply supported, the moments at two supports, M (0) = M (L) = 0

3. Use the y(x) = 0 at x = 0 & x = L in subsequent integrations to determine the integration constants.