User:Eml4500.f08.delta 6.castrillon/HW 7

=The Two-Bar Frame=

Modifying the familiar two-bar truss system, we accomodate it as a frame and varying cross-sectional area and modulus. This two-bar frame can be found below.



This arrangement is effective to accomodate the general $$\mathbf{k}^{(i)}$$ equation:


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

Frame elements can be seen as a combination between a truss and a beam element.

=Model Frame Problem with 2 Elements=

We have the following 2-bar fram system:



This frame system difers from a typical truss system since the only two pivoting nodes are the ones located at the ends of the global picture. In other words, frame systems have rigid elbows at element joints. The elbow angle will not change after deformation takes place.

The free-body diagrams of the elements are found below.





Notice the additional dof at each node. This dof comes from the ability of the node to rotate, therefore we say that the $$d$$ labels in the elements fbds above not only represent displacement but also rotation. We still refer to this label as a dof, regardless of whether it is a displacement or a rotation.

We can also relate these dofs to forces and moments at each node. As a general rule,


 * $$d_i^{(e)}$$→$$f_i^{(e)}$$

Where $$d_i^{(e)}$$ is the displacement or rotation of node i of element e, and $$f_i^{(e)}$$ is the force or moment at node i of element e.

Furthermore, the rotational dofs:


 * $$d_3^{(e)}, d_6^{(e)}$$

correspond to the bending moments:


 * $$f_3^{(e)}, f_6^{(e)}$$

=Assembling The Global Stiffness Matrix=

The frame's global dofs are shown below.



There are two element stiffness matrices:


 * $$\mathbf{k}_{6x6}^{(e)}$$, where $$e$$= 1 and 2.

The global stiffness matrix can be assembled by following the same procedure as for the truss system:


 * $$\mathbf{K}_{(9x9)}=A_{e=1}^{e=2}\mathbf{k}_{6x6}^{(e)}$$

The topography of the global stiffness matrix of the above 2-bar frame is:


 * $$\begin{bmatrix}

K_{11}^{(1)} & K_{12}^{(1)} & K_{13}^{(1)} & K_{14}^{(1)} & K_{15}^{(1)} & K_{16}^{(1)} & 0 & 0 & 0 \\ K_{21}^{(1)} & K_{22}^{(1)} & K_{23}^{(1)} & K_{24}^{(1)} & K_{25}^{(1)} & K_{26}^{(1)} & 0 & 0 & 0 \\ K_{31}^{(1)} & K_{32}^{(1)} & K_{33}^{(1)} & K_{34}^{(1)} & K_{35}^{(1)} & K_{36}^{(1)} & 0 & 0 & 0 \\ K_{41}^{(1)} & K_{42}^{(1)} & K_{43}^{(1)} & (K_{44}^{(1)}+K_{11}^{(2)}) & (K_{45}^{(1)}+K_{12}^{(2)})& (K_{46}^{(1)}+K_{13}^{(2)}) & K_{14}^{(1)} & K_{15}^{(1)} & K_{16}^{(1)}\\ K_{51}^{(1)} & K_{52}^{(1)} & K_{53}^{(1)} & (K_{54}^{(1)}+K_{21}^{(2)}) & (K_{55}^{(1)}+K_{22}^{(2)})& (K_{56}^{(1)}+K_{23}^{(2)}) & K_{24}^{(1)} & K_{25}^{(1)} & K_{26}^{(1)}\\ K_{61}^{(1)} & K_{62}^{(1)} & K_{63}^{(1)} & (K_{64}^{(1)}+K_{31}^{(2)}) & (K_{65}^{(1)}+K_{32}^{(2)})& (K_{66}^{(1)}+K_{33}^{(2)}) & K_{34}^{(1)} & K_{35}^{(1)} & K_{36}^{(1)}\\ 0 & 0 & 0 & K_{41}^{(2)} & K_{42}^{(2)} & K_{43}^{(2)} & K_{44}^{(2)} & K_{45}^{(2)} & K_{46}^{(2)} \\ 0 & 0 & 0 & K_{51}^{(2)} & K_{52}^{(2)} & K_{53}^{(2)} & K_{54}^{(2)} & K_{55}^{(2)} & K_{56}^{(2)} \\ 0 & 0 & 0 & K_{61}^{(2)} & K_{62}^{(2)} & K_{63}^{(2)} & K_{64}^{(2)} & K_{65}^{(2)} & K_{66}^{(2)} \end{bmatrix}_{9x9}$$