Question

2) (30 points) The 2-segment bar shown below is loaded by an axial force f(x) and is fixed at both ends. Develop the finite element model of the bar using two linear elements. Note that the 3 nodes are shown and labeled. Use the following data: Ej = E2 = 200 GPa Aj = 0.01 m² A2 = 0.005 m² Li = L2 = 1 m f(x) = 3e7.x N/m HX f(x) - - 1 E1, A1,47 - - 2 E2, A2 , L2 a) (7 points) Formulate the element equations. Substitute values into the equations.

Answer 1

a) For a general bar, a small clement can be for invest gated as follows: F(x), AG) F(x)ox (x+4x) A tax ) for equilibrium, {fxo. -) 6(x+4x) Axtax) + f(x)ax -6 )A) 20 >> GCx+x)A(x+4x) – 6(x)A(x) + f(x)0 ax For Axmo, we get :- (6 (4) A(x)) + f(x) 20 Also, 6(x) = EE where Theo, we get a (EAC) du + f(x)=0 weak-foron using weighted residual technique,

Swan (sans ) a fos]deco er, any og integration by party, ww) sada - setembre + wexsficos de 20 Encode Jwantros de + WG) for n elements, this equation takes the following formi ent dw EAC) dhe dx - swex.fbdx te Terromat Term 2 + 2WxEA dy je K-1 where, en Teron-3 Terron I : It tells about met force on the system. Terran2 It contains information about the body forces Terms in It takes care about the Reaction forces at nodes.

our system is as follows: Ą, A, 1, 2 E2, A2, 23 here, and represent the 10 sle reds Using ucx) = (1 - Z) 4x + Zukty as the interpolation function, equation (i) reduces the following forsonin (Note: Here we are sirona Galeskim method. Hence W, = (1 - 2 and W₂ a ). for Element Here, a LoN Also, for element @ Regatiom forces. SAN - Puertos de Wex) f(x) dx Неве,

b) we have, W, (x) = (art W₂ (x) = ( 2 ). and Then, Swadfx) dx = 8 (2-2) 3xb0x dx =345[* 3xohl! - toru A, sự soi , x = 4 Also, | - 10 1² This, processes ] Assembling the system equations, We getin ora 2074] [1/7 Lo u lil lemi

Here, there are no external reaction forces for applied forces at nede 2. Hence, Them, 20. to 2 -2 0784 77007 1-2 3 -11 -107 3 t o to loo d) Note; - 4(+20). 4, 20 4 (xal) = 4220. and Then, 102 - of 0 -2 lo 3 - - t Нење, - 2x10°42 3x10 4 -100 a toto = 3 x 107 = 1 x 107 +0 -> 4, 0-125x162 mm >> 429 d. 25 woon Then, o 107 + 2*10° x 0.125x to? 0,8 x Lotn -X107