In this article, related to the calculation of internal forces, we will deal with the determination of axial forces in truss members. The main topic of this entry is the nodes equilibrium method.
In this article, we will cover the following subjects:
Axial forces – since they’re “normal,” they must be fine 😀
From the article on the basics of truss statics, you learned what trusses are and that they consist of pin-jointed rods. As you already know, these rods are capable of carrying only those loads that act along the rod’s axis—these forces are called axial forces (or normal forces). The name comes from the fact that axial forces are perpendicular (in physics, when something is perpendicular, it is said to be “normal”) to the rod’s cross-section.
I think we can initially put forward the thesis that calculating axial forces in truss members is very simple compared to determining the full set of internal forces in beams or frames. In cases where trusses are not loaded with mass (treated as distributed load), the forces in the truss rods have constant values along the entire length of the rod. This eliminates the need to write normal force equations in the form of functions or to determine the values at the beginning and end of the rod. Thanks to this, the calculation of internal forces in a truss can be reduced solely to examining the interactions of rods at their points of connection—the truss nodes. This is precisely what the nodes equilibrium method is based on.
In general, the most commonly used analytical methods for determining forces in truss members are:
- Nodes equilibrium method – which will be discussed in this article
- Ritter’s method – which will be the subject of a separate article.
In addition to these, there are also other methods, such as graphical ones like Cremona’s diagram method. Perhaps I will write more about it someday, but for now, let’s move on to the nodes equilibrium method.
Coplanar concurrent force system – everything in one place
As I wrote above—in the method of joints we will examine how the forces in the truss rods act on the truss nodes. What we should first pay attention to is the fact that all truss rods (by definition) meet exactly at the nodes. This means that when analyzing a given truss node, all the axial forces of the rods (that meet at this node) intersect precisely at the point of the truss node.
A system of concentrated forces that intersect at a single point is called
a coplanar concurrent force system!
Such a system has one fundamental property—the moment generated by the concentrated forces is equal to 0. The easiest way to explain this is to take the analyzed node itself as the reference point for calculating the resultant moment. If the line of action of a concentrated force passes through that point, then the force does not generate a moment with respect to it.
This condition is fulfilled by all the forces in a coplanar concurrent force system. As a result, when writing the sum of moments with respect to the node, as well as with respect to any other point in the system, the moment of forces will always automatically balance (become zero).
Moreover, for such a node to be static (stationary), the remaining equilibrium conditions must be satisfied:
∑F_{ix} =0,∑F_{iy} =0 \to ∑\overrightarrow{F_i} =\overrightarrow0That is, the sums of the forces along the given directions of the chosen coordinate system must equal zero. This can also be expressed in vector form—which is precisely what is used in graphical methods. It should therefore be said that for the entire truss to remain stationary, in each of its nodes we must determine whether the above two equilibrium equations are satisfied. The set of such equations gives us a system that allows us to determine the unknown support reactions as well as the forces in the members of a statically determinate truss.
To summarize—thanks to the fact that in each truss node we can analyze a coplanar concurrent force system, for every node we obtain 2 equilibrium conditions (equations). These form a system of equations that makes it possible to determine all the unknowns in the structure. As is easy to see, since the number of equations available is twice the number of nodes, only that many unknowns can, by default, be solved. On this basis, we obtain the condition of static determinacy of a truss, which can also be expressed mathematically:
2w-k-r=0
where: w – number of truss nodes, k – number of truss members, r – number of support reactions. Additionally, let me remind you that in single-panel systems (where the rod system behaves like a rigid panel), the maximum number of reactions we can calculate is 3.
Let us move on to the final part of the article, which concerns writing the equilibrium equations for a truss node.
Equilibrium equations of nodes – balance is everything
Now, since we know that all we need to do with our truss in order to determine the axial forces in the members is to write two equations for each node (and then transform them mathematically), the question remains: how should we write these equations correctly? In practice, to properly write the force equilibrium equations, it is enough to have a basic knowledge of trigonometry and to choose a suitable coordinate system.
As for the coordinate system—in most cases, we don’t even worry about it—we select the classic system with the horizontal x-axis pointing to the right and the vertical y-axis pointing upward. This is most often due to our habits connected with how trigonometric relations were presented in school.
Sometimes, however, it is worth slightly rotating the coordinate system in order to simplify the equilibrium equations we need to write. An example of this is shown in the figure below.
Let us remember that in writing the equilibrium equations, a force is taken as positive when its direction (or its projection onto an axis) is consistent with the positive direction of that coordinate axis!
As for trigonometry… well… only the basics are needed—namely, the ability to decompose a vector into components along the axes of the coordinate system. Here I can only add that we will use the sine and cosine functions, where: the sine corresponds to the leg opposite the analyzed angle, the cosine corresponds to the leg adjacent to the analyzed angle.
At the end of the article, here’s a small tip for young students of mechanics. Always try to choose the order of the analyzed nodes so that there are at most two unknowns in the equations. This way, we can avoid solving coupled systems of equations, and from each node we immediately obtain the appropriate solutions 😊. Unfortunately, this is not always possible… for some trusses, especially if we don’t calculate the support reactions first, it will be necessary to build larger systems of equations.
To summarize—the nodes equilibrium method is a very simple way to determine the axial forces in truss members. It is especially helpful when our task is to determine all the forces in a truss!
If you’ve made it this far, thank you for reading the article, and I invite you to try out the EquiTruss module, which will let you calculate the forces in the members of any truss using the method of joints! 😊 I also encourage you to check out the article on calculating axial forces using Ritter’s method!
