Gaussian Row Eschelon Form

The method of joints is a simple extension of traditional statics. You simply create a free-body diagram for each of the joints, assuming that the members are connected via pins that allow rotation but not translational motion. (This places the emphasis on the strength of the design, not the materials; also, using pins at the joints is a fair approximation, since the beams move minimally under a load.)

After completing free-body diagrams for each joint in the truss, it becomes a matter of attempting to solve a large number of equations. In most trusses, this is a simple matter: find a joint or reaction whose forces can be determined, and continue from that point.

You could also start at any arbitrary joint and solve for the forces in terms of the other forces. Then, continue through the truss substituting in for various forces what you know.

The truss analysis module uses this type of idea. It starts at the first force at the first joint and solves it in terms of the other forces, and continues through all the forces. For a statically determinate truss, the final force (the nth force) will have a determinate value (even if it is zero), and can be substituted back into the equation for the (n-1)th force. This then continues backwards until all the forces have a definite numerical value.

Finding the Gaussian reduced eschelon form (REF) of an augmented matrix makes this process efficient and simple. Yeah, it sounds bad here, but that's where the computer comes in. (The REF is actually a pretty easy method to use in solving simultaneous equations, which is all the method of joints really is at heart.)

Let's take for example this simple two joint system. It happens to have one force acting in the vertical, one in the horizontal, and two acting at angles. We just have to create a simple augmented matrix for the system, and find its row eschelon form.

So, what's an augmented matrix? It's a square matrix with an extra column tacked onto the end. The matrix that the module uses consists of the coefficients of each force in the x or y direction at a joint and the sum of the forces in that direction. Say you had [(Sum Ax:) F1-.1736F4=0] as one equation; then, the corresponding row would be 1.0, 0, 0, -.1736, 0. (Since F2 and F3 are not involved, they have coefficients of zero, and no external forces are present.) For example, the system shown above could be represented with the following augmented matrix:

F1F2F3F4-R
Ax1.00.0-.17360.00.0
Ay0.00.0.98480.00.0
Bx0.00.00.0.93970.0
By0.01.00.0-.34200.0

The next step in the REF method is to reduce the leading coefficient in the first row to a one. This is done by dividing each element in the row by the coeffient, namely 1 (leaving this row as it is). Why is this important? This action reduces the row to an equation in terms of the first force. Plus, it makes it much easier to work with in the next step.

Next, the force needs to be substituted into each other equation. How? Take the coefficient of the first force from the next row down, multiply all the coefficients in the first row by it, and subtract the two rows. This effectively removes the first force from the rest of the equations, leaving a zero as the coefficient of the first force in that row. The same multiplication and subtraction is repeated down through the rows.

The process then continues down the matrix again, this time starting in the second row with the second force. If the leading force is not in a row, then that row is swapped with one where the force does appear, simply changing the order that the equations are solved.

In the end, the last force is solved for numerically. Then, this value is substitued into the previous equation, and the row is solved for the single unknown. This repeats up the matrix, until each of the forces has a definite value. The sytem above is pretty dull, since no external forces are acting on it. Plus, it's no truss. However, if we repeat everything for a real truss, like the one shown below, it's a lot more interesting. For this truss, the base of each triangular frame is 4 meters, and the height is 3 meters. The load is 4.0 kN downwards.

Matix of the Truss

F-ABF-BCF-CEF-DEF-ADF-BDF-CDS1S2S3-R
Ax.55470001.00001.000
Ay.83210000001.0000
Bx-.55471000.554700000
By-.83210000-.832100000
Cx0-1.0.5547000-.55470000
Cy00-.8321000-.83210000
Dx0001.0-1.0-.5547.55470000
Dy00000.8321.83210004.0
Ex00-.5547-1.00000000
Ey00.83210000001.00

Matrix, REF applied

F-ABF-BCF-CEF-DEF-ADF-BDF-CDS1S2S3-R
Ax1.00000001.202000
Ay01.0000.55470.6666000
Bx001.00001.00000
By0001.0-1-.5547.55470000
Cx00001.000-.66661.000
Cy000001.00-1.202000
Dx0000001.0-1.202000
Dy00000001.0002.0
Ex000000001.000
Ey0000000001.02.0

Solution Set:


F-ABF-BCF-CEF-DEF-ADF-BDF-CDS1S2S3
-2.404-2.667-2.4041.3331.3332.4042.4042.00.02.0

We were originally using the assumption that all the forces acted on the joints in tension. Those forces with a negative value must be acting opposite our assuption, thus putting those members into compression. The diagram generated from the truss analysis module shows this, with the tensioned members in blue and the compressed members in red.

In general, the REF method will work for any static truss. However, to solve for all variables, there needs to be one equation per variable. In the truss analysis module, only trusses where this specification is met are accepted (unless loaded from disk). However, there is no good way to tell the difference between a statically determinate and indeterminate model until you run the analysis. If any of the forces cannot be determined, all are left empty (zero). There is a very small chance that, due to the limited precision of the computer, all the forces may be found even in an indeterminate case; however, these analyses return with a large number of broken members and huge forces compared to the loads. (IE, they are extremely easy to spot.)

The REF method is overall very stable, and is an extremely fast algorithm for solving multiple equations simultaneously.