User:EML5526.S11.Team3.risher/homework3

3.1
Repeat homework problem 2.9 w/ poly. basis functions $$ (x + k)^j, j = 1,2,...n $$

Choose k=1 to avoid $$ b_j(0) = 0 $$

3.2
FB, p. 37, problem 2.1

For the spring system given in Figure 2.16 on page 37,HH1488

A.) Number of elements and nodes

B.) Assemble the global stiffness and force matrix

C.) Partition the system and solve for the nodal displacements

D.) Compute the reaction forces

3.3
FB, p. 37, problem 2.3

Consider the truss structure given in Figure 2.18 on page 37. Nodes A and B are fixed. A force equal to 10N acts in the positive x-direction at node C. Coordinates of joints are given in meters. Young's modulus is $$E = 10^{11}$$ Pa and the cross-sectional area for all bars are $$ A = 2*10^{-2} m^{2}$$.

A.) Number the elements and nodes

B.) Assemble the global stiffness and force matrix

C.) Partition the system and solev for nodal displacements

D.) Computer the stresses and reactions

3.4
$$ \alpha = b_{i} \left[ a^{'}_{2} b^{'}_{j} + a_{2} b^{''}_{j} \right] $$

$$ \beta = b_{j} \left[ a^{'}_{2} b^{'}_{i} + a_{2} b^{''}_{i} \right] $$

Show $$ \alpha \ne \beta $$

Ex:

$$ \left. \begin{matrix} \begin{matrix} b_i(x) = cos(ix) \\ \end{matrix} \\ \begin{matrix} b_j(x) = cos(jx) \\ \end{matrix} \\ \end{matrix} \right\} i \ne j $$

3.5
FB, p. 37, problem 2.2

Show that the equivalent stiffness of a spring aligned in the x direction for the bar of thickness t with centered hole shown in Figure 2.17 on page 37 is:

$$ k = \frac{5Etab}{(a+b)l'} $$

where E is the Young's modulus and t is the width of the bar (Hint: subdivide the bar with a square hole into 3 elements).

3.6
FB, p. 38, problem 2.4

Given the three-bar structure subjected to the prescribed load at point C equal to $$ 10^{3} $$ N as shown in Figure 2.19 on page 38. The Young's modulus is $$ E = 10^{11} $$ Pa, the cross-sectional area of the bar BC is $$ 2*10^{-2} m^{2} $$ and that of BD and BF is $$ 10^{-2} m^{2} $$. Note that point D is free to move in the x-direction. Coordinates of the joints are given in meters.

A.) Construct the global stiffness matrix and load matrix

B.) Partition the matrices and solve for the unknown displacements at point B and displacement in the x-direction at point D

C.) Find the stresses in the three bars

D.) Find the reactions at the nodes C, D, and F

3.7
Do problem 3.1 with k = 0

3.8
FB, p. 72, problem 3.1

3.9
FB, p. 72, problem 3.3

3.10
FB, p. 72, problem 3.4

3.11
Do problem 2.9 with full Fourier basis {$$ 1, cosx, sinx, cos2x, sin2x,... $$}

$$ b_j (0) \ne 0 $$ for some j

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