Solid and hollow shafts

  1. 1. The problem statement, all variables and given/known data
    assuming that the maximum shear stress and torsion are the same in both shafts design a hollow shaft to replace the solid one . the ratio diameters is to be 0.6 for the maximum shear stress i calculated 79.95 x 10 ^6 mn/m2 and for the torsion i calculated 122.78x10^3 in the previous question.


    2. Relevant equations
    (T) torque/ (J) polar second moment of area and (t) shear stress/ ( r) radius

    polar second moment of area = pie (d4-d4)/32

    radius = D/2


    3. The attempt at a solution
    (j)= pie ((0.6)4-d4)/32 = 0.08545 d4

    (r) = 0.6/2 = 0.3

    solid shaft = j/r = 152.41x10^6 / 99.25 x 10 ^-3 = 1.535 x 10^-3values from previous question

    hollow shaft = j/r = 0.08545/0.3 = 284.83 x 10^ -3

    d 3 square root with 1.535 x 10^-3/284.83 x 10^-3 under it = 175.32 x 10^-3

    D= 0.6 x D = 0.6 x 175.32 x 10^-3 = 105.19 x 10^-3

    therefore the dimensions are 175mm and 105mm

    i think these values are wrong as the solid shaft is 198.5mm and i always thought the hollow shaft would be larger in diameter please help as i dont no what i have done wrong..
     
  2. jcsd
  3. PhanthomJay

    PhanthomJay 6,270
    Science Advisor
    Homework Helper
    Gold Member

    that's pie(d_o^4 - d_i^4)/32
    please watch your subscripts, r = d_o/2
    the equation should be pie(d_o^4 - (0.6d_o)^4))/32, but your answer is correct
    r is d_o/2
    Yes, it should be larger in diameter; please correct your errors and try again. Be careful with the math, there are a lot of places where you can go wrong..
     
  4. Had another look today and relised where i have went wrong here is my new answers :

    torsion / polar second moment of area = tau/ radius

    : (t) 122.78 x 10^3/(j) 0.08545 = (tau) 79.95 x 10 ^6/d/2

    : (t) 122.78 x 10 ^3/(tau) 79.95 x 10 ^6 = (j) 0.08545 / d/2


    D= 3 square root with 122.78 x 10^3/79.95 x 10 ^6 divide by 2 x 0.08545 = 207.90x10 ^-3m


    d= 0.6 x D = 0.6 x 207.90 x 10 ^-3 = 124.74 x 10 -3m

    the dimensions for the hollow shaft are D=208mm and d=125mm

    hopefully i am correct this time!!!!!
     
  5. PhanthomJay

    PhanthomJay 6,270
    Science Advisor
    Homework Helper
    Gold Member

    Yes, that looks about right.:approve:
     
  6. thanks for checking :smile:
     
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