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
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.
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.