Considering a very slender elliptical solid cross section subjected to a torque

Question

Considering a very slender elliptical solid cross section subjected to a torque T. using both the exact solution you obtained from Problem 3.3 and the approximate solution for narrow cross-sections with variable thickness to compute torsional coefficient (constant) J and the maximum shear stress . Derive and show your approximate solution of J and in terms of a and b. compare the exact and approximate solutions with a table for J/b4 and *(b3/T) when a/b = 1, 2, 5, 10, 20, and 100. Consider the straight bar of a uniform elliptical cross section. The semimajor and semiminor axes are a and b, respectively. Show that the stress function of the form phi = (x2/a2 + y2/b2 - 1) Provides the solution for torsion of the bar. Find the expression of c and show that J = pi a2 b2/a2 + b2 tauzx = -2Ty/piab3 , tauzy = 2Tx/pia3b And the warping displacement. w = T(b2 - a2)/pia3b3G xy phi = c(x2/a2 + y2/b2 -1) T = 2 phidA = 2c (x2/a2 + y2/b2 - 1)dxdy = 2c (x2/a2 + y2/b2)dxdy - 2CA A = pi ab let X = x/a Y = y/b dy = dy/b Y2 = X2 + Y2 dxdy = rdrdtheta T = 2c (X2 + Y2)abdXdY - 2CA = 2abc r3drdtheta - 2CA r4/4 2 Tv(2abc) - 2CA = Tv absc - 2piabc = -piabc = GJtheta J = delta/delta2 phi phi dA = - 2 phidA/((b2 + a2/a2b2))C = pi/abc(a2b2)/(b2 + a2)c = TVa3b3/a2 + b2 tauxz = dphi/dy, tauyz = -dphi/dx

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