User:Eml4500.f08.ateam.boggs.t/Lecture 35

Two-Bar Tapered Truss
The class was presented with a two-bar truss with tapered elements. The Material properties are applied at the nodes instead of an average value applied over the length of the element. The general set up for the problem is presented below.





The class was asked to compute the solution to the two-bar truss with tapered elements and to plot and compare the deformed shape to that of a two-bar truss with average material properties applied across the entire length of the element.

Frame Element
A frame element is the combination of a truss, or bar element, and a beam element. The truss element corresponds to the axial deformation experienced by the frame and the beam element corresponds to the transverse deformation. A model of a frame with two elements bound together by a rigid connection is shown below.



A rigid connection requires that the angle between the two elements will not change after deformation. The free body diagrams of each element are presented below.



The general forces associated with the above elements are presented in the following form and include forces and bending moments.

fie $$\rightarrow$$ Generalized forces

Where e is equal to 1 or 2 and i can range from 1 to 6.

Looking at the above free body diagrams d3e and d6e are the rotational degrees of freedom. Correspondingly f3e and f6e are the bending moments.

A diagram of the frame global degrees of freedom are shown below.



Here d3,6,9 are the rotational degrees of freedom and the rest displacements. For this frame the 2 element stiffness matrix Ke is a 6x6 matrix where e can be equal to 1 or 2. Therefore the global stiffness matrix, K, becomes a 9x9 matrix where,


 * $$ K = A_{e=1}^{e=2}K^{(e)}$$