GaussSeidel Method Matlab To find the maximum stresses in a

Gauss-Seidel Method Matlab

To find the maximum stresses in a compound cylinder, the following four simultaneous linear equations need to be solved. [4.2857 times 10^7 4.2857 times 10^7 -6.5 0 -9.2307 times 10^5 -5.4619 times 10^5 -0.15384 0 0 -4.2857 times 10^7 6.5 4.2857 times 10^7 0 5.4619 times 10^5 0.15384 -3.6057 times 10^5][c_1 c_2 c_3 c_4] = [-7.887 times 10^3 0 0.007 0] In the compound cylinder, the inner cylinder has an internal radius of a = 5\", and an outer radius c = 6.5\", while the outer cylinder has an internal radius of c = 6.5\" and an outer radius of b = 8\". Given E = 30 times 10^6 psi, v = 0.3, and that the hoop stress in the outer cylinder is given by sigma_theta = E/1 + V)c_4 (1 - V/r^2)], find the stress on the inside radius of the outer cylinder. Find the values of c_1, c_2, c_3 and c_4 using the Gauss-Seidel Method. Use [c_1 c_2 c_3 c_4] = [-0.005 0.001 0.0002 0.03] as the initial guess and conduct two iterations.

Solution

Rewriting the equations for c1, c2, c3 and c4

c1= (-7.887*10³ - (-9.2307*10⁵)*c2 - 0*c3 - 0*c4)/ 4.2857*10⁷

c2= (0-4.2857*10⁷c1 - (-4.2857*10⁷)*c3 - 5.4619*10⁵c4)/ -5.4619*10⁵

c3= (0.007 - (-6.5)c1 - (-0.15384)c2 - 0.15384c4)/ 6.5

c4 = ( 0 - 0c1 - 0c2 - 4.2857*10⁷c3)/ -3.6057*10⁵

For 1 iteration, substituting initial guesses (-0.005, 0.001, 0.0002, 0.03) given in question for (c1, c2, c3 c4) in above equations, we get

c1 = -1.6249*10^-4

c2 = 1.5569*10^-3

c3 = 2.4125*10^-4

c4 = 2.8675*10^-2

For relative absolute approximate error, E = |(c_new - c_old)/c_new|*100

E₁ = 2977.1%, E₂ = 35.77%, E₃ = 17.09%, E₄ = 4.62%

Maximum relative error is 2977.1%

Similarly for iteration 2, using values (c1 = -1.6249*10^-4, c2 = 1.5569*10^-3, c3 = 2.4125*10^-4, c4 = 2.8675*10^-2)

c1 = -1.5*10^-4, c2 = -2.06*10^-3, c3 = 1.98*10⁴, c4= 2.36*10^-2

Maximum absolute relative approximate error = 175.4%

The stress on the inside radius of outer cylinder is calculated at the end of second iteration is by putting values in given equation, Stress = 21439 psi