Beam Stress

  1. 1. The problem statement, all variables and given/known data
    Cast iron has a strength in compression of about three to four times the strength in tension, depending upon the grade.
    using the stress as a prime consideration, design the optimum T section for a cast iron beam using a uniform section thickness such that the compressive stress will be related to the tensile stress by a factor of 4


    2. Relevant equations
    TStress = (M*c1)/I
    CStress = -(M*c2)/I


    3. The attempt at a solution
     
  2. jcsd
  3. TSectionStress
    Please see attached Diagram of T Beam

    Let 1 = Top Rectangle
    Let 2 = Bottom Rectangle
    Let T = Thickness = 10
    Let H1 = T = Height of top
    Let H2 = height of bottom
    Let HT = Total height of T section = T + H2
    Let B1 = Length of Top
    Let B2 = length of bottom = T
    Let A1 = Top Area = B1 * T
    Let A2 = Bottom area = H2 * T
    Let AT = Total Area = A1 + A2
    Let C1 = centroid from the top of 1 = ((T * B1 * (T / 2)) + ((T * H2) * (T + (H2 / 2)))) / AT
    Let C2 = Centroid from the bottom of 2 = HT - C1
    Let I1 = Second moment of area of top rectangle = (((B1 / 10) * ((H1 ^ 3) / 10)) / 12) / 100
    Let I2 = Second moment of area of bottom rectangle = (((B2 / 10) * ((H2 ^ 3) / 10)) / 12) / 100
    Let D1 = Distance from C1 to 1 axis = C1 - (T / 2)
    Let D2 = Distance from C2 to 2 axis = C2 - (H2 / 2)
    Let IX1 = Second moment of area about any parallel axis to the c1 axis a distance d1 removed = (I1 + ((A1 / 100) * ((D1 / 10) ^ 2)))
    Let IX2 = Second moment of area about any parallel axis to the c2 axis a distance d2 removed = (I2 + ((A2 / 100) * ((D2 / 10) ^ 2)))
    let I = IX1 + IX2
    let StressT = Tensile Stress = ((1600 * (C1 / 10)) / I)
    Let StressC = Compressive Stress = ((1600 * (C2 / 10)) / I)




    Attempt at a solution

    StressC/StressT = 4

    ((1600 * (C2 / 10)) / I)/((1600 * (C1 / 10)) / I) = 4
    Therefore
    C1 = C2/4 and C2 =4*C1

    But from above C2 = HT-C1
    So
    ((T * B1 * (T / 2)) + ((T * H2) * (T + (H2 / 2)))) / AT= HT-C1/4

    The above equation should give me H2
    but I can only solve it to get 0=0
    What am I doing wrong.
     

    Attached Files:

  4. Solution
    ((T * B1 * (T / 2)) + ((T * H2) * (T + (H2 / 2)))) / (B1 * T) + (H2 * T)
    Put the above equation for C1 into the form of a quadratic equation in terms of H2
    Then put in values for T and B1.
    Use the Quadratic formula the find H2.
     
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