Mech 417 Homework 6 assigned 1/24/19 due 1/31/19

# 1 Reference: Page 3-5 of notes at
https://www.clear.rice.edu/mech517/Class_notes/SolvedBars.pdf

Beginning with the assembled (and singular) 4 by 4 matrix
equilibrilum equations for the three bar problem find the
displacements, reaction, and element stresses in each bar
for the boundary condition:
  a) U_1 = 1.0 at node 1
  b) U_3 = 0.0 at node 3


# 2 References
https://www.clear.rice.edu/mech517/Books/PFEA/Chap_3_NumericInt.pdf
https://www.clear.rice.edu/mech517/Akin_FEA_Lib/qp_rule_unit_Gauss.txt
https://www.clear.rice.edu/mech517/Akin_FEA_Lib/Lagrange_1D_library.txt

 The Gaussian one-dimensional quadrature points are symmetrically 
placed with respect to the center of the interval, and n_q points in 1D
will EXACTLY integrate a  1D integrand that is a polynomial of degree 
(2*n_q − 1).  In other words 
            (1D Integrand polynomial degree] <= 2*n_q - 1

The mass matrix for any element is defined as the integral of the
tranpose of the interpolation funtions times the interpolation 
functions times the mass density times the differential volume. In
1D:
   M = density*area* Length_Integral H_transpose * H dx
for a four noded cubic line element the interpolation functions are
  H(r) = [H_1   H_2   H_3   H_4]
  H(r) = [(1-r*11/2+9*r^2-9*r^3/2), (9*r-45*r^2/2+27*r^3/2), ...
         (-9*r/2+18*r^2-27*r^3/2), (r-9*r^2/2+9*r^3/2)].
  a) what is the polynomial degree of the interpolation functions?
  b) what is the polynomial degree of the element mass matrix?
  c) how many integration (quadrature) points, n_q, are needed to 
     exactly integrate the element mass matrix?

Mech 517: In addition to the above problems compute the integral of
     the cubic interpolation functions over the domain 0 to 1.