User:Eml4500.f08.delta 6.krueger/HW dimensions of ktilde dtilde

Verify the dimensions of all terms of $$\tilde{k_{ij}}\tilde{d_{j}}$$

Start with the matrices of $$\tilde{k_{ij}}$$ and $$\tilde{d_{j}}$$.

$$\tilde{k_{ij}}\tilde{d_{j}}=\begin{bmatrix} \frac{EA}{L} & 0 & 0 & -\frac{EA}{L} & 0 & 0 \\ 0 & \frac{12EI}{L^{3}} & \frac{6EI}{L^{2}} & 0 & -\frac{12EI}{L^{3}} & -\frac{6EI}{L^{2}} \\ 0 & \frac{6EI}{L^{2}} & \frac{4EI}{L} & 0 & -\frac{6EI}{L^{2}} & \frac{2EI}{L} \\ -\frac{EA}{L} & 0 & 0 & \frac{EA}{L} & 0 & 0 \\ 0 & -\frac{12EI}{L^{3}} & \frac{6EI}{L^{2}} & 0 & \frac{12EI}{L^{3}} & \frac{6EI}{L^{2}} \\ 0 & -\frac{6EI}{L^{2}} & \frac{2EI}{L} & 0 & \frac{6EI}{L^{2}} & -\frac{4EI}{L} \end{bmatrix} \begin{pmatrix} \tilde{d_{1}} \\ \tilde{d_{2}} \\ \tilde{d_{3}} \\ \tilde{d_{4}} \\ \tilde{d_{5}} \\ \tilde{d_{6}} \end{pmatrix}$$

The displacements $$\tilde{d_{j}}$$ where i=1,2,4 and 5 are transverse displacements. The dimensions of transverse displacements are length. Shown below is the dimensional analysis. $$\left[\tilde{d_{i}}\right] = \left[\tilde{d_{1}}\right] = \left[\tilde{d_{2}}\right] = \left[\tilde{d_{4}}\right] = \left[\tilde{d_{5}}\right] = L$$ The square brackets, in dimensional analysis, indicate that the dimension of the enclosed are being found. The displacements $$\tilde{d_{j}}$$ where i=3 and 6 are rotational displacements. The dimensions of rotational displacements are 1. Shown below is the dimensional analysis. $$\left[\tilde{d_{3}}\right] = \left[\tilde{d_{6}}\right] = 1$$ From the class notes, it is known that: $$\left[E\right] = \frac{F}{L^{2}}$$ and $$\left[A\right] = L^{2}$$ and $$\left[I\right] = L^{4}$$ Therefore: $$\left[\frac{EA}{L}\right] = \frac{F}{L}$$ and $$\left[\frac{EI}{L}\right] = FL$$ and $$\left[\frac{EI}{L^{2}}\right] = F$$ and $$\left[\frac{EI}{L^{3}}\right] = \frac{F}{L}$$ Since zero is dimensionless, it is known that the dimensions of all $$\tilde{k_{ij}}\tilde{d_{j}}$$ that equal to zero have dimensions of 1. Zero is obviously dimensionless, and a dimensionless parameter is 1. Below are all $$\tilde{k_{ij}}\tilde{d_{i}}$$ that have a dimension of 1.

$$\left[\tilde{k_{12}}\tilde{d_{2}}\right] = \left[\tilde{k_{13}}\tilde{d_{3}}\right] = \left[\tilde{k_{15}}\tilde{d_{5}}\right] = \left[\tilde{k_{16}}\tilde{d_{6}}\right] = \left[\tilde{k_{21}}\tilde{d_{1}}\right] = \left[\tilde{k_{24}}\tilde{d_{4}}\right] = \left[\tilde{k_{31}}\tilde{d_{1}}\right] = \left[\tilde{k_{34}}\tilde{d_{4}}\right] = \left[\tilde{k_{42}}\tilde{d_{2}}\right] = \left[\tilde{k_{43}}\tilde{d_{3}}\right] = \left[\tilde{k_{45}}\tilde{d_{5}}\right] = \left[\tilde{k_{46}}\tilde{d_{6}}\right]$$ $$ = \left[\tilde{k_{51}}\tilde{d_{1}}\right] = \left[\tilde{k_{54}}\tilde{d_{4}}\right] = \left[\tilde{k_{61}}\tilde{d_{1}}\right] =  \left[\tilde{k_{64}}\tilde{d_{4}}\right] = 1 $$

When the rest of the stiffness parameters are multiplied out with the corresponding displacements the following dimensional parameters are found:

$$\left[\tilde{k_{11}}\tilde{d_{1}}\right] = \left[\tilde{k_{14}}\tilde{d_{4}}\right] = \left[\tilde{k_{22}}\tilde{d_{2}}\right] = \left[\tilde{k_{25}}\tilde{d_{5}}\right] = \left[\tilde{k_{41}}\tilde{d_{1}}\right] = \left[\tilde{k_{44}}\tilde{d_{4}}\right] = \left[\tilde{k_{52}}\tilde{d_{2}}\right] = \left[\tilde{k_{55}}\tilde{d_{5}}\right] = F$$

$$\left[\tilde{k_{23}}\tilde{d_{3}}\right] = \left[\tilde{k_{26}}\tilde{d_{6}}\right] = \left[\tilde{k_{32}}\tilde{d_{2}}\right] = \left[\tilde{k_{33}}\tilde{d_{3}}\right] = \left[\tilde{k_{35}}\tilde{d_{5}}\right] = \left[\tilde{k_{36}}\tilde{d_{6}}\right] = \left[\tilde{k_{53}}\tilde{d_{3}}\right] = \left[\tilde{k_{56}}\tilde{d_{6}}\right] = \left[\tilde{k_{62}}\tilde{d_{2}}\right] = \left[\tilde{k_{63}}\tilde{d_{3}}\right] = \left[\tilde{k_{65}}\tilde{d_{5}}\right] = \left[\tilde{k_{66}}\tilde{d_{6}}\right] = FL$$