User:Eml4500.f08/HW report table/The best of HW3

 Under construction; not final. The intention here is to document the best features in any HW report for the readers (including you); if you see excellent features in any HW report (including your team's) that I may have missed noticing, don't hesitate to let me know. (I don't have time to read all HW reports in detail.) Eml4500.f08 13:57, 13 October 2008 (UTC)


 * derivation of element stiffness matrix $$\displaystyle \mathbf k ^{(e)}_{4 \times 4}$$ in global coordinates: Axial dofs $$\displaystyle \mathbf q^{(e)}_{2 \times 1}$$, 2x4 transformation matrix $$\displaystyle \mathbf T ^{(e)}_{2 \times 4}$$, transformation of dofs $$\displaystyle \mathbf q ^{(e)} _{2 \times 1} = \mathbf T ^{(e)}_{2 \times 4} \mathbf d^{(e)}_{4 \times 1}$$, axial force-displacement (FD) relation $$\displaystyle \mathbf p ^{(e)} _{2 \times 1} = \hat{\mathbf k}^{(e)} _{2 \times 2} \mathbf q^{(e)} _{2 \times 1}$$, FD relation in global coordinates $$\displaystyle \mathbf f ^{(e)} _{4 \times 1} = {\mathbf k}^{(e)} _{4 \times 4} \mathbf d^{(e)} _{4 \times 1}$$, verification of expression for element stiffness matrix $$\displaystyle \mathbf k ^{(e)} = {\mathbf T ^{(e)}}^T \hat{\mathbf k}^{(e)} \mathbf T ^{(e)}$$. Best explanation: Team Bike
 * singularity of global stiffness matrix: solve eigenvalue problem on global stiffness matrix and observe the zero eigenvalues. Best solution: Team Bike Team Ramrod, but the explanation of the zero eigenvalues was wrong for both teams.


 * closing the loop: For statically determinate problems, such as the 2-bar truss system, first solve for the reactions using statics, then solve for the displacements at the global node 2 using infinitesimal displacements. Best solution: Team Wiki1 did everything right, even detailed the solution of 2 simultaneous equations for the coordinates $$\displaystyle (x_D, y_D)$$ of point $$\displaystyle D$$.  See also Team Ramrod, who got almost perfect solution, where there was a misprint in the second equation: $$\displaystyle \cdots \tan \left( \frac{2\pi}{3} \right) - (x_D-5.917) \cdots $$ should be $$\displaystyle \cdots \tan \left( \frac{2\pi}{3} \right) (x_D-5.917) \cdots $$, i.e., no minus sign; also the solution process for $$\displaystyle (x_D , y_D)$$ was not described in detail as in Team Wiki1, but that is minor point.  Only these two teams got this problem right.
 * figure for conceptual steps in "closing the loop". Best figure: Team Echo (did not point out where "closing the loop" was) Team FEABBQ (did not point out where "closing the loop" was) Team Wiki (did point out where "closing the loop" was, but had an extra arrow in lower left corner that I did not intend to have in my lecture)
 * computation of the reactions using method 2, i.e., multiply the reduced global stiffness matrix $$\displaystyle \mathbf K _{6 \times 2}$$ by the reduced displacement matrix $$\displaystyle \mathbf d _{2 \times 1}$$, and ignoring rows 3 and 4, which correspond to the known applied forces. Best solution: Team Bike Team Ramrod


 * 3-bar truss system: This system is statically indeterminate; 3 unknown reactions, but only 2 equations, $$\displaystyle \sum F_x = 0$$ and $$\displaystyle \sum F_y = 0$$. The equation $$\displaystyle \sum M_A = 0$$, i.e., sum-of-moments-about-point-A equals zero, leads to a trivial equation $$\displaystyle 0 = 0$$, and thus is not useful (point A is the intersection of the 3 bars).  Show that the sum of moments about any point in the plane leads to a trivial (not-useful) equation $$\displaystyle 0 = 0$$, and thus the moment equation cannot provide a 3rd independent equation to solve for the 3 unknown reactions.  The theory was explained in detail in class; see my lecture presentation.  Best explanation: Team Bike
 * figure for assembly of element stiffness matrices into global stiffness matrix. Best figure: Team Wiki Team Echo
 * 3-bar truss global stiffness matrix. Best expression for $$\displaystyle \mathbf K = \overset{e=nel}{\underset{e=1}{\mathbb A}} \mathbf k^{(e)}$$ : Team Jamama


 * 2-bar truss system: matlab plot of deformed shape. Best code, plot, documentation: Team Wiki plotted a square around each global node number using matlab and not other software; Team Team for complete documentation, including sample matlab codes.

EAS 4200C, The best of HW3