User:Eml4500.f08.team.foskey.ckf/hw5e

Is the space truss statically indeterminate?


Looking at the three bar space truss problem, the problem is easily solve using the finite element method. However, the question is raised as to whether or not the truss can be solved using statics.

When first looking at the problem, one sees that there will be nine reactions, three for each bar. However, statics only gives a limited number of equations.

The problem is solved statically by using the Euler cut principle.



In doing this, the bars are all cut in half, and the equilibrium of node 4 is solved for. When the bars are cut, their associated axial forces are left. The axial forces must then be resolved into their components for each directions.

To resolve the axial forces into components we first need the lengths of the bars. This is done by using the equation

$$ L= \sqrt{(X_2 - X_1)^2 + (Y_2 -Y_1)^2 + (Z_2 -Z_1)^2}$$

where L is the bar length, X2 is the x coordinate of node 4, Y2 is the y coordinate of node 4, Z2 is the z coordinate of node 4, and X1, Y1, Z1 are the coordinates of the pinned node for a bar.

Using the above formula, the lengths of the bars are found to be L1 =2.9339 L2 = 2.8543 L3 = 2

From there the forces are resolved into components by using a scaling factor. This scaling factor is found by $$ \frac{L_D}{L}$$, where LD is the length in a coordinate direction (x, y, or z) and L is the total length. LD could also be considered the distance between the nodes of a bar in one direction (x, y,or z)

So to find the x component force in the axial force of bar 1, you would use the formula $$ F^1_x = \frac{(X_2-X_1)}{L^1}*F^1$$ where F1 is the axial force in bar 1. This process is repeated for all of the forces in all of the bars, until 9 force components are obtained.

Summing the forces acting on node 4 for the x, y, and z directions will result in the axial forces. These equations are

$$ \Sigma F_x = F^1_x +F^2_x +F^3_x =0$$ $$ \Sigma F_y = F^1_y +F^2_y +F^3_y =-20000$$ $$ \Sigma F_z = F^1_z +F^2_z +F^3_z =0$$

Keep in mind that F1x, F1y, and F1z are all in terms of the axial force F1 (and likewise with F2 and F3) so that there are only three true unknowns in the above equations. Solving those three equations will give the axial forces F1, F2) and F3).

From there, the axial forces are resolved into their components to find the reactions. This is accomplished by using the same scaling factors calculated earlier.

Using this method, the reactions were found to be

The reaction forces calculated by the MATLAB code are:

There is a significant amount of discrepancy between the two methods. This can possibly be attributed to the statics method not taking into account the deformations the bars undergo, the differences in physical properties of the bars. Errors are also possible simply from errors in calculating the values by hand.

To conclude, the three bar space truss can be determined using statics, however there were errors in our calculations, although we believe our methodology to be sound.