Hyperbolic disk centrifugal stress calculation

[jamorga37]

Please help -

I am trying to figure out the stress in a hyperbolic disk rotating at some RPM. The best equations I could find (and example) is from the book "advanced Strength of Materials" by J. P. Den Hartog. Google books shows page 62 with these formulas:
Thickness (t) = t_i/(r$$^{q}$$) where r is radius.
q = 1 shows a "ordinary hyperabola"
formula 48:
n₁,n₂ = -0.5* q +/- sqrt(0.25*q² + mu*q + 1) where mu is poissons ratio

formula 49 after dividing t*r out is:
radial stress (s_r) = c₁*r^n₁ + c₂*r^n₂ - [(3+mu)*rho*omega²*r²/(8-(3+mu)*q)] where omega is rotational velocity in radians per unit time, and rho is disk density
tangential stress (s_t) = r * s_r' + rho * omega² * r²

I put these equations in EES (engineering equation solver) and matched the books numbers for the example (more or less since the author did the calculations without a calculator).

My problem is that as q goes to 0 (disk evens out to a flat disk), max stress: s_t (evaluated at the bore of the disk) becomes less and less. The opposite happens as q increases past 1. I've looked over the equations and I can't find a mistake.

I've attached the EES plots. The bold lines show the q=1 solution for radial and tangential stresses (both divided by rho*omega²) which look just like the ones in the textbook (tangential larger then radial at the bore - r = 3in). The other lines are tangential and radial stresses for q = 0.5 and q = 0. The tangential stress at the bore drops as q drops. I would expect the tangential stress at the center to increase as q drops.


Using the equations for flat disks:
s_t=rho*omega² * [(3+v)/(8)] * [(r_i² +r_o²+[(r_i²*r_o²)/r²]-[(1+3*mu) / (3+mu)] * r²)]

which gives a value of 190 at the bore (again divided by rho * omega²) - far above the value of 36 shown in the plot for q = 0.

I've attached my EES code for reference.

{start}


function s_r(r ) 
  	$Common mu, omega, rho,q, n1, n2, c1 , c2
	s_r = c1*r^n1 + c2*r^n2 - (r^2*(3+mu)*rho * omega^2)/((8-(3+mu)*q) * 386.4 [lbm-in/lbf-s^2])
end

function s_t(r  ) 
  	$Common mu, omega, rho,q, n1, n2, c1 , c2
	s_r_prime = c1*n1*r^(n1-1) + c2*n2*r^(n2-1) - (2* r*(3+mu)*rho * omega^2)/((8-(3+mu)*q) * 386.4 [lbm-in/lbf-s^2])
	s_t = s_r_prime* r + r^2 * rho * omega^2/ (386.4 [lbm-in/lbf-s^2])
end

function u(r ) 
  	$Common mu, omega, rho,q, n1, n2, c1 , c2, E
	u = r * (s_t(r) - mu * s_r(r)) / E 
end


n1 = -0.5 * q + (0.25*q^2 + mu * q + 1)^0.5
n2 = -0.5 * q - (0.25*q^2 + mu * q + 1)^0.5

Factor_Safety = Strength_Yield / s_t(r_i)

{book example - hyperbolic profile steel disk}
Strength_Yield = 11500 [lbf / in^2]
E = 30e6 [lbf / in^2]
rho = 0.28 [lbm / in^3]
r_i = 3 [in]
r_o = 15 [in]
mu = 0.3
omega =632 [1/sec]
q=1

0 = c1*r_i^n1 + c2*r_i^n2 - (r_i^2*(3+mu)*rho * omega^2)/((8-(3+mu)*q) * 386.4 [lbm-in/lbf-s^2])
0 = c1*r_o^n1 + c2*r_o^n2 - (r_o^2*(3+mu)*rho * omega^2)/((8-(3+mu)*q)* 386.4 [lbm-in/lbf-s^2])

{max tangential stress @ r_i  and max hoop stress since s_r_r_i is 0}
s_t_i = s_t(r_i)
s_t_o = s_t(r_o)
s_r_i = s_r(r_i)
s_r_o = s_r(r_o)

{s_t_r = s_t(r) * 386.4 [lbm-in/lbf-s^2]/ (r_o^2 * rho * omega^2)
s_r_r = s_r(r) * 386.4 [lbm-in/lbf-s^2]/ (r_o^2  * rho * omega^2)
t = t_i/(r^q)
t_i = 15}

bore_displacement = u(r_i)

tip_displacement = u(r_o)

{bore_displacementa = u(r_i) *  E / (rho * omega^2)}

Am I mis-interpreting the equations or what??

Thanks for any help in advance,

-Jeff

 

[Illmatic1]

Answer:It looks like the equations you are using are correct, but the results you are getting are not. It could be because of a mistake in the calculation or an issue with the parameters used. You should double-check your calculations and make sure all the parameters you used are correct. If that doesn't help, then you may need to look for other equations or methods to calculate the stress in a hyperbolic disk.