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Cantilever related questions and clarifications

  1. Dec 8, 2012 #1
    Hi,

    I am trying to understand the force and stress calculation in a cantilever beam, with point load at free end and fixed end at other.

    So, Bending Force F=3EI.d/L3, E-Elastic Modulus, I Second moment of Inertia along XX axis, d deflection, L distance between point of consideration and point of force application. ......(1)

    Stress = F.L.y/I, y being distance from neutral axis. ........................................(2)

    From 1 and 2, stress = 3.E.y.d/L2................................................................(3)


    So here is my confusion. Pls help.
    1. Let us say I am suspending a weight of 1kg, which is approx 10N force. From 1, L at the point of force application is zero. Does that mean F is inifinite where L=0?

    2. From 2, stress is inversely proportional to Moment of Inertia. I beam has less MI than a retangular beam of same hieght and weight. So I select a 3x3 retangular beam and carve out a I beam, I reducing the MI therefore increasing the stress, Does not sound right...Pls help.

    3. Materials have different tensile and compressive strengths. Lets say I have a material with compressive strength half the tensile strength. When I bend, the top portion is subject to tension and the lowest is subject to compression. From a design view, should I consider the compressive strength to decide the cross section area?

    Thoroughly confused. Pls enlighten.
    Neophysicist
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Dec 8, 2012 #2

    SteamKing

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    1. The equation for 'bending force' is a rearranged formula for the deflection in a cantilever beam resulting when a force F is applied at the free end. Thus,
    deflection = F*L^3 / (3 E I). L is the length of the beam, not the location of the force or the position where the deflection is calculated.

    For the deflection at a location 'x' when F is applied at the free end, the formula is
    deflection at 'x' = F * L^2 * (3*L - x) / (6 E I), and x is measured from the fixed end.
    You can see that when x = L, the first formula above is obtained.

    2. If you take a 3x3 beam and carve out an I-beam section from that, the inertia of the I-beam section will be less than the original rectangular section. The deflection of the beam is dependent on the actual cross section of the beam, not what it might have been at some other time in the past.

    3. Most materials used for construction of beams have similar strength properties in tension and compression. If you were trying to construct a cantilever beam from concrete which had not been reinforced, then it would be proper to consider the effect of compression and tension separately when designing the beam.
     
  4. Dec 8, 2012 #3
    Mr Steamking,


    much. You clarified my doubts like sun shine on mist!. My heartfelt thanks for your time and kind responses.

    For #2, you are essentially saying "Hey if you use a I beam, the deflection is function of the current beam's MI". I understand. But then for a given 3X3 perimeter an I beam fitting into it will have less MI and there fore more stress. If then, why do we say I beam is better?

    As to 3, I am trying to build a model with Balsa. Balsa has 4400 PSI in compression and 8700 PSI in tension. So when I subject a beam made of Balas, the top part is in tension and lower part is in compression. For design purpose, I assume I should go with compression strength. Am I right?

    Thanks,
     
  5. Dec 8, 2012 #4

    SteamKing

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    There are different factors which come into play when selecting the cross section for a given beam. Often, it is using a cross section which has a certain moment of inertia but which also has the least weight per unit length. Other times, it could be which cross section provides access for inspection and maintenance.

    The I-beam section locates material at the upper and lower surfaces where bending stresses are a maximum. It is also an open section, which allows for ease of inspection and maintenance of the structure. This is not to say that all beams must be constructed using open sections.

    Since your project is using a material which has different strength properties in tension and compression, yes, I would use the compressive stress as a guide for determining the best cross section to use. I don't know what your design will encompass, but if the compressive stress is the limiting factor, the you should also check for buckling if loads will be applied axially along the length of the member, in addition to any bending loads.
     
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