User:Eml4500.f08.Lulz.Layton.Eric/Hw7

Frame Elements
A basic frame element consists of a truss element combined with a beam element, which gives the frame element the capacity to handle both axial and transverse deformations. Consider Figure 1, which shows a model problem consisting of a system of two frame elements. The point of connection between the two elements, where the force $$\;P$$ is applied, is a rigid connection. This means that the angle between the two elements remains constant after deformation.



The free body diagrams for both elements are shown in Figures 2 and 3. Generally speaking, the degrees of freedom for such a system correspond directly to the general forces, or $$d_i^{(e)} \Rightarrow f_i^{(e)}$$. However, for this problem, two new degrees of freedom have been introduced for both elements. These degrees of freedom correspond as follows from the force vector:


 * $$\underbrace{\left\{\begin{matrix} d_3^{(e)} \\ d_6^{(e)} \end{matrix}\right\}}_{\mbox{rotational degrees of freedom}} \Rightarrow \underbrace{\left\{\begin{matrix} f_3^{(e)} \\ f_6^{(e)} \end{matrix}\right\}}_{\mbox{bending moments}}$$

These new degrees of freedom correspond to rotations, and the associated forces are actually bending moments. Figure 4 shows the global view of the system, with the global degrees of freedom. Once element stiffness matrices $$\mathbf{k}^{(e)}\;$$ are generated, they can be assembled into the global stiffness matrix as follows:


 * $$\mathbf{K} = A\mathbf{k}^{(e)}$$

$$A\;$$ is the assembly operator. An overview of the global stiffness matrix is given in Figure 5.



In order to generate the element stiffness matrices, consider the local degrees of freedom for element 1, as shown in Figure 6. The stiffness matrices can be determined from the following familiar relationship:


 * $$\mathbf{\tilde{k}}^{(e)}\mathbf{\tilde{d}}^{(e)} = \mathbf{\tilde{f}}^{(e)}$$

Where $$\mathbf{\tilde{d}}^{(e)}$$ and $$\mathbf{\tilde{f}}^{(e)}$$ are given as:


 * $$\mathbf{\tilde{d}}^{(e)} = \begin{Bmatrix} \tilde{d}^{(e)}_1\\ \vdots \\\tilde{d}^{(e)}_6 \end{Bmatrix}$$


 * $$\mathbf{\tilde{f}}^{(e)} = \begin{Bmatrix} \tilde{f}^{(e)}_1\\ \vdots \\\tilde{f}^{(e)}_6 \end{Bmatrix}$$

For the rotational degrees of freedom, the following relationships are true:


 * $$\tilde{f}^{(e)}_3 = f^{(e)}_3$$, $$\tilde{f}^{(e)}_6 = f^{(e)}_6$$

Using these considerations, the general element stiffness matrix $$\mathbf{\tilde{k}}^{(e)}$$ can be constructed as follows:


 * $$\mathbf{\tilde{k}}^{(e)} = \begin{bmatrix} \frac{EA}{L}& 0 & 0 & \frac{-EA}{L} & 0 & 0\\ & \frac{12EI}{L^2} & \frac{6EI}{L^2} & 0 & \frac{-6EI}{L^2} & \frac{6EI}{L^2}\\  &  & \frac{4EI}{L} & 0 & \frac{-6EI}{L^2} & \frac{-2EI}{L}\\  &  &  & \frac{EA}{L} & 0 & 0\\  &  &  &  & \frac{12EI}{L^2} & \frac{-6EI}{L^2}\\ sym. &  &  &  &  & \frac{4EI}{L}\end{bmatrix}$$

Dimensional Analysis
It is useful to look at the dimensions of some of the variables involveed in the frame system. Consider first the non-rotational degrees of freedom:


 * $$[\tilde{d}_1] = L = [\tilde{d}_i]$$, for $$i=1,2,4,5\;$$

Conversely, consider the rotational degrees of freedom:


 * $$[\tilde{d}_3] = 1 = [\tilde{d}_6]$$

Figure 7 displays the arc length relationship for circular sections. This relationship is summarized as:


 * $$\widehat{AB} = R\theta \Rightarrow \theta = \frac{\widehat{AB}}{R}$$

Performing a dimensional analysis on this concept results in:


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

Furthermore, consider the stress strain relation:


 * $$\sigma = E\varepsilon \Rightarrow [\sigma] = [E]\underbrace{[\varepsilon]}_{1} = [E]$$


 * $$[\sigma] = [E] = \frac{F}{L^2}$$

Consider the axial displacement $$q\;$$:


 * $$[q] = \frac{[du]}{[dx]} = \frac{L}{L} = 1$$

For some more fundamental variables, consider the area and moment of inertia:


 * $$[A] = L^2\;$$


 * $$[I] = L^4\;$$

With the dimensions of these more basic terms defined, it is possible to analyze the dimensions of more complex variables. For example, consider the stiffness factor $$k\;$$:


 * $$\left[\frac{EA}{L}\right] = [\tilde{k}_1] = \frac{(\frac{F}{L^2})(L^2)}{L} = \frac{F}{L}$$

Analyzing the product of the global stiffness coefficient and the global degree of freedom proves that this product generates the global force components:


 * $$[\tilde{k}_{11}\tilde{d}_1] = [\tilde{k}_{11}][\tilde{d}_1] = (\frac{F}{L})(L) = F$$


 * $$[\tilde{k}_{23}\tilde{d}_3] = [\tilde{k}_{23}]\underbrace{[\tilde{d}_3]}_1 = \frac{[6][E][I]}{[L^2]} = \frac{(1)(\frac{F}{L^2})(L^4)}{L^2} = F$$

Knowing that the stiffness matrix looks as follows:

$$\tilde{\mathbf{k^e}}=\begin{bmatrix} \frac{EA}{L} & 0 & 0 & \frac{-EA}{L} & 0 & 0\\ 0 & \frac{12EI}{L^{3}} & \frac{6EI}{L^2} & 0 & \frac{-12EI}{L^{3}} & \frac{6EI}{L^2}\\ 0 & \frac{6EI}{L^2} & \frac{4EI}{L} & 0 & \frac{-6EI}{L^2} & \frac{2EI}{L}\\ \frac{-EA}{L} & 0 & 0 & \frac{EA}{L} & 0 & 0\\ 0 & \frac{-12EI}{L^3} & \frac{-6EI}{L^2} & 0 & \frac{12EI}{L^{3}} & \frac{-6EI}{L^2}\\ 0 & \frac{6EI}{L^2} & \frac{2EI}{L} & 0 & \frac{-6EI}{L^2} & \frac{4EI}{L} \end{bmatrix}$$

Using the dimensional analysis demonstrated above, the stiffness matrix becomes:

$$\tilde{\mathbf{k^e}}=\begin{bmatrix} \frac{F}{L} & 0 & 0 & \frac{-F}{L} & 0 & 0\\ 0 & \frac{F}{L} & F & 0 & \frac{-F}{L} & F\\ 0 & F & FL & 0 & -F & FL\\ \frac{-F}{L} & 0 & 0 & \frac{F}{L} & 0 & 0\\ 0 & \frac{-F}{L} & -F & 0 & \frac{F}{L} & -F\\ 0 & F & FL & 0 & -F & FL \end{bmatrix}$$

Element Displacement Relation
Returning to the local degrees of freedom, the following relation can be written to convert $$\mathbf{d}^{(e)}\;$$ to $$\mathbf{\tilde{d}}^{(e)}\;$$:


 * $$\begin{Bmatrix} \tilde{d}_1\\ \tilde{d}_2\\ \tilde{d}_3\\ \tilde{d}_4\\ \tilde{d}_5\\ \tilde{d}_6 \end{Bmatrix} = \underbrace{\begin{bmatrix} l^{(e)}& m^{(e)} & 0 & 0 & 0 & 0\\ -m^{(e)} & l^{(e)} & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & l^{(e)} & m^{(e)} & 0\\ 0 & 0 & 0 & -m^{(e)} & l^{(e)} & 0\\ 0 & 0 & 0 & 0 & 0 & 1\end{bmatrix}}_{\mathbf{\tilde{T}^{(e)}}} \begin{Bmatrix} d_1\\ d_2\\ d_3\\ d_4\\ d_5\\ d_6 \end{Bmatrix}$$

Where $$\begin{bmatrix} l^{(e)} & m^{(e)} \\ -m^{(e)} & l^{(e)}\end{bmatrix}$$ is the previously discussed $$\mathbf{R}\;$$.

Derivation of Element Stiffness Matrix from the Principle of Virtual Work
Consider the Principle of Virtual Work (PVW) for beams given as Equation 1 below:


 * $$\int_0^L w(\underbrace{x}_\tilde{x}) [\frac{-\partial^2}{\partial x^2}[(EI)\frac{\partial v^2}{\partial x^2}] + f_t(x) - m(x)\ddot{v}] = 0\;,\;\mbox{for all possible} \;w(x)\;\;(1)$$

Isolating the first term and performing an integration by parts yields the following:


 * $$\alpha := \int_0^L \underbrace{w(x)}_{s(x)} \underbrace{\frac{\partial^2}{\partial x^2}\left\{(EI)\frac{\partial v^2}{\partial x^2}\right\}}_{\underbrace{\frac{\partial}{\partial x} \underbrace{\left(\frac{\partial}{\partial x}\left\{(EI)\frac{\partial v^2}{\partial x^2}\right\}\right)dx}_{r(x)}}_{r'(x)}}$$


 * $$\alpha := \underbrace{\left[w(x)\frac{\partial}{\partial x}\left\{(EI)\frac{\partial v^2}{\partial x^2}\right\}\right]^L_0}_{\beta_1} - \int_0^L \underbrace{\frac{\partial w}{\partial x}}_{s'(x)}\underbrace{\left(\frac{\partial}{\partial x}\left\{(EI)\frac{\partial v^2}{\partial x^2}\right\}\right)dx}_{r(x)}$$

Performing another integration by parts on the second term of the above yields the following:


 * $$\alpha := \underbrace{\left[w(x)\frac{\partial}{\partial x}\left\{(EI)\frac{\partial v^2}{\partial x^2}\right\}\right]^L_0}_{\beta_1} - \int_0^L \underbrace{\frac{\partial w}{\partial x}}_{s'(x)}\underbrace{\left(\frac{\partial}{\partial x}\left\{(EI)\frac{\partial v^2}{\partial x^2}\right\}\right)dx}_{r(x)}$$


 * $$\alpha := \beta_1 - \underbrace{\left[\frac{-\partial w}{\partial x}(EI)\frac{\partial v^2}{\partial x^2}\right]^L_0}_{\beta_2} + \int_0^L \underbrace{\frac{\partial^2 w}{\partial x^2}\left\{(EI)\frac{\partial v^2}{\partial x^2}\right\}dx}_{\gamma}$$

Thus, Equation 1 can be rewritten as:


 * $$ -\beta_1 + \beta_2 - \gamma + \int^L_0 w f_t dx - \int^L_0 w m \ddot{v} dx = 0$$

Consider the beam diagram in Figure 7. Focusing on the $$\gamma\;$$ term for now, the beam stiffness matrix and shape functions can be derived as follows. First, consider the definitions for the deformation components $$v(\tilde{x})\;$$ and $$u(\tilde{x})\;$$:


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


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

Figure 8 shows the drawings for the various shape functions. The functions themselves are defined as:


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


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


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


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

Recall the general element transformation relation:


 * $$\mathbf{\tilde{d}^{(e)}} = \mathbf{\tilde{T}^{(e)}}\mathbf{d^{(e)}}$$



Consider Figure 9, the deformation components, and then the following relation for computing them:


 * $$\mathbf{u}(\tilde{x}) = u(\tilde{x}) \vec{\tilde{i}} + v(\tilde{x}) \vec{\tilde{j}} = u_x(\tilde{x}) \vec{\tilde{i}} + u_y(\tilde{x}) \vec{\tilde{j}}$$

Using the above and the previously discussed shape-function relationships, the following relationships can be derived:


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


 * $$\begin{Bmatrix} u(\tilde{x})\\v(\tilde{x})\end{Bmatrix} = \underbrace{\begin{bmatrix} N_1 & 0 & 0 & N_4 & 0 & 0 \\ 0 & N_2 & N_3 & 0 & N_5 & N_6 \end{bmatrix}}_{\mathbb{N}(\tilde{x})} \mathbf{\tilde{d}^{(e)}}$$

Combining the results of this section, the following overall relation can be determined:


 * $$\begin{Bmatrix} u_x(\tilde{x})\\u_y(\tilde{x})\end{Bmatrix} = \mathbf{R}^T \mathbb{N}(\tilde{x}) \mathbf{\tilde{T}^{(e)}}\mathbf{d^{(e)}}$$

Dimensional Analysis
As before, it is useful to revisit the dimensional analysis of some of the results for the previous section:


 * $$[u] = L\;$$


 * $$ [N_1] = \cdots = [N_6] = 1 $$


 * $$[N_1 \tilde{d}_1] = [N_1][\tilde{d}_1] = L$$


 * $$[N_4 \tilde{d}_4] = [N_4][\tilde{d}_4] = L$$


 * $$[v] = L\;$$


 * $$[N_2][\tilde{d}_2] = [N_5][\tilde{d}_5] = L$$


 * $$[N_3][\tilde{d}_3] = [N_6][\tilde{d}_6] = 1$$

Solving the Beam PDE
Consider again the PDE for beams:


 * $$\frac{\partial^2}{\partial x^2} \left\{(EI)\frac{\partial^2 v}{\partial x^2}\right\} = 0$$

Making the assumption that $$EI\;$$ is a constant leads to the following:


 * $$\frac{\partial v^4}{\partial x^4} = 0$$

This is an easily integrable equation which yields the following relationship and its derivative:


 * $$v(x) = C_0 + C_1x + C_2x^2 + C_3x^3\;$$


 * $$v'(x) = C_1 + 2C_2x + 3C_3x^2\;$$

In order to solve for the coefficients of integration, certain boundary conditions must be used. The following set are used to obtain $$N_2(x)\;$$, where $$\tilde{x} = x\;$$ for simplicity:


 * $$v(0) = 1\;$$


 * $$v'(0) = 1\;$$


 * $$v(L) = 1\;$$


 * $$v'(L) = 0\;$$

Application of the first two boundary condition leads directly to $$C_0\;$$ and $$C_1\;$$


 * $$v(0) = 1 = C_0\;$$


 * $$v'(0) = C_1 = 0\;$$

Evaluating the equation and its derivative at $$x = L$$ yields the following:


 * $$v(L) = \underbrace{C_0}_1 + \underbrace{C_1L}_0 + C_2L^2 + C_3L^3 = 0\;$$


 * $$v'(L) = \underbrace{C_1}_0 + 2C_2L + 3C_3L^2 = 0\;$$

Evaluating for the determined coefficients allows the above to be rewritten as:


 * $$C_2L^2 + C_3L^3 = -1\;$$


 * $$2C_2L + 3C_3L^2 = 0\;$$

These equations form a simple system of two equations, which can be easily solved to yields expressions for the remaining two coefficients of integration:


 * $$C_2 = \frac{-3}{L^2}$$


 * $$C_3 = \frac{2}{L^3}$$

Thus, the original equation can be rewritten as follows:


 * $$v(x) = 1 - \frac{3}{L^2} + \frac{2}{L^3}$$

Obtain $$N_3(x)\;$$, $$N_5(x)\;$$, and $$N_6(x)\;$$ can be achieved with an identical solution method and the following boundary conditions (the order of sets corresponds to the order of the list):


 * $$v(0) = 0\;$$


 * $$v'(0) = 1\;$$


 * $$v(L) = 0\;$$


 * $$v'(L) = 0\;$$


 * $$v(0) = 0\;$$


 * $$v'(0) = 1\;$$


 * $$v(L) = 0\;$$


 * $$v'(L) = 0\;$$


 * $$v(0) = 0\;$$


 * $$v'(0) = 0\;$$


 * $$v(L) = 0\;$$


 * $$v'(L) = 1\;$$

Derive Coefficients in Element Stiffness Matrix
A sample coefficient of the element stiffness matrix can be written as follows:


 * $$\tilde{k}_{23} = \frac{6EI}{L^2} = \int_0^L \frac{\partial^2 N_2}{\partial x^2}(EI)\frac{\partial^2 N_3}{\partial x^2} dx $$

In general, the following relationship is true:


 * $$\tilde{k}_{ij} = \int_0^L \frac{\partial^2 N_i}{\partial x^2}(EI)\frac{\partial^2 N_j}{\partial x^2} dx $$

Elastodynamics (Vibrations via FEA)
Consider the following discrete PVW equation:


 * $$\mathbf{\bar{w}} \cdot [\mathbf{\bar{M}}\mathbf{\ddot{\bar{d}}} + \mathbf{\bar{k}}\mathbf{\bar{d}} - \mathbf{\bar{F}}] = 0\; \mbox{for all}\; \mathbf{\bar{w}}$$

This can easily be rewritten as Equation 1, with the associated initial conditions:


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


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


 * $$\mathbf{\dot{\bar{d}}}(0) = \bar{v}_0$$

Solving Equation 1
Consider the following "unforced" vibrations problem, given as Equation 2. The problem can be considered unforced because the right hand side is the zero vector:


 * $$\mathbf{\bar{M}}\mathbf{\ddot{v}} + \mathbf{\bar{k}}\mathbf{v} = \mathbf{0}\;\;(2)$$

Assume that $$\mathbf{v}(t) = (\sin \omega t) \boldsymbol{\phi}$$, where $$\boldsymbol{\phi}$$ is not time dependent. Applying this relationship to Equation 2 yields:


 * $$-\omega^2 (\sin \omega t)\mathbf{\bar{M}} \boldsymbol{\phi} + (\sin \omega t)\mathbf{\bar{k}}\boldsymbol{\phi} = \mathbf{0}$$

This leads to the following generalized eigenvalue problem given as Equation 3:


 * $$\mathbf{\bar{k}}\boldsymbol{\phi} = \omega^2 \mathbf{\bar{M}} \boldsymbol{\phi}\;\;(3)$$


 * {| class="collapsible collapsed"

!Standard Eigenvalue Problems
 * 
 * 

The general form of the eigenvalue problem is as follows:

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

This can be reduced to the standard eigenvalue problem by making the assumption that $$\mathbf{B} = \mathbf{I}\;$$, where $$\mathbf{I}\;$$ is the identity matrix:

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

$$\mathbf{I} = \left[\begin{matrix} 1\\ 0 \end{matrix}\right. \ddots \left.\begin{matrix} 0\\ 1 \end{matrix}\right]$$
 * }


 * $$\lambda = \omega^2\;$$ is defined as the eigenvalue, and the eigenpair is given as $$(\lambda_i,\phi_i)\;$$, for $$i = 1, \cdots, n\;$$. A general eigen mode is defined as:


 * $$\mathbf{v}_i(t) = (\sin \omega_i t) \boldsymbol{\phi_i}$$, for $$i = 1, \cdots, n\;$$

Now, considering the following method based on superposition. The orthogonal project of eigenpairs is given as:


 * $$\boldsymbol{\phi_i}^T \mathbf{\bar{M}}\boldsymbol{\phi_j} = \delta_{ij} = \left\{ \begin{matrix} 1, \mbox{if}\; i = j \\ 0, \mbox{if}\; i \neq j \end{matrix} \right.$$


 * $$\delta_{ij}\;$$ refers to the Kronecker delta. The mass orthogonality of eigenvectors is given as:


 * $$\mathbf{\bar{M}}\boldsymbol{\phi_j} = \lambda_j^{-1} \mathbf{\bar{k}}\boldsymbol{\phi_j}$$


 * $$\underbrace{\boldsymbol{\phi_i}^T \mathbf{\bar{M}}\boldsymbol{\phi_j}}_{\delta_{ij}} = \lambda_j^{-1} \boldsymbol{\phi_i}^T \mathbf{\bar{k}}\boldsymbol{\phi_j}$$

This leads to the following relationship:


 * $$\boldsymbol{\phi_i}^T \mathbf{\bar{k}}\boldsymbol{\phi_j} = \lambda_j \delta_{ij}$$

From Equation 1, the following can be derived:


 * $$\mathbf{\bar{d}}(t) = \Sigma \xi_i(t) \boldsymbol{\phi_i}$$

Combining these results yields the following:


 * $$\mathbf{\bar{M}} \underbrace{\left(\sum_j \ddot{\xi_j} \boldsymbol{\phi_j}\right)}_{\mathbf{\ddot{\bar{d}}}} + \mathbf{\bar{k}} \underbrace{\left(\sum_j \xi_j \boldsymbol{\phi_j}\right)}_{\mathbf{\bar{d}}} = \mathbf{\bar{F}}$$

Rearranging and multiplying by $$\boldsymbol{\phi_i}^T\;$$ yields:


 * $$\sum_j \ddot{\xi_j} \underbrace{\left(\boldsymbol{\phi_i}^T \mathbf{\bar{M}}\boldsymbol{\phi_j}\right)}_{\delta_{ij}} + \sum_j \xi_j \underbrace{\left(\boldsymbol{\phi_i}^T \mathbf{\bar{k}}\boldsymbol{\phi_j}\right)}_{\lambda_j \delta_{ij}} = \boldsymbol{\phi_i}^T \mathbf{\bar{F}}$$

Finally, the overall result can be written as:


 * $$\ddot{\xi_j} + \lambda_j\xi_j = \boldsymbol{\phi_i}^T \mathbf{\bar{F}}$$ for $$i = 1, \cdots, n\;$$

Solve 2-element frame system and plot undeformed shape, deformed shape of a 2-bar truss, and deformed shape of a 2-element frame
To accomplish this, the Matlab script our group has been using was modified to include calculation and plotting of the 2-element frame system. This requires use of the TwoBarTrussSystemDebugged.m function to calculate the displacements for the truss condition, which in turn requires the NodalSoln.m and PlaneFrameElement.m functions that are provided from our textbook's publisher's website and are herein. Our code and the publisher-provided code are reproduced for extra rigor. The plot is provided below showing the dotted line as the undeformed shape, the red line as the deformed frame shape, and the blue line as the deformed truss shape.



TwoBarTrussDisplayMod.m
This code plots the image required in the assignment.

TwoBarTrussSystemDebugged.m
This code calculates the displacement values for the standard two-bar truss system and outputs them as a matrix, d.

ElectricPylonDisplay.m
Command Window Output

Truss

>> ElectricPylonSystem

d =

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

results =

1.0781e-005 2.1562e+006       862.48 -6.5356e-006 -1.3071e+006     -522.85 6.7339e-006 1.3468e+006       538.71 6.4556e-006 1.2911e+006       516.45 -3.5152e-006 -7.0304e+005     -281.22 -2.3798e-005 -4.7596e+006     -1903.8 -1.2325e-005 -2.465e+006      -986.01 7.1182e-006 1.4236e+006       569.46 1.9269e-005 3.8538e+006       1541.5 -2.8828e-005 -5.7655e+006     -2306.2 -1.8941e-007      -37882      -15.153 8.778e-009      1755.6      0.70224 1.9328e-005 3.8655e+006       1546.2 1.4698e-005 2.9395e+006       1175.8 -1.8494e-005 -3.6988e+006     -1479.5 -2.883e-005 -5.7661e+006     -2306.4 1.7948e-006 3.5896e+005       143.58 2.621e-005  5.242e+006       2096.8 9.8986e-006 1.9797e+006       791.89 -1.8874e-005 -3.7749e+006       -1510 -3.258e-005 -6.5159e+006     -2606.4 2.7892e-006 5.5784e+005       223.14 3.2194e-006 6.4388e+005       257.55 2.9828e-005 5.9656e+006       2386.3 -4.0852e-006 -8.1704e+005     -326.82 -1.9808e-005 -3.9616e+006     -1584.6 1.16e-005   2.32e+006       928.02 -4.0653e-006 -8.1307e+005     -325.23 -2.8778e-005 -5.7556e+006     -2302.2 -1.4237e-007      -28474       -11.39 -4.9677e-006 -9.9353e+005     -397.41 -7.6072e-008      -15214      -6.0857 2.9608e-005 5.9216e+006       2368.6 2.7907e-007       55813       22.325 -2.4358e-005 -4.8717e+006     -1948.7 6.9573e-006 1.3915e+006       556.59 -1.5116e-008     -3023.3      -1.2093 -2.8794e-005 -5.7588e+006     -2303.5 -2.427e-005 -4.854e+006      -1941.6 6.9522e-006 1.3904e+006       556.17 -9.4154e-007 -1.8831e+005     -75.323 1.129e-005  2.258e+006       903.22 -3.16e-006  -6.32e+005       -252.8 -2.0051e-005 -4.0102e+006     -1604.1 -1.0133e-019 -2.0266e-008 -8.1063e-012 7.3233e-019 1.4647e-007  5.8586e-011 3.2073e-007       64147       25.659 1.3501e-006 2.7001e+005       108.01 1.3575e-006 2.7149e+005        108.6 -1.0389e-005 -2.0778e+006     -831.13 -2.2136e-005 -4.4271e+006     -1770.9 -2.9388e-005 -5.8777e+006     -2351.1 -3.6774e-005 -7.3549e+006     -2941.9 -4.331e-005 -8.662e+006      -3464.8 -4.3478e-005 -8.6957e+006     -3478.3 0           0            0  3.0942e-019  6.1883e-008  2.4753e-011 -2.6985e-019 -5.397e-008 -2.1588e-011 3.0512e-019 6.1024e-008  2.4409e-011 -8.8522e-007 -1.7704e+005     -70.818 1.1034e-006 2.2069e+005       88.276 -9.5965e-007 -1.9193e+005     -76.772 9.9436e-007 1.9887e+005       79.549 1.0732e-005 2.1464e+006       858.56 -1.0415e-005 -2.083e+006      -833.19 1.1448e-005 2.2895e+006       915.81 -1.1321e-005 -2.2643e+006     -905.71 1.0517e-005 2.1035e+006       841.38 -1.0732e-005 -2.1464e+006     -858.56 7.0689e-006 1.4138e+006       565.51 -6.8859e-006 -1.3772e+006     -550.87 7.9177e-006 1.5835e+006       633.42 -1.8464e-005 -3.6928e+006     -1477.1 -4.8696e-007      -97393      -38.957 -1.1845e-007      -23690      -9.4762 1.5604e-007       31208       12.483 4.5239e-005 9.0479e+006       3619.2 4.6964e-019 9.3927e-008  3.7571e-011 4.5158e-005 9.0317e+006       3612.7 -7.9385e-019 -1.5877e-007 -6.3508e-011 4.5256e-005 9.0511e+006       3620.5 -6.5877e-007 -1.3175e+005     -52.702 -1.4822e-006 -2.9645e+005     -118.58 8.9829e-006 1.7966e+006       718.63 2.2438e-005 4.4876e+006         1795 3.3117e-005 6.6233e+006       2649.3 3.8878e-005 7.7755e+006       3110.2 -5.3704e-020 -1.0741e-008 -4.2963e-012 0           0            0 -4.8034e-020 -9.6069e-009 -3.8428e-012 0           0            0

max_compressive_stress =

-8.6957e+006

element_number =

55

max_tensile_stress =

9.0511e+006

element_number =

81

min_eigenvalues =

132.16      2468.4       2940.3

min_eigen_num =

82   85    86

lowest_vibrational_periods =

0.54654     0.12647      0.11587

Frame

dF =

0           0            0            0            0            0  -0.00019023   0.00017464  9.1757e-006 -0.00012769 -1.4086e-006 -2.4036e-005 -0.00016005  -0.0002967  9.1254e-006 -0.0001267  0.00032601 -4.1467e-005 -0.00012797   -0.000128 -6.1763e-005 -0.00012791 -0.00054327 -3.4931e-005 0.00044932    0.000309  -0.00011737   0.00046381  -0.00061057  -0.00011826     0.001017   0.00064582  -0.00016353    0.0010325  -0.00019427  -0.00013889    0.0010502  -0.00097027  -0.00015648    0.0017167    0.0010338  -0.00024125    0.0017162    0.0002326  -0.00017832    0.0016833  -0.00064959  -0.00016107     0.001683   -0.0013293  -0.00019661    0.0023452   0.00064079  -0.00028233    0.0021862   -0.0010297  -0.00022023     0.004712    0.0023901   -0.0003608    0.0039576   -0.0024164  -0.00033295    0.0063604    0.0057326  -0.00022721    0.0063604    0.0037875  -0.00022759    0.0063604    0.0029802  -0.00022392    0.0063612    0.0023839  -0.00023194    0.0063655    0.0016463  -0.00021304    0.0063709   0.00086187  -0.00020968    0.0063181   -0.0002427  -0.00024756    0.0062287   -0.0013181  -0.00031262    0.0061362   -0.0023672  -0.00036099    0.0060411    -0.003323  -0.00041992    0.0058837   -0.0051276  -0.00057906    0.0055151    -0.012586   -0.0010709    0.0067824    0.0042646  -0.00022701    0.0069958   -0.0063358  -0.00065704    0.0070118    0.0034572  -0.00022802    0.0074004    -0.004359  -0.00048349    0.0071503    0.0029771    -0.000229    0.0071476    0.0020507  -0.00022519    0.0071419    0.0012277  -0.00021034    0.0071818   0.00032022  -0.00022274    0.0072814  -0.00073249  -0.00027283    0.0074071   -0.0018632  -0.00034933    0.0075641   -0.0033871  -0.00041223    0.0090003    0.0029771  -0.00023027     0.010455   -0.0033871  -0.00035946

fa =

12.113           0      -1520.2       13.647       4.7645      -1520.2       15.182       9.5289      -1520.2       12.113            0      -654.71       18.291       4.7645      -654.71       24.468       9.5289      -654.71       15.182       9.5289        538.7       19.825       9.5289        538.7       24.468       9.5289        538.7       24.468       9.5289       662.83       30.525       4.7645       662.83       36.581            0       662.83       24.468       9.5289      -281.21       29.071       9.5289      -281.21       33.674       9.5289      -281.21       36.581            0      -1285.2       35.128       4.7645      -1285.2       33.674       9.5289      -1285.2       15.182       9.5289       404.52       19.825       13.571       404.52       24.468       17.612       404.52       33.674       9.5289       209.49       29.071       13.571       209.49       24.468       17.612       209.49       15.182       9.5289       598.85       16.474       13.571       598.85       17.766       17.612       598.85       33.674       9.5289       302.98       32.382       13.571       302.98        31.09       17.612       302.98       17.766       17.612      -15.156       21.117       17.612      -15.156       24.468       17.612      -15.156       24.468       17.612      0.70819       27.779       17.612      0.70819        31.09       17.612      0.70819       17.766       17.612         5419       19.098        21.65         5419       20.431       25.688         5419       24.468       17.612       5115.2        22.45        21.65       5115.2       20.431       25.688       5115.2       24.468       17.612       5243.7       26.487        21.65       5243.7       28.506       25.688       5243.7        31.09       17.612       5583.1       29.798        21.65       5583.1       28.506       25.688       5583.1       20.431       25.688       143.57       24.468       25.688       143.57       28.506       25.688       143.57       20.431       25.688        10514       19.664       27.707        10514       18.896       29.725        10514       20.431       25.688       8170.8        22.45       27.707       8170.8       24.468       29.725       8170.8       28.506       25.688       7967.7       26.487       27.707       7967.7       24.468       29.725       7967.7       28.506       25.688        10932       29.233       27.707        10932        29.96       29.725        10932       18.896       29.725        223.1       21.682       29.725        223.1       24.468       29.725        223.1       24.468       29.725       257.54       27.214       29.725       257.54        29.96       29.725       257.54       18.896       29.725        13045        18.17       31.744        13045       17.443       33.763        13045       18.896       29.725        12088       20.027       31.744        12088       21.157       33.763        12088       24.468       29.725        10475       22.813       31.744        10475       21.157       33.763        10475       24.468       29.725       9970.7       26.124       31.744       9970.7       27.779       33.763       9970.7        29.96       29.725        11038       28.869       31.744        11038       27.779       33.763        11038        29.96       29.725        11797       30.686       31.744        11797       31.413       33.763        11797       17.443       33.763      -11.502         19.3       33.763      -11.502       21.157       33.763      -11.502       21.157       33.763      -397.41       24.468       33.763      -397.41       27.779       33.763      -397.41       27.779       33.763       -6.051       29.596       33.763       -6.051       31.413       33.763       -6.051       17.443       33.763        28144       15.989       37.764        28144       14.536       41.766        28144       17.443       33.763        16535        18.25       35.051        16535       19.058       36.339        16535       21.157       33.763        15140       20.108       35.051        15140       19.058       36.339        15140       27.779       33.763        12107       28.829       35.051        12107       29.879       36.339        12107       31.413       33.763        13426       30.646       35.051        13426       29.879       36.339        13426       31.413       33.763        21371       32.867       37.764        21371        34.32       41.766        21371       19.058       36.339        26805       16.797       39.052        26805       14.536       41.766        26805       29.879       36.339        20207         32.1       39.052        20207        34.32       41.766        20207       14.536       41.766        18542       13.324       45.109        18542       12.113       48.452        18542       14.536       41.766        17643       16.231       45.109        17643       17.927       48.452        17643        34.32       41.766        24351       32.665       45.109        24351       31.009       48.452        24351        34.32       41.766        23438       35.532       45.109        23438       36.743       48.452        23438            0       48.452 -4.3248e-005 4.2799      48.452 -4.3248e-005 8.5599      48.452 -4.3248e-005 8.5599      48.452    0.0012651       10.336       48.452    0.0012651       12.113       48.452    0.0012651       12.113       48.452       25.644       13.405       48.452       25.644       14.697       48.452       25.644       14.697       48.452       108.06       16.312       48.452       108.06       17.927       48.452       108.06       17.927       48.452       108.68       19.906       48.452       108.68       21.884       48.452       108.68       21.884       48.452      -831.05       24.428       48.452      -831.05       26.972       48.452      -831.05       26.972       48.452      -1770.8       28.991       48.452      -1770.8       31.009       48.452      -1770.8       31.009       48.452        -2351       32.584       48.452        -2351       34.159       48.452        -2351       34.159       48.452      -2941.8       35.451       48.452      -2941.8       36.743       48.452      -2941.8       36.743       48.452      -3464.7        38.56       48.452      -3464.7       40.377       48.452      -3464.7       40.377       48.452      -3477.9       44.616       48.452      -3477.9       48.856       48.452      -3477.9            0       48.452       5022.9       3.2301       49.381       5022.9       6.4603        50.31       5022.9       8.5599       48.452        12045       7.5101       49.381        12045       6.4603        50.31        12045       8.5599       48.452        16214       9.2867       49.886        16214       10.013       51.319        16214       12.113       48.452        14666       11.063       49.886        14666       10.013       51.319        14666       12.113       48.452        18199       12.113       50.188        18199       12.113       51.925        18199       14.697       48.452        14585       13.405       50.188        14585       12.113       51.925        14585       14.697       48.452        16713       15.424       50.188        16713       16.151       51.925        16713       17.927       48.452        16041       17.039       50.188        16041       16.151       51.925        16041       17.927       48.452        15305       18.977       50.188        15305       20.027       51.925        15305       21.884       48.452        15662       20.956       50.188        15662       20.027       51.925        15662       21.884       48.452        14986       23.176       50.188        14986       24.468       51.925        14986       26.972       48.452        16141        25.72       50.188        16141       24.468       51.925        16141       26.972       48.452        19380       27.941       50.188        19380        28.91       51.925        19380       31.009       48.452        20754        29.96       50.188        20754        28.91       51.925        20754       31.009       48.452        24395       31.857       50.188        24395       32.705       51.925        24395       34.159       48.452        27010       33.432       50.188        27010       32.705       51.925        27010       34.159       48.452        26392       35.451       50.188        26392       36.743       51.925        26392       36.743       48.452        35087       36.743       50.188        35087       36.743       51.925        35087       36.743       48.452        30192       37.833       49.886        30192       38.923       51.319        30192       40.377       48.452        37749        39.65       49.886        37749       38.923       51.319        37749       40.377       48.452        32431       41.386       49.381        32431       42.396        50.31        32431       48.856       48.452        17622       45.626       49.381        17622       42.396        50.31        17622       6.4603        50.31       4967.8       8.2369       50.814       4967.8       10.013       51.319       4967.8       42.396        50.31       8950.5       40.659       50.814       8950.5       38.923       51.319       8950.5       10.013       51.319       5070.9       11.063       51.622       5070.9       12.113       51.925       5070.9       38.923       51.319         5787       37.833       51.622         5787       36.743       51.925         5787       12.113       51.925      -52.735       14.132       51.925      -52.735       16.151       51.925      -52.735       16.151       51.925      -118.65       18.089       51.925      -118.65       20.027       51.925      -118.65       20.027       51.925       718.56       22.248       51.925       718.56       24.468       51.925       718.56       24.468       51.925         1795       26.689       51.925         1795        28.91       51.925         1795        28.91       51.925       2649.3       30.808       51.925       2649.3       32.705       51.925       2649.3       32.705       51.925       3110.1       34.724       51.925       3110.1       36.743       51.925       3110.1       12.113       51.925        18327       12.113       55.962        18327       12.113           60        18327       16.151       51.925        16416       14.132       55.962        16416       12.113           60        16416       32.705       51.925        27005       34.724       55.962        27005       36.743           60        27005       36.743       51.925        28638       36.743       55.962        28638       36.743           60        28638

bma =

12.113           0     0.022994       13.647       4.7645    0.0024442       15.182       9.5289    -0.018106       12.113            0    0.0081231       18.291       4.7645   -0.0041078       24.468       9.5289    -0.016339       15.182       9.5289    -0.029398       19.825       9.5289   -0.0095366       24.468       9.5289     0.010324       24.468       9.5289      0.01673       30.525       4.7645    0.0041588       36.581            0   -0.0084124       24.468       9.5289    -0.033187       29.071       9.5289    0.0096057       33.674       9.5289     0.052398       36.581            0    -0.052715       35.128       4.7645    0.0024426       33.674       9.5289     0.057601       15.182       9.5289     -0.01314       19.825       13.571    -0.015365       24.468       17.612     -0.01759       33.674       9.5289     0.036926       29.071       13.571     -0.01543       24.468       17.612    -0.067786       15.182       9.5289     0.048155       16.474       13.571    -0.015913       17.766       17.612    -0.079982       33.674       9.5289    -0.044294       32.382       13.571    -0.013844        31.09       17.612     0.016607       17.766       17.612    -0.046563       21.117       17.612    -0.008075       24.468       17.612     0.030413       24.468       17.612    -0.023904       27.779       17.612     0.010805        31.09       17.612     0.045515       17.766       17.612      0.12186       19.098        21.65    -0.023801       20.431       25.688     -0.16946       24.468       17.612      0.22808        22.45        21.65    -0.016423       20.431       25.688     -0.26092       24.468       17.612     0.048107       26.487        21.65    -0.016687       28.506       25.688     -0.08148        31.09       17.612      0.10335       29.798        21.65    -0.026208       28.506       25.688     -0.15577       20.431       25.688    0.0075088       24.468       25.688  -0.00029532       28.506       25.688   -0.0080994       20.431       25.688       0.7806       19.664       27.707    -0.028498       18.896       29.725      -0.8376       20.431       25.688      0.10201        22.45       27.707    -0.010053       24.468       29.725     -0.12211       28.506       25.688      0.55492       26.487       27.707   -0.0096357       24.468       29.725     -0.57419       28.506       25.688      0.17591       29.233       27.707    -0.023751        29.96       29.725     -0.22341       18.896       29.725     0.013049       21.682       29.725     0.011789       24.468       29.725     0.010529       24.468       29.725     0.010025       27.214       29.725   -0.0085411        29.96       29.725    -0.027107       18.896       29.725       1.0434        18.17       31.744      -0.0483       17.443       33.763        -1.14       18.896       29.725      0.27371       20.027       31.744   -0.0085272       21.157       33.763     -0.29076       24.468       29.725      0.71637       22.813       31.744    -0.020138       21.157       33.763     -0.75665       24.468       29.725      0.18105       26.124       31.744    -0.011324       27.779       33.763     -0.20369        29.96       29.725      0.79461       28.869       31.744   -0.0026649       27.779       33.763     -0.79994        29.96       29.725      0.32132       30.686       31.744    -0.024934       31.413       33.763     -0.37119       17.443       33.763       0.0198         19.3       33.763     0.045175       21.157       33.763      0.07055       21.157       33.763     0.095071       24.468       33.763    0.0069498       27.779       33.763    -0.081172       27.779       33.763    -0.062273       29.596       33.763     -0.02608       31.413       33.763     0.010113       17.443       33.763      0.82758       15.989       37.764    -0.037442       14.536       41.766     -0.90246       17.443       33.763      0.66146        18.25       35.051    -0.036031       19.058       36.339     -0.73352       21.157       33.763       1.6168       20.108       35.051     -0.08346       19.058       36.339      -1.7837       27.779       33.763      0.31977       28.829       35.051    -0.047475       29.879       36.339     -0.41472       31.413       33.763       1.6245       30.646       35.051    -0.021012       29.879       36.339      -1.6665       31.413       33.763      0.21494       32.867       37.764    -0.042703        34.32       41.766     -0.30034       19.058       36.339       1.2596       16.797       39.052    -0.029621       14.536       41.766      -1.3189       29.879       36.339      0.13701         32.1       39.052    -0.042864        34.32       41.766     -0.22274       14.536       41.766      0.89574       13.324       45.109     0.051327       12.113       48.452     -0.79309       14.536       41.766      0.45314       16.231       45.109     0.052554       17.927       48.452     -0.34803        34.32       41.766       1.0151       32.665       45.109    0.0072661       31.009       48.452      -1.0006        34.32       41.766       0.5275       35.532       45.109    -0.032612       36.743       48.452     -0.59272            0       48.452   0.00020202       4.2799       48.452  -0.00011786       8.5599       48.452  -0.00043774       8.5599       48.452   -0.0038971       10.336       48.452    0.0027601       12.113       48.452    0.0094173       12.113       48.452    -0.025728       13.405       48.452   -0.0082818       14.697       48.452    0.0091644       14.697       48.452    -0.013445       16.312       48.452     0.015603       17.927       48.452      0.04465       17.927       48.452     0.055359       19.906       48.452    0.0022659       21.884       48.452    -0.050827       21.884       48.452     0.016315       24.428       48.452    -0.019854       26.972       48.452    -0.056022       26.972       48.452     0.011473       28.991       48.452    -0.042972       31.009       48.452    -0.097416       31.009       48.452    -0.022087       32.584       48.452    -0.040959       34.159       48.452    -0.059832       34.159       48.452     0.066558       35.451       48.452    -0.060811       36.743       48.452     -0.18818       36.743       48.452     -0.10405        38.56       48.452     -0.11678       40.377       48.452      -0.1295       40.377       48.452      -0.2579       44.616       48.452     -0.15468       48.856       48.452    -0.051457            0       48.452     0.020854       3.2301       49.381  7.9562e-005 6.4603       50.31    -0.020695       8.5599       48.452       2.2694       7.5101       49.381   0.00055068       6.4603        50.31      -2.2683       8.5599       48.452      0.62211       9.2867       49.886  -0.00035528       10.013       51.319     -0.62282       12.113       48.452        1.619       11.063       49.886   -0.0030813       10.013       51.319      -1.6251       12.113       48.452       1.0355       12.113       50.188   -0.0039046       12.113       51.925      -1.0433       14.697       48.452       1.3603       13.405       50.188    0.0018119       12.113       51.925      -1.3567       14.697       48.452      0.60014       15.424       50.188    0.0047791       16.151       51.925     -0.59058       17.927       48.452       1.3159       17.039       50.188    -0.008309       16.151       51.925      -1.3325       17.927       48.452      0.42979       18.977       50.188    0.0017763       20.027       51.925     -0.42624       21.884       48.452       1.2303       20.956       50.188  -0.00044645       20.027       51.925      -1.2312       21.884       48.452      0.32859       23.176       50.188   -0.0080473       24.468       51.925     -0.34469       26.972       48.452       1.3859        25.72       50.188     0.015458       24.468       51.925       -1.355       26.972       48.452      0.53437       27.941       50.188    -0.016946        28.91       51.925     -0.56826       31.009       48.452       1.7494        29.96       50.188     0.026152        28.91       51.925      -1.6971       31.009       48.452        0.761       31.857       50.188    -0.025335       32.705       51.925     -0.81167       34.159       48.452       2.0868       33.432       50.188    0.0082603       32.705       51.925      -2.0703       34.159       48.452      0.52651       35.451       50.188    -0.031565       36.743       51.925     -0.58964       36.743       48.452        1.838       36.743       50.188    0.0059089       36.743       51.925      -1.8262       36.743       48.452      0.68174       37.833       49.886    -0.047064       38.923       51.319     -0.77587       40.377       48.452       3.9142        39.65       49.886     0.079287       38.923       51.319      -3.7556       40.377       48.452      0.96027       41.386       49.381    -0.075804       42.396        50.31      -1.1119       48.856       48.452        4.434       45.626       49.381      0.16418       42.396        50.31      -4.1056       6.4603        50.31     0.037974       8.2369       50.814  -0.00072706       10.013       51.319    -0.039428       42.396        50.31       5.0698       40.659       50.814      0.12798       38.923       51.319      -4.8139       10.013       51.319     0.063037       11.063       51.622    -0.001194       12.113       51.925    -0.065425       38.923       51.319       6.2874       37.833       51.622     0.083975       36.743       51.925      -6.1194       12.113       51.925   -0.0066914       14.132       51.925    0.0025132       16.151       51.925     0.011718       16.151       51.925     0.032676       18.089       51.925     0.010221       20.027       51.925    -0.012235       20.027       51.925      0.03654       22.248       51.925   -0.0074467       24.468       51.925    -0.051434       24.468       51.925     0.008706       26.689       51.925    -0.030071        28.91       51.925    -0.068849        28.91       51.925    0.0017025       30.808       51.925    -0.053755       32.705       51.925     -0.10921       32.705       51.925    -0.028189       34.724       51.925    -0.041539       36.743       51.925    -0.054889       12.113       51.925      0.45456       12.113       55.962  -0.00041857       12.113           60      -0.4554       16.151       51.925       0.5839       14.132       55.962   -0.0014983       12.113           60      -0.5869       32.705       51.925      0.32595       34.724       55.962   -0.0029912       36.743           60     -0.33193       36.743       51.925      0.78191       36.743       55.962     0.017425       36.743           60     -0.74706

Va =

12.113           0   -0.0041055       13.647       4.7645   -0.0041055       15.182       9.5289   -0.0041055       12.113            0   -0.0015678       18.291       4.7645   -0.0015678       24.468       9.5289   -0.0015678       15.182       9.5289    0.0042773       19.825       9.5289    0.0042773       24.468       9.5289    0.0042773       24.468       9.5289   -0.0016314       30.525       4.7645   -0.0016314       36.581            0   -0.0016314       24.468       9.5289    0.0092967       29.071       9.5289    0.0092967       33.674       9.5289    0.0092967       36.581            0     0.011073       35.128       4.7645     0.011073       33.674       9.5289     0.011073       15.182       9.5289  -0.00036143       19.825       13.571  -0.00036143       24.468       17.612  -0.00036143       33.674       9.5289   -0.0085471       29.071       13.571   -0.0085471       24.468       17.612   -0.0085471       15.182       9.5289    -0.015099       16.474       13.571    -0.015099       17.766       17.612    -0.015099       33.674       9.5289    0.0071763       32.382       13.571    0.0071763        31.09       17.612    0.0071763       17.766       17.612     0.011485       21.117       17.612     0.011485       24.468       17.612     0.011485       24.468       17.612     0.010483       27.779       17.612     0.010483        31.09       17.612     0.010483       17.766       17.612    -0.034258       19.098        21.65    -0.034258       20.431       25.688    -0.034258       24.468       17.612    -0.054161        22.45        21.65    -0.054161       20.431       25.688    -0.054161       24.468       17.612    -0.014353       26.487        21.65    -0.014353       28.506       25.688    -0.014353        31.09       17.612    -0.030561       29.798        21.65    -0.030561       28.506       25.688    -0.030561       20.431       25.688   -0.0019328       24.468       25.688   -0.0019328       28.506       25.688   -0.0019328       20.431       25.688     -0.37464       19.664       27.707     -0.37464       18.896       29.725     -0.37464       20.431       25.688    -0.039249        22.45       27.707    -0.039249       24.468       29.725    -0.039249       28.506       25.688     -0.19774       26.487       27.707     -0.19774       24.468       29.725     -0.19774       28.506       25.688    -0.093054       29.233       27.707    -0.093054        29.96       29.725    -0.093054       18.896       29.725  -0.00045216       21.682       29.725  -0.00045216       24.468       29.725  -0.00045216       24.468       29.725    -0.006762       27.214       29.725    -0.006762        29.96       29.725    -0.006762       18.896       29.725     -0.50877        18.17       31.744     -0.50877       17.443       33.763     -0.50877       18.896       29.725     -0.12198       20.027       31.744     -0.12198       21.157       33.763     -0.12198       24.468       29.725      -0.2821       22.813       31.744      -0.2821       21.157       33.763      -0.2821       24.468       29.725    -0.073683       26.124       31.744    -0.073683       27.779       33.763    -0.073683        29.96       29.725     -0.34749       28.869       31.744     -0.34749       27.779       33.763     -0.34749        29.96       29.725     -0.16137       30.686       31.744     -0.16137       31.413       33.763     -0.16137       17.443       33.763     0.013662         19.3       33.763     0.013662       21.157       33.763     0.013662       21.157       33.763    -0.026616       24.468       33.763    -0.026616       27.779       33.763    -0.026616       27.779       33.763     0.019919       29.596       33.763     0.019919       31.413       33.763     0.019919       17.443       33.763     -0.20319       15.989       37.764     -0.20319       14.536       41.766     -0.20319       17.443       33.763      -0.4588        18.25       35.051      -0.4588       19.058       36.339      -0.4588       21.157       33.763      -1.0232       20.108       35.051      -1.0232       19.058       36.339      -1.0232       27.779       33.763     -0.22101       28.829       35.051     -0.22101       29.879       36.339     -0.22101       31.413       33.763      -1.0976       30.646       35.051      -1.0976       29.879       36.339      -1.0976       31.413       33.763    -0.060518       32.867       37.764    -0.060518        34.32       41.766    -0.060518       19.058       36.339     -0.36503       16.797       39.052     -0.36503       14.536       41.766     -0.36503       29.879       36.339    -0.051302         32.1       39.052    -0.051302        34.32       41.766    -0.051302       14.536       41.766     -0.23747       13.324       45.109     -0.23747       12.113       48.452     -0.23747       14.536       41.766     -0.10686       16.231       45.109     -0.10686       17.927       48.452     -0.10686        34.32       41.766     -0.27015       32.665       45.109     -0.27015       31.009       48.452     -0.27015        34.32       41.766     -0.15752       35.532       45.109     -0.15752       36.743       48.452     -0.15752            0       48.452 -7.4739e-005 4.2799      48.452 -7.4739e-005 8.5599      48.452 -7.4739e-005 8.5599      48.452    0.0037472       10.336       48.452    0.0037472       12.113       48.452    0.0037472       12.113       48.452     0.013503       13.405       48.452     0.013503       14.697       48.452     0.013503       14.697       48.452     0.017985       16.312       48.452     0.017985       17.927       48.452     0.017985       17.927       48.452    -0.026835       19.906       48.452    -0.026835       21.884       48.452    -0.026835       21.884       48.452    -0.014219       24.428       48.452    -0.014219       26.972       48.452    -0.014219       26.972       48.452    -0.026968       28.991       48.452    -0.026968       31.009       48.452    -0.026968       31.009       48.452    -0.011985       32.584       48.452    -0.011985       34.159       48.452    -0.011985       34.159       48.452    -0.098578       35.451       48.452    -0.098578       36.743       48.452    -0.098578       36.743       48.452   -0.0070024        38.56       48.452   -0.0070024       40.377       48.452   -0.0070024       40.377       48.452     0.024347       44.616       48.452     0.024347       48.856       48.452     0.024347            0       48.452   -0.0061812       3.2301       49.381   -0.0061812       6.4603        50.31   -0.0061812       8.5599       48.452      -1.6187       7.5101       49.381      -1.6187       6.4603        50.31      -1.6187       8.5599       48.452     -0.38732       9.2867       49.886     -0.38732       10.013       51.319     -0.38732       12.113       48.452     -0.91295       11.063       49.886     -0.91295       10.013       51.319     -0.91295       12.113       48.452     -0.59865       12.113       50.188     -0.59865       12.113       51.925     -0.59865       14.697       48.452     -0.62772       13.405       50.188     -0.62772       12.113       51.925     -0.62772       14.697       48.452     -0.31631       15.424       50.188     -0.31631       16.151       51.925     -0.31631       17.927       48.452       -0.679       17.039       50.188       -0.679       16.151       51.925       -0.679       17.927       48.452     -0.21096       18.977       50.188     -0.21096       20.027       51.925     -0.21096       21.884       48.452     -0.62508       20.956       50.188     -0.62508       20.027       51.925     -0.62508       21.884       48.452     -0.15555       23.176       50.188     -0.15555       24.468       51.925     -0.15555       26.972       48.452     -0.64028        25.72       50.188     -0.64028       24.468       51.925     -0.64028       26.972       48.452     -0.27728       27.941       50.188     -0.27728        28.91       51.925     -0.27728       31.009       48.452     -0.84936        29.96       50.188     -0.84936        28.91       51.925     -0.84936       31.009       48.452     -0.40696       31.857       50.188     -0.40696       32.705       51.925     -0.40696       34.159       48.452      -1.1043       33.432       50.188      -1.1043       32.705       51.925      -1.1043       34.159       48.452     -0.25787       35.451       50.188     -0.25787       36.743       51.925     -0.25787       36.743       48.452      -1.0552       36.743       50.188      -1.0552       36.743       51.925      -1.0552       36.743       48.452      -0.4047       37.833       49.886      -0.4047       38.923       51.319      -0.4047       40.377       48.452      -2.3862        39.65       49.886      -2.3862       38.923       51.319      -2.3862       40.377       48.452     -0.75536       41.386       49.381     -0.75536       42.396        50.31     -0.75536       48.856       48.452      -1.2704       45.626       49.381      -1.2704       42.396        50.31      -1.2704       6.4603        50.31    -0.020955       8.2369       50.814    -0.020955       10.013       51.319    -0.020955       42.396        50.31      -2.7332       40.659       50.814      -2.7332       38.923       51.319      -2.7332       10.013       51.319    -0.058787       11.063       51.622    -0.058787       12.113       51.925    -0.058787       38.923       51.319      -5.4827       37.833       51.622      -5.4827       36.743       51.925      -5.4827       12.113       51.925    0.0045594       14.132       51.925    0.0045594       16.151       51.925    0.0045594       16.151       51.925    -0.011587       18.089       51.925    -0.011587       20.027       51.925    -0.011587       20.027       51.925    -0.019807       22.248       51.925    -0.019807       24.468       51.925    -0.019807       24.468       51.925    -0.017462       26.689       51.925    -0.017462        28.91       51.925    -0.017462        28.91       51.925    -0.029223       30.808       51.925    -0.029223       32.705       51.925    -0.029223       32.705       51.925   -0.0066128       34.724       51.925   -0.0066128       36.743       51.925   -0.0066128       12.113       51.925     -0.11268       12.113       55.962     -0.11268       12.113           60     -0.11268       16.151       51.925     -0.12968       14.132       55.962     -0.12968       12.113           60     -0.12968       32.705       51.925    -0.072866       34.724       55.962    -0.072866       36.743           60    -0.072866       36.743       51.925     -0.18934       36.743       55.962     -0.18934       36.743           60     -0.18934

max_bending_moment =

48.856

ans =

55

max_shear_force =

48.856

ans =

55

min_eigenvalues =

132.16      2468.4       2940.3

min_eigen_num =

82   85    86

lowest_vibrational_periods =

0.54654     0.12647      0.11587



In the image above, there exists two lines in the deformed shape that overlap each other. Although it is hard to tell, the truss deformation (solid blue) differs only slightly from the frame deformation (dotted black). The difference is on the order of less than a millimeter. The magnification factor (m) for the deformation is 1000.

This system is not statically determinate. With 91 elements, there are way more unknown values than possible equilibrium statics equations.

The lowest eigenvalues and vibrational periods are almost identical for both cases. Rounding and calculation errors cause the differences.

Essay: Wikiversity vs. E-Learning
In the debate between Mediawiki and E-learning, the advantages of using the Wikiversity system through Mediawiki are far superior to other methods of presenting class notes. Mediawiki has allowed students to be more flexible in their presentation of notes, as well as a future location for their files. Mediawiki is well known, fairly easy to use for engineering students, and allows students to learn more about HTML coding or equation editing.

E-learning is a local system to UF, but there are still many disadvantages of using the E-learning system. One of the major complaints of the E-learning system is the lag time due to many students being on the same server. Some of the Team Lulz group members have experienced E-learning “outages” where the server has lagged so much that incoming students cannot log into the server. Mediawiki does not have this problem; there is literally no limit to the number of people on one page in the Wikipedia servers.

Another major disadvantage of the E-learning system is the lack of creative tools for editing pages. E-learning is seen more of a file “dump”, where professors or students can upload files to be viewed by others. The E-learning style of a system is not designed for constant editing or the creative assembly from Mediawiki. Once the semester is over, students are usually not allowed to log back into their course website. You must also have a UF account, so students who eventually graduate cannot access their work in the future. Mediawiki allows for students to access the Wikiversity site anytime, worldwide and in the future. Mediawiki allows students to even edit their own posts in the future.

A major detriment to the Mediawiki experience as implemented in EML4500 was the extreme amount of work it required. Massive reproduction of notes in LaTeX and wiki markup replaced actual learning exercises with rote reproduction and syntax debugging. This has caused a majority of the time working outside of the classroom in this course to be devoted to tasks which don't actually utilize any FEA methods applied to a real-world example. As the only course focusing on FEA in the undergraduate mechanical engineering curriculum, this almost fanatical concern with syntax and reproduction of notes instead of actually working on FEA problems to familiarize ourselves with engineering applications of our curriculum was far from what was expected in the course syllabus or catalog description. Mediawiki is a fine collaboration system, but its implementation in this course has been detrimental to the overall learning experience with regard to FEA methods.

Essay: Tools Used
During the course of using Wikiversity, the Latex equation editor was used to model equations in our Wikiversity pages. We found Latex to be extremely easy to use and not difficult to comprehend. There were a few websites that we used to edit equations, but several members of the group utilized several different methods of creating equations.

After Homework 4, our team found that we could use the “/math” html code already built into the Wiki editor to make all of our equations. Some matrices that were completed in the homework were done in other sites, but a majority of the notes were completed using the built-in math functions. The built-in html coding was the easiest to use in regards to actually editing the Wikiversity homework page. We did find issues when were doing special cases with figures or matrices, and these had to be completed using an outside editor.

One request from one of the members is the ability to write all of the notes, including equations, into Microsoft Word. A Word plug-in for Mediawiki use would be of extreme use since we will not have to connect to the internet to write the equations. Word allows for easy editing of any kind of notes, so a plug-in for even any word processor would help immensely in the future. Mediawiki was not always friendly with the built-in equation editor. We often had errors or difficulty typing FEA related equations and had to go back to fix our errors. With a Word built-in FEA equation editor, all of our equations could easily be fixed and translated into Mediawiki.

Contributing Team Members
Aaron Fisher - Eml4500.f08.Lulz.fisher 17:12, 9 December 2008 (UTC)

Eric Layton - Eml4500.f08.Lulz.Layton.Eric 20:44, 9 December 2008 (UTC)

Sam Miorelli - Eml4500.f08.lulz.abcd 19:47, 9 December 2008 (UTC)

Benjamin Mitchell - Eml4500.f08.Lulz.mitchell.bm 18:23, 9 December 2008 (UTC)

John Saxon - Eml4500.f08.Lulz.js 21:04, 9 December 2008 (UTC)

Andrew Strack - Eml4500.f08.lulz.strack 21:57, 9 December 2008 (UTC)