The Method of Bar Conversions is a procedure for solving the internal bar forces of complex trusses. It is a unique procedure that makes use of a number of analysis concepts.
Method of Bar Conversions is a procedure for solving the internal bar forces of complex trusses.
Key Idea: Superposition
To solve complex trusses, we have to take advantage of superposition. To freshen our minds, say you have a photo of a mountain and a picture of a sun. If you put the sun's image on top of the other picture, you'll get a new photo combining the mountain and the sun. What does this have to do with solving internal bar forces?
Imagine your complex truss as the combined photo of the mountain and sun in our analogy. Remember that the combined object is a composition of multiple smaller objects. Similarly, we can imagine a complex truss as simple trusses combined.
What we are going to do with the complex truss is to break it down into structures that we can solve. Afterward, we solve these individual structures like simple trusses, and we will combine their solutions to compute the internal bar forces of the complex truss. This powerful idea in classical analysis is called superposition.
Now that we got a general idea, how should we break down the complex truss?
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For this method, we will break down the complex truss into two simple structures: a simple truss and a proxy truss. When we add these two structures together, you'll get the bar forces for each complex truss.
The outline below is best understood using this example.
Convert to a Simple Truss
The first step is to convert our complex truss into a simple one by removing any member; however, in doing so, we made the truss unstable. To reverse that effect, we add another member. There is no rule on what member to remove or add. There is also no rule about what should be the two joints of our new substitute member. Our goal here is to make our truss simple.
We load this simple truss with the supports and loads like that of our original complex truss. The result is a new structure with the same reactions and loads \(S_p\), serving as an element for the superposition process.
Like any other simple truss, the next step is to thoroughly analyze the reactions and bar forces of this simple truss using the method of joints or sections.
At the end of the analysis, you will have each member's axial forces, including the substitute member.
Create Proxy Truss
The next step is to create a proxy truss for the superposition process \(S_x\). Remember that we could not solve for the missing member in the simple truss because we removed it; hence, to counteract that effect, we must create the same simple truss as \(S_p\) and represent the missing force with a tensile force \(x\).
Our goal here is to represent all bar forces in terms of this variable \(x\) using either method of joints or section.
Apply Superposition
At this point, you have completely solved the bar forces of both structures \(S_p\) and \(S_x\); hence, we are ready to apply the idea of superposition.
Remember that we are solving for the bar forces of the complex truss \(S_T\). Equation-wise, it is:
\(S_T=S_p+S_x\)
We first apply superposition to solve for \(x\). With it, you can solve the bar forces of the complex truss. To find for \(x\), let's take an interest in the substitute member at \(S_p\).
We know that it does not exist in our truss example; hence, \(S_T\) for this substitute member must be zero:
\(0=S_p+S_x(x)\)
Solving for \(x\) should be possible using the equation above. When you solve for it, you already got the axial force of the original member you removed. With \(x\) cracked, you can complete your analysis by performing superposition to all members.
Tips
We should use this method for complex trusses only. We can't use it for simple or compound trusses; we cannot further break them into simpler ones. If you do so, you're changing the truss itself. For example, suppose you removed a member of a simple Warren truss and replaced it with another substitute member to create a simpler truss. In that case, you're converting it into a compound truss which is a different structure than the original. As a consequence, superposition would give the wrong result.
After applying this method, you can check your answers by using the method of joints or sections in the complex truss. In the end, it must comply with the equilibrium principle.
Summarise Your Results
After analysis, it is always a good idea to summarise your results using a table.
Application
Let's illustrate how to use the method of bar conversions in a complex truss example.
Summary
Let's summarize:
Method of Bar Conversions is a procedure for solving the internal bar forces of complex trusses.
We break the complex truss into structures that we can solve. Afterward, we solve these individual structures like simple trusses, and we will combine their solutions to compute the internal bar forces of the complex truss.
We first break it into a simple truss by removing one member of the complex truss and replacing it with another. Next, we load this new structure with the original loads and supports. Then, we solve for the bar forces of this structure.
We create a proxy truss similar to the simple truss. We represent the missing bar force with a variable and express every member in terms of said variable.
We apply the principle of superposition for both the simple and proxy truss. The result would be the internal bar forces of the complex truss.
