Question

Use the Rayleigh Ritz Method to determine the unknown coefficients.

1. The rod is subjected to a distributed load po and two point forces (F applied at x 3L/4 and x -L). a. Select an appropriate two-term approximate solution of the form b. Use the Rayleigh Ritz Method to determine the unknown coefficients a and a LU4

Answer 1

"The rod is subjected to distributed load ""p_0"" and this point forces x = 3 L/4 x = L Governing equation A e d^2 u/d x^2 + P_0 x = 0 d^2 u/d x^2 = -p_0 I/A E under condition x = 0: u = 0 x = L , Sigma = d u/d x = f/A \r\n Therefore exact solution u = -p_0 x^3/6 elementof_p + c_1 z + c_2 Apply condition x = 0 u = 0 c_2 = 0 x = L f d u /d x = f/A elementof(-3 p_0 x^2/6 elementof_A) + c_1 elementof = f/A therefore C_1 = f/A elementof + p_0 x^2/2 elementof A Therefore c_1 = f/A elementof + p_0 x^2/2 elementof A Therefore u = -P_0 x^3/6 elementof A x^3 + (f/A elementof + p_0 L^3/64 times L0 Now deflection at x = L/4 h = 1/ elementof A) -P_0/6 L^3/69 + FL/4 + P_0 L^3/64 times L) H = 1/ elementof A (FL/4 + P_0 L^3 (1/128 - 1/384)) \r\n h = 1/elementof A (FL/4 + 2 P_0 L^3/3) bending moment at distances x = w/L from A m = -(FL/4 + 1/2 P_0 L 6 2/16) x Deflection Sigma = 1/ elementof A (f times 1/4 + 1/2 P_0 L^2/16) L ="