User:Eml 4500.f08.delta 6.specht/HW6

=Homework 6=

Monday 11/3
PVW (Continuous case) Dynamics of Elastic Bar

PDE:

$$ \frac{d}{dx}[ (EA)\frac{\partial u}{\partial x} ] + \mathit{f} = m\ddot{u} $$ (1)

Note: Equation (1) is known as the discrete equation of motion (EOM)

The first term in equation (1) as seen below,

$$ \frac{d}{dx}[ (EA)\frac{\partial u}{\partial x} ] $$

will give rise to

$$ -Kd $$

Likewise, the second term,

$$ \mathit{f} = m\ddot{u} $$

will give rise to

$$ +F = M\ddot{d} $$

where M represents the mass matrix.

Thus,

$$ -Kd + F = M\ddot{d} \Rightarrow M\ddot{d} + Kd = F $$ (2)

Note: Equation (2) is a multi-degree of freedom equation

Goal: Derive (2) from (1)

Review: Associate a single degree of freedom system to a multi-degree of freedom system.



Note: Equation from single degree of freedom system looks very similar to the multi-degree of freedom system

Now deriving (2) from (1) results in equation (3),

$$ \int_{x = 0}^{x = L}{w(x)\left\{\frac{d}{dx} [ (EA) \frac{\partial u}{\partial x}] + f - m\ddot{u}\right\}dx = 0} $$

for all possible w(x), where w(x) represents the weighting function.

Note: (1) → (2) is trivial, but (3) → (1) is not trivial

Further, equation (3) can be rewritten as,

$$ \int w(x) g(x) dx = 0 $$

for all possible w(x)

Since equation (3) holds for all w(x), select w(x) = g(x). Thus, equation (3) becomes,

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

Since g(x)2 > 0, therefore, g(x) = 0.

Wednesday 11/5
Integration by Parts

Recalling the method of integration by pats for integration, it is described below.

We have two functions, $$r(x)$$ and $$s(x)$$. When we want to determine the derivative of the product of these two functions, we have:

$$(rs)'=r's+rs'$$

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

Then, $$(rs)=\int (rs)'dx=\int rs'dx+\int r'sdx$$ $$\int r'sdx=rs-\int rs'dx$$

Recalling the Principle of Virtual Work, we can apply this method of integration to this principle.

Let $$r(x)=EA \frac{\partial u}{\partial x}$$ and $$s(x)=w(x)$$

Now applying Integration by Parts,


 * $$\int _{x=0}^{L}w(x) \frac{\partial }{\partial x} \left[ EA \frac {\partial u}{\partial x} \right] dx$$
 * $$\left[w(EA)\frac {\partial u}{\partial x} \right]_{0}^{L}-\int_{0}^{L}\frac{dw}{dx}EA\frac{\partial u}{\partial x}dx$$
 * $$w(L)EA(L) \frac{\partial u(L,t)}{\partial x}-w(0)EA(0) \frac{\partial u(0,t)}{\partial x}- \int_{0}^{L} \frac{dw}{dx}EA\frac{\partial u}{\partial x}dx$$

It is noticeable that
 * $$N(L,t)=EA(L) \frac{\partial u(L,t)}{\partial x}$$

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

Now we need to account for the boundary conditions (BCs). We use the BC defined in the diagram below:



At the wall, (x=0), we have to select w(x) such that w(0)=0. This requirement would be kinematically admissible as it would adhere to the constraints.

Descrete PVW applied to the reduction of the global FD rel

 * $$w$$ $$_{6x1}$$•$$\left( \begin{bmatrix}

K \end{bmatrix}_{6x2}

\begin{pmatrix} d \end{pmatrix}_{2x1}-F_{6x1} \right)=0_{1x1}$$; for all $$w$$

We have the transpose of F:


 * $$F^T$$$$=\begin{bmatrix}

F_1 & F_2 & F_5 & F_6 \end{bmatrix}$$

where F_1, F_2, F_5, and F_6 are all unknown reactions.

Since $$w$$ can be selected arbitrarily, we say that $$w$$ is such that:
 * $$w_1=w_2=w_5=w_6=0$$

Therefore, to eliminate the equation involving unknown reactions (element rows 1, 2, 5, and 6), we say that:
 * $$\bar{w}_{2x1}$$•$$\left( \mathbf{\bar{K} \bar{d} \bar{F}} \right)=0$$; for all $$w$$.

This yields to,
 * $$\bar{K}_{2x2} \bar{d}_{2x1}$$ = $$\bar{F}_{2x1}$$

where,
 * $$\bar{d}_{2x1}$$=$$\begin{pmatrix}

d_3\\ d_4 \end{pmatrix}$$ and
 * $$\bar{F}_{2x1}$$=$$

\begin{pmatrix} F_3\\ F_4 \end{pmatrix}$$

Turning our attention back to the Principle of Virtual Work, and recalling the reaction terms,


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

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

we can build the following expression:


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

Rewriting in row form,


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

Stiffness Term


Note: Nodes do not have to be equidistant.

Plot


Note: Since you are drawing a straight line between nodes you are assuming the their displacement was linear.

Assume displacement $$ u\bigl(\begin{matrix}x\end{matrix}\bigr)$$ for $$ x_{i}\leq x\leq x_{i+1} $$ (i.e. $$x \epsilon \left [ x_i, x_{i+1} \right ]$$)

Motivation for linear interpolation of $$ u\bigl(\begin{matrix}x\end{matrix}\bigr)$$


In this plot can be easily observed that the deformed shape is a straight line confirming that there was an implicit assumption of the linear interpolation of the displacement between the two nodes. Consider the case where these are only axial displacements (i.e. 0 translational displacement).

Question: Express $$ u\bigl(\begin{matrix}x\end{matrix}\bigr)$$ in terms of $$d_{i}=u\bigl(\begin{matrix}x_{i}\end{matrix}\bigr)$$ and $$d_{i+1}=u\bigl(\begin{matrix}x_{i+1}\end{matrix}\bigr)$$ as a linear function in x (i.e. linear interpolation)

$$u\bigl(\begin{matrix}x_{i}\end{matrix}\bigr)=N_{i}(x)d_{i}+N_{i+1}(x)d_{i+1}$$

where $$N_{i}\bigl(\begin{matrix}x\end{matrix}\bigr)$$ and $$N_{i+1}\bigl(\begin{matrix}x\end{matrix}\bigr)$$ are linear functions of x.



Note: From this plot it can be easily observed that $$ N_{i+1}(x)=\frac{x-x_{i}}{x_{i+1}-x_{i}} $$.

Monday 11/10
Now three motivations will be given for the equation being used in the form $$N_{i}(x)+N_{i+1}(x)$$. 1st Motivation: It can be observed from earlier discussion that $$N_{i}(x)+N_{i+1}(x)$$ are linear or represented by straight lines. Therefore the expression for U(x) is linear. This is true because the combination of $$N_{i}(x)$$ and $$N_{i+1}(x)$$ is linear. $$N_{i}(x)$$ and $$N_{i+1}(x)$$ can be written in the typical linear format:

$$N_{i}(x)=\alpha_{i}+\beta_{i}x$$

$$N_{i+1}(x)=\alpha_{i+1}+\beta_{i+1}x$$

where $$\alpha_{i}$$ is the y-intercept of the line and $$\beta_{i}$$ is the slope of the line and they both represent real numbers.

So the linear combination of $$N_{i}(x)$$ and $$N_{i+1}(x)$$ is:

$$N_{i}d_{i}+N_{i+1}d_{i+1}=(\alpha_{i}+\beta_{i}x)d_{i}+(\alpha_{i+1}+\beta_{i+1}x)d_{i+1}$$

Now the like terms are combined to give:

$$U_{x}=(\alpha_{i}d_{i}+\alpha_{i+1}d_{i+1})+(\beta_{i}d_{i}+\beta_{i+1}d_{i+1})x$$

Where $$d_{i}$$ is the weighting coefficient. It can clearly be seen that this is a linear function of x.

2nd Motivation:

Recall the equation for the interpolation of U(x):$$U(x)=N_{i}(x)d_{i}+N_{i+1}(x)d_{i+1}$$

Now replace x with $$x_{i}$$ and the equation becomes:$$U(x)=N_{i}(x_{i})d_{i}+N_{i+1}(x_{i})d_{i+1}$$

In the drawing above it was shown that $$N_{i}(x_{i})d_{i}=1$$ and that $$N_{i+1}(x_{i})d_{i+1}=0$$. The eqaution for $$U(x_{i})$$ becomes:

$$U(x)=d_{i}$$

The equation shows that $$U(x_{i})$$ is equal to the weighting coefficient $$d_{i}$$.

FEM via PVW (cont.)
Apply same interpolation for W(X). For example:

$$ W(x)=N_{i}(x)W_{i}+N_{i+1}(x)W_{i+1} $$

Elemental Stiffness Matrix for Element i
Now the element stiffness matrix can be written as:

$$ \beta = \int_{x_{i}}^{x_{i+1}}{[N_{i}'W_{i}+N_{i+1}'W_{i+1}](EA)}[N_i'd_i+N_{i+1}'d_{i+1}]dx$$ where $$N_{i}'W_{i}+N_{i+1}'W_{i+1}=W'(x)$$ and $$N_i'd_i+N_{i+1}'d_{i+1}=U'(x)$$

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

Likewise, the relation is the same for Ni+1':

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


 * NOTE

$$ u(x)=\begin{bmatrix} N_i(x) & N_{i+1}(x) \end{bmatrix} \begin{Bmatrix} d_i\\ d_{i+1} \end{Bmatrix} =\mathbf{N(x)} \begin{Bmatrix} d_i\\ d_{i+1} \end{Bmatrix} $$

$$ \frac{du(x)}{dx}=\begin{bmatrix} N_i'(x) & N_{i+1}'(x) \end{bmatrix} \begin{Bmatrix} d_i\\ d_{i+1} \end{Bmatrix} =\mathbf{B(x)} \begin{Bmatrix} d_i\\ d_{i+1} \end{Bmatrix} $$

Similarly:

$$ W(x)=\mathbf{N(x)}\begin{Bmatrix} W_i\\ W_{i+1} \end{Bmatrix} $$

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

Now, let us recall the element degrees of freedom:



$$ \begin{Bmatrix} d_i\\ d_{i+1}

\end{Bmatrix} = \begin{Bmatrix} d_1^{(i)}\\ d_2^{(i)} \end{Bmatrix} = \mathbf{d^{(i)}} $$

$$ \begin{Bmatrix} W_i\\ W_{i+1}

\end{Bmatrix} = \begin{Bmatrix} W_1^{(i)}\\ W_2^{(i)} \end{Bmatrix} = \mathbf{W^{(i)}} $$

The element stiffness matrix now becomes: $$ \beta =\int_{x_{i}}^{x_{i+1}}{\left(\textbf{BW}^{(i)}\right)\left(EA \right)\left(\textbf{Bd}^{(i)}\right)}dx$$

$$ =\textbf{W}^{(i)}* \left(\textbf{k}^{(i)} \textbf{d}^{(i)}\right) $$

$$ \beta =\int_{x_i}^{X_{i+1}}(EA)(\mathbf{BW^{(i)}})*(\mathbf{Bd^{(i)}})dx-(\mathbf{BW^{(i)}})^T(\mathbf{Bd^{(i)}}) $$

where

$$ \mathbf{BW^{(i)}}= \mathbf{W^{(i)T}B^T} = \mathbf{W^{(i)}*B^T} $$

$$ \beta= \mathbf{W^{(i)}}*\int \mathbf{B^T}EA\mathbf{B}dx\mathbf{d^{(i)}} $$

$$ \mathbf{k^{(i)}}=\int_{x_i}^{x^{i+1}}{\mathbf{B(x)^T}}(EA)\mathbf{B(x)}dx $$

Transformation of variable coordinate system from $$x$$ to $$ \tilde{x} $$
$$ d\tilde{x}=dx $$

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

$$ \mathbf{k^{(i)}}=\int_{\tilde{x}=0}^{\tilde{x}=L^{(i)}}{\mathbf{B^T(\tilde{x})}}EA(\tilde{x})\mathbf{B(\tilde{x}})d(\tilde{x}) $$

Friday 11/14
The equation for area of a beam with a cross sectional area that changes linearly is shown below.

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

The same type of equation can be established for a beam with a non-constant modulus of elasticity. This is shown below.

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

The value for N that is used in each of these equations is shown in the equation below.

$$ N_{i}(x)=N_1^{(i)}(\tilde{x})=1-\frac{\tilde{x}}{L^{(i)}} $$

This equation is designed to satisfy the two boundary conditions:

$$ (1)N_1^{(i)}(\tilde{x})= 1 $$ at $$\tilde{x}=0$$ $$ (2)N_1^{(i)}(\tilde{x})= 0 $$ at $$\tilde{x}=L^{(i)}$$

$$ N_{i+1}(x)=N_2^{(i)}(\tilde{x})=\frac{\tilde{x}}{L^{(i)}} $$

This equation is designed to satisfy the two boundary conditions:

$$ (1)N_2^{(i)}(\tilde{x})= 0 $$ at $$\tilde{x}=0$$ $$ (2)N_2^{(i)}(\tilde{x})= 1 $$ at $$\tilde{x}=L^{(i)}$$

$$ k^{(i)}=\frac{E\left( \frac{A_1+A_2}{2}\right)}{L^{(i)}}\begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix} $$

$$ k^{(i)}_{ave}=\frac{\left(\frac{E_1+E_2}{2} \right)\left( \frac{A_1+A_2}{2}\right)}{L^{(i)}}\begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix} $$

Mean Value Theorem
$$ \int_{x=a}^{x=b}{f(x)dx}=f(\bar{x})[b-a] $$

The above integral is true for any $$\bar{x}\epsilon [a,b]$$. This statement means that it is true within the interval $$a\leq \bar{x}\leq b$$.

In order to find $$\bar{x}$$ the centroid of the mass in question should be found, which is located at $$ (\bar{x},\bar{y},\bar{z})$$. The x-component of the centroid can be found by the following relation:

$$ \int_{A}{xdA}=\bar{x}\int_A{dA}=\bar{x} A $$

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

This statement is true within the interval $$a\leq \bar{x}\leq b$$. The following general equations show that each individual component cannot be evaluated in this same manner.

$$ f(\bar{x})\neq \frac{1}{b-a}\int_{a}^{b}{f(x)dx} $$

$$ g(\bar{x})\neq \frac{1}{b-a}\int_{a}^{b}{g(x)dx} $$

Electric Pylon MATLAB Code Used for Analysis
% Matlab Script %********************************************************************* % filename: electric_pylon.m % % PURPOSE: %  Plot an electric pylon ("superman") structure with matlab %  Perform required FEA analysis % % AUTHOR: %   Team Delta_6 %   EML 4500 Finite Element Analysis and Design  Fall 2008. % % Modified on: vql, Fri, 21 Nov 2008, 08:37:46 EDT % Created on : Spring 2000 % % DEPENDENCIES: %  call: % % REMARKS: % %*********************************************************************

% Plot the "superman" structure %

clear; close;

% model with 3-D beam elements dof = 2;         %  dof per node:  1= disp x,  2= disp y A=.0004;          %  Area of the elements in (m^2) e=200000000000;  %  Young's modulus of the elements in (Pa) p=7800;          %  Density of the elements in (kg/m^3) P=-1000;         %  Downward Vertical Force applied at node #33

% obtain the coordinates of all nodes %  n_node = 46;             % number of nodes n_elem = 91;            % number of elements total_dof = 2 * n_node; % total dof of system

position(:, 1) = (60/7.43)*[ 1.5; 0; 0]; position(:, 2) = (60/7.43)*[ 4.53; 0; 0];

position(:, 3) = (60/7.43)*[ 1.88; 1.18; 0]; position(:, 4) = (60/7.43)*[ 3.03; 1.18; 0]; position(:, 5) = (60/7.43)*[ 4.17; 1.18; 0];

position(:, 6) = (60/7.43)*[ 2.2; 2.181; 0]; position(:, 7) = (60/7.43)*[ 3.03; 2.181; 0]; position(:, 8) = (60/7.43)*[ 3.85; 2.181; 0];

position(:, 9) = (60/7.43)*[ 2.53; 3.181; 0]; position(:, 10) = (60/7.43)*[ 3.53; 3.181; 0];

position(:, 11) = (60/7.43)*[ 2.34; 3.681; 0]; position(:, 12) = (60/7.43)*[ 3.03; 3.681; 0]; position(:, 13) = (60/7.43)*[ 3.71; 3.681; 0];

position(:, 14) = (60/7.43)*[ 2.16; 4.181; 0]; position(:, 15) = (60/7.43)*[ 2.62; 4.181; 0]; position(:, 16) = (60/7.43)*[ 3.44; 4.181; 0]; position(:, 17) = (60/7.43)*[ 3.89; 4.181; 0];

position(:, 18) = (60/7.43)*[ 2.36; 4.5; 0]; position(:, 19) = (60/7.43)*[ 3.7; 4.5; 0];

position(:, 20) = (60/7.43)*[ 1.8; 5.172; 0]; position(:, 21) = (60/7.43)*[ 4.25; 5.172; 0];

position(:, 22) = (60/7.43)*[ 0; 6; 0]; position(:, 23) = (60/7.43)*[ 1.06; 6; 0]; position(:, 24) = (60/7.43)*[ 1.5; 6; 0]; position(:, 25) = (60/7.43)*[ 1.82; 6; 0]; position(:, 26) = (60/7.43)*[ 2.22; 6; 0]; position(:, 27) = (60/7.43)*[ 2.71; 6; 0]; position(:, 28) = (60/7.43)*[ 3.34; 6; 0]; position(:, 29) = (60/7.43)*[ 3.84; 6; 0]; position(:, 30) = (60/7.43)*[ 4.23; 6; 0]; position(:, 31) = (60/7.43)*[ 4.55; 6; 0]; position(:, 32) = (60/7.43)*[ 5; 6; 0]; position(:, 33) = (60/7.43)*[ 6.05; 6; 0];

position(:, 34) = (60/7.43)*[ .8; 6.23; 0]; position(:, 35) = (60/7.43)*[ 5.25; 6.23; 0];

position(:, 36) = (60/7.43)*[ 1.24; 6.355; 0]; position(:, 37) = (60/7.43)*[ 4.82; 6.355; 0];

position(:, 38) = (60/7.43)*[ 1.5; 6.43; 0]; position(:, 39) = (60/7.43)*[ 2; 6.43; 0]; position(:, 40) = (60/7.43)*[ 2.48; 6.43; 0]; position(:, 41) = (60/7.43)*[ 3.03; 6.43; 0]; position(:, 42) = (60/7.43)*[ 3.58; 6.43; 0]; position(:, 43) = (60/7.43)*[ 4.05; 6.43; 0]; position(:, 44) = (60/7.43)*[ 4.55; 6.43; 0];

position(:, 45) = (60/7.43)*[ 1.5; 7.43; 0]; position(:, 46) = (60/7.43)*[ 4.55; 7.43; 0];

% print the node coord. for i = 1 : n_node; x(i) = position(1,i); y(i) = position(2,i); z(i) = position(3,i);

end

node_connect(1, 1) = 1;  % element 1 node_connect(2, 1) = 3;

node_connect(1, 2) = 1;  % element 2 node_connect(2, 2) = 4;

node_connect(1, 3) = 3;  % element 5 node_connect(2, 3) = 4;

node_connect(1, 4) = 4;  % element 3 node_connect(2, 4) = 2;

node_connect(1, 5) = 4;  % element 6 node_connect(2, 5) = 5;

node_connect(1, 6) = 2;  % element 4 node_connect(2, 6) = 5;

node_connect(1, 7) = 3;  % element 8 node_connect(2, 7) = 7;

node_connect(1, 8) = 5;  % element 9 node_connect(2, 8) = 7;

node_connect(1, 9) = 3;  % element 7 node_connect(2, 9) = 6;

node_connect(1, 10) = 5;  % element 10 node_connect(2, 10) = 8;

node_connect(1, 11) = 6;  % element 11 node_connect(2, 11) = 7;

node_connect(1, 12) = 7;  % element 12 node_connect(2, 12) = 8;

node_connect(1, 13) = 6;  % element 13 node_connect(2, 13) = 9;

node_connect(1, 14) = 7;  % element 14 node_connect(2, 14) = 9;

node_connect(1, 15) = 7;  % element 15 node_connect(2, 15) = 10;

node_connect(1, 16) = 8;  % element 16 node_connect(2, 16) = 10;

node_connect(1, 17) = 9;  % element 17 node_connect(2, 17) = 10;

node_connect(1, 18) = 9;  % element 18 node_connect(2, 18) = 11;

node_connect(1, 19) = 9;  % element 19 node_connect(2, 19) = 12;

node_connect(1, 20) = 10;  % element 20 node_connect(2, 20) = 12;

node_connect(1, 21) = 10;  % element 21 node_connect(2, 21) = 13;

node_connect(1, 22) = 11;  % element 22 node_connect(2, 22) = 12;

node_connect(1, 23) = 12;  % element 23 node_connect(2, 23) = 13;

node_connect(1, 24) = 11;  % element 24 node_connect(2, 24) = 14;

node_connect(1, 25) = 11;  % element 25 node_connect(2, 25) = 15;

node_connect(1, 26) = 12;  % element 26 node_connect(2, 26) = 15;

node_connect(1, 27) = 12;  % element 27 node_connect(2, 27) = 16;

node_connect(1, 28) = 13;  % element 28 node_connect(2, 28) = 16;

node_connect(1, 29) = 13;  % element 29 node_connect(2, 29) = 17;

node_connect(1, 30) = 14;  % element 30 node_connect(2, 30) = 15;

node_connect(1, 31) = 15;  % element 31 node_connect(2, 31) = 16;

node_connect(1, 32) = 16;  % element 32 node_connect(2, 32) = 17;

node_connect(1, 33) = 14;  % element 33 node_connect(2, 33) = 20;

node_connect(1, 34) = 14;  % element 34 node_connect(2, 34) = 18;

node_connect(1, 35) = 15;  % element 35 node_connect(2, 35) = 18;

node_connect(1, 36) = 16;  % element 36 node_connect(2, 36) = 19;

node_connect(1, 37) = 17;  % element 37 node_connect(2, 37) = 19;

node_connect(1, 38) = 17;  % element 38 node_connect(2, 38) = 21;

node_connect(1, 39) = 18;  % element 39 node_connect(2, 39) = 20;

node_connect(1, 40) = 19;  % element 40 node_connect(2, 40) = 21;

node_connect(1, 41) = 20;  % element 41 node_connect(2, 41) = 24;

node_connect(1, 42) = 20;  % element 42 node_connect(2, 42) = 26;

node_connect(1, 43) = 21;  % element 43 node_connect(2, 43) = 29;

node_connect(1, 44) = 21;  % element 44 node_connect(2, 44) = 31;

node_connect(1, 45) = 22;  % element 45 node_connect(2, 45) = 23;

node_connect(1, 46) = 23;  % element 46 node_connect(2, 46) = 24;

node_connect(1, 47) = 24;  % element 47 node_connect(2, 47) = 25;

node_connect(1, 48) = 25;  % element 48 node_connect(2, 48) = 26;

node_connect(1, 49) = 26;  % element 49 node_connect(2, 49) = 27;

node_connect(1, 50) = 27;  % element 50 node_connect(2, 50) = 28;

node_connect(1, 51) = 28;  % element 51 node_connect(2, 51) = 29;

node_connect(1, 52) = 29;  % element 52 node_connect(2, 52) = 30;

node_connect(1, 53) = 30;  % element 53 node_connect(2, 53) = 31;

node_connect(1, 54) = 31;  % element 54 node_connect(2, 54) = 32;

node_connect(1, 55) = 32;  % element 55 node_connect(2, 55) = 33;

node_connect(1, 56) = 22;  % element 56 node_connect(2, 56) = 34;

node_connect(1, 57) = 23;  % element 57 node_connect(2, 57) = 34;

node_connect(1, 58) = 23;  % element 58 node_connect(2, 58) = 36;

node_connect(1, 59) = 24;  % element 59 node_connect(2, 59) = 36;

node_connect(1, 60) = 24;  % element 60 node_connect(2, 60) = 38;

node_connect(1, 61) = 25;  % element 61 node_connect(2, 61) = 38;

node_connect(1, 62) = 25;  % element 62 node_connect(2, 62) = 39;

node_connect(1, 63) = 26;  % element 63 node_connect(2, 63) = 39;

node_connect(1, 64) = 26;  % element 64 node_connect(2, 64) = 40;

node_connect(1, 65) = 27;  % element 65 node_connect(2, 65) = 40;

node_connect(1, 66) = 27;  % element 66 node_connect(2, 66) = 41;

node_connect(1, 67) = 28;  % element 67 node_connect(2, 67) = 41;

node_connect(1, 68) = 28;  % element 68 node_connect(2, 68) = 42;

node_connect(1, 69) = 29;  % element 69 node_connect(2, 69) = 42;

node_connect(1, 70) = 29;  % element 70 node_connect(2, 70) = 43;

node_connect(1, 71) = 30;  % element 71 node_connect(2, 71) = 43;

node_connect(1, 72) = 30;  % element 72 node_connect(2, 72) = 44;

node_connect(1, 73) = 31;  % element 73 node_connect(2, 73) = 44;

node_connect(1, 74) = 31;  % element 74 node_connect(2, 74) = 37;

node_connect(1, 75) = 32;  % element 75 node_connect(2, 75) = 37;

node_connect(1, 76) = 32;  % element 76 node_connect(2, 76) = 35;

node_connect(1, 77) = 33;  % element 77 node_connect(2, 77) = 35;

node_connect(1, 78) = 34;  % element 78 node_connect(2, 78) = 36;

node_connect(1, 79) = 35;  % element 79 node_connect(2, 79) = 37;

node_connect(1, 80) = 36;  % element 80 node_connect(2, 80) = 38;

node_connect(1, 81) = 37;  % element 81 node_connect(2, 81) = 44;

node_connect(1, 82) = 38;  % element 82 node_connect(2, 82) = 39;

node_connect(1, 83) = 39;  % element 83 node_connect(2, 83) = 40;

node_connect(1, 84) = 40;  % element 84 node_connect(2, 84) = 41;

node_connect(1, 85) = 41;  % element 85 node_connect(2, 85) = 42;

node_connect(1, 86) = 42;  % element 86 node_connect(2, 86) = 43;

node_connect(1, 87) = 43;  % element 87 node_connect(2, 87) = 44;

node_connect(1, 88) = 38;  % element 88 node_connect(2, 88) = 45;

node_connect(1, 89) = 39;  % element 89 node_connect(2, 89) = 45;

node_connect(1, 90) = 43;  % element 90 node_connect(2, 90) = 46;

node_connect(1, 91) = 44;  % element 91 node_connect(2, 91) = 46;

% connect all beam elements by connectivity array for i = 1 : n_elem; node_1 = node_connect(1,i); node_2 = node_connect(2,i); xx = [x(node_1),x(node_2)]; yy = [y(node_1),y(node_2)]; zz = [z(node_1),z(node_2)]; plot3(xx,yy,zz,'--') hold on  end

title('Electric Pylon') xlabel('x') ylabel('y') zlabel('z')

view([0 0 1])     % xy plane view % view([0 1 0])    % xz plane view %view([1 0 0])    % yz plane view

% produce postscript file for later printing % print -dps electric_pylon.ps  print -deps electric_pylon.ps

R=zeros(dof*n_node,1); R(66)=P; R; %Location Master Matrix Array

for i = 1:n_elem lmm(i,1)= (2*node_connect(1,i)-1); lmm(i,2)= (2*node_connect(1,i)); lmm(i,3)= (2*node_connect(2,i)-1); lmm(i,4)= (2*node_connect(2,i));

end K=zeros(92); %Element Stiffness Matrix for i=1:n_elem node1=node_connect(1,i); node2=node_connect(2,i); coordinates=[x(node1), y(node1);x(node2), y(node2)];

lm=lmm(i,:); con=node_connect(:,i);

k=PlaneTrussElement(e, A, coordinates); K(lm, lm) = K(lm, lm) + k; end

K; R;

%Computing reactions debc=[1,2,3,4]; ebcVals=zeros(length(debc),1); [d, reactions] = NodalSoln(K, R, debc, ebcVals); results=[]; for i=1:n_elem results = [results; PlaneTrussResults(e, A, ...         coordinates, d(lmm(i,:)))]; end format short g results;

%Finding the Max Stresses %Compressive max_comp_sigma = 0; max_tens_sigma=0; max_comp_elem=0; max_tens_elem=0; Stresses=[]; for i=1:n_elem node1=node_connect(1,i); node2=node_connect(2,i); coordinates=[x(node1), y(node1);x(node2), y(node2)]; x1=coordinates(1,1); y1=coordinates(1,2); x2=coordinates(2,1); y2=coordinates(2,2); L=sqrt((x2-x1)^2+(y2-y1)^2);

ls=(x2-x1)/L; ms=(y2-y1)/L; T=[ls,ms,0,0; 0,0,ls,ms]; disps=d(lmm(i,:)); d_def = T*disps; eps= (d_def(2)-d_def(1))/L; sigma = e.*eps; Stresses(i,1)=sigma;

force = sigma.*A;

if (sigmamax_tens_sigma) max_tens_sigma=sigma; max_tens_elem=i;

end end end Stresses; max_comp_sigma; max_tens_sigma; max_comp_elem; max_tens_elem;

%Lumped Mass Matrix

M=zeros(2*n_node); m=zeros(n_node); for i=1:n_node for j=1:n_elem if(node_connect(1,j)==i || node_connect(2,j)==i) node1=node_connect(1,j); node2=node_connect(2,j); coordinates=[x(node1), y(node1);x(node2), y(node2)]; x1=coordinates(1,1); y1=coordinates(1,2); x2=coordinates(2,1); y2=coordinates(2,2); L=sqrt((x2-x1)^2+(y2-y1)^2); mass=A*L*p; m(i,i)=m(i,i)+(mass/2);

end end

end

for i=1:n_node M((2*i)-1,(2*i)-1)=m(i,i); M((2*i),(2*i))=m(i,i); end M;

%Computation of K_bar and M_bar K_bar=zeros(88); M_bar=zeros(88); z=1;                   %row number for the K_bar and M_bar matrix for i=5:92             %row number for addition n=1;               %column number for the K_bar and M_bar matrix for j=5:92         %column number for addition M_bar(z,n)=M(i,j); K_bar(z,n)=K(i,j); n=n+1; end z=z+1;             %row number for the K_bar matrix end

K_bar; M_bar; Eig_sol=inv(M_bar)*K_bar; [V,D]=eig(Eig_sol);

%Find the 3 Lowest Eigenvalues lambda1=1e9; lambda2=1e9; lambda3=1e9; for i=1:88 if (D(i,i)lambda1) lambda2=D(i,i); v2=V(:,i); vect_col2=i; else if(D(i,i)lambda2) lambda3=D(i,i); v3=V(:,i); vect_col3=i; end end end end

lambda1; lambda2; lambda3; v1; v2; v3; %Computation of the 3 Vibrational Periods Corresponding to the 3 Lowest %Eigenvalues omega1=sqrt(lambda1); f1=omega1/(2*pi); T1=1/f1;

omega2=sqrt(lambda2); f2=omega2/(2*pi); T2=1/f2;

omega3=sqrt(lambda3); f3=omega3/(2*pi); T3=1/f3;

%Finding the 3 Lowest Vibrational Periods vp1=1e100; vp2=1e100; vp3=1e100; Vib_Per=[]; for i=1:88 omega=sqrt(D(i,i)); f=omega/(2*pi); T=1/f; Vib_Per(i,1)=T; if (Tvp1) vp2=T; %vect_col2=i; else if (Tvp2) vp3=T; %vect_col3=i; end end end end vp1; vp2; vp3;

%Printing of Eigenvectors over the original Structure % Original Structure for i = 1 : n_elem; node_1 = node_connect(1,i); node_2 = node_connect(2,i); xx = [x(node_1),x(node_2)]; yy = [y(node_1),y(node_2)];

%plot(xx,yy,'--') hold on  end % produce postscript file for later printing % print -dps electric_pylon.ps  print -deps electric_pylon.ps

%First Eigenvector FirstRows=zeros(4,88); V_plot=20*[FirstRows;V];

%new positions for the nodes for the first (smallest) eigenvector for i=1:n_node position_vect1(:,i)=[position(1,i)-V_plot((2*i)-1,vect_col1), position(2,i)-V_plot(2*i,vect_col1)]; end

%Nodal Coordinates for i=1:n_node x_vect1(i)=position_vect1(1,i); y_vect1(i)=position_vect1(2,i); end for i = 1 : n_elem; node_1 = node_connect(1,i); node_2 = node_connect(2,i); xx = [x_vect1(node_1),x_vect1(node_2)]; yy = [y_vect1(node_1),y_vect1(node_2)];

%plot(xx,yy,'r-') hold on  end

%title('Electric Pylon (Undeformed and Eigenvector Associated with Smallest Eigenvalue')  %xlabel('x')   %ylabel('y')

%Second Eigenvector FirstRows=zeros(4,88); V_plot=20*[FirstRows;V];

%new positions for the nodes for the second smallest eigenvector for i=1:n_node position_vect2(:,i)=[position(1,i)-V_plot((2*i)-1,vect_col2), position(2,i)-V_plot(2*i,vect_col2)]; end

%Nodal Coordinates for i=1:n_node x_vect2(i)=position_vect2(1,i); y_vect2(i)=position_vect2(2,i); end for i = 1 : n_elem; node_1 = node_connect(1,i); node_2 = node_connect(2,i); xx = [x_vect2(node_1),x_vect2(node_2)]; yy = [y_vect2(node_1),y_vect2(node_2)];

%plot(xx,yy,'r-') hold on  end

%title('Electric Pylon (Undeformed and Eigenvector Associated with Second Smallest Eigenvalue')  %xlabel('x')   %ylabel('y')

%Third Eigenvector FirstRows=zeros(4,88); V_plot=20*[FirstRows;V];

%new positions for the nodes for the second smallest eigenvector for i=1:n_node position_vect3(:,i)=[position(1,i)-V_plot((2*i)-1,vect_col3), position(2,i)-V_plot(2*i,vect_col3)]; end

%Nodal Coordinates for i=1:n_node x_vect3(i)=position_vect3(1,i); y_vect3(i)=position_vect3(2,i); end for i = 1 : n_elem; node_1 = node_connect(1,i); node_2 = node_connect(2,i); xx = [x_vect3(node_1),x_vect3(node_2)]; yy = [y_vect3(node_1),y_vect3(node_2)];

%plot(xx,yy,'r-') hold on  end

%title('Electric Pylon (Undeformed and Eigenvector Associated with Third Smallest Eigenvalue')  %xlabel('x')   %ylabel('y')

%Displaced Structure for i=1:n_node %d_def=zeros(46,2); disp1=d((2*i)-1,1); disp2=d((2*i),1);

position(1, i) = (position(1,i))+(150*disp1); position(2, i) = (position(2,i))+(150*disp2);

end

% print the node coord. for i = 1 : n_node; x(i) = position(1,i); y(i) = position(2,i); z(i) = position(3,i);

end % connect all beam elements by connectivity array for i = 1 : n_elem; node_1 = node_connect(1,i); node_2 = node_connect(2,i); xx = [x(node_1),x(node_2)]; yy = [y(node_1),y(node_2)]; zz = [z(node_1),z(node_2)]; plot3(xx,yy,zz,'r-') hold on  end

title('Electric Pylon') xlabel('x') ylabel('y') zlabel('z')

view([0 0 1])     % xy plane view % view([0 1 0])    % xz plane view %view([1 0 0])    % yz plane view

% produce postscript file for later printing % print -dps electric_pylon.ps  print -deps electric_pylon.ps

Plane Truss Element
function k = PlaneTrussElement(e, A, coord) % PlaneTrussElement(e, A, coord) % Generates stiffness matrix for a plane truss element % e = modulus of elasticity % A = area of cross-section % coord = coordinates at the element ends

x1=coord(1,1); y1=coord(1,2); x2=coord(2,1); y2=coord(2,2); L=sqrt((x2-x1)^2+(y2-y1)^2); ls=(x2-x1)/L; ms=(y2-y1)/L; k = e*A/L*[ls^2, ls*ms,-ls^2,-ls*ms; ls*ms, ms^2,-ls*ms,-ms^2; -ls^2,-ls*ms,ls^2,ls*ms; -ls*ms,-ms^2,ls*ms,ms^2];

Plane Truss Results
function [results] = PlaneTrussResults(e, A, coord, disps) % [eps, sigma, force] = PlaneTrussResults(e, A, coord, disps) % Compute plane truss element results % e = modulus of elasticity % A = Area of cross-section % coord = coordinates at the element ends % disps = displacements at element ends % The output quantities are eps = axial strain % sigma = axial stress and force = axial force. results=[]; x1=coord(1,1); y1=coord(1,2); x2=coord(2,1); y2=coord(2,2); L=sqrt((x2-x1)^2+(y2-y1)^2); ls=(x2-x1)/L; ms=(y2-y1)/L; T=[ls,ms,0,0; 0,0,ls,ms]; d = T*disps; eps= (d(2)-d(1))/L; sigma = e.*eps; force = sigma.*A; results=[eps, sigma, force];

Nodal Solution
function [d, rf] = NodalSoln(K, R, debc, ebcVals) % [nd, rf] = NodalSoln(K, R, debc, ebcVals) % Computes nodal solution and reactions % K = global coefficient matrix % R = global right hand side vector % debc = list of degrees of freedom with specified values % ebcVals = specified values dof = length(R); df = setdiff(1:dof, debc); Kf = K(df, df); Rf = R(df) - K(df, debc)*ebcVals; dfVals = Kf\Rf; d = zeros(dof,1); d(debc) = ebcVals; d(df) = dfVals; rf = K(debc,:)*d - R(debc);

Global Stiffness Matrix
The global stiffness matrix is obtained by summing the individual element stiffness matrices together. This results in the matrix $$K_{92x92}$$, since there are 46 nodes and 2 degrees of freedom per node. The value for the global stiffness matrix that was obtained is shown below. K =

Columns 1 through 7

3.9658e+006 4.8112e+006            0            0 -7.5088e+005 -2.3317e+006 -3.2149e+006 4.8112e+006 9.1527e+006            0            0 -2.3317e+006 -7.2404e+006 -2.4795e+006 0           0  3.8903e+006 -4.7637e+006            0            0 -3.2065e+006 0           0 -4.7637e+006  9.3306e+006            0            0  2.5224e+006 -7.5088e+005 -2.3317e+006           0            0  1.3936e+007  8.2837e+006 -8.6145e+006 -2.3317e+006 -7.2404e+006           0            0  8.2837e+006  1.8594e+007            0 -3.2149e+006 -2.4795e+006 -3.2065e+006 2.5224e+006 -8.6145e+006            0  2.3726e+007 -2.4795e+006 -1.9123e+006 2.5224e+006 -1.9843e+006            0            0       -42945 0           0 -6.8377e+005  2.2412e+006            0            0 -8.6901e+006 0           0  2.2412e+006 -7.3463e+006            0            0            0 0           0            0            0 -8.7405e+005 -2.7341e+006            0 0           0            0            0 -2.7341e+006 -8.5527e+006            0 0           0            0            0 -3.6968e+006 -3.2178e+006            0 0           0            0            0 -3.2178e+006 -2.8009e+006            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            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            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Columns 8 through 14

-2.4795e+006           0            0            0            0            0            0 -1.9123e+006           0            0            0            0            0            0 2.5224e+006 -6.8377e+005 2.2412e+006            0            0            0            0 -1.9843e+006 2.2412e+006 -7.3463e+006            0            0            0            0 0           0            0 -8.7405e+005 -2.7341e+006 -3.6968e+006 -3.2178e+006 0           0            0 -2.7341e+006 -8.5527e+006 -3.2178e+006 -2.8009e+006 -42945 -8.6901e+006           0            0            0            0            0 3.8966e+006           0            0            0            0            0            0 0 1.3935e+007  -8.213e+006            0            0 -3.6872e+006  3.2376e+006 0 -8.213e+006  1.8742e+007            0            0  3.2376e+006 -2.8428e+006 0           0            0  1.3734e+007  5.5338e+006 -1.1936e+007            0 0           0            0  5.5338e+006  1.7037e+007            0            0 0 -3.6872e+006 3.2376e+006 -1.1936e+007            0  3.4945e+007       -19744 0 3.2376e+006 -2.8428e+006            0            0       -19744  1.9821e+007 0 -8.7405e+005 2.7341e+006            0            0 -1.2081e+007            0 0 2.7341e+006 -8.5527e+006            0            0            0            0 0           0            0 -9.2388e+005 -2.7996e+006 -1.7722e+006  3.5443e+006 0           0            0 -2.7996e+006 -8.4838e+006  3.5443e+006 -7.0886e+006 0           0            0            0            0 -1.7722e+006 -3.5443e+006 0           0            0            0            0 -3.5443e+006 -7.0886e+006 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            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Columns 15 through 21

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            0            0            0            0            0            0            0            0            0            0            0 -8.7405e+005  2.7341e+006            0            0            0            0            0 2.7341e+006 -8.5527e+006           0            0            0            0            0 0           0 -9.2388e+005 -2.7996e+006            0            0            0 0           0 -2.7996e+006 -8.4838e+006            0            0            0 -1.2081e+007           0 -1.7722e+006  3.5443e+006 -1.7722e+006 -3.5443e+006            0 0           0  3.5443e+006 -7.0886e+006 -3.5443e+006 -7.0886e+006            0 1.3832e+007 -5.473e+006            0            0 -8.7643e+005  2.7389e+006            0 -5.473e+006 1.7112e+007            0            0  2.7389e+006 -8.5589e+006            0 0           0  2.1945e+007  1.1041e+005 -9.9067e+006            0  -2.337e+006 0           0  1.1041e+005  3.8762e+007            0            0    6.15e+006 -8.7643e+005 2.7389e+006 -9.9067e+006            0  2.1699e+007 -2.5842e+005            0 2.7389e+006 -8.5589e+006           0            0 -2.5842e+005  3.9156e+007            0 0           0  -2.337e+006    6.15e+006            0            0   2.296e+007 0           0    6.15e+006 -1.6184e+007            0            0 -4.7215e+006 0           0 -7.0051e+006 -7.0051e+006 -7.0051e+006  7.0051e+006 -1.4357e+007 0           0 -7.0051e+006 -7.0051e+006  7.0051e+006 -7.0051e+006            0 0           0            0            0 -2.1388e+006 -5.9412e+006            0 0           0            0            0 -5.9412e+006 -1.6503e+007            0 0           0            0            0            0            0 -2.1388e+006 0           0            0            0            0            0  5.9412e+006 0           0            0            0            0            0  -4.127e+006 0           0            0            0            0            0 -7.3697e+006 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       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Columns 22 through 28

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Columns 29 through 35

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Columns 36 through 42

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Columns 43 through 49

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Columns 50 through 56

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Columns 57 through 63

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Columns 64 through 70

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Columns 71 through 77

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Columns 78 through 84

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Columns 85 through 91

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0            0            0            0            0            0            0 -3.9867e+006 -8.1632e+006            0            0            0            0            0 -8.1632e+006 -1.6715e+007           0            0            0            0            0 -3.1687e+006 7.5697e+006 -6.5876e+006  -8.852e+006            0            0            0 7.5697e+006 -1.8083e+007 -8.852e+006 -1.1895e+007            0            0            0 0           0            0            0            0            0            0            0            0            0 -2.3039e+007            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 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  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 -2.1078e+007            0            0            0            0            0            0 0           0            0            0            0            0            0  4.9819e+007  4.1378e+006 -1.9813e+007            0            0            0 -1.7722e+006 4.1378e+006 4.1887e+007            0            0            0            0 -3.5443e+006 -1.9813e+007           0  5.9221e+007 -2.6474e+005            0            0            0 0           0 -2.6474e+005  4.7373e+007            0            0            0 0           0            0            0  1.7722e+006 -3.5443e+006            0 0           0            0            0 -3.5443e+006  1.6995e+007            0 -1.7722e+006 -3.5443e+006           0            0            0            0  1.7722e+006 -3.5443e+006 -7.0886e+006           0 -9.9067e+006            0            0  3.5443e+006

Column 92

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            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 -3.5443e+006 -7.0886e+006 0 -9.9067e+006 0           0  3.5443e+006 1.6995e+007

Force Vector
The force vector, R, is the vector that contains all of the forces that act on the electric pylon structure. For this case there is a single force that is acting at the rightmost node of the upper arm of the electric pylon structure. The plots for the undeformed and deformed structure are shown in the section containing the displacement vector. The force vector is given as follows:

R =

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           0           0           0           0           0           0           0           0           0           0           0           0           0           0           0           0           0           0           0           0       -1000           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

The force on the structure is in the 66th row of this vector. The force vector contains all x-direction force in the odd numbered rows, and all y-direction forces in the even numbered rows. The individual x and y forces on each node are paired together one row after another. From this it can be seen that the force of 1000N is acting down in the vertical direction on the 33rd node.

Displacements
The displacements that are obtained from the loading vector, R, given in the previous section are contained within the displacement vector, d. The displacement vector is given below.

d =

0           0            0            0  -0.00019023   0.00017465   -0.0001277 -1.4074e-006 -0.00016006  -0.0002967  -0.00012671   0.00032602  -0.00012798    -0.000128  -0.00012792  -0.00054327   0.00044932   0.00030901   0.00046381  -0.00061057     0.001017   0.00064583    0.0010325  -0.00019427    0.0010502  -0.00097027    0.0017167    0.0010338    0.0017162   0.00023259    0.0016833  -0.00064959     0.001683   -0.0013293    0.0023452   0.00064079    0.0021863   -0.0010297    0.0047122    0.0023902    0.0039576   -0.0024165    0.0063605    0.0057328    0.0063605    0.0037877    0.0063605    0.0029803    0.0063614     0.002384    0.0063657    0.0016463    0.0063711   0.00086194    0.0063182  -0.00024268    0.0062289   -0.0013182    0.0061363   -0.0023672    0.0060413   -0.0033231    0.0058839   -0.0051278    0.0055152    -0.012587    0.0067826    0.0042648     0.006996   -0.0063359     0.007012    0.0034574    0.0074006   -0.0043591    0.0071505    0.0029772    0.0071478    0.0020508    0.0071421    0.0012278     0.007182   0.00032026    0.0072816  -0.00073249    0.0074073   -0.0018633    0.0075643   -0.0033872    0.0090005    0.0029772     0.010455   -0.0033872

Plot of the Deformed and Undeformed Structures
= The deformed structure is shown below (solid lines), as well as the undeformed structure (dotted lines).



Reactions at the Fixed Nodes and Static Analysis
The reactions at the fixed nodes at the bottom of the structure are shown below. These results were obtained using the PlaneTrussResults function. reactions =

149.64     -501.65      -149.64       1501.7

The system is statically indeterminate, because the points at which the structure is fixed (nodes 1 and 2) are at the same height. The sum of the forces in the x direction will give the result that the force in the x direction at node one is the negative value of the force in the x direction at node two. If a moment equation is set up in order to solve for one of these values, it will not be solvable, because the distance in the y direction from that point to nodes 1 and 2 will be the same. These terms will therefore cancel each other out automatically, making the system statically indeterminate.

The values for the forces in the y direction are obtainable, and turned out to have the same value as the y components of the reactions vector.

Axial Stresses
The axial stresses were determined using the deformed and undeformed shapes of the electric pylon structure. The function Plane Truss Results that is shown in an earlier section was used to help determine these values. The stresses of all the elements are contained within the vector $$Stresses_{92x92}$$. Stresses =

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 -2.0266e-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 9.2825e-008 -1.0794e-007 -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 4.6964e-008 9.0317e+006 -2.3815e-007 9.0511e+006 -1.3175e+005 -2.9645e+005 1.7966e+006 4.4876e+006 6.6233e+006 7.7755e+006 -1.0741e-008 -4.8034e-009 1.441e-008 -1.0741e-008 Once the vector containing all of the stresses within the structure was obtained, the maximum compressive and tensile stresses were found. These values, as well as the elements in which they occur are listed below, where sigma is used to signify stress. max_comp_sigma =

-8.6957e+006

max_tens_sigma =

9.0511e+006

max_comp_elem =

55

max_tens_elem =

81

Lumped Mass Matrix
The lumped mass matrix is constructed creating a nodal mass vector by adding half of the mass of each element that has an endpoint at the node together. This results in a column vector, $$m_{46x1}$$. The values in this column vector then get added to the diagonal of the lumped mass matrix, $$M_{92x92}$$. The lumped mass matrix is twice the size of the nodal mass vector, which corresponds to the fact that each node has two degrees of freedom. Therefore each row in the nodal mass vector corresponds to both the (2*i)th row and column, and the ((2*i)-1)th row and column. These values are all along the diagonal of the lumped mass matrix. The lumped mass matrix is given below.

M =

Columns 1 through 7

39.958           0            0            0            0            0            0            0       39.958            0            0            0            0            0            0            0       39.584            0            0            0            0            0            0            0       39.584            0            0            0            0            0            0            0        62.55            0            0            0            0            0            0            0        62.55            0            0            0            0            0            0            0       77.232            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        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Columns 8 through 14

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Columns 15 through 21

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Columns 22 through 28

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Columns 29 through 35

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            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            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Columns 36 through 42

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            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            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Columns 43 through 49

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            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            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Columns 50 through 56

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            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            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Columns 57 through 63

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            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            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Columns 64 through 70

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Columns 71 through 77

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Columns 78 through 84

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            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            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Columns 85 through 91

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Column 92

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Reduced K and M Matrices
The global stiffness matrix and the lumped mass matrix can both be reduced by the same method. This is due to the zero displacements that occur at the fixed nodes at the bottom of the electric pylon. These displacements can also be seen above in the displacement vector, $$d$$. The first four rows and columns of each matrix can thus be ignored in order to obtain the reduced K and M matrices, $$\bar{K}$$ and $$\bar{M}$$ respectively. These two matrices are shown below.

Reduced K Matrix
K_bar =

Columns 1 through 7

1.3936e+007 8.2837e+006 -8.6145e+006            0            0            0 -8.7405e+005 8.2837e+006 1.8594e+007            0            0            0            0 -2.7341e+006 -8.6145e+006           0  2.3726e+007       -42945 -8.6901e+006            0            0 0           0       -42945  3.8966e+006            0            0            0 0           0 -8.6901e+006            0  1.3935e+007  -8.213e+006            0 0           0            0            0  -8.213e+006  1.8742e+007            0 -8.7405e+005 -2.7341e+006           0            0            0            0  1.3734e+007 -2.7341e+006 -8.5527e+006           0            0            0            0  5.5338e+006 -3.6968e+006 -3.2178e+006           0            0 -3.6872e+006  3.2376e+006 -1.1936e+007 -3.2178e+006 -2.8009e+006           0            0  3.2376e+006 -2.8428e+006            0 0           0            0            0 -8.7405e+005  2.7341e+006            0 0           0            0            0  2.7341e+006 -8.5527e+006            0 0           0            0            0            0            0 -9.2388e+005 0           0            0            0            0            0 -2.7996e+006 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            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0           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Columns 8 through 14

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Columns 15 through 21

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Columns 22 through 28

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Columns 29 through 35

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Columns 36 through 42

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Columns 43 through 49

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Columns 50 through 56

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Columns 57 through 63

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Columns 64 through 70

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Columns 71 through 77

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Columns 78 through 84

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-1.8083e+007  -8.852e+006 -1.1895e+007 0           0            0            0            0            0            0            0            0            0            0            0            0 -2.3039e+007 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            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  -3.282e+007  9.1168e+006 0           0            0            0            0  9.1168e+006 -2.5324e+006 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       -13374 -1.8012e+007            0            0            0            0            0 2.4192e+007           0            0            0            0            0            0 0 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Columns 85 through 88

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Reduced M Matrix
M_bar =

Columns 1 through 7

62.55           0            0            0            0            0            0            0        62.55            0            0            0            0            0            0            0       77.232            0            0            0            0            0            0            0       77.232            0            0            0            0            0            0            0       62.253            0            0            0            0            0            0            0       62.253            0            0            0            0            0            0            0       36.961            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         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Columns 8 through 14

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            0            0            0            0       36.961            0            0            0            0            0            0            0       87.274            0            0            0            0            0            0            0       87.274            0            0            0            0            0            0            0       36.796            0            0            0            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Columns 15 through 21

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            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            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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            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

Columns 22 through 28

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            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            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0       30.515            0            0            0            0            0            0            0       30.515            0            0            0            0            0            0            0       36.674            0            0            0            0            0            0            0       36.674            0            0            0            0            0            0            0       36.487            0            0            0            0            0            0            0       36.487            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            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Columns 29 through 35

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            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            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Columns 36 through 42

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            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            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Columns 43 through 49

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            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            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Columns 50 through 56

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            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            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Columns 57 through 63

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            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            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Columns 64 through 70

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            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            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Columns 71 through 77

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            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            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Columns 78 through 84

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Columns 85 through 88

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Eigenvalues and Eigenvectors
The reduced matrices are given as $$\bar{K}$$ and $$\bar{M}$$. These matrices can then be used in order to find the eigenvalues, $$\lambda$$, and corresponding eigenvectors, $$v$$. The equation to accomplish this is shown below.

$$ \bar{K}v=\lambda \bar{M} v $$

Three Lowest Eigenvalues
The three lowest eigenvalues acquired from this set of calculations are shown below. lambda1 =

132.16

lambda2 =

2468.4

lambda3 =

2940.3 The corresponding eigenvectors were obtained in order to find the displacement of the structure due to these eigenvectors. The values of these eigenvectors are given below. v1 =

0.0020791  -0.0071291    0.0014821  1.6378e-005 0.0019837   0.0069693   -0.0022959    -0.011897   -0.0022509 -3.2055e-006 -0.0022517     0.01179    -0.021615    -0.011731    -0.021669     0.011739    -0.037067    -0.020184    -0.036979  1.9512e-005 -0.037032     0.01981    -0.054017    -0.028795    -0.053987     -0.01073    -0.053953     0.010723    -0.053969     0.028403    -0.067939    -0.020094    -0.067913     0.020109     -0.11676    -0.056498     -0.11728     0.056322     -0.16183     -0.15112     -0.16195    -0.095279     -0.16199    -0.072626     -0.16204    -0.055582     -0.16239    -0.035052     -0.16301    -0.013353     -0.16304     0.013567     -0.16246       0.0356     -0.16213     0.055448     -0.16209     0.072318     -0.16205      0.09529     -0.16193      0.15012     -0.17386     -0.10875     -0.17385      0.10813     -0.18021    -0.085996     -0.18015     0.086087     -0.18401    -0.072481     -0.18356    -0.046205     -0.18277    -0.023356     -0.18241   0.00033152     -0.18273     0.023971     -0.18347      0.04616     -0.18391     0.072183     -0.23697    -0.072676     -0.23638     0.072377

v2 =

-0.0052058   -0.026405   0.00058503  6.7794e-006 0.0067449   -0.025514   0.00030902    -0.044291   0.00031417    -0.054647   0.00021457    -0.043576     0.003487    -0.061119   -0.0036908    -0.060326    0.0028748     -0.07199  -0.00076852     -0.05835   -0.0046442    -0.070418   -0.0056146    -0.088945   -0.0053339    -0.062651    0.0026949    -0.061685    0.0029198    -0.087172    -0.026856    -0.075838      0.02418    -0.074674     -0.14724     -0.16674      0.14852     -0.16678    -0.028212     0.040085    -0.027156    -0.087693    -0.026932     -0.13626    -0.025422     -0.18603     -0.01856     -0.23273   -0.0075241     -0.26537    0.0080155     -0.26527     0.019209     -0.23158     0.025889     -0.18573     0.027401     -0.13596     0.027644    -0.086162     0.028704     0.040844   -0.0007524    -0.058022    0.0011713    -0.057552     0.013069     -0.10767    -0.012682     -0.10619     0.022847       -0.141     0.018957     -0.21005     0.011208     -0.25367  7.6934e-005 -0.27176   -0.011039     -0.25286    -0.018548     -0.20976    -0.022429     -0.14069      0.16576     -0.13973     -0.16538     -0.13941

v3 =

0.045081    0.018913     0.033357   0.00039039     0.045511     -0.01965     0.098674     0.016365     0.097717  -0.00091773     0.098553    -0.017541      0.17325    0.0062508      0.17316   -0.0082824      0.19254    0.0038869      0.18938   -0.0010393      0.19268   -0.0059587      0.19749   -0.0059468       0.1963    0.0044001      0.19632   -0.0065813      0.19749    0.0038072      0.18839   0.00015745      0.18813   -0.0021818       0.1049    -0.063525      0.10208     0.062271     0.067357     -0.37236     0.057138     -0.15137     0.051727     -0.09449     0.048496    -0.067951     0.045128    -0.037234     0.042026    -0.012707     0.041875     0.013602     0.044809     0.037899     0.047913     0.066758     0.051026      0.09221     0.056303      0.14837     0.066022      0.35912     0.023294     -0.19037     0.023719      0.18442      0.01011     -0.12642     0.010893      0.12429    0.0057056    -0.098237     0.011336     -0.05318      0.01618    -0.023341     0.018417   0.00071086     0.016554     0.024441     0.012064     0.052578     0.006687     0.095825    -0.086975     -0.10041    -0.082225     0.097902

Plots of the Three Smallest Eigenvectors
= The plots that correspond to each of these eigenvectors respectively are shown below, with the undeformed shape as well.



Three Lowest Vibrational Periods
The formulas that were used to find the vibrational periods are given as:

$$ \lambda = \omega ^2 $$

$$ f = \omega / 2 \pi $$

$$ T=1/f $$

The values for the vibrational periods that are associated with these eigenvalues are shown below.

T1 =

0.54654

T2 =

0.12647

T3 =

0.11587

The vibrational periods that are given above correspond to the three lowest eigenvalues. The three lowest vibrational periods in the structure actually correspond to the three highest eigenvalues. The values for the three lowest vibrational periods can be shown to have the following values.

vp1 =

0.0028107

vp2 =

0.0028224

vp3 =

0.0030428

The Three-Bar Square Truss
The following truss was assigned in homework 4, and an analysis of its behavior is performed.



The elements of this truss have the following properties:


 * $$A^{(1)}=A^{(2)}=A^{(3)}=1$$
 * $$E^{(1)}=E^{(2)}=E^{(3)}=1$$
 * $$L^{(1)}=L^{(2)}=L^{(3)}=1$$

The following is the MATLAB code for determining the global stiffness matrix, $$\mathbf{K}$$, and the dof values.

It is evident that the Finite Element Method is not enough to solve the behavior of this truss. It is evident that this truss is very inestable, and so there is an undetermined number of ways this truss will behave under any load. It is necessary that the engineer add another bar element for support of the structure.

The Eigenvalues and Eigenvectors
The following command yielded the eigenvalues (D) and eigenvectors (V) for the global stiffness matrix assembled above.

We can see that there are five zero-value eigenvalues. Below are the plots of these five eigenvalues of the three-bar square truss and the generic code for obtaining the plots below.

First Zero Eigenvalue



Second Zero Eigenvalue



Third Zero Eigenvalue



Fourth Zero Eigenvalue



Fifth Zero Eigenvalue



The Four-Bar Square Truss
The following truss is a more rigid version of the above three-bar square truss. Another bar element has been placed diagonally from the global node 1 to the global node 3.



This new truss system has slightly different properties:


 * $$A^{(1)}=A^{(2)}=A^{(3)}=3$$
 * $$E^{(1)}=E^{(2)}=E^{(3)}=2$$
 * $$L^{(1)}=L^{(2)}=L^{(3)}=1$$ and $$L^{4}=\sqrt{2}$$

The source code for assembling the global stiffness matrix, and determining the dof values is below:

It is now evident that by the addition of the fourth bar element the truss system is now stable.

The Eigenvalues and Eigenvectors
The eigenvalues (D) and eigenvectors (V) were determined with the code below. Also, the actual results are displayed.

We can see that there are four zero eigenvalues. The plots of these eigenvalues can be found below, as well as the code for the fourth zero eigenvalue, as an example.

First Zero Eigenvalue



Second Zero Eigenvalue



Third Zero Eigenvalue



Fourth Zero Eigenvalue



Proof of N^(i)(x)
The formula for the above diagram is: $$N_{i}(x)={x-x_{i+1}\over x_{i}-x_{i+1}}$$ Wasn't sure if we were supposed to find an actual value for N(x). So, below are the values for N(x). The total length of the bar is $$L^{(i)}$$ and at the origin x is equal to zero. To find $$N_{i}(x)$$ the following formula is used: $$N_{i}(x)=1-{x\over L^{(i)}}$$ Now let x=0 and x=$$L^{(i)}$$ and it can be seen that when: x=0 $$\Rightarrow$$ $$N_{i}=1$$ and when x=$$L^{(i)}$$ $$\Rightarrow$$ $$N_{i}=0$$

Proof of N^(i)(~x)
From the diagram above it can be seen that $$\tilde{x}$$ is equal to $$x-x_{i}$$.Therefore the formula for $$N_{i}(\tilde{x})$$ from the above diagram is: $$N_{i}(\tilde{x})={x-x_{i} \over x_{i}-x_{i-1}}$$ It is also know that: $$N_{i}(\tilde{x})=N_{i}(x)=1-{x\over L^{(i)}}$$ Now let x=0 and x=$$L^{(i)}$$ and it can be seen that when: x=0 $$\Rightarrow$$ $$N_{i}(\tilde{x})=1$$ and when x=$$L^{(i)}$$ $$\Rightarrow$$ $$N_{i}(\tilde{x})=0$$

Proof of k^(i)
Show that, if EA is constant,

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

Knowing that

$$ \beta = \int_{0}^{L}{\frac{dw}{dx}EA\frac{du}{dx}dx}\; \; \; \; \; \; \; \; \; \; Equation (1) $$

$$ \frac{dW}{dx}=[N_i'W_i + N_{i+1}'W_{i+1}] $$

$$ \frac{du}{dx}=[N_i'd_i + N_{i+1}'d_{i+1}] $$

Equation 1 reduces to

$$ \beta = \int_{x_i}^{x_{i+1}}{\mathbf{B}W^{(i)}EA\mathbf{B}d^{(i)}dx}\; \; \; \; \; \; \; \; \; \; Equation (2) $$

where

$$ W^{(i)} = \begin{bmatrix} W_i\\ W_{i+1} \end{bmatrix},

d^{(i)} = \begin{bmatrix} d_i\\ d_{i+1} \end{bmatrix},

\mathbf{B} = \begin{bmatrix} N_i' &N_{i+1}' \end{bmatrix} $$

Manipulating Equation (2) with transpose definitions, we determine:

$$ \mathbf{B} = W^{(i)}*\left(\int_{x_i}^{x_{i+1}}{\mathbf{B^T}EA\mathbf{B}dx} \right)d^{(i)}\; \; \; \; \; \; \; \; \; \; Equation (3) $$

Equation (3) shows that

$$ k^{(i)} = \int_{x_i}^{x_{i+1}}{\mathbf{B^T}EA\mathbf{B}dx} $$

Since EA remains constant, this reduces to

$$ k^{(i)} = EA\int_{x_i}^{x_{i+1}}{\mathbf{B^T}\mathbf{B}dx} $$

Now $$\mathbf{B(x)}$$ must be analyzed.

$$ \mathbf{B(x)}= \begin{bmatrix} N_i'(x) & N_{i+1}'(x) \end{bmatrix} = \begin{bmatrix} \frac{1}{x_i-x_{i+1}} & \frac{1}{x_{i+1}-x_i} \end{bmatrix}= \begin{bmatrix} \frac{1}{-L^{(i)}} & \frac{1}{L^{(i)}} \end{bmatrix} $$

$$ \mathbf{B^T}=\begin{bmatrix} \frac{1}{-L^i}\\ \frac{1}{L^i} \end{bmatrix} $$

Hence,

$$ k^{(i)} = EA\int_{x_i}^{x_{i+1}}{\begin{bmatrix} \frac{1}{-L^i}\\ \frac{1}{L^i} \end{bmatrix}\begin{bmatrix} \frac{1}{-L^{(i)}} & \frac{1}{L^{(i)}} \end{bmatrix}dx} $$

Multiplying matrices gives

$$ k^{(i)} = EA\begin{bmatrix} \int_{x_i}^{x_{i+1}}{\frac{1}{L^2}dx} & \int_{x_i}^{x_{i+1}}{\frac{-1}{L^2}dx}\\ \int_{x_i}^{x_{i+1}}{\frac{-1}{L^2}dx} & \int_{x_i}^{x_{i+1}}{\frac{1}{L^2}dx} \end{bmatrix} $$


 * NOTE

One integer will be used for example. Others differ by a positive or negative sign.

$$ \int_{x_i}^{x_{i+1}}{\frac{1}{L^2}dx} = \frac{x_{i+1}-x_i}{L^2} = \frac{L}{L^2} = \frac{1}{L} $$

Thus,

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

Extracting $$\frac{1}{L}$$ gives us

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

Proof of k^(i) for $$\tilde{x}$$


where

$$ A(\tilde{x})=N^{(i)}_{1}(\tilde{x})A_{1}+N^{(i)}_{2}(\tilde{x})A_{2} $$

and

$$ E(\tilde{x})=N^{(i)}_{1}(\tilde{x})E_{1}+N^{(i)}_{2}(\tilde{x})E_{2} $$

We will find an expression for $$\mathbf{k^{(i)}}$$ using

$$ \mathbf{k^{(i)}}=\int_{\tilde{x}=0}^{\tilde{x}=L^{(i)}}\mathbf{B^T}(\tilde{x})EA(\tilde{x})\mathbf{B}(\tilde{x})d\tilde{x} $$

By plugging the above equations in for $$E(\tilde{x})$$ and $$A(\tilde{x})$$, we get

$$ \mathbf{k^{(i)}}=\int_{\tilde{x}=0}^{\tilde{x}=L^{(i)}}\mathbf{B^T(\tilde{x})}(N^{(i)}_{1}(\tilde{x})E_{1}+N^{(i)}_{2}(\tilde{x})E_{2})(N^{(i)}_{1}(\tilde{x})A_{1}+N^{(i)}_{2}(\tilde{x})A_{2})\mathbf{B(\tilde{x})}d\tilde{x} $$

Element stiffness matrices for elements 1 and 2
Finding the element stiffness matrices for the 2 bar system The following is an equation that was derived earlier. It will ultimately be used to find the two element stiffness matrices for the system. Eqn (0) $$\mathbf{k^{(i)}}=\int_{\tilde{x}=0}^{\tilde{x}=L^{(i)}}{\mathbf{B^T(\tilde{x})}}EA(\tilde{x})\mathbf{B(\tilde{x}})d(\tilde{x}) $$ Eqn (1)

$$ N_{i}(x)=N_1^{(i)}(\tilde{x}) $$

Eqn (2)

$$ N_{i+1}(x)=N_2^{(i)}(\tilde{x}) $$ Eqn (3)

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

Eqn (4)

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

Eqn (5) $$B(x)=\begin{bmatrix} N_i(x) & N_{i+1}(x) \end{bmatrix}$$ Eqn (6) $$ B(x)^{T}=\begin{Bmatrix} N_{i}(x)\\ N_{i+1}(x) \end{Bmatrix}$$ To transform the matrices of B(x) and $$B(x)^{T}$$ into a function of $$\tilde{x}$$ Eqn (1) and (2) must be substituted into the B(x) and the $$B(x)^T$$ equation. Then the $$B(\tilde{x})$$, $$B(\tilde{x})^T$$, Eqn (3) and Eqn (4) have to be substituted into Eqn (0).This gives the following equations: Eqn (7) $$k^{(i)}=(N_{1}^{(i)}(\tilde{x})E_{1}+N_{2}^{(i)}(\tilde{x})E_{2})\begin{bmatrix}N_{1}^{(i)}(\tilde{x}) & N_{2}^{(i)}(\tilde{x})\end{bmatrix}\begin{Bmatrix}N_{1}^{(i)}(\tilde{x}) \\ N_{2}^{(i)}(\tilde{x}) \end{Bmatrix}(N_{1}^{(i)}(\tilde{x})A_{1}+N_{2}^{(i)}(\tilde{x})A_{2})$$ $$-(N_{1}^{(i)}(\tilde{x})E_{1}+N_{2}^{(i)}(\tilde{x})E_{2})\begin{bmatrix}N_{1}^{(i)}(\tilde{x}) & N_{2}^{(i)}(\tilde{x})\end{bmatrix}\begin{Bmatrix}N_{1}^{(i)}(\tilde{x}) \\ N_{2}^{(i)}(\tilde{x}) \end{Bmatrix}(N_{1}^{(i)}(\tilde{x})A_{1}+N_{2}^{(i)}(\tilde{x})A_{2})$$ The following values are known for the given variables. $$ (1)N_1^{(i)}(\tilde{x})= 1 $$ at $$\tilde{x}=0$$ $$ (1)N_2^{(i)}(\tilde{x})= 0 $$ at $$\tilde{x}=0$$ $$ (2)N_2^{(i)}(\tilde{x})= 1 $$ at $$\tilde{x}=L^{(i)}$$ $$ (2)N_1^{(i)}(\tilde{x})= 0 $$ at $$\tilde{x}=L^{(i)}$$ The first term in the element stiffness formula, Eqn (7), is evaluated at$$\tilde{x}=L^{(i)}$$ and the second term is evaluated at $$\tilde{x}=0$$. The above values are substituted into Eqn (7) and the $$B(\tilde{x})$$ and $$B(\tilde{x})^{T}$$ matrices are multiplied out to simplify the equation. This gives: Eqn (8) $$k^{(i)}=(E_{2}^{(i)}A_{2}^{(i)})\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}-(E_{1}^{(i)}A_{1}^{(i)})\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$$ For element 1 the following values are substituted into the element stiffness matrix, Eqn (8). $$k^{(i)}$$. $$E_{1}^{(1)}=2$$ $$E_{2}^{(1)}=4$$ $$A_{1}^{(1)}=.5$$ $$A_{2}^{(1)}=1.5$$ This gives the element stiffness matrix for element 1. $$k^{(1)}=\begin{bmatrix}-1 & 0 \\ 0 & 6\end{bmatrix}$$ For element 2 the same process is used, but the following is substituted into Eqn (8). $$E_{1}^{(2)}=3$$ $$E_{2}^{(2)}=7$$ $$A_{1}^{(2)}=1$$ $$A_{2}^{(2)}=3$$

This gives the element stiffness matrix for element 2. $$k^{(1)}=\begin{bmatrix}-3 & 0 \\ 0 & 21\end{bmatrix}$$

Contributing Team Members
Iain Specht --Eml 4500.f08.delta 6.specht 20:05, 16 November 2008 (UTC) Sean Lopez --Eml4500.f08.delta 6.lopez 02:34, 17 November 2008 (UTC) Diana Guzman -- Eml4500.f08.delta 6.guzman 03:46, 17 November 2008 (UTC) Eric Ramirez --Eml4500.f08.delta 6.ramirez 21:16, 17 November 2008 (UTC) Micheal Krueger --Eml4500.f08.delta 6.krueger 02:24, 18 November 2008 (UTC) Nicolas Castrillon --Eml4500.f08.delta 6.castrillon 05:37, 18 November 2008 (UTC)