A uniform bar is subjected to a body force per unit length f a concentrated force at the end R, as shown. X Constant cross-sectional area A Young's modulus E Differential formulation fB(x) = fB(x) = ax EA A differential formulation gives the strong form mathematical model with the following differential equation and Boundary Conditions (see Study Notes 1): du dx + f = 0 EA u|x=0 du dx R x=L = ax and Figure 3.2 Uniform bar subjected to body load f (force/unit length) and tip load R in the bar = R (3.18) (3.19) (3.20) (a) Use the Galerkin method to determine an approximate solution Use a second order polynomial for . Determine the displacement u, the strain &, and the stress o. Hint: Student A (in Study Notes 1) has started the process for you. Express your answers in terms of E, A, L, R and a . Draw a block around an answer to highlight it. (b) Repeat part (a) except this time use a third order polynomial for . Hinr: Student B (in Study Notes 1) has started the process for you. (c) This problem has exact analytic solution. Determine the exact solution for u, e and o. Hint: Start with the differential equation du dx + fB = 0 where fB = ax with its 2 boundary conditions: u|x=0 = 0 EA du dx x=L Show the steps to get: U Eexact ? Texact ? = R = SO, u exact a 1 al -68 A X + = A (R+ /) x 6EA EA 2 - (2) x + ( R + 1/{ L ) x a EA Express your answers in terms of E, A, L, R and a. (d) Plot and compare the stress o obtained from (a), (b), (c), using N L = 10 cm, A = 1 cm, cm R = 100 N J a = 1 In the plot, the Y - axis is stress o, and X - axis is x (0 x 10) (e) Plot and compare the quantity Eu obtained from (a), (b), (c), using N R = 100 N, cm In the plot, the Y- axis is the quantity Eu, and X - axis is x (0 x 10) L 10 cm, A = 1 cm. = a = : 1