User:EML4500.f08.A-team.VandenBerg/HW6

 Comment: This is HW6 located in the user namespace of EML4500.f08.A-team.VandenBerg 16:08, 12 November 2008 (UTC)

Meeting 29: Monday, 3 November 2008. EML4500
At the beginning of class comments were made on mechanisms and eigenvalues from HW4, and then HW6 was talked about and how to redo the vector truss.

PVW (continuous) of Dynamics of Elastic Bar
PDE:


 * (1) $$ \frac{\partial}{\partial x}$$$$\begin{bmatrix}

(EA) \frac{\partial u}{\partial x} \end{bmatrix} + f = m\ddot{u} $$

Discrete EOM:


 * - Kd + F = M \underline{\ddot{u}}


 * $$ \rightarrow (2) \qquad \underline{M}\underline{\ddot{u}} + \underline{K}\underline{d} = \underline{F} $$


 * This is a multiple degree of freedom system, M.D.O.F.
 * A single degree of freedom system, S.D.O.F., is a system that involves something like a mass and spring (This is an identical representation of what is being taught in the University of Florida Vibrations course).

deriving the equation (2) from equation (1), the following equation (3) is retrieved:


 * (3) $$ \int\limits_{0}^{\alpha = L}w(x)$$

$$ \{ $$ $$ \frac{\partial}{\partial x} $$ $$ \begin{bmatrix} (EA) \frac{\partial u}{\partial x} \end{bmatrix}$$ $$ + f - m\ddot{u} \} = 0 \rightarrow $$ for all possible weighting functions, w(x).

(1) \rightarrow </math (3) trivial (3) \rightarrow </math (1) not trivial


 * Equation (3) can be rewritten as:


 * $$ \int w(x)g(x)dx = 0$$ for all w(x)

Since equation (3) holds true for all w(x), select W(x) = g(x), then equation (3) would become:


 * $$ \int g^2, dx = 0 \rightarrow g(x) = 0 $$

where: $$ g^2 is \geqq 0 $$

Integration by Parts: r(x), s(x)
$$ (rs)' = r's + rs' $$

$$ r' = \frac{dr}{dx} \quad, \quad s' = \frac{ds}{dx}$$

$$ \underbrace{\int(rs)'}_{rs} = \int r's + \int rs' $$

$$\Rightarrow \int r's = rs - \int rs' $$

Recall continuous PVW (Eqn. (3) from meeting 29)


 * 1st term: $$ \qquad r(x) = (EA)\frac{\partial u}{\partial x} $$
 * 2nd term: $$ \qquad s(x) = W(x) $$

By integration by parts:

$$ \int\limits_{x=0}^{x=L} \underbrace{W(x)}_{s} \frac{\partial }{\partial x} \underbrace{\left [(EA) \frac{\partial u}{\partial x}\right ]}_{r} dx = \left [W (EA) \frac{\partial u}{\partial x}  \right]_{x=0}^{x=L} - \int\limits_{0}^{L}\frac{dW}{dx}(EA)\frac{\partial u}{\partial x}dx $$

$$ \Rightarrow W(L) \underbrace{(EA)(L)\frac{\partial u}{\partial x}(L,t)}_{F=N(L,t)} - W(0) \underbrace{(EA)(0)\frac{\partial u}{\partial x}(0,t)}_{N(0,t)}- \int\limits_{0}^{L}\frac{dW}{dx}(EA)\frac{\partial u}{\partial x}dx $$

Model Problem
Image

At x = 0, select W(x) so that W(0) = 0, meaning the system is kinematically admissible.

Motivation: Discrete PVW applied to the equation shown below:

$$ \mathbf{W}_{6x1}\left(\left[ \mathbf{K} \right]_{6x2}\begin{Bmatrix} d_{3} \\ d_{4}\end{Bmatrix}_{2x1}-\mathbf{F}_{6x1}\right)=\mathbf{0}_{1x1} $$ for all $$ \mathbf{W} $$

where $$ \mathbf{F}^{T}=\left[F_{1 }F_{2} F_{3} F_{4} F_{5} F_{6}\right] $$

$$ F_{3}, F_{4} $$ are known reactions and $$ F_{1}, F_{2}, F_{5}, F_{6} $$ are unknown reactions.

Since $$ \mathbf{W} $$ can be selected arbitrarily, select $$ \mathbf{W} $$ such that $$ W_{1}=W_{2}=W_{5}=W_{6}=0 $$ so to eliminate equations involving unknown reactions $$\Rightarrow$$ This eliminates rows 1, 2, 5, and 6. The result is shown below:

$$ \mathbf{K}_{2x2}\mathbf{d}_{2x1}=\mathbf{F}_{2x1} \Rightarrow$$ Equation (1)

Note: $$ \mathbf{W}\left(\mathbf{K}\mathbf{d}-\mathbf{F}\right)=0 $$ for all $$ \mathbf{W} $$

Back to the continuous principle of virtual work

The unknown reaction is:

$$ N(0,t)=(EA)(0)\frac{\partial u}{\partial x}(0,t) $$

Due to continuous PVW, the equation then becomes:

$$ W\left(L\right)F(t)-\int_{0}^{L}\frac{\partial W}{\partial x}(EA)\frac{\partial u}{\partial x}dx+\int W(x)\left[f-m\ddot{u}\right]dx=0 $$ for all $$ W\left(x\right) $$ such that $$ W\left(0\right)=0 $$

The final equation can be written as:

$$ \int_{0}^{L}W\left(m\ddot{u}\right)dx+\int_{0}^{L} \frac{\partial W}{\partial x}(EA)\frac{\partial u}{\partial x}dx=W(L)F(t)+\int_{0}^{L}Wfdx $$ for all $$ W\left(x\right) $$ such that $$ W\left(0\right)=0 $$

Meeting 31: Friday, 7 November 2008. EML4500
Displacement within element $$i$$ is linear displacement $$u(x)$$ for $$x_{i}\le x\le x_{i+1}$$, (i.e., $$x\in \left[ x_{i},x_{i+1} \right]$$)

Motivation for linear interpolation of $$u(x)$$

Deformed shape is a straight line (linear interpolation) of the displacement of 2-nodes.

Consider the case where there are only axial displacements:

Q: Express $$x$$ in terms of $$u_{i}$$ $$(u(x_{i})\text{ and }d_{i+1}=u(x_{i+1}))$$ as a linear function of $$x$$ (i.e. linear interpolation)

$$u(x)=N_{i}(x)d_{i}+N_{i+1}(x)d_{i+1}$$

express $$N_{i}(x)$$ and $$N_{i+1}(x)$$ as linear functions of $$x$$.

$$N_{i}(x)=N_{1}^{(i)}(\tilde{x})$$

$$N_{i+1}(x)=\frac{x-x_{i}}{x_{i+1}-x_{i}}$$

This linear interpolation utilizes the geometry properties of similar triangles in order to provide the unknown value, which lies in between two known values.

Meeting 32: Monday, 10 November 2008. EML4500
32-1 In the beginning of class, a video was watched about a Rude-Goldberg machine as well as Walter Benjamin (Pronounced Valter Benyamin in German; another word like this is Volkswagen, which is pronounced like Folksvagen).

After this, there was a long discussion of "Honesty, imagination, and ethics" as well as narration in homework reports.

After this, previous slides from meeting 31 were shown and ta;led about

Cont. of PVW to Discrete PVW (continued)
Legrangian Interpolation

The equation: u(x) = Ni(x)di + Ni+1(x)di+1 was talked about from page 31-3.

Motivation
Motivation for the form of Ni(x) and Ni+1(x)

(1) Ni(x) and Ni+1(x) are linear (slight lines), thus any linear combination of Ni and Ni+1 is also linear, and in particular for the expression of u(x) on page 31-3.

Ni(x) = $$\alpha$$i + $$\beta$$i

Ni+1(x) = $$\alpha$$i+1 + $$\beta$$i+1  (Numbers: $$\alpha$$i+1, $$\beta$$i+1 32-2

Linear Combinations
Linear combinations of Ni and Ni+1:


 * Nidi + Ni+1di+1 = ($$\alpha$$i + $$\beta$$ix)di + ($$\alpha$$i+1 + $$\beta$$i+1x)di+1


 * ($$\alpha$$idi + $$\alpha$$i+1di+1) + $$\beta$$idi + $$\beta$$i+1di+1)x

as you can see from this result, this is clearly a linear function in x.

(2) If you recall, the interpretation of the equation of u(x) from page 31-2:


 * u(xi) = Ni(xi)di + Ni+1(xi)di+1 = di

because:


 * Ni(xi) = 1


 * Ni+1(xi) = 0

Meeting 33: Wednesday, 12 November 2008. EML4500
FEM vs. PVW (cont.)

$$ u(x_{i+1}) {=} d_{i+1} $$

p. 31-1: interpolation for u(x)

Apply same interpolation for w(x), i.e.

$$ w(x) {=} N_i(x)w_i + N_{i+1}(x)w_{i+1} $$

Element stiffness matrix for element i:

$$ \int_{x_i}^{x_{i+1}} [N_i^'w_i + N_{i+1}^'w_{i+1}](EA)[N_i^'d_i + N_{i+1}^'d_{i+1}]\, dx $$

$$ N_i^' :{=} \frac{dN_i(x)}{dx} $$

Likewise for $$ N_{i+1}^' $$:

$$ N_{i+1}^' :{=} \frac{dN_{i+1}(x)}{dx} $$

Note: $$ u(x) {=} \underbrace{ \begin{vmatrix} N_i(x) & N_{i+1}(x) \end{vmatrix}}_{N(x)_{1x2}} \begin{Bmatrix} d_i \\ d_{i+1} \end{Bmatrix}_{2x1} $$

$$ \frac{du(x)}{dx} = \underbrace{ \begin{vmatrix} N_i^'(x) & N_{i+1}^'(x) \end{vmatrix}}_{\underline{B}(x)_{1x2}} \begin{Bmatrix} d_i \\ d_{i+1} \end{Bmatrix}_{2x1} $$

Similarly: $$ w(x) {=} \underline{N}(x) \begin{Bmatrix} W_i \\ W_{i+1} \end{Bmatrix} $$

$$ \frac{dW(x)}{dx} {=} \underline{B}(x) \begin{Bmatrix} W_i \\ W_{i+1} \end{Bmatrix} $$

Recall the element degree of freedom conventions:



$$ \begin{Bmatrix} d_i \\ d_{i+1} \end{Bmatrix} {=} \begin{Bmatrix} d_1^{(i)} \\ d_2^{(i)} \end{Bmatrix} {=} \underline{d}^{(i)} $$

$$ \begin{Bmatrix} w_i \\ w_{i+1} \end{Bmatrix} {=} \begin{Bmatrix} w_1^{(i)} \\ w_2^{(i)} \end{Bmatrix} {=} \underline{w}^{(i)} $$

$$ \beta {=} \int_{x^i}^{x^{i+1}} \underbrace{(\underline{B} \underline{w}^{(i)})}_{1x1} \underbrace{ (EA)}_{1x1} \underbrace{(\underline{B}\underline{d}^{(i)})}_{1x1}, dx (scalar) $$

$$ {=} \underline{w}^i \cdot (\underline{k}^{(i)} \underline{d}^{(i)}) $$

$$ \beta {=} \int_{x^i}^{x^{i+1}} (EA) \underbrace{(\underline{B} \underline{w}^{(i)})}_{1x1} \cdot \underbrace{(\underline{B}\underline{d}^{(i)})}_{1x1} dx $$

$$ (\underline{B}\underline{w}^{(i)})^T(\underline{B}\underline{d}^{(i)}) $$

$$ (\underline{B}\underline{w}^{(i)})^T {=} \underline{w}^{(i)T} \underline{B}^T {=} \underline{w}^{(i)} \cdot \underline{B}^T $$

\beta {=} \underline{w}^{(i)} \cdot (\int \underline{B}^T(EA)\underline{B} dx )d^{(i)}

$$ \underline{k}^{(i)}_{2x2} {=} \int_{x_i}^{x_{i+1}} \underbrace{\underline{B}^T(x)}_{2x1} \underbrace{(EA)}_{\underbrace{(x)}_{1x1}} \underbrace{\underline{B}(x)}_{1x2} dx $$

$$ B(x) {=} \begin{vmatrix} HW & \frac{1}{L^{(i)}} \end{vmatrix} $$

$$ L^{(i)} {=} x_{i+1} - x_i $$ (Length of element i)

HW 6: Consider EA = const.

$$ k^{(i)} {=} \frac{EA}{L^{(i)}} \begin{vmatrix} 1 & -1 \\ -1 & 1 \end{vmatrix} $$

Transfer of variable coordinates from x to $$ \tilde{x} $$

$$ \tilde{x} :{=} x - x_i $$

$$ d\tilde{x} {=} dx $$

$$ k^{(i)} {=} \int^{\tilde{x} {=} L^{(i)}}_{\tilde{x} {=} 0} \underline{B}^T(\tilde{x})(EA)(\tilde{x})(\underline{B}(\tilde{x}) d\tilde{x} $$

HW6: Find expression for $$ \underline{k}^{(i)} $$ using above:



$$ A(\tilde{x}) {=} N_1^{(i)}(\tilde{x})A_1 + N_2^{(i)}(\tilde{x})A_2 $$

$$ E(\tilde{x}) {=} N_1^{(i)}(\tilde{x})E_1 + N_2^{(i)}(\tilde{x})E_2 $$

$$ \underline{k}^{(i)} {=} ? $$

Meeting 34: Friday, 14 November 2008. EML4500
p.31-4: N1(1) $$ (\tilde{x}) $$ = HW6

N2(1) $$ (\tilde{x}) = \frac{\tilde{x}}{L^{(i)}} = \begin{cases} 0, & \mbox{at }\tilde{x}\mbox{ = 0} \\ 1, & \mbox{at }\tilde{x}\mbox{ = L} \end{cases} $$

shape function N1(i), N2(i)

(basis)

HW6 : book p.159

set E1 = E2 = E

Let A($$ \tilde{x} $$) be linear as on p.33-5

Obtain k (i) from previous problem (p.33-5) and compare to expression given in book.

$$ \frac{E}{L^{(i)}} \underbrace{\frac{(A_1 + A_2}{2}}_{Ave. \ Area} \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} $$ = k (i)



Next, compare the general k (i) on p.33-5 to the stiffness matrix obtained by using $$ \frac{1}{2} (A_1 + A_2) $$. Note: $$ E_1 \ne E_2 $$

$$ \frac{(E_1 + E_2)(A_1 + A_2)}{4L^{(i)}} \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} $$ = k (i)ave.

Find k (i) - k (i)ave.

Rem : Recall the Mean Value Theorem (MVT) and its relation to centroid:

MVT: $$ \int_{x=a}^{x=b} f(x)\, dx = f(\overline{x})[b-a] $$

for $$ \overline{x} \ \epsilon \ [b-a] $$

$$ \epsilon \ $$= "belongs to"

$$ a \le \overline{x} \le b $$

$$ \int_{A}^{} x\, dx = \overline{x} \int_{A}^{} \,dA = \overline{x} A $$

$$ \int_{x=a}^{x=b} f(x)g(x)\,dx = f(\overline{x})g(\overline{x}) [b-a] $$

$$ a \le \overline{x} \le b $$

But $$ \ f(\overline{x}) \ \ne \ \underbrace{\frac{1}{b-a} \ \int_{a}^{b} f(x) \, dx}_{Ave. \ value \ of \ f} $$

In general

$$ \ g(\overline{x}) \ \ne \ \underbrace{\frac{1}{b-a} \ \int_{a}^{b} g(x) \, dx}_{Ave. \ value \ of \ g} $$

Modify 2-bar truss code to accommodate general k (i) on p.33-5



Three Bar Truss Redo
Below are the plots of the original undeformed 3 bar truss followed by an individual plot of each of the five zero eigenvectors.

Here is the code for the Unstable 3 bar truss:

And here are the results showing the stiffness matrix K, and the eigenvalues and eigenvectors:

K =

0    0     0     0     0     0     0     0     0     6     0    -6     0     0     0     0     0     0     6     0    -6     0     0     0     0    -6     0     6     0     0     0     0     0     0    -6     0     6     0     0     0     0     0     0     0     0     6     0    -6     0     0     0     0     0     0     0     0     0     0     0     0     0    -6     0     6

V =

Columns 1 through 7

1.0000        0         0         0         0         0         0         0   -0.7071         0         0         0         0         0         0         0   -0.7071         0         0   -0.7071         0         0   -0.7071         0         0         0         0         0         0         0   -0.7071         0         0    0.7071         0         0         0         0         0   -0.7071         0   -0.7071         0         0         0    1.0000         0         0         0         0         0         0         0   -0.7071         0    0.7071

Column 8

0  -0.7071         0    0.7071         0         0         0         0

D =

0    0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0    12     0     0     0     0     0     0     0     0    12     0     0     0     0     0     0     0     0    12

3 Bar Truss Stabalized by Addition of Another Bar Element Redo
Below are the plots of the original Stabalized 4 bar truss, along with individual plots of each of the zero eigenvectors.

Here is the code for the Stable 4 Bar Truss

Below once again are the stiffness matrix and the eigenvectors and eigenvalues

K =

Columns 1 through 7

2.1213   2.1213         0         0   -2.1213   -2.1213         0    2.1213    8.1213         0   -6.0000   -2.1213   -2.1213         0         0         0    6.0000         0   -6.0000         0         0         0   -6.0000         0    6.0000         0         0         0   -2.1213   -2.1213   -6.0000         0    8.1213    2.1213         0   -2.1213   -2.1213         0         0    2.1213    8.1213         0         0         0         0         0         0         0         0         0         0         0         0         0   -6.0000         0

Column 8

0        0         0         0         0   -6.0000         0    6.0000

V =

Columns 1 through 7

0.2372   0.1828   -0.6190   -0.3565    0.5948   -0.0000    0.0000    0.2512   -0.5293    0.1627    0.1742    0.1586    0.2656    0.5127    0.5834    0.0231    0.0449   -0.2390   -0.4362   -0.5767   -0.0263    0.2512   -0.5293    0.1627    0.1742    0.4362   -0.2656   -0.5127    0.5834    0.0231    0.0449   -0.2390   -0.1586    0.5767    0.0263   -0.0950   -0.3695   -0.5012    0.0566   -0.1586   -0.3112    0.4863    0.3447    0.3636   -0.2394    0.8317    0.0000    0.0000    0.0000   -0.0950   -0.3695   -0.5012    0.0566   -0.4362    0.3112   -0.4863

Column 8

0.2150   0.4914    0.2764   -0.2764   -0.4914   -0.4914    0.0000    0.2764

D =

Columns 1 through 7

-0.0000        0         0         0         0         0         0         0   -0.0000         0         0         0         0         0         0         0   -0.0000         0         0         0         0         0         0         0    0.0000         0         0         0         0         0         0         0    3.8183         0         0         0         0         0         0         0   12.0000         0         0         0         0         0         0         0   12.0000         0         0         0         0         0         0         0

Column 8

0        0         0         0         0         0         0   16.6670

EDU>>

91 Bar Electric Pylon (Superman)
Below are the plots of the undeformed and deformed electric pylon, along with plots of the three lowest eigenvectors





Here is the code used to solve for the reactions of the Electric Pylon

Below are the results as caluclated by MatLab

d =

0        0         0         0   -0.0002    0.0002   -0.0001   -0.0000   -0.0002   -0.0003   -0.0001    0.0003   -0.0001   -0.0001   -0.0001   -0.0005    0.0004    0.0003    0.0005   -0.0006    0.0010    0.0006    0.0010   -0.0002    0.0011   -0.0010    0.0017    0.0010    0.0017    0.0002    0.0017   -0.0006    0.0017   -0.0013    0.0023    0.0006    0.0022   -0.0010    0.0047    0.0024    0.0040   -0.0024    0.0064    0.0057    0.0064    0.0038    0.0064    0.0030    0.0064    0.0024    0.0064    0.0016    0.0064    0.0009    0.0063   -0.0002    0.0062   -0.0013    0.0061   -0.0024    0.0060   -0.0033    0.0059   -0.0051    0.0055   -0.0126    0.0068    0.0043    0.0070   -0.0063    0.0070    0.0035    0.0074   -0.0044    0.0072    0.0030    0.0071    0.0021    0.0071    0.0012    0.0072    0.0003    0.0073   -0.0007    0.0074   -0.0019    0.0076   -0.0034    0.0090    0.0030    0.0105   -0.0034

reactions =

1.0e+003 *

0.1496  -0.5017   -0.1496    1.5017

results =

1.0781e-005 2.1562e+006       862.48 -6.5356e-006 -1.3071e+006     -522.85 6.7339e-006 1.3468e+006       538.71 6.4556e-006 1.2911e+006       516.45 -3.5152e-006 -7.0304e+005     -281.22 -2.3798e-005 -4.7596e+006     -1903.8 -1.2325e-005 -2.465e+006      -986.01 7.1182e-006 1.4236e+006       569.46 1.9269e-005 3.8538e+006       1541.5 -2.8828e-005 -5.7655e+006     -2306.2 -1.8941e-007      -37882      -15.153 8.778e-009      1755.6      0.70224 1.9328e-005 3.8655e+006       1546.2 1.4698e-005 2.9395e+006       1175.8 -1.8494e-005 -3.6988e+006     -1479.5 -2.883e-005 -5.7661e+006     -2306.4 1.7948e-006 3.5896e+005       143.58 2.621e-005  5.242e+006       2096.8 9.8986e-006 1.9797e+006       791.89 -1.8874e-005 -3.7749e+006       -1510 -3.258e-005 -6.5159e+006     -2606.4 2.7892e-006 5.5784e+005       223.14 3.2194e-006 6.4388e+005       257.55 2.9828e-005 5.9656e+006       2386.3 -4.0852e-006 -8.1704e+005     -326.82 -1.9808e-005 -3.9616e+006     -1584.6 1.16e-005   2.32e+006       928.02 -4.0653e-006 -8.1307e+005     -325.23 -2.8778e-005 -5.7556e+006     -2302.2 -1.4237e-007      -28474       -11.39 -4.9677e-006 -9.9353e+005     -397.41 -7.6072e-008      -15214      -6.0857 2.9608e-005 5.9216e+006       2368.6 2.7907e-007       55813       22.325 -2.4358e-005 -4.8717e+006     -1948.7 6.9573e-006 1.3915e+006       556.59 -1.5116e-008     -3023.3      -1.2093 -2.8794e-005 -5.7588e+006     -2303.5 -2.427e-005 -4.854e+006      -1941.6 6.9522e-006 1.3904e+006       556.17 -9.4154e-007 -1.8831e+005     -75.323 1.129e-005  2.258e+006       903.22 -3.16e-006  -6.32e+005       -252.8 -2.0051e-005 -4.0102e+006     -1604.1 -2.0266e-019 -4.0531e-008 -1.6213e-011 7.3233e-019 1.4647e-007  5.8586e-011 3.2073e-007       64147       25.659 1.3501e-006 2.7001e+005       108.01 1.3575e-006 2.7149e+005        108.6 -1.0389e-005 -2.0778e+006     -831.13 -2.2136e-005 -4.4271e+006     -1770.9 -2.9388e-005 -5.8777e+006     -2351.1 -3.6774e-005 -7.3549e+006     -2941.9 -4.331e-005 -8.662e+006      -3464.8 -4.3478e-005 -8.6957e+006     -3478.3 -1.2903e-019 -2.5807e-008 -1.0323e-011 1.5471e-019 3.0942e-008  1.2377e-011 0           0            0  1.2205e-019  2.4409e-008  9.7638e-012 -8.8522e-007 -1.7704e+005     -70.818 1.1034e-006 2.2069e+005       88.276 -9.5965e-007 -1.9193e+005     -76.772 9.9436e-007 1.9887e+005       79.549 1.0732e-005 2.1464e+006       858.56 -1.0415e-005 -2.083e+006      -833.19 1.1448e-005 2.2895e+006       915.81 -1.1321e-005 -2.2643e+006     -905.71 1.0517e-005 2.1035e+006       841.38 -1.0732e-005 -2.1464e+006     -858.56 7.0689e-006 1.4138e+006       565.51 -6.8859e-006 -1.3772e+006     -550.87 7.9177e-006 1.5835e+006       633.42 -1.8464e-005 -3.6928e+006     -1477.1 -4.8696e-007      -97393      -38.957 -1.1845e-007      -23690      -9.4762 1.5604e-007       31208       12.483 4.5239e-005 9.0479e+006       3619.2 0           0            0  4.5158e-005  9.0317e+006       3612.7 -1.5877e-018 -3.1754e-007 -1.2702e-010 4.5256e-005 9.0511e+006       3620.5 -6.5877e-007 -1.3175e+005     -52.702 -1.4822e-006 -2.9645e+005     -118.58 8.9829e-006 1.7966e+006       718.63 2.2438e-005 4.4876e+006         1795 3.3117e-005 6.6233e+006       2649.3 3.8878e-005 7.7755e+006       3110.2 0           0            0  2.4017e-020  4.8034e-009  1.9214e-012 -1.9214e-019 -3.8428e-008 -1.5371e-011 1.0741e-019 2.1482e-008  8.5927e-012

Element_Stress =

2.1562e+006 -1.3071e+006 1.3468e+006 1.2911e+006 -7.0304e+005 -4.7596e+006 -2.465e+006 1.4236e+006 3.8538e+006 -5.7655e+006 -37882      1755.6  3.8655e+006 2.9395e+006 -3.6988e+006 -5.7661e+006 3.5896e+005 5.242e+006 1.9797e+006 -3.7749e+006 -6.5159e+006 5.5784e+005 6.4388e+005 5.9656e+006 -8.1704e+005 -3.9616e+006 2.32e+006 -8.1307e+005 -5.7556e+006 -28474 -9.9353e+005 -15214 5.9216e+006 55813 -4.8717e+006 1.3915e+006 -3023.3 -5.7588e+006 -4.854e+006 1.3904e+006 -1.8831e+005 2.258e+006 -6.32e+005 -4.0102e+006 -4.0531e-008 1.4647e-007 64147 2.7001e+005 2.7149e+005 -2.0778e+006 -4.4271e+006 -5.8777e+006 -7.3549e+006 -8.662e+006 -8.6957e+006 -2.5807e-008 3.0942e-008 0 2.4409e-008 -1.7704e+005 2.2069e+005 -1.9193e+005 1.9887e+005 2.1464e+006 -2.083e+006 2.2895e+006 -2.2643e+006 2.1035e+006 -2.1464e+006 1.4138e+006 -1.3772e+006 1.5835e+006 -3.6928e+006 -97393      -23690        31208  9.0479e+006 0 9.0317e+006 -3.1754e-007 9.0511e+006 -1.3175e+005 -2.9645e+005 1.7966e+006 4.4876e+006 6.6233e+006 7.7755e+006 0 4.8034e-009 -3.8428e-008 2.1482e-008

Tstress =

9.0511e+006

Tstress_Element =

81

Cstress =

-8.6957e+006

Cstress_Element =

55

lamda =

132.16      2468.4       2940.3

column =

79   82    83

T =

0.54654     0.12647      0.11587

The first column of results d, is the displacement experienced by the structure before a magnifying factor was applied. The next set is the reaction forces. Next is all of the strains and stresses applied to each element of the structure, it was observed that the highest tensile stress occurred in element number 81 and was equal to about 9.05e6 Pa. The highest compressive stress observed was in element 55 and was equal to about -8.7e6 Pa.

The electric pylon is not statically determinant since there are not enough equations to satisfy all of the unknowns.

Below are the three lowest eigenvalues of the structure as determined by MatLab:

132.16

2468.4

2940.3

Finally the three lowest vibrational periods are as follows:

0.54654

0.12647

0.11587

Below are the additional MatLab m-files created to assist in the analysis of the electric pylon:

ReducedMassM.m

and

ReducedStiffM.m

Contributions
--EML4500.f08.A-team.rieth 21:24, 10 November 2008 (UTC)

--EML4500.f08.A-team.melvin 23:04, 14 November 2008 (UTC)

--Eml4500.f08.a-team.robinson 22:17, 20 November 2008 (UTC)

--EML4500.f08.A-team.VandenBerg 22:32, 20 November 2008 (UTC)

--EML4500.f08.A-team.morford 21:05, 21 November 2008 (UTC)

EML4500.f08.A-team.kirley 23:20, 22 November 2008 (UTC)