Calculating the flex of a solid bar under force

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It is all explained here if you search simply supported beams.

You will find the equation $$w = \frac{F \ell^3}{48 E I}$$

Here $\ell$ is the distance between the supports, $F$ is the force applied, $E$ is the elastic modulus of the material and $I$ is area moment of the section. Rectangular sections have $I=\frac{1}{12} b h^3$ where $b$ is width and $h$ is height. The caveat here is the use of consistent units. You cannot mix metric with inches with feet.

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Findus
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Findus

Updated on January 11, 2021

Comments

  • Findus
    Findus almost 3 years

    I want to calculate how much a solid bar will flex when force is applied to it. The set up looks like this:

    Sketch

    The rod (in green) rests on two stationary points, and the force is applied in the center between the two points. The rod has a rectangular cross section.

    What I want to know is the length that the middle of the rod will be moved in the direction of the force. The force can be assumed to be small enough not to deform the rod.

    What properties of the material in the rod do I need to know to calculate this? And how do I do the actual calculation?

    • JamalS
      JamalS almost 9 years
      A general bar's deformation is governed by the Euler-Bernoulli equation, $(EI u''(x))'' \sim f(x)$, under certain assumptions.
    • Findus
      Findus almost 9 years
      I don't understand how to use that. Could you describe what all the variables are?
    • paisanco
      paisanco almost 9 years
      Wait a second, you want to know "how much a solid bar will flex when force is applied to it", yet "The force can be assumed to be small enough not to deform the rod" ? That doesn't seem consistent.
    • Findus
      Findus almost 9 years
      Okay, maybe I am using the wrong terms here (English is not my native language). What I mean is that force is applied, which makes the rod bend, but it is not permanently deformed. I.e. when the force is no longer applied, the rod returns to straight.
    • paisanco
      paisanco almost 9 years
      OK, you meant deformations are small enough to be elastic, not plastic.
  • John Alexiou
    John Alexiou almost 9 years
    This would be better received with some math formatting. See physics.stackexchange.com/help/notation
  • Findus
    Findus almost 9 years
    Thanks! Exactly what I needed, and also a very helpful link! I think the reason I did not fins this on google myself is that I did not know the correct terms in english for this.
  • John Alexiou
    John Alexiou almost 9 years
    The two common types of supports are called: simply supported beam and cantelever beam. Those are the correct terms.