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Statics 2D Virtual Work problem regarding calculation of Normal force

  1. Oct 27, 2011 #1
    PROBLEM 1


    1. The problem statement, all variables and given/known data

    VIRT_LR_009.jpg

    The frame in the figure is supported by a hinge in A and a roller in G. It is loaded by a couple = 14 kN*m in B, a force = 12 kN in D and a distributed force = 4 kN/m on section EG. = 1.4 m.
    Calculate the normal force in C. Use the correct signs for tension and compression. Hint: Solve using the principle of virtual work and use previously mastered methods to check your answer.


    This is a problem from mastering engineering, the statics book. I can solve the problem using equilibrium equations, but i want to know what i am doing wrong with when I am doing using the virtual work theorem.

    2. Relevant equations

    [itex]\delta[/itex]W = 0
    [itex]\delta\theta small → tan ( \delta\theta) ≈ \delta\theta ; [/itex]


    3. The attempt at a solution

    DSC00011.jpg


    [itex]\delta\theta_{1} = \delta\theta_{2} = \delta\theta[/itex]


    [itex]\delta\theta_{1} = \delta u_{1}/a[/itex]
    [itex]\delta\theta_{2} = \delta u_{2}/2a[/itex]


    LET
    [itex]\delta u_{2} = \delta u[/itex]

    [itex]\delta u_{2}/2a = \delta u_{1}/a → \delta u_{1} = \delta u/2[/itex]
    [itex]\delta u_{3}/(a/2) = \delta u/2a → \delta u_{3} = \delta u/4[/itex]
    [itex]\delta u_{5} = \delta u_{1} = \delta u/2[/itex]
    [itex]\delta u_{4} = \delta u_{2} = \delta u[/itex]

    [itex]\delta W = 0 [/itex]

    Therefore
    [itex] -M*\delta\theta-N*\delta u_{5}-N*\delta u_{4}-F*\delta u_{2}-qa*\delta u_{3} = 0 [/itex]
    [itex]\delta u ≠ 0 [/itex]
    [itex]-M*\delta u/2a-N*\delta u/2-N*\delta u-F*\delta u-qa*\delta u/4 = 0 [/itex]
    [itex]-\frac{3}{2}N = M/2a + F + qa/4 [/itex]
    N = -12.2666... kN


    But the answer should be -17.6 kN according to equilibrium equations.



    PROBLEM 2

    1. The problem statement, all variables and given/known data

    attachment.php?attachmentid=40380&stc=1&d=1319724126.jpg

    The frame in the figure is supported by a hinge in A and a roller in G. It is loaded by a couple = 4 kN*m in D, a force = 6 kN in B and a distributed force = 1 kN/m on section EG. = 1.5 m
    Calculate the shear force in C with the sign convention as shown in the figure. Hint: Solve using the principle of virtual work and use previously mastered methods to check your answer.

    2. Relevant equations

    [itex]\delta[/itex]W = 0
    [itex]\delta\theta small → tan ( \delta\theta) ≈ \delta\theta ; [/itex]


    3. The attempt at a solution

    attachment.php?attachmentid=40381&stc=1&d=1319724126.jpg

    I have no idea how to calculate the virtual displacement for the force F at B.
     

    Attached Files:

    Last edited: Oct 27, 2011
  2. jcsd
  3. Oct 28, 2011 #2
    For problem 1, why are you sure that δθ1=δθ2 ?
     
  4. Oct 28, 2011 #3
    i induce the same amount of rotation in both, and than follow to measure the change in the distances
     
  5. Oct 28, 2011 #4
    I would try the other way around : assuming a deplacement δu2=δu1 and then calculate the rotation
     
  6. Oct 30, 2011 #5
    tried it, doesnt work.

    the new virtual work equation becomes...

    -M/a - F - qa/4 = 2N
    => N = -11.7
     
    Last edited: Oct 30, 2011
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