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I have this HW and I need someone to answer it all

Class is ME 599



src="/qa/attachment/10754857/" alt="Screen Shot 2019-10-14 at 5.03.33 PM.png" />Screen Shot 2019-10-14 at 5.03.25 PM.png

Screen Shot 2019-10-14 at 5.03.25 PM.png

1. (30 pts) The governing equation for deflection w(x) of a beam of length L subjected to
transverse loading p(x) is given by
d4 P ( x) = 0, 0< x< 1
Assume the following boundary conditions:
w(0) = 0,
dw
(0) =odw
d' w
dx
dx2
(L) = M.
(L) =-V
and use the following approximate solution (Galerkin Method)
W(x) = _c,p. (x)
in order to show that the governing equation can be written as
K. C. = F,
where
Ko = Jo dx2 dx-
-dx, and F, = [ p(x)edx_d'w
d'w do.
dx dx
2. (35 pts) (a) Solve the fourth order differential equation given in problem 1 using L=1,
p(x)=1,V=1, M=2, and w(x) = Eco, (x) = qx+c,x +cax+cax*.
(b) Plot the approximate solution w(x) as a function of x and compare it with the exact
solution, w(x) :
24

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3. (35 pts) Consider the rod element stiffness matrix. K() =
1
. and the force vector
F(@) = ['9) [ P(x)
[N (5)
N2 (5)
d5 .
Use a) two and b) three finite elements of equal length in order to solve the following
differential equation:
d'u
dx2
-+p=0, 0<x<L, u(0)=u(L)=0, p=10 N/m, L=0.3 m, E=10 GPa, and A=104 m'.
c) Solve the differential equation exactly and compare it with the numerical solutions of part a
and part b.

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