User:Uf.team.wiki/HW7

Problem Setup
Shown below is the Frame element that consists of a truss element (axial deformation) and a beam element (transverse deformation)



General Frame Breakdown with Local DOF's
In general, $$d_i^{(e)} \Rightarrow f_i^{(e)}$$ (generalized forces)

e=1,2 i=1,2,...6

Here, $$d_3^{(e)}, d_6^{(e)} \Rightarrow $$ represent rotational degrees of freedom and $$f_3^{(e)}, f_6^{(e)} \Rightarrow $$ correspond to DOF's that are bending moments



Frame with Global Degrees of Freedom



 * 2 element stiffness matrices: $$ \underset{6 \times 6} \mathbf k^{(e)}, \quad e=1,2 $$


 * Global stiffness matrix: $$ \underset{6 \times 6} \mathbf K^{(e)} = A \underset{6 \times 6} \mathbf k^{(e)}, \quad e=1,2 $$

Local DOF's shown with Element 1


$$ \mathbf \tilde d^{(e)} = \begin{pmatrix} \tilde d^{(e)}_1 \\ \tilde d^{(e)}_2 \\ \tilde d^{(e)}_3 \\ \tilde d^{(e)}_4 \\ \tilde d^{(e)}_5 \\ \tilde d^{(e)}_6 \end{pmatrix} \qquad \qquad \mathbf \tilde f^{(e)} = \begin{pmatrix} \tilde f^{(e)}_1 \\ \tilde f^{(e)}_2 \\ \tilde f^{(e)}_3 \\ \tilde f^{(e)}_4 \\ \tilde f^{(e)}_5 \\ \tilde f^{(e)}_6 \end{pmatrix}$$

Note, $$\tilde f^{(e)}_3 = f^{(e)}_3 \qquad \tilde f^{(e)}_6 = f^{(e)}_6 \qquad \qquad (moments \quad about \quad\tilde z = z)$$

$$ \underset{6 \times 6}\mathbf \tilde k = \begin{bmatrix} \tfrac{EA}{L} & 0 & 0 & -\tfrac{EA}{L}& 0 & 0 \\ & -\tfrac{12EI}{L^3} & \tfrac{6EI}{L^2} & 0 & -\tfrac{12EI}{L^3}& \tfrac{6EI}{L^2}\\ & & \tfrac{4EI}{L} & 0 & -\tfrac{6EI}{L^2} & \tfrac{2EI}{L} \\ & & & \tfrac{EA}{L} & 0 & 0 \\ & SYM. & & & \tfrac{12EI}{L^3} & -\tfrac{6EI}{L^2} \\ & & & & & \tfrac{4EI}{L} \\ \end{bmatrix} $$

Dimensional Analysis
Here we are analyzing the dimensions of the force displament relationship. We are loking to see how the units will emerge in the element stiffness matrix.

Arc Length

 * [x] = square brackets represent a dimensiokn

$$[\tilde d_1] = L = [\tilde d_i] \qquad i=1,2,4,5$$

$$[\tilde d_3] = [\tilde d_6] = 1 \qquad $$ (rotational degree of freedom, no dimension)

$$\underbrace{\overline {AB}}_{arc length} = R \underbrace{\theta}_{radians}$$

$$\theta= \frac{\overline {AB}}{R} $$

$$[\theta] = \frac{\overline {[AB]}}{[R]} = \frac{L}{L} = 1$$



Dimensional Analysis on Other Useful Parameters
$$ \theta = E \epsilon \quad \Rightarrow \quad [\theta] = [E] \underbrace{[\epsilon]}_{=1}$$

$$	\left [ E \right ] = \left [ \frac{du}{dx} \right ] = \left [ \frac{L}{L} \right ] = 1$$

$$\left [ \sigma' \right ] = \left [ E \right ] = \left [ \frac{F}{L^2} \right ]$$

$$	\left [ A \right ] = L^2$$

$$	\left [ I \right ] = L^4$$

$$\left [ \frac{EA}{L} \right ] = \left [ \tilde{K_{11}} \right ] = (\frac{\frac{F}{L^2}) * ({L^2})}{L} = \frac{F}{L}$$

$$\left [ \tilde{K_{11}}\tilde{d_1} \right ] = \left [ \tilde{K_{11}}\tilde{d_{11}} \right ] = \left [ F \right ]$$

$$\left [ \tilde{K_{11}}\tilde{d_1} \right ] = \left [ \tilde{K_{11}} \right ]\left [ \tilde{d_{11}} \right ] = F$$

$$\left [ \tilde{K_{23}}\tilde{d_3} \right ] = \left [ \tilde{K_{23}} \right ]\left [ \tilde{d_{3}} \right ] = \frac{\left [ 6 \right ]\left [ E \right ]\left [ I \right ]}{\left [ {L^2} \right ]}  = \frac{1(\frac{F}{L^2}){L^4}}{L^2} = F$$

$$\left [ \tilde{K_{23}}\tilde{d_3} \right ] = \left [ \tilde{K_{23}} \right ]\left [ \tilde{d_{3}} \right ] = \frac{\left [ 6 \right ]\left [ E \right ]\left [ I \right ]}{\left [ {L^2} \right ]}  = \frac{1(\frac{F}{L^2}){L^4}}{L^2} = F$$

Force Displacement Relationship Proofs
The matrix $$ \tilde{k_{ij}}\tilde{d_j} $$ is discussed above and shown below

$$ \underset{6 \times 6}\mathbf \tilde K = \begin{bmatrix} \tfrac{EA}{L} & 0 & 0 & -\tfrac{EA}{L}& 0 & 0 \\ & -\tfrac{12EI}{L^3} & \tfrac{6EI}{L^2} & 0 & -\tfrac{12EI}{L^3}& \tfrac{6EI}{L^2}\\ & & \tfrac{4EI}{L} & 0 & -\tfrac{6EI}{L^2} & \tfrac{2EI}{L} \\ & & & \tfrac{EA}{L} & 0 & 0 \\ & SYM. & & & \tfrac{12EI}{L^3} & -\tfrac{6EI}{L^2} \\ & & & & & \tfrac{4EI}{L} \\ \end{bmatrix} $$

The transfer matrix, $$\mathbf \tilde T^{(e)}$$ is given as:

$$\underset{6 \times 6}\mathbf \tilde T^{(e)} = \begin{bmatrix}

l^{(e)} & m^{(e)} & 0 & 0 & 0 & 0 \\

-m^{(e)} & l^{(e)} & 0 & 0 & 0 & 0 \\

0 & 0 & 1 & 0 & 0 & 0 \\

0 & 0 & 0 & l^{(e)} & m^{(e)} & 0 \\

0 & 0 & 0 & -m^{(e)} & l^{(e)} & 0 \\

0 & 0 & 0 & 0 & 0 & 1 \\ \end{bmatrix}$$

The transpose of the transfer matrix, $$\mathbf \tilde T^{(e)^T}$$ is given as:

$$\underset{6 \times 6}\mathbf \tilde T^{(e)^T} = \begin{bmatrix}

l^{(e)} & -m^{(e)} & 0 & 0 & 0 & 0 \\

m^{(e)} & l^{(e)} & 0 & 0 & 0 & 0 \\

0 & 0 & 1 & 0 & 0 & 0 \\

0 & 0 & 0 & l^{(e)} & -m^{(e)} & 0 \\

0 & 0 & 0 & m^{(e)} & l^{(e)} & 0 \\

0 & 0 & 0 & 0 & 0 & 1 \\ \end{bmatrix}$$

$$ \mathbf \underset{6 \times 6}\mathbf K = \mathbf \underset{6 \times 6}\mathbf \tilde T^{(e)^T} \mathbf \underset{6 \times 6}\mathbf \tilde K \mathbf \underset{6 \times 6}\mathbf \tilde T^{(e)} = \begin{bmatrix}

l^{(e)} & -m^{(e)} & 0 & 0 & 0 & 0 \\

m^{(e)} & l^{(e)} & 0 & 0 & 0 & 0 \\

0 & 0 & 1 & 0 & 0 & 0 \\

0 & 0 & 0 & l^{(e)} & -m^{(e)} & 0 \\

0 & 0 & 0 & m^{(e)} & l^{(e)} & 0 \\

0 & 0 & 0 & 0 & 0 & 1 \\ \end{bmatrix} \begin{bmatrix} \tfrac{EA}{L} & 0 & 0 & -\tfrac{EA}{L}& 0 & 0 \\ 0 & -\tfrac{12EI}{L^3} & \tfrac{6EI}{L^2} & 0 & -\tfrac{12EI}{L^3}& \tfrac{6EI}{L^2}\\ 0 & \tfrac{6EI}{L^2} & \tfrac{4EI}{L} & 0 & -\tfrac{6EI}{L^2} & \tfrac{2EI}{L} \\ \tfrac{-EA}{L} & 0 & 0 & \tfrac{EA}{L} & 0 & 0 \\ 0 & \tfrac{12EI}{L^3} & -\tfrac{6EI}{L^2} & 0 & \tfrac{12EI}{L^3} & -\tfrac{6EI}{L^2} \\ 0 & \tfrac{6EI}{L^2} & \tfrac{2EI}{L} & 0 & -\tfrac{6EI}{L^2} & \tfrac{4EI}{L} \\ \end{bmatrix}\begin{bmatrix}

l^{(e)} & m^{(e)} & 0 & 0 & 0 & 0 \\

-m^{(e)} & l^{(e)} & 0 & 0 & 0 & 0 \\

0 & 0 & 1 & 0 & 0 & 0 \\

0 & 0 & 0 & l^{(e)} & m^{(e)} & 0 \\

0 & 0 & 0 & -m^{(e)} & l^{(e)} & 0 \\

0 & 0 & 0 & 0 & 0 & 1 \\ \end{bmatrix}$$

The preceding matricies will be beneficial in developing the following relationship between the element force-displacement relationship in global coordinates from the element force-displacement relationship in local coordinates and is represented as:

$$ \underline{k}^{(e)}_{6x6}\underline{d}^{(e)}_{6x1} = \underline{f}^{(e)}_{6x1} $$


 * where $$ \underline{k}^{(e)}_{6x6} = {\underline{\tilde{T}}^{(e)}_{6x6}}^T\underline{\tilde{k}}^{(e)}_{6x6}\underline{\tilde{T}}^{(e)}_{6x6} $$


 * knowing $$ \underline{\tilde{k}}^{(e)}_{6x6}\underline{\tilde{d}}^{(e)}_{6x1} = \underline{\tilde{f}}^{(e)}_{6x1} $$

$$\mathbf \underset{6 \times 6}\mathbf K =\begin{bmatrix}

l^{(e)}*l^{(e)}*\tfrac{EA}{L} & & 0 & 0 & 0 & 0 \\

0 & l^{(e)}*l^{(e)}*-\tfrac{12EI}{L^3} & 0 & 0 & 0 & 0 \\

0 & 0 & \tfrac{4EI}{L} & 0 & 0 & 0 \\

0 & 0 & 0 & l^{(e)}*l^{(e)}*\tfrac{EA}{L} & 0 & 0 \\

0 & 0 & 0 & 0 & l^{(e)}*l^{(e)}*-\tfrac{12EI}{L^3} & 0 \\

0 & 0 & 0 & 0 & 0 & \tfrac{4EI}{L} \\ \end{bmatrix}$$

$$\begin{Bmatrix} \tilde d^{(1)}&\\

\tilde d^{(2)}&\\

\tilde d^{(3)}&\\

\tilde d^{(4)}&\\

\tilde d^{(5)}&\\

\tilde d^{(6)}&\\ \end{Bmatrix}

= \begin{bmatrix}

\mathbf \underset{2 \times 2}\mathbf R && 0 & 0 & 0 & 0 \\

&& 0 & 0 & 0 & 0 \\

0 & 0 & 1 & 0 & 0 & 0 \\

0 & 0 & 0 &\mathbf \underset{2 \times 2}\mathbf R && 0 \\

0 & 0 & 0 &&& 0 \\

0 & 0 & 0 & 0 & 0 & 1 \\ \end{bmatrix}

\begin{Bmatrix} d^{(1)}&\\

d^{(2)}&\\

d^{(3)}&\\

d^{(4)}&\\

d^{(5)}&\\

d^{(6)}&\\ \end{Bmatrix} $$

Where $$\mathbf \underset{2 \times 2}\mathbf R  = \begin{bmatrix} l^{(e)}&m^{(e)}&\\

-m^{(e)}&l^{(e)}&\\

\end{bmatrix} $$

PVW for Beams
Motivation: find displacement V(ʎ) of element given deformed shape of truss

$$\int_{0}^{L}W(x)[\frac{-d^2}{dy^2}(EI)\frac{d^2y}{dx^2}+(f_x)-m\ddot{v]}$$ for all W(x)

Integration by parts for the first term yields:

$$\alpha = \int_{0}^{L}{W(x)}[\frac{d^2}{dx^2}(EI)\frac{d^2y}{dx^2}]dx$$


 * $$S(x) = \int_{0}^{L}{W(x)}$$


 * $$V'(x)= \frac{d}{dx}(\frac{d}{dx}(EI)\frac{d^2v}{dx^2})$$

$$\beta_1 = [W\frac{d}{dx}(EI)\frac{d^2v}{dx^2}]\mid^L_0$$

$$\beta_2 = \beta_1-[\frac{dW}{dx}(EI)\frac{d^2y^2}{dx^2}]\mid^L_0$$

$$\gamma=\int_{0}^{L}{\frac{d^2W}{dx^2}(EI)\frac{d^2V}{dx^2}}$$

$$\alpha=\beta_2+\gamma$$

Now, looking at the figure below, focus on the stiffness term $$\gamma$$  to derive the beam stiffness matrix and to identify the beam shape functions:



$$v(\tilde{x})=N_2(\tilde{x})\tilde{d}_2+N_3(\tilde{x})\tilde{d}_3+N_5(\tilde{x})\tilde{d}_5+N_6(\tilde{x})\tilde{d}_6$$

Recall: $$u(\tilde{x})=N_1(\tilde{x})\tilde{d}_1+N_4(\tilde{x})\tilde{d}_4$$\

Knowing the following interpolations shown below, their modal shapes are shown in the figure that follows.

$$N_2(\tilde{x})=1-\frac{3\tilde{x}^2}{L^2}+\frac{2\tilde{x}^3}{L^3}\tilde{d}_2 $$

$$N_3(\tilde{x})=\tilde{x}-\frac{2\tilde{x}^2}{L}+\frac{\tilde{x}^3}{L^2}\tilde{d}_3 $$

$$N_5(\tilde{x})=\frac{3\tilde{x}^2}{L^2}-\frac{2\tilde{x}^3}{L^3}\tilde{d}_5$$

$$N_6(\tilde{x})=-\frac{\tilde{x}^2}{L}+\frac{\tilde{x}^3}{L^2}\tilde{d}_6$$



Determination of Applied Forces
The other applied internal forces $$ N_{5}\left(\tilde{x}\right) $$ and $$ N_{6}\left(\tilde{x}\right) $$ are illustrated in figure 9 below.



Looking at the equation for a vector
Recall: $$\tilde{d}^{(e)}=\tilde{T}^{(e)}d^{(e)}$$ when determining beam shape functions

Compute: $$u(\tilde{x}),v(\tilde{x})$$



In order to compute $$u(\tilde{x}),v(\tilde{x})$$ we use the following equations:

$$u(\tilde{x})=N_1(\tilde{x})\tilde{d}_1+N_4(\tilde{x})\tilde{d}_4$$

$$v(\tilde{x})=N_2(\tilde{x})\tilde{d}_2+N_3(\tilde{x})\tilde{d}_3+N_5(\tilde{x})\tilde{d}_5+N_6(\tilde{x})\tilde{d}_6$$

The following relationship can be established and represent both vectors as:

$$\begin{Bmatrix} u(\tilde{x})\\ v(\tilde{x}) \end{Bmatrix}=\begin{bmatrix} N_1 & 0 & 0 & N_4 & 0 & 0\\ 0 & N_2 & N_3 & 0 & N_5 & N_6 \end{bmatrix}\begin{Bmatrix} \tilde{d}_1^{(e)}\\ \tilde{d}_2^{(e)}\\ \tilde{d}_3^{(e)}\\ \tilde{d}_4^{(e)}\\ \tilde{d}_5^{(e)}\\ \tilde{d}_6^{(e)} \end{Bmatrix}$$

Further simplifying this result, we take the following:

$$\begin{Bmatrix} u_x(\tilde{x})\\ u_y(\tilde{x}) \end{Bmatrix}= R^T\begin{Bmatrix} u(\tilde{x})\\ v(\tilde{x}) \end{Bmatrix}$$

$$\begin{Bmatrix} u(\tilde{x})\\ v(\tilde{x}) \end{Bmatrix}=R^T(N(\tilde{x}))\tilde{T}^{(e)}d^{(e)}$$

Using Dimensional Analysis with the Shape Functions
$$\begin{bmatrix} u \end{bmatrix}=L$$

$$\begin{bmatrix} N_{1} \end{bmatrix}=\begin{bmatrix} N_{4} \end{bmatrix}=1$$

$$\begin{bmatrix} V \end{bmatrix}=L$$

$$\begin{bmatrix} N_{2} \end{bmatrix}\begin{bmatrix} \tilde{d}_{2} \end{bmatrix}=L$$ (displacement translational)

$$\begin{bmatrix} N_{3} \end{bmatrix}\begin{bmatrix} \tilde{d}_{3} \end{bmatrix}=L$$ (rotational)

$$\begin{bmatrix} N_{5} \end{bmatrix}\begin{bmatrix} \tilde{d}_{5} \end{bmatrix}=L$$ (displacement translational)

$$\begin{bmatrix} N_{6} \end{bmatrix}\begin{bmatrix} \tilde{d}_{6} \end{bmatrix}=L$$ (rotational)

Deriving Beam Shape Functions
From the m\ddot{v} for the static case: $$\frac{\delta^{2}}{\delta x^{2}}\begin{Bmatrix} (EI)\frac{\delta^{2}v}{\delta x^{2}} \end{Bmatrix}=0$$

"EI" is a constant so the above equation becomes:

$$\frac{\delta^{4} }{\delta x^{4}}v=0$$

$$V(X)=C_{0}+C_{1}X^{1}+C_{2}X^{2}+C_{3}X^{3}$$

Use the following boundary conditions to solve for C1, C2, C3 $$V(0)=1, V(L)=0$$

$$V'(0)= V'(L)=0$$

Solving for C1, C2, C3: $$V(L)=1+C_{1}L^{1}+C_{2}L^{2}+C_{3}L^{3}$$$$(1)$$

$$V'(X)=C_{1}+2C_{2}X+3C_{3}X^{2}$$

$$V'(0)=C_{1}=0$$

$$V'(L)=2C_{2}L+3C_{3}L^{2}=0$$$$(2)$$

$$(2)\Rightarrow \frac{C_{3}}{C_{2}}=-\frac{2C_{2}}{3L}$$

$$(1)\Rightarrow 0=1+C_{2}L^{2}+(-\frac{2C_{2}}{3L})L^{3}$$

$$C_{2}=-\frac{3}{L^{2}}$$ $$C_{3}=\frac{2}{L^{3}}$$

General Cases: The Simple Form
$$\hat{k}_{23} = \frac{6EI}{L^2} = \int_{0}^{L}{\frac{d^2N_2}{dx^2}(EI)\frac{d^2N_3}{dx^2}dx}$$

In general,

$$\hat{k}_{ij} = \int_{0}^{L}{\frac{d^2N_i}{dx^2}(EI)\frac{d^2N_j}{dx^2}dx}$$

With i,j = 2,3,5,6

Elastodynamics (trusses, 2-D frames, 3-D elasticity)
Looking at the Modal problem (lecture 31), we use the discrete Principle of Virtual Work:

$$ \mathbf{\bar{w}} = [\mathbf{\bar{M}}\mathbf{\ddot{\bar{d}}} + \mathbf{\bar{K}}\mathbf{\bar{d}} - \mathbf{\bar{F}}] = 0$$

It is important to note the following:
 * the boundary conditions are already applied
 * K matrix represents the reduced stiffness matrix.

The above equation is true for all $$\mathbf{\bar{w}}$$.

$$\mathbf{\bar{M}}\mathbf{\ddot{\bar{d}}} + \mathbf{\bar{k}}\mathbf{\bar{d}} = \mathbf{\bar{F}}(t) (1)$$

$$\mathbf{\bar{d}}(0) = \mathbf{\bar{d}}_0 (2)$$

$$\mathbf{\dot{\bar{d}}}(0) = \mathbf{\bar{V}}_0 (3)$$

Equations (1), (2), and (3) complete the ordinary differential equations (ODEs). These differential equations are second order in time, and have their initial conditions governed by the elastodynamics of the discretized continuous problem.

Equations (1), (2), and (3) are solved by taking the following steps:

Step 1: Using the unforced vibrations problem
$$\mathbf{\bar{M}}_{nxn}\mathbf{\ddot{v}}_{nx1} + \mathbf{\bar{K}}_{nxn}\mathbf{v}_{nx1} = \mathbf{0}_{nx1} (4)$$

This equation is equal to zero because there are no fores acting on the system. This equation will be referred as (4).

Assuming:

$$\mathbf{v}(t)_{nx1} = (sinwt)\mathbf{\phi _{nx1}}$$

Where,


 * the phi matrix is not dependent on time

Therefore:

$$\mathbf{\ddot{v}} = -\omega ^2 sin\omega t\mathbf{\phi }$$

This means equation (4) translates into:

$$-\omega ^2 sin\omega t\mathbf{\bar{M}}\mathbf{\phi } + \omega ^2 sin\omega t\mathbf{\bar{K}}\mathbf{\phi } = \mathbf{0}$$

Concluding that,

$$\mathbf{\bar{k}}\mathbf{\phi } = \omega ^2\mathbf{\bar{M}}\mathbf{\phi }$$

Which is the generalized eigenvalue problem that was looked at in HW 4 and HW 5, as has the general form:

$$\mathbf{A}\mathbf{x} = \lambda \mathbf{B}\mathbf{x}$$

Where lambda represents an eigen value.

Setting the B matrix to zero corresponds to a general eigen value problem and is equal to the identity matrix. After this transformation the equation becomes:

$$\mathbf{A}\mathbf{x} = \lambda \mathbf{x}$$

Indicating that one of the eigenvalue is:

$$\lambda =\omega ^2$$

Using,

$$(\lambda _i,\mathbf{\phi _i})$$

to represent the eigen pairs, i = 1 through n

Step 2: Modal Superposition Method
Knowing the orthogonal properties of the eigen pairs, we can calculate the following:

$$\mathbf{\phi _i}^T_{1xn}\mathbf{\bar{M}}_{nxn}\mathbf{\phi }_{nx1} = \delta _{ij} = \begin{cases} & \text{1 if } i=j \\ & \text{0 if } i\neq j \end{cases}$$

The delta is taken to be known as the Kronecker delta.

This simplification is made possible due to the mass orthogonality properties of eigenvectors.

This is now applied to equations (1), (2), (3), and (4) and results:

$$\mathbf{\bar{M}}\mathbf{\phi }_j = \lambda _j \cdot \mathbf{\bar{k}}\mathbf{\phi }_j$$

$$ \mathbf{\phi }_i^T\mathbf{\bar{M}}\mathbf{\phi }_j = \lambda _j\phi _i^T\mathbf{\bar{k}}\mathbf{\phi }_j$$

Here, the left hand side of the equation is equal to the Kronecker delta.

Therefore,

$$\phi _i^T\mathbf{\bar{k}}\mathbf{\phi }_j = \frac{1}{\lambda _j}\delta _{ij}$$

$$\mathbf{\bar{d}}_{nx1}(t) = \sum{\zeta _i(t)\mathbf{\phi }_{inx1}}$$

Equations (1)-(3) simplify to:

$$\mathbf{\bar{M}}(\sum_{j}^{}{\ddot{\zeta }_j\mathbf{\phi }_j}) + \mathbf{\bar{K}}(\sum_{j}^{}{\zeta _j\mathbf{\phi }_j}) = \mathbf{F}$$

Looking at the equation above, the first parenthesis term is equal to the second derivative of the d matrix, while the second parenthesis term is equal to the d matrix itself.

$$\sum_{j}^{}{\ddot{\zeta }_j(\mathbf{\phi }_i^T\mathbf{\bar{M}}\mathbf{\phi }_j} + \sum_{j}^{}{\zeta _j(\mathbf{\phi }_i^T\mathbf{\bar{K}}\mathbf{\phi }_j} = \mathbf{\phi }_i^T\mathbf{F}$$

Looking at this next equation above, the first parenthesis term is equal to the Kronecker delta, while the second parenthesis term is equal to the product of an eigenvalue and the Kronecker delta. Therefore, it is known that this equation can be simplified into an ordinary differential equation (ODE) shown below:

$$\ddot{\zeta } + \lambda _i\zeta _i = \mathbf{\phi }_i^T\mathbf{F}$$

with i ranges from 1 to n.

Wikiversity vs. E-learning Comparison
In concluding this semester, this team agrees that there are both drawbacks and benefits of using MediaWiki and WebCT. The comparisons made are in the areas of collaboration, layout and privacy.

As a means of collaboration, Mediawiki is far superior. When working in terms of collaboration, it is known for its purpose in collaboration. It benefits from common internet accessibility, the history of edits and a common document. When all group members are able to see the same document real time and edit as necessary, it makes things much smoother. The ability to track user edits was very useful for assessing team member contributions and correcting mistakes. WebCT uses discussion boards for collaboration, which severely limits the scope of projects capable. Since there is no common document is established, it is very hard to understand edits rather than constantly exchanging a master document via email or messaging. Ultimately, for this class, the scope of collaboration did not extend as far as was perhaps intended. In general, the assignment was divided based on lecture days and each person made their own posts with little editing. In particular, collaboration across groups was very limited, and hardly existed.

The file structure is very straightforward for WebCT when compared to MediaWiki. The layout of WebCT lends itself to a more logical repository of common files, such as syllabus, homework assignments, solutions, grades, etc. The file structure is very straightforward and easily located for WebCT. However, Mediawiki doesn't have as good a hierarchical structure. This can be confusing and circular at times.

Because WebCT requires an individual login and provides the ability to privately upload individual assignments, it outperforms Mediawiki in terms of privacy due to this "double wall". WebCT is a much more individualized platform as compared to Mediawiki. Also, the universal accessibility of Mediawiki reduces the problems of limited bandwidth often encountered during high traffic periods by WebCT.

Regardless of the platform, we feel that submitting homework and notes as was done in this class is not the most effective use of time. It was often frustrating to type out the Latex code for equations, taking far longer than just writing it by hand. Furthermore, drawing figures by means of Visio or Inkscape was also troublesome and frustrating and extremely time consuming. The return on investment for this time was very small. It served more to frustrate and waste time than to reinforce learning. Mediawiki certainly represents a great resource, but may be better used for presenting a final report or project rather than day-to-day homework assignments in a class such as this one.

Analysis of the 2-Element System
This section analyzes the 2-Element System found in HW2. Various frame/truss and boundary conditions will be explored.

2-Bar Truss
This part is a recap of the 2-Bar Truss System found in HW3

Required MATLAB Functions
Before running the 2-Bar Truss System program, the following three MATLAB program functions must be in the same directory. They are obtained from the textbook website (See HW1 for textbook information). Matlab Truss Files

PlaneTrussElement.m
This function takes the Modulus of Elasticity (e), the Cross-Sectional Area (A) and the Nodal Coordinates (coord) of the element and generates the local stiffness matrix.

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];

NodalSoln.m
This function takes the Global Stiffness Matrix (K), the Global Force vector (R), and two inputs that are the boundary conditions. The output is an array of the displacement and reaction forces.

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);

PlaneTrussResults.m
This function inputs the Modulus of Elasticity (e), Cross-Sectional Area (A), Nodes of Each Element (coord), and the Displacement calculated in the previous function (disps). The output are the Strain, Stress, and Axial Force of each element.

function results = PlaneTrussResults(e, A, coord, disps) % results = 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.

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];

Main MATLAB Function
The following is the main MATLAB function (bar2.m) to solve for the 2-Bar Truss System

e = [3 5]; A = [1 2]; P = 7; L=[4 2]; theta = [30 45]*pi/180; nodes = [0, 0; L(1)*cos(theta(1)), L(1)*sin(theta(1)); L(1)*cos(theta(1))+L(2)*cos(theta(1)), L(1)*sin(theta(1))-L(2)*sin(theta(2))]; dof=2*length(nodes); conn=[1,2; 2,3]; lmm = [1, 2, 3, 4; 3, 4, 5, 6]; elems=size(lmm,1); K=zeros(dof); debc = [1, 2, 5, 6]; ebcVals = zeros(length(debc),1);

%load vector R = zeros(dof,1); R(4) = P;

% Assemble global stiffness matrix K=zeros(dof); for i=1:elems lm=lmm(i,:); con=conn(i,:); k=PlaneTrussElement(e(i), A(i), nodes(con,:)); K(lm, lm) = K(lm, lm) + k; end K R % Nodal solution and reactions [d1, reactions] = NodalSoln(K, R, debc, ebcVals) results=[]; for i=1:elems results = [results; PlaneTrussResults(e(i), A(i), nodes(conn(i,:),:), d1(lmm(i,:)))]; end format short g results

Results
The following are the results to bar2.m

K =

0.5625     0.32476      -0.5625     -0.32476            0            0      0.32476       0.1875     -0.32476      -0.1875            0            0      -0.5625     -0.32476       3.2458      -1.8661      -2.6833       2.1909     -0.32476      -0.1875      -1.8661       1.9764       2.1909      -1.7889            0            0      -2.6833       2.1909       2.6833      -2.1909            0            0       2.1909      -1.7889      -2.1909       1.7889

R =

0    0     0     7     0     0

d1 =

0           0       4.4547       7.7482            0            0

reactions =

-5.0221     -2.8995       5.0221      -4.1005

results =

1.933       5.799        5.799      0.64835       3.2417       6.4835

2-Element Frame
The following analyzes the 2-Element Frame System. The properties of the frame system is the same as the bar truss system. The cross-sectional area is assumed to be square.

Required MATLAB Functions
Before running the 2-Bar Truss System program, the following three MATLAB program functions must be in the same directory. They are obtained from the textbook website (See HW1 for textbook information). Matlab Truss Files

PlaneFrameElement.m
function [ke, rq] = PlaneFrameElement(modulus, inertia, A, qs, qt, coord) % [ke, rq] = PlaneFrameElement(modulus, inertia, A, qs, qt, coord) % Generates equations for a plane frame element % modulus = modulus of elasticity % inertia = moment of inertia % A = area of cross-section % qs = distributed load along the element axis % qt = distributed load normal to the element axis % coord = coordinates at the element ends

EI=modulus*inertia; EA = modulus*A; 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;

ke = [(EA*L^2*ls^2 + 12*EI*ms^2)/L^3, ((-12*EI + EA*L^2)*ls*ms)/L^3, ...       (-6*EI*ms)/L^2, -((EA*L^2*ls^2 + 12*EI*ms^2)/L^3), ... ((12*EI - EA*L^2)*ls*ms)/L^3, (-6*EI*ms)/L^2; ((-12*EI + EA*L^2)*ls*ms)/L^3, (12*EI*ls^2 + EA*L^2*ms^2)/L^3, ...       (6*EI*ls)/L^2, ((12*EI - EA*L^2)*ls*ms)/L^3, ... -((12*EI*ls^2 + EA*L^2*ms^2)/L^3), (6*EI*ls)/L^2; (-6*EI*ms)/L^2, (6*EI*ls)/L^2, (4*EI)/L, ... (6*EI*ms)/L^2, (-6*EI*ls)/L^2,(2*EI)/L; -((EA*L^2*ls^2 + 12*EI*ms^2)/L^3), ((12*EI -EA*L^2)*ls*ms)/L^3, ...       (6*EI*ms)/L^2, (EA*L^2*ls^2 + 12*EI*ms^2)/L^3, ... ((-12*EI + EA*L^2)*ls*ms)/L^3, (6*EI*ms)/L^2; ((12*EI - EA*L^2)*ls*ms)/L^3, -((12*EI*ls^2 + EA*L^2*ms^2)/L^3),...       (-6*EI*ls)/L^2, ((-12*EI + EA*L^2)*ls*ms)/L^3, ... (12*EI*ls^2 + EA*L^2*ms^2)/L^3, (-6*EI*ls)/L^2; (-6*EI*ms)/L^2, (6*EI*ls)/L^2, (2*EI)/L, (6*EI*ms)/L^2, ... (-6*EI*ls)/L^2, (4*EI)/L]; rq = [(L*(ls*qs - ms*qt))/2; (L*(ms*qs + ls*qt))/2;(L^2*qt)/12; (L*(ls*qs - ms*qt))/2; (L*(ms*qs + ls*qt))/2; -(L^2*qt)/12];

NodalSoln.m
NodalSoln.m

PlaneFrameResults.m
function [f, bm, V] = PlaneFrameResults(modulus, inertia, A, ...   qs, qt, coord, dn) % [f, bm, V] = PlaneFrameResults(modulus, inertia, A, ... %   qs, qt, coord, dn) % Generates frame element results % modulus = modulus of elasticity % inertia = moment of inertia % A = area of cross-section % qs = distributed load along the element axis % qt = distributed load normal to the element axis % coord = coordinates at the element ends % dn = nodal solution % [f, bm, V] = [axial force, bending moment, shear]

EI=modulus*inertia; EA = modulus*A; 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;

u = dn([1,4]); v = dn([2,3,5,6]);

f=[]; bm=[]; V=[]; % Change increment to get results at more points for s=0:L/2:L x = x1 + s*ls; y = y1+ s*ms; f = [f; [x,y,EA*(u(2)-u(1))/L]]; dn2 = [(12*s)/L^3 - 6/L^2, (6*s)/L^2 - 4/L, 6/L^2 - (12*s)/L^3, ... (6*s)/L^2 - 2/L]; bm = [bm; [x, y, EI*dn2*v+(qt*(L^2 - 6*s*L + 6*s^2))/(12)]]; dn3 = [12/L^3, 6/L^2, -(12/L^3), 6/L^2]; V = [V; [x, y, EI*dn3*v+((qt*(12*s - 6*L))/(12))]]; end

Main MATLAB Function
The following is the main MATLAB function, frame2.m

e = [3 5]; a = [1 2]; s = [sqrt(a(1)) sqrt(a(2))]; inertia = [(s(1)^4)/12 (s(2)^4)/12]; L = [4 2]; P = 7; theta = [30 45]*pi/180; nodes = [0, 0; L(1)*cos(theta(1)), L(1)*sin(theta(1)); L(1)*cos(theta(1))+L(2)*cos(theta(1)), L(1)*sin(theta(1))-L(2)*sin(theta(2))]; conn=[1,2; 2,3]; lmm=[1,2,3,4,5,6; 4,5,6,7,8,9]; n=3*length(nodes); debc=[1,2,7,8]; ebcVals=zeros(length(debc),1); K=zeros(n); R = zeros(n,1); R(5) = P;

% Generate equations for each element and assemble them. for i=1:2 lm=lmm(i,:); con=conn(i,:); [ke, rq] = PlaneFrameElement(e(i), inertia(i), a(i), 0, 0, nodes(con,:)); K(lm, lm) = K(lm, lm) + ke; R(lm) = R(lm) + rq; end

K R

% Nodal solution and reactions d2 = NodalSoln(K, R, debc, ebcVals) fa=[]; bma=[]; Va=[]; for i=1:2 lm=lmm(i,:); con=conn(i,:); [f, bm, V]=PlaneFrameResults(e(i), inertia(i), a(i), 0, 0, nodes(con,:), d2(lm)); fa = [fa; f]; bma = [bma; bm]; Va = [Va; V]; end fa bma Va

Results
The following are the results of frame2.m

K =

Columns 1 through 6

0.57422     0.30446    -0.046875     -0.57422     -0.30446    -0.046875      0.30446      0.22266      0.08119     -0.30446     -0.22266      0.08119    -0.046875      0.08119         0.25     0.046875     -0.08119        0.125     -0.57422     -0.30446     0.046875        3.973      -1.0101       1.3118     -0.30446     -0.22266     -0.08119      -1.0101       3.0848        1.468    -0.046875      0.08119        0.125       1.3118        1.468       3.2314            0            0            0      -3.3988       1.3145      -1.2649            0            0            0       1.3145      -2.8622      -1.5492            0            0            0       1.2649       1.5492       1.4907

Columns 7 through 9

0           0            0            0            0            0            0            0            0      -3.3988       1.3145       1.2649       1.3145      -2.8622       1.5492      -1.2649      -1.5492       1.4907       3.3988      -1.3145      -1.2649      -1.3145       2.8622      -1.5492      -1.2649      -1.5492       2.9814

R =

0    0     0     0     7     0     0     0     0

d2 =

0           0        3.198       4.0789       7.1629      -3.2732            0            0      -3.8159

fa =

0           0       3.0592       1.7321            1       3.0592       3.4641            2       3.0592       3.4641            2      -18.241       4.3301       1.2929      -18.241       5.1962      0.58579      -18.241

bma =

0           0      0.28116       1.7321            1     -0.40445       3.4641            2      -1.0901       3.4641            2       1.1215       4.3301       1.2929     -0.40445       5.1962      0.58579      -1.9304

Va =

0           0     -0.34281       1.7321            1     -0.34281       3.4641            2     -0.34281       3.4641            2      -1.3648       4.3301       1.2929      -1.3648       5.1962      0.58579      -1.3648

2-Element Frame/Truss Hybrid
The following analyzes the 2-element frame/truss. Element 1 is a frame element and element 2 is a truss element.

Required MATLAB Functions
Before running the 2-Bar Truss System program, the following three MATLAB program functions must be in the same directory. They are obtained from the textbook website (See HW1 for textbook information). Matlab Truss Files

PlaneFrameElement.m
PlaneFrameElement.m

PlaneTrussElement.m
PlaneTrussElement.m

NodalSoln.m
NodalSoln.m

Main MATLAB Function
The following is the main MATLAB function, hybrid.m

e = [3 5]; a = [1 2]; s = [sqrt(a(1)) sqrt(a(2))]; inertia = [(s(1)^4)/12 (s(2)^4)/12]; L = [4 2]; P = 7; theta = [30 45]*pi/180; nodes = [0, 0; L(1)*cos(theta(1)), L(1)*sin(theta(1)); L(1)*cos(theta(1))+L(2)*cos(theta(1)), L(1)*sin(theta(1))-L(2)*sin(theta(2))]; conn=[1,2; 2,3]; lmm=[1,2,3,4,5,6; 4,5,6,7,8,9]; n=3*length(nodes); debc=[1,2,3,7,8]; ebcVals=zeros(length(debc),1); K=zeros(n); R = zeros(n,1); R(5) = P;

% Generate equations for each element and assemble them. % Frame Element lm=lmm(1,:); con=conn(1,:); [ke, rq] = PlaneFrameElement(e(1), inertia(1), a(1), 0, 0, nodes(con,:)); K(lm, lm) = K(lm, lm) + ke; R(lm) = R(lm) + rq; %Truss Element lm=lmm(2,:); con=conn(2,:); k=PlaneTrussElement(e(2), a(2), nodes(con,:)); k1=[]; for i=1:4 k1 = [k1;k(i,:) 0 0]; end k1 = [k1; 0 0 0 0 0 0; 0 0 0 0 0 0]; K(lm, lm) = K(lm, lm) + k1;

K R

% Nodal solution d3 = NodalSoln(K, R, debc, ebcVals)

Results
The following is the results of hybrid.m. The NaN results are unknown and ignored in the next step

K =

Columns 1 through 6

0.57422     0.30446    -0.046875     -0.57422     -0.30446    -0.046875      0.30446      0.22266      0.08119     -0.30446     -0.22266      0.08119    -0.046875      0.08119         0.25     0.046875     -0.08119        0.125     -0.57422     -0.30446     0.046875       3.2575      -1.8864      -2.6364     -0.30446     -0.22266     -0.08119      -1.8864       2.0115       2.1097    -0.046875      0.08119        0.125      -2.6364       2.1097       2.9333            0            0            0       2.1909      -1.7889      -2.1909            0            0            0            0            0            0            0            0            0            0            0            0

Columns 7 through 9

0           0            0            0            0            0            0            0            0       2.1909            0            0      -1.7889            0            0      -2.1909            0            0       1.7889            0            0            0            0            0            0            0            0

R =

0    0     0     0     7     0     0     0     0

d3 =

0           0            0      -0.1556       14.169       -10.33            0            0          NaN

Plot
The following is a plot of the undeformed 2-element, deformed 2-bar truss deformed 2-frame, and 2-element truss/frame hybrid systems. The frames are incomplete since the moment displacement are not taken into account.

The following is the MATLAB code used to plot.

clear all close all

bar2; frame2; hybrid;

% Number of DOFs (Total of 2 for x and y displacement) dof = 2;

% Nodal Coordinates n_node = 3; % # of Nodes n_elem = 2; % # of Elements total_dof = dof * n_node; % Total DOFs of System

L1 = 4; % Length of Element 1 L2 = 2; % Length of Element 2 theta1 = 30*pi/180; % Angle of Element 1 wrt x-axis in rad theta2 = 45*pi/180; % Angle of Element 2 wrt x-axis in rad

% Initial conditions of Truss System position(:,1) = [0; 0]; position(:,2) = [L1*cos(theta1); L1*sin(theta1)]; position(:,3) = [(L1*cos(theta1))+(L2*cos(theta2)); (L1*sin(theta1))-(L2*sin(theta2))];

% Deformed locations of Truss System position2(:,1) = position(:,1); % Node Location Fixed (Boundary Condtion) position2(:,2) = position(:,2)+[d1(3);d1(4)]; % Solved in HW2 position2(:,3) = position(:,3); % Node Location Fixed (Boundary Condtion)

% Deformed locations of Frame System position3(:,1) = position(:,1); % Node Location Fixed (Boundary Condtion) position3(:,2) = position(:,2)+[d2(4);d2(5)]; position3(:,3) = position(:,3); % Node Location Fixed (Boundary Condtion)

% Deformed locations of Hybrid System position4(:,1) = position(:,1); % Node Location Fixed (Boundary Condtion) position4(:,2) = position(:,2)+[d3(4);d3(5)]; position4(:,3) = position(:,3); % Node Location Fixed (Boundary Condtion)

% Set Up Nodal Loccatons for i = 1 : n_node x(i) = position(1,i); y(i) = position(2,i); x2(i) = position2(1,i); y2(i) = position2(2,i); x3(i) = position3(1,i); y3(i) = position3(2,i); x4(i) = position4(1,i); y4(i) = position4(2,i); end

% Connect the Nodes node_connect(1, 1) = 1; % Element 1 node_connect(2, 1) = 2; node_connect(1, 2) = 2; % Element 2 node_connect(2, 2) = 3;

% Plot The Truss System 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)]; xx2 = [x2(node_1),x2(node_2)]; yy2 = [y2(node_1),y2(node_2)]; xx3 = [x3(node_1),x3(node_2)]; yy3 = [y3(node_1),y3(node_2)]; xx4 = [x4(node_1),x4(node_2)]; yy4 = [y4(node_1),y4(node_2)]; plot(xx,yy,'--',xx2,yy2,xx3,yy3,'-r',xx4,yy4,'-g') hold on % Hold the current figure for the next element end

legend('Undeformed','Deformed Truss','Deformed Frame','Deformed Frame/Truss',2)

% Add Global Nodes and Element Numbers text(0.2,0.7,'1','EdgeColor','black'); % Node 1 text(3.4,2.3,'2','EdgeColor','black'); % Node 2 (Initial) text(4.8,1.1,'3','EdgeColor','black'); % Node 3 text(2.0,0.8,'1'); % Element 1 (Initial) text(4.0,1.0,'2'); % Element 2 (Initial)

% Label the Graphs and Axies title('Two-Element System') xlabel('x') ylabel('y')

Analysis of the Electric Pylon System
This section analyzes the Electric Pylon System found in HW6

Electric Pylon Truss System
Analysis of the Electric Pylon truss system was solved in the last homework and can be found here

Electric Pylon Frame System
The Electric Pylon frame system is similar to the truss system, but the analysis is different. The cross-sectional area is assumed to be square

Required MATLAB Functions
Before running the 2-Bar Truss System program, the following three MATLAB program functions must be in the same directory. They are obtained from the textbook website (See HW1 for textbook information). Matlab Truss Files

PlaneFrameElement.m
PlaneFrameElement.m

NodalSoln.m
NodalSoln.m

PlaneFrameResults.m
PlaneFrameResults.m

Main MATLAB Function
The following is the main MATLAB Function to analyze the Electric Pylon Frame, electricframe.m

a = 400; % Cross-sectional area (mm^2) e = 200000; % Modulus (MPa) P = 1000; % Applied load (N) s = sqrt(a); I = (s^4)/12;

% Node locations position(:, 1) = [ 1.5; 0; 0]; % Left Leg position(:, 2) = [ 4.53; 0; 0]; % Right Leg position(:, 3) = [ 1.88; 1.18; 0]; position(:, 4) = [ 3.03; 1.18; 0]; position(:, 5) = [ 4.17; 1.18; 0]; position(:, 6) = [ 2.2; 2.181; 0]; position(:, 7) = [ 3.03; 2.181; 0]; position(:, 8) = [ 3.85; 2.181; 0]; position(:, 9) = [ 2.53; 3.181; 0]; position(:, 10) = [ 3.53; 3.181; 0]; position(:, 11) = [ 2.34; 3.681; 0]; position(:, 12) = [ 3.03; 3.681; 0]; position(:, 13) = [ 3.71; 3.681; 0]; position(:, 14) = [ 2.16; 4.181; 0]; position(:, 15) = [ 2.62; 4.181; 0]; position(:, 16) = [ 3.44; 4.181; 0]; position(:, 17) = [ 3.89; 4.181; 0]; position(:, 18) = [ 2.36; 4.5; 0]; position(:, 19) = [ 3.7; 4.5; 0]; position(:, 20) = [ 1.8; 5.172; 0]; position(:, 21) = [ 4.25; 5.172; 0]; position(:, 22) = [ 0; 6; 0]; % Left Arm position(:, 23) = [ 1.06; 6; 0]; position(:, 24) = [ 1.5; 6; 0]; position(:, 25) = [ 1.82; 6; 0]; position(:, 26) = [ 2.22; 6; 0]; position(:, 27) = [ 2.71; 6; 0]; position(:, 28) = [ 3.34; 6; 0]; position(:, 29) = [ 3.84; 6; 0]; position(:, 30) = [ 4.23; 6; 0]; position(:, 31) = [ 4.55; 6; 0]; position(:, 32) = [ 5; 6; 0]; position(:, 33) = [ 6.05; 6; 0]; % Right Arm position(:, 34) = [ .8; 6.23; 0]; position(:, 35) = [ 5.25; 6.23; 0]; position(:, 36) = [ 1.24; 6.355; 0]; position(:, 37) = [ 4.82; 6.355; 0]; position(:, 38) = [ 1.5; 6.43; 0]; position(:, 39) = [ 2; 6.43; 0]; position(:, 40) = [ 2.48; 6.43; 0]; position(:, 41) = [ 3.03; 6.43; 0]; position(:, 42) = [ 3.58; 6.43; 0]; position(:, 43) = [ 4.05; 6.43; 0]; position(:, 44) = [ 4.55; 6.43; 0]; position(:, 45) = [ 1.5; 7.43; 0]; position(:, 46) = [ 4.55; 7.43; 0];

% Scale to 60m positionM = max(position,[],2); xmax = positionM(1); ymax = positionM(2); position(1,:) = position(1,:)/xmax*60; position(2,:) = position(2,:)/ymax*60; % No need to scale the z-axis since all values are Zero

% Convert from m to mm  position = position*1000;

% Change position array to work in this function nodes = position';

% Build the conn Array 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;

% Adjust the node_connect array to work on this function conn = node_connect';

% Build the lmm array

% Create dof for each node dofs=[]; j = 1; for i=1:size(nodes,1) dofs = [dofs;j j+1 j+2]; j = j + 3; end

% Add the dofs to the lmm array in right order lmm=[]; for i=1:size(conn,1) lmm = [lmm;dofs(conn(i,1),:) dofs(conn(i,2),:)]; end

dof=3*length(nodes); % Total number of DOFs debc = [1:6]; % Boundary Conditions on the Legs ebcVals = zeros(length(debc),1);

%load vector R = zeros(dof,1); R(dofs(33,2)) = -P; % Applies on Right Arm

% Assemble global stiffness matrix K=zeros(dof); for i=1:size(conn,1) lm=lmm(i,:); con=conn(i,:); [ke, rq] = PlaneFrameElement(e, I, a, 0, 0, nodes(con,:)); K(lm, lm) = K(lm, lm) + ke; R(lm) = R(lm) + rq; end

K R

% Nodal solution and reactions [d2, reactions] = NodalSoln(K, R, debc, ebcVals) fa=[]; bma=[]; Va=[]; for i=1:size(conn,1) lm=lmm(i,:); con=conn(i,:); [f, bm, V]=PlaneFrameResults(e, I, a, 0, 0, nodes(con,:), d2(lm)); fa = [fa; f]; bma = [bma; bm]; Va = [Va; V]; end fa bma Va

Results
The following are the results to eletricframe.m

K =

Columns 1 through 6

4258.1      4680.9       -168.2            0            0            0       4680.9         8014       98.246            0            0            0       -168.2       98.246  1.6363e+006            0            0            0 0           0            0       4178.7      -4639.8      -172.35            0            0            0      -4639.8       8211.5      -97.383            0            0            0      -172.35      -97.383   1.652e+006 -1056       -2670        141.7            0            0            0        -2670      -6751.1       -56.04            0            0            0       -141.7        56.04  5.2047e+005            0            0            0 -3202.1     -2010.9       26.505      -3210.9       2056.8       27.652      -2010.9      -1262.8      -42.206       2056.8      -1317.5       43.169      -26.505       42.206  2.9766e+005      -27.652      -43.169  3.0189e+005 0           0            0      -967.81         2583        144.7            0            0            0         2583        -6894       54.214            0            0            0       -144.7      -54.214  5.2412e+005 0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            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 7 through 12

-1056       -2670       -141.7      -3202.1      -2010.9      -26.505        -2670      -6751.1        56.04      -2010.9      -1262.8       42.206        141.7       -56.04  5.2047e+005       26.505      -42.206  2.9766e+005 0           0            0      -3210.9       2056.8      -27.652            0            0            0       2056.8      -1317.5      -43.169            0            0            0       27.652       43.169  3.0189e+005 13110      8503.5      -103.14      -7014.5            0            0       8503.5        16647        211.3            0    -0.021571       123.01      -103.14        211.3  3.9676e+006            0      -123.01  4.6763e+005 -7014.5           0            0        20504      -45.887       54.157            0    -0.021571      -123.01      -45.887       2580.4       3.1306            0       123.01  4.6763e+005       54.157       3.1306  3.0778e+006 0           0            0        -7076            0            0            0            0            0            0    -0.022144      -125.18            0            0            0            0       125.18  4.7173e+005 -1230.3     -3133.7       197.49            0            0            0      -3133.7        -7982      -77.534            0            0            0      -197.49       77.534  6.1415e+005            0            0            0 -3809.3     -2699.9       47.346            0            0            0      -2699.9      -1913.6        -66.8            0            0            0      -47.346         66.8  3.8152e+005            0            0            0 0           0            0            0            0            0            0            0            0            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 13 through 18

0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0      -967.81         2583       -144.7            0            0            0         2583        -6894      -54.214            0            0            0        144.7       54.214  5.2412e+005            0            0            0 0           0            0      -1230.3      -3133.7      -197.49            0            0            0      -3133.7        -7982       77.534            0            0            0       197.49      -77.534  6.1415e+005 -7076           0            0            0            0            0            0    -0.022144       125.18            0            0            0            0      -125.18  4.7173e+005            0            0            0 13083       -8440      -100.97            0            0            0        -8440        16823      -215.88            0            0            0      -100.97      -215.88  3.9875e+006            0            0            0 0           0            0        12245       6329.7       2.1773            0            0            0       6329.7        15868       237.76            0            0            0       2.1773       237.76  3.7483e+006 -3809      2723.3       48.176      -9718.9            0            0       2723.3      -1947.1       67.381            0    -0.057376      -236.14      -48.176      -67.381  3.8374e+005            0       236.14  6.4793e+005 -1230.3      3133.7       197.49            0            0            0       3133.7        -7982       77.534            0            0            0      -197.49      -77.534  6.1415e+005            0            0            0 0           0            0      -1295.3        -3196       195.31            0            0            0        -3196      -7886.1      -79.154            0            0            0      -195.31       79.154  6.1209e+005 0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            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 19 through 24

0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0      -3809.3      -2699.9      -47.346            0            0            0      -2699.9      -1913.6         66.8            0            0            0       47.346        -66.8  3.8152e+005            0            0            0 0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0        -3809       2723.3      -48.176      -1230.3       3133.7      -197.49       2723.3      -1947.1      -67.381       3133.7        -7982      -77.534       48.176       67.381  3.8374e+005       197.49       77.534  6.1415e+005 -9718.9           0            0            0            0            0            0    -0.057376       236.14            0            0            0            0      -236.14  6.4793e+005            0            0            0 31798      -23.47      -208.15      -9837.4            0            0       -23.47        16122       6.3754            0    -0.059501       241.94      -208.15       6.3754  6.3893e+006            0      -241.94  6.5583e+005 -9837.4           0            0        12301      -6272.4     -0.31588            0    -0.059501      -241.94      -6272.4        15969      -242.14            0       241.94  6.5583e+005     -0.31588      -242.14  3.7693e+006 -2311.6      3764.4       151.83            0            0            0       3764.4      -6130.6       93.233            0            0            0      -151.83      -93.233  5.6281e+005            0            0            0 -2311.6     -3764.4       151.83      -1233.5       3138.7        197.8      -3764.4      -6130.6      -93.233       3138.7      -7986.7       77.736      -151.83       93.233  5.6281e+005       -197.8      -77.736  6.1468e+005 0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            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 25 through 30

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Columns 31 through 36

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Columns 37 through 42

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Columns 43 through 48

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Columns 49 through 54

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Columns 55 through 60

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Columns 61 through 66

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Columns 67 through 72

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Columns 73 through 78

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Columns 79 through 84

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Columns 85 through 90

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Columns 91 through 96

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Columns 97 through 102

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Columns 103 through 108

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Columns 109 through 114

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Columns 115 through 120

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506.15  1.3005e+006        686.7      -509.92  1.2331e+006 0           0            0      -5804.7       8835.9      -774.76            0            0            0       8835.9       -13452      -508.93            0            0            0       774.76       508.93  1.2837e+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     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1587.2      -11.602   1.085e+007            0      -706.07  1.1204e+006 -16806           0            0        43851       17.199       1461.5            0     -0.29665      -706.07       17.199        25375      -169.28            0       706.07  1.1204e+006       1461.5      -169.28    9.23e+006 0           0            0       -14667            0            0            0            0            0            0     -0.19719      -537.78            0            0            0            0       537.78  9.7778e+005 0           0            0            0            0            0            0            0            0            0            0            0            0            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 121 through 126

0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            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 127 through 132

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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            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            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        51597       2198.6       2083.7            0     -0.26245      -650.71       2198.6        43632       932.12            0       650.71  1.0756e+006       2083.7       932.12  1.2697e+007 0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0      -2311.6      -3764.4       151.83    -0.060766            0       245.36      -3764.4      -6130.6      -93.233            0      -9906.7            0      -151.83       93.233  5.6281e+005      -245.36            0  6.6044e+005

Columns 133 through 138

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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            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            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            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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           0           0           0           0           0           0           0           0           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           0           0           0           0           0           0           0           0           0           0           0           0           0           0

d2 =

0           0            0            0            0            0     -0.24614      0.21895   1.417e-005 -0.15703  -0.0027938 -2.4686e-005 -0.211    -0.34319  1.6905e-005 -0.21575     0.38648 -3.2663e-005 -0.21762    -0.14083 -6.0656e-005 -0.21753    -0.62551 -2.6214e-005 0.30459     0.35724  -0.00011182      0.33289      -0.6888  -0.00011361      0.84792      0.74728  -0.00016371      0.87753      -0.2446  -0.00012624      0.91042      -1.1024   -0.0001487       1.5418       1.2099  -0.00024631       1.5409      0.23912  -0.00017152         1.48     -0.75545  -0.00014212       1.4796      -1.4891  -0.00017223        2.174       0.7244  -0.00027969       1.9197      -1.1672  -0.00018714       4.5897       2.8772  -0.00034985       3.4349        -2.63  -0.00032074       6.0713        6.519  -0.00020147       6.0713        4.401  -0.00020181       6.0713       3.5219  -0.00019771       6.0736       2.8632  -0.00020731        6.086       2.0637  -0.00018169        6.106       1.2662  -0.00017283       6.0416      0.11615  -0.00022184       5.9189      -1.1413  -0.00031855       5.7851      -2.4771  -0.00038829       5.6429      -3.7847  -0.00047491       5.4055      -6.3545  -0.00069162       4.8495      -18.248   -0.0014319       6.4455       4.9205  -0.00020122       6.7538      -8.1535  -0.00081383       6.6489       4.0414  -0.00020254       7.1971      -5.2394  -0.00056907       6.7723       3.5161   -0.0002042       6.7647       2.4997  -0.00019862       6.7484       1.6335  -0.00017478       6.7998      0.71144  -0.00018876       6.9413     -0.44373  -0.00025705       7.1234      -1.8244  -0.00036432       7.3565      -3.8516  -0.00046347       8.4199       3.5161  -0.00020588       10.425      -3.8516  -0.00038093

reactions =

183.78     -501.65      -51.164      -183.78       1501.7      -52.192

fa =

14876           0      -1921.6        16760       4764.5      -1921.6        18645       9528.9      -1921.6        14876            0      -701.14        22463       4764.5      -701.14        30050       9528.9      -701.14        18645       9528.9       625.03        24347       9528.9       625.03        30050       9528.9       625.03        30050       9528.9       711.11        37488       4764.5       711.11        44926            0       711.11        30050       9528.9       -381.9        35702       9528.9       -381.9        41355       9528.9       -381.9        44926            0      -1658.9        43140       4764.5      -1658.9        41355       9528.9      -1658.9        18645       9528.9        163.2        24347        13571        163.2        30050        17612        163.2        41355       9528.9      -38.088        35702        13571      -38.088        30050        17612      -38.088        18645       9528.9       279.92        20231        13571       279.92        21818        17612       279.92        41355       9528.9      -60.131        39769        13571      -60.131        38182        17612      -60.131        21818        17612      -18.156        25934        17612      -18.156        30050        17612      -18.156        30050        17612      0.88143        34116        17612      0.88143        38182        17612      0.88143        21818        17612       4777.5        23455        21650       4777.5        25091        25688       4777.5        30050        17612       4408.6        27570        21650       4408.6        25091        25688       4408.6        30050        17612       4647.5        32529        21650       4647.5        35008        25688       4647.5        38182        17612         5075        36595        21650         5075        35008        25688         5075        25091        25688       228.26        30050        25688       228.26        35008        25688       228.26        25091        25688       9755.2        24149        27707       9755.2        23207        29725       9755.2        25091        25688       7167.7        27570        27707       7167.7        30050        29725       7167.7        35008        25688       6813.7        32529        27707       6813.7        30050        29725       6813.7        35008        25688        10466        35901        27707        10466        36793        29725        10466        23207        29725       346.15        26628        29725       346.15        30050        29725       346.15        30050        29725       390.13        33421        29725       390.13        36793        29725       390.13        23207        29725        12573        22314        31744        12573        21421        33763        12573        23207        29725        11313        24595        31744        11313        25983        33763        11313        30050        29725       9261.1        28017        31744       9261.1        25983        33763       9261.1        30050        29725       8411.3        32083        31744       8411.3        34116        33763       8411.3        36793        29725       9405.4        35455        31744       9405.4        34116        33763       9405.4        36793        29725        10314        37686        31744        10314        38579        33763        10314        21421        33763      -15.427        23702        33763      -15.427        25983        33763      -15.427        25983        33763      -598.82        30050        33763      -598.82        34116        33763      -598.82        34116        33763      -7.3223        36347        33763      -7.3223        38579        33763      -7.3223        21421        33763        27825        19636        37764        27825        17851        41766        27825        21421        33763        15557        22413        35051        15557        23405        36339        15557        25983        33763        13897        24694        35051        13897        23405        36339        13897        34116        33763       9651.2        35405        35051       9651.2        36694        36339       9651.2        38579        33763        11032        37636        35051        11032        36694        36339        11032        38579        33763        17850        40364        37764        17850        42149        41766        17850        23405        36339        24888        20628        39052        24888        17851        41766        24888        36694        36339        15754        39421        39052        15754        42149        41766        15754        17851        41766        16196        16364        45109        16196        14876        48452        16196        17851        41766        15196        19934        45109        15196        22017        48452        15196        42149        41766        25393        40116        45109        25393        38083        48452        25393        42149        41766        24136        43636        45109        24136        45124        48452        24136            0        48452 -6.6526e-005 5256.2       48452 -6.6526e-005 10512       48452 -6.6526e-005 10512       48452      0.00165        12694        48452      0.00165        14876        48452      0.00165        14876        48452       59.344        16463        48452       59.344        18050        48452       59.344        18050        48452       249.91        20033        48452       249.91        22017        48452       249.91        22017        48452       328.99        24446        48452       328.99        26876        48452       328.99        26876        48452      -825.09        30000        48452      -825.09        33124        48452      -825.09        33124        48452      -1979.2        35603        48452      -1979.2        38083        48452      -1979.2        38083        48452      -2768.3        40017        48452      -2768.3        41950        48452      -2768.3        41950        48452      -3584.6        43537        48452      -3584.6        45124        48452      -3584.6        45124        48452        -4255        47355        48452        -4255        49587        48452        -4255        49587        48452      -4271.3        54793        48452      -4271.3        60000        48452      -4271.3            0        48452       3673.9       3966.9        49381       3673.9       7933.9        50310       3673.9        10512        48452       9420.5       9223.1        49381       9420.5       7933.9        50310       9420.5        10512        48452        13682        11405        49886        13682        12298        51319        13682        14876        48452        11984        13587        49886        11984        12298        51319        11984        14876        48452        16149        14876        50188        16149        14876        51925        16149        18050        48452        11881        16463        50188        11881        14876        51925        11881        18050        48452        14159        18942        50188        14159        19835        51925        14159        22017        48452        13239        20926        50188        13239        19835        51925        13239        22017        48452        12251        23306        50188        12251        24595        51925        12251        26876        48452        12369        25736        50188        12369        24595        51925        12369        26876        48452        11798        28463        50188        11798        30050        51925        11798        33124        48452        13078        31587        50188        13078        30050        51925        13078        33124        48452        17097        34314        50188        17097        35504        51925        17097        38083        48452        18911        36793        50188        18911        35504        51925        18911        38083        48452        23797        39124        50188        23797        40165        51925        23797        41950        48452        27421        41058        50188        27421        40165        51925        27421        41950        48452        26724        43537        50188        26724        45124        51925        26724        45124        48452        39480        45124        50188        39480        45124        51925        39480        45124        48452        31697        46463        49886        31697        47802        51319        31697        49587        48452        42442        48694        49886        42442        47802        51319        42442        49587        48452        34819        50826        49381        34819        52066        50310        34819        60000        48452        18696        56033        49381        18696        52066        50310        18696       7933.9        50310       3632.6        10116        50814       3632.6        12298        51319       3632.6        52066        50310       8093.3        49934        50814       8093.3        47802        51319       8093.3        12298        51319       3726.7        13587        51622       3726.7        14876        51925       3726.7        47802        51319       4643.8        46463        51622       4643.8        45124        51925       4643.8        14876        51925      -121.95        17355        51925      -121.95        19835        51925      -121.95        19835        51925       -274.4        22215        51925       -274.4        24595        51925       -274.4        24595        51925       753.77        27322        51925       753.77        30050        51925       753.77        30050        51925       2075.7        32777        51925       2075.7        35504        51925       2075.7        35504        51925       3124.9        37835        51925       3124.9        40165        51925       3124.9        40165        51925       3761.5        42645        51925       3761.5        45124        51925       3761.5        14876        51925        16323        14876        55962        16323        14876        60000        16323        19835        51925        13973        17355        55962        13973        14876        60000        13973        40165        51925        27871        42645        55962        27871        45124        60000        27871        45124        51925        30396        45124        55962        30396        45124        60000        30396

bma =

14876           0       25.987        16760       4764.5       3.6877        18645       9528.9      -18.612        14876            0       7.2088        22463       4764.5       -3.674        30050       9528.9      -14.557        18645       9528.9      -28.985        24347       9528.9      -9.0853        30050       9528.9       10.814        30050       9528.9       15.048        37488       4764.5       3.7263        44926            0      -7.5958        30050       9528.9      -27.294        35702       9528.9       9.8101        41355       9528.9       46.914        44926            0       -61.89        43140       4764.5       4.4302        41355       9528.9       70.751        18645       9528.9      -17.129        24347        13571      -14.274        30050        17612      -11.419        41355       9528.9       27.064        35702        13571      -14.882        30050        17612      -56.827        18645       9528.9       38.198        20231        13571      -14.381        21818        17612      -66.961        41355       9528.9      -64.564        39769        13571      -13.241        38182        17612       38.082        21818        17612      -42.892        25934        17612      -9.0686        30050        17612       24.755        30050        17612      -20.512        34116        17612       11.294        38182        17612         43.1        21818        17612       102.27        23455        21650      -24.226        25091        25688      -150.72        30050        17612       219.95        27570        21650      -14.398        25091        25688      -248.75        30050        17612       34.583        32529        21650      -14.902        35008        25688      -64.387        38182        17612       88.612        36595        21650      -26.861        35008        25688      -142.33        25091        25688       11.202        30050        25688     -0.48104        35008        25688      -12.164        25091        25688       777.99        24149        27707      -31.054        23207        29725       -840.1        25091        25688       56.326        27570        27707      -6.0118        30050        29725      -68.349        35008        25688        468.6        32529        27707      -5.2657        30050        29725      -479.13        35008        25688       114.58        35901        27707      -21.194        36793        29725      -156.97        23207        29725       14.661        26628        29725       14.603        30050        29725       14.544        30050        29725        15.48        33421        29725       -8.881        36793        29725      -33.242        23207        29725       1072.9        22314        31744      -49.894        21421        33763      -1172.7        23207        29725       204.44        24595        31744      -4.2486        25983        33763      -212.93        30050        29725       630.32        28017        31744      -21.071        25983        33763      -672.46        30050        29725       118.34        32083        31744      -7.3924        34116        33763      -133.13        36793        29725       720.33        35455        31744       3.6185        34116        33763      -713.09        36793        29725       249.93        37686        31744      -14.213        38579        33763      -278.36        21421        33763       30.105        23702        33763        43.72        25983        33763       57.335        25983        33763       77.557        30050        33763       9.6387        34116        33763       -58.28        34116        33763      -43.823        36347        33763      -17.988        38579        33763       7.8472        21421        33763       860.16        19636        37764      -31.508        17851        41766      -923.17        21421        33763       532.04        22413        35051      -27.378        23405        36339       -586.8        25983        33763       1495.7        24694        35051      -79.142        23405        36339        -1654        34116        33763        193.9        35405        35051      -32.936        36694        36339      -259.77        38579        33763       1393.9        37636        35051      -12.461        36694        36339      -1418.8        38579        33763       167.12        40364        37764      -45.194        42149        41766      -257.51        23405        36339       1195.8        20628        39052      -24.095        17851        41766        -1244        36694        36339       86.404        39421        39052      -46.303        42149        41766      -179.01        17851        41766       846.56        16364        45109       55.437        14876        48452      -735.69        17851        41766       386.98        19934        45109       56.924        22017        48452      -273.13        42149        41766       1043.2        40116        45109      0.74729        38083        48452      -1041.7        42149        41766       468.62        43636        45109      -56.174        45124        48452      -580.97            0        48452      0.17061       5256.2        48452    -0.085796        10512        48452     -0.34221        10512        48452        -3.78        12694        48452       2.5092        14876        48452       8.7983        14876        48452       -33.49        16463        48452       -8.073        18050        48452       17.344        18050        48452      -11.123        20033        48452       17.226        22017        48452       45.575        22017        48452       48.122        24446        48452       4.8617        26876        48452      -38.399        26876        48452        13.06        30000        48452       -20.92        33124        48452        -54.9        33124        48452       1.5903        35603        48452      -52.006        38083        48452       -105.6        38083        48452      -14.753        40017        48452      -48.087        41950        48452       -81.42        41950        48452       25.811        43537        48452      -72.779        45124        48452      -171.37        45124        48452      -102.82        47355        48452      -129.49        49587        48452      -156.17        49587        48452      -313.11        54793        48452      -189.57        60000        48452      -66.037            0        48452       10.244       3966.9        49381     0.084684       7933.9        50310      -10.074        10512        48452       1838.2       9223.1        49381      0.50096       7933.9        50310      -1837.2        10512        48452       452.72        11405        49886     -0.57028        12298        51319      -453.86        14876        48452       1386.2        13587        49886      -3.3391        12298        51319      -1392.9        14876        48452       913.27        14876        50188      -4.9857        14876        51925      -923.24        18050        48452       1173.7        16463        50188        1.766        14876        51925      -1170.1        18050        48452       456.17        18942        50188       5.9411        19835        51925      -444.28        22017        48452       1145.6        20926        50188      -11.007        19835        51925      -1167.6        22017        48452       295.65        23306        50188       4.2567        24595        51925      -287.14        26876        48452       1008.6        25736        50188      -1.2552        24595        51925      -1011.1        26876        48452        204.8        28463        50188      -9.0334        30050        51925      -222.87        33124        48452       1170.1        31587        50188        19.02        30050        51925      -1132.1        33124        48452        382.3        34314        50188      -22.304        35504        51925      -426.91        38083        48452       1699.2        36793        50188       37.914        35504        51925      -1623.4        38083        48452       652.44        39124        50188      -30.142        40165        51925      -712.72        41950        48452       2243.5        41058        50188       16.376        40165        51925      -2210.8        41950        48452        412.1        43537        50188      -42.617        45124        51925      -497.33        45124        48452         2082        45124        50188       8.7813        45124        51925      -2064.4        45124        48452        552.6        46463        49886      -64.014        47802        51319      -680.62        49587        48452       4647.7        48694        49886       96.765        47802        51319      -4454.1        49587        48452       783.27        50826        49381       -105.2        52066        50310      -993.66        60000        48452       4839.7        56033        49381       202.27        52066        50310      -4435.1       7933.9        50310       19.161        10116        50814     -0.78544        12298        51319      -20.732        52066        50310       5101.3        49934        50814       148.94        47802        51319      -4803.4        12298        51319       28.789        13587        51622      -1.6753        14876        51925      -32.139        47802        51319       6057.5        46463        51622       102.58        45124        51925      -5852.4        14876        51925       -8.493        17355        51925       3.0026        19835        51925       14.498        19835        51925       29.296        22215        51925        13.35        24595        51925      -2.5968        24595        51925       30.496        27322        51925      -6.8345        30050        51925      -44.165        30050        51925     -0.74301        32777        51925      -33.386        35504        51925       -66.03        35504        51925      -11.635        37835        51925      -61.365        40165        51925      -111.09        40165        51925      -36.974        42645        51925      -53.323        45124        51925      -69.673        14876        51925        405.7        14876        55962      -0.5557        14876        60000      -406.81        19835        51925       520.53        17355        55962      -2.0447        14876        60000      -524.62        40165        51925       263.27        42645        55962      -4.6761        45124        60000      -272.62        45124        51925       863.78        45124        55962       27.256        45124        60000      -809.27

Va =

14876           0   -0.0043524        16760       4764.5   -0.0043524        18645       9528.9   -0.0043524        14876            0   -0.0012148        22463       4764.5   -0.0012148        30050       9528.9   -0.0012148        18645       9528.9    0.0034897        24347       9528.9    0.0034897        30050       9528.9    0.0034897        30050       9528.9   -0.0012818        37488       4764.5   -0.0012818        44926            0   -0.0012818        30050       9528.9    0.0065637        35702       9528.9    0.0065637        41355       9528.9    0.0065637        44926            0     0.013035        43140       4764.5     0.013035        41355       9528.9     0.013035        18645       9528.9   0.00040842        24347        13571   0.00040842        30050        17612   0.00040842        41355       9528.9    -0.006036        35702        13571    -0.006036        30050        17612    -0.006036        18645       9528.9    -0.012109        20231        13571    -0.012109        21818        17612    -0.012109        41355       9528.9      0.01182        39769        13571      0.01182        38182        17612      0.01182        21818        17612    0.0082181        25934        17612    0.0082181        30050        17612    0.0082181        30050        17612    0.0078221        34116        17612    0.0078221        38182        17612    0.0078221        21818        17612    -0.029035        23455        21650    -0.029035        25091        25688    -0.029035        30050        17612     -0.04946        27570        21650     -0.04946        25091        25688     -0.04946        30050        17612    -0.010444        32529        21650    -0.010444        35008        25688    -0.010444        38182        17612    -0.026617        36595        21650    -0.026617        35008        25688    -0.026617        25091        25688   -0.0023561        30050        25688   -0.0023561        35008        25688   -0.0023561        25091        25688     -0.36315        24149        27707     -0.36315        23207        29725     -0.36315        25091        25688    -0.019497        27570        27707    -0.019497        30050        29725    -0.019497        35008        25688     -0.14821        32529        27707     -0.14821        30050        29725     -0.14821        35008        25688    -0.061511        35901        27707    -0.061511        36793        29725    -0.061511        23207        29725 -1.7051e-005 26628       29725 -1.7051e-005 30050       29725 -1.7051e-005 30050       29725   -0.0072246        33421        29725   -0.0072246        36793        29725   -0.0072246        23207        29725     -0.50866        22314        31744     -0.50866        21421        33763     -0.50866        23207        29725    -0.085171        24595        31744    -0.085171        25983        33763    -0.085171        30050        29725     -0.22735        28017        31744     -0.22735        25983        33763     -0.22735        30050        29725    -0.043885        32083        31744    -0.043885        34116        33763    -0.043885        36793        29725     -0.29586        35455        31744     -0.29586        34116        33763     -0.29586        36793        29725     -0.11966        37686        31744     -0.11966        38579        33763     -0.11966        21421        33763    0.0059689        23702        33763    0.0059689        25983        33763    0.0059689        25983        33763    -0.016704        30050        33763    -0.016704        34116        33763    -0.016704        34116        33763     0.011578        36347        33763     0.011578        38579        33763     0.011578        21421        33763     -0.20351        19636        37764     -0.20351        17851        41766     -0.20351        21421        33763     -0.34413        22413        35051     -0.34413        23405        36339     -0.34413        25983        33763     -0.86414        24694        35051     -0.86414        23405        36339     -0.86414        34116        33763     -0.12447        35405        35051     -0.12447        36694        36339     -0.12447        38579        33763     -0.88128        37636        35051     -0.88128        36694        36339     -0.88128        38579        33763    -0.048457        40364        37764    -0.048457        42149        41766    -0.048457        23405        36339     -0.31422        20628        39052     -0.31422        17851        41766     -0.31422        36694        36339    -0.034495        39421        39052    -0.034495        42149        41766    -0.034495        17851        41766      -0.2162        16364        45109      -0.2162        14876        48452      -0.2162        17851        41766    -0.083795        19934        45109    -0.083795        22017        48452    -0.083795        42149        41766     -0.26642        40116        45109     -0.26642        38083        48452     -0.26642        42149        41766     -0.14342        43636        45109     -0.14342        45124        48452     -0.14342            0        48452 -4.8782e-005 5256.2       48452 -4.8782e-005 10512       48452 -4.8782e-005 10512       48452    0.0028825        12694        48452    0.0028825        14876        48452    0.0028825        14876        48452     0.016018        16463        48452     0.016018        18050        48452     0.016018        18050        48452     0.014293        20033        48452     0.014293        22017        48452     0.014293        22017        48452    -0.017805        24446        48452    -0.017805        26876        48452    -0.017805        26876        48452    -0.010877        30000        48452    -0.010877        33124        48452    -0.010877        33124        48452    -0.021617        35603        48452    -0.021617        38083        48452    -0.021617        38083        48452    -0.017236        40017        48452    -0.017236        41950        48452    -0.017236        41950        48452    -0.062132        43537        48452    -0.062132        45124        48452    -0.062132        45124        48452    -0.011954        47355        48452    -0.011954        49587        48452    -0.011954        49587        48452     0.023727        54793        48452     0.023727        60000        48452     0.023727            0        48452   -0.0024935       3966.9        49381   -0.0024935       7933.9        50310   -0.0024935        10512        48452      -1.1566       9223.1        49381      -1.1566       7933.9        50310      -1.1566        10512        48452     -0.26845        11405        49886     -0.26845        12298        51319     -0.26845        14876        48452     -0.72075        13587        49886     -0.72075        12298        51319     -0.72075        14876        48452     -0.52889        14876        50188     -0.52889        14876        51925     -0.52889        18050        48452     -0.49823        16463        50188     -0.49823        14876        51925     -0.49823        18050        48452     -0.23063        18942        50188     -0.23063        19835        51925     -0.23063        22017        48452     -0.56408        20926        50188     -0.56408        19835        51925     -0.56408        22017        48452     -0.13475        23306        50188     -0.13475        24595        51925     -0.13475        26876        48452     -0.48615        25736        50188     -0.48615        24595        51925     -0.48615        26876        48452    -0.090914        28463        50188    -0.090914        30050        51925    -0.090914        33124        48452     -0.49639        31587        50188     -0.49639        30050        51925     -0.49639        33124        48452     -0.19222        34314        50188     -0.19222        35504        51925     -0.19222        38083        48452     -0.76823        36793        50188     -0.76823        35504        51925     -0.76823        38083        48452     -0.33715        39124        50188     -0.33715        40165        51925     -0.33715        41950        48452      -1.1409        41058        50188      -1.1409        40165        51925      -1.1409        41950        48452     -0.19333        43537        50188     -0.19333        45124        51925     -0.19333        45124        48452      -1.1941        45124        50188      -1.1941        45124        51925      -1.1941        45124        48452     -0.31437        46463        49886     -0.31437        47802        51319     -0.31437        49587        48452      -2.6951        48694        49886      -2.6951        47802        51319      -2.6951        49587        48452      -0.5736        50826        49381      -0.5736        52066        50310      -0.5736        60000        48452      -1.1382        56033        49381      -1.1382        52066        50310      -1.1382       7933.9        50310   -0.0089071        10116        50814   -0.0089071        12298        51319   -0.0089071        52066        50310      -2.2602        49934        50814      -2.2602        47802        51319      -2.2602        12298        51319    -0.023003        13587        51622    -0.023003        14876        51925    -0.023003        47802        51319      -4.3383        46463        51622      -4.3383        45124        51925      -4.3383        14876        51925    0.0046365        17355        51925    0.0046365        19835        51925    0.0046365        19835        51925   -0.0066998        22215        51925   -0.0066998        24595        51925   -0.0066998        24595        51925    -0.013688        27322        51925    -0.013688        30050        51925    -0.013688        30050        51925    -0.011969        32777        51925    -0.011969        35504        51925    -0.011969        35504        51925    -0.021338        37835        51925    -0.021338        40165        51925    -0.021338        40165        51925   -0.0065944        42645        51925   -0.0065944        45124        51925   -0.0065944        14876        51925     -0.10062        14876        55962     -0.10062        14876        60000     -0.10062        19835        51925     -0.11029        17355        55962     -0.11029        14876        60000     -0.11029        40165        51925    -0.056551        42645        55962    -0.056551        45124        60000    -0.056551        45124        51925     -0.20718        45124        55962     -0.20718        45124        60000     -0.20718

Plot
The following is the plot of the Electric Pylon with its undeformed shape, deformed truss, and deformed frame. The deformation is magnified by 1000 for better viewing.

The deformed truss and frame are very similar. The following is a zoomed portion of the right side. The differences are made clear.

The following is the MATLAB function used to plot

close all clear all electric % Must be run to get the displacement array (d) electricframe

% model with 3-D beam elements dof = 2;         % x and y displacement only

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

position(:, 1) = [ 1.5; 0; 0]; position(:, 2) = [ 4.53; 0; 0]; position(:, 3) = [ 1.88; 1.18; 0]; position(:, 4) = [ 3.03; 1.18; 0]; position(:, 5) = [ 4.17; 1.18; 0]; position(:, 6) = [ 2.2; 2.181; 0]; position(:, 7) = [ 3.03; 2.181; 0]; position(:, 8) = [ 3.85; 2.181; 0]; position(:, 9) = [ 2.53; 3.181; 0]; position(:, 10) = [ 3.53; 3.181; 0]; position(:, 11) = [ 2.34; 3.681; 0]; position(:, 12) = [ 3.03; 3.681; 0]; position(:, 13) = [ 3.71; 3.681; 0]; position(:, 14) = [ 2.16; 4.181; 0]; position(:, 15) = [ 2.62; 4.181; 0]; position(:, 16) = [ 3.44; 4.181; 0]; position(:, 17) = [ 3.89; 4.181; 0]; position(:, 18) = [ 2.36; 4.5; 0]; position(:, 19) = [ 3.7; 4.5; 0]; position(:, 20) = [ 1.8; 5.172; 0]; position(:, 21) = [ 4.25; 5.172; 0]; position(:, 22) = [ 0; 6; 0]; position(:, 23) = [ 1.06; 6; 0]; position(:, 24) = [ 1.5; 6; 0]; position(:, 25) = [ 1.82; 6; 0]; position(:, 26) = [ 2.22; 6; 0]; position(:, 27) = [ 2.71; 6; 0]; position(:, 28) = [ 3.34; 6; 0]; position(:, 29) = [ 3.84; 6; 0]; position(:, 30) = [ 4.23; 6; 0]; position(:, 31) = [ 4.55; 6; 0]; position(:, 32) = [ 5; 6; 0]; position(:, 33) = [ 6.05; 6; 0]; position(:, 34) = [ .8; 6.23; 0]; position(:, 35) = [ 5.25; 6.23; 0]; position(:, 36) = [ 1.24; 6.355; 0]; position(:, 37) = [ 4.82; 6.355; 0]; position(:, 38) = [ 1.5; 6.43; 0]; position(:, 39) = [ 2; 6.43; 0]; position(:, 40) = [ 2.48; 6.43; 0]; position(:, 41) = [ 3.03; 6.43; 0]; position(:, 42) = [ 3.58; 6.43; 0]; position(:, 43) = [ 4.05; 6.43; 0]; position(:, 44) = [ 4.55; 6.43; 0]; position(:, 45) = [ 1.5; 7.43; 0]; position(:, 46) = [ 4.55; 7.43; 0];

% Scale to 60m positionM = max(position,[],2); xmax = positionM(1); ymax = positionM(2); position(1,:) = position(1,:)/xmax*60; position(2,:) = position(2,:)/ymax*60; % No need to scale the z-axis since all values are Zero

% Convert from m to mm  position = position*1000;

% The displacement of nodes (electric.m must be run beforehand to get  % the d array) d = d * 1000; %Magnification Factor to see displacement position2=position; j=1; for i=1:size(position2,2) position2(1,i)=position2(1,i)+d(j); position2(2,i)=position2(2,i)+d(j+1); j = j + 2; end

% The displacement of nodes for the frame d2 = d2 * 1000; %Magnification Factor to see displacement position3=position; j=1; for i=1:size(position3,2) position3(1,i)=position3(1,i)+d2(j); position3(2,i)=position3(2,i)+d2(j+1); j = j + 3; 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); x2(i) = position2(1,i); y2(i) = position2(2,i); z2(i) = position2(3,i); x3(i) = position3(1,i); y3(i) = position3(2,i); z3(i) = position3(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)]; xx2 = [x2(node_1),x2(node_2)]; yy2 = [y2(node_1),y2(node_2)]; zz2 = [z2(node_1),z2(node_2)]; xx3 = [x3(node_1),x3(node_2)]; yy3 = [y3(node_1),y3(node_2)]; zz3 = [z3(node_1),z3(node_2)]; plot3(xx,yy,zz,'--',xx2,yy2,zz2,xx3,yy3,zz3,'-r') hold on  end

title('Electric Pylon') xlabel('x (mm)') ylabel('y (mm)') zlabel('z (mm)') legend('Undeformed','Deformed Truss','Deformed Frame',4)

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

Contributing Members
The following Team Wiki members contributed to this report.

Thomas Kehoe --Eml4500.f08.wiki.Kehoe 17:15, 9 December 2008 (UTC)

Gonzalo Barcia -- Eml4500.f08.wiki.Barcia 14:32, 9 December 2008

Ricardo Lopez -- Eml4500.f08.wiki.Lopez 15:06, 9 December 2008

Rodney Dagulo --Eml4500.f08.wiki.dagulo.r 20:18, 9 December 2008 (UTC)

Renee Hood --Eml4500.f08.wiki.hood 17:01, 9 December 2008 (UTC)

Cortland Glowacki --Eml4500.f08.wiki.cort 15:08, 10 December 2008 (UTC)