In the next article related to the basics of statics, we will deal with truss calculations. The main topics of this entry will be the kinematic analysis and the calculation of support reactions in trusses.
In this article, we will cover the following subjects:
- What trusses and pin-jointed rods are
- Determining rigid panels and free rods in a truss
- Calculation of support reactions
Trusses and pin-jointed rods – no bending here!
I think our discussion of trusses should begin with the statement that truss calculations are among the simplest analyses we can encounter in mechanics or strength of materials. Why this is so—I will try to explain later in this article.
What you should know at the outset is that trusses are a system/structure composed of pin-jointed rods.
So, what are these pin-jointed rods?
Pin-jointed rods are straight members (rods) connected by hinges at their ends and loaded only with forces acting along their axis.
Such could be the simplified definition, from which one very important fact (from the point of view of strength) follows: in pin-jointed rods only axial forces occur—that is, forces perpendicular to the cross-section. Such a member is not subjected to bending, shear, or torsion, which in itself gives it very high stiffness and allows it to carry large loads. In truss analyses, apart from calculating reactions, we will be looking precisely for the values of these axial forces—they have a direct impact on the stresses in the truss members.
As a side note to this definition, it can be added that in reality such rods are also loaded, for example, by mass forces (gravity), wind forces, etc., which have some effect on their strength. However, for preliminary calculations, the presented “rod model” is fully sufficient.
Returning to the truss—it is a set of pin-jointed rods connected exactly at their ends, which we will further call the nodes of the truss. From the very definition of pin-jointed rods, several important issues regarding loads and supports in a truss also follow. For a truss to remain a “theoretical truss,” all loads and supports must be applied at the truss nodes. Moreover, only forces can be applied at the nodes! Thus, the only mechanical loads present in a theoretical truss are forces (at any angle), and the only supports that can be used are hinged supports (fixed and movable).
To sum up this short introduction—truss calculations most often consist of determining the support reactions and axial forces in the truss members. No other internal forces occur in the members, which already greatly simplifies the calculations (for example, compared to the analysis of beams or frames). In truss analysis, the set of possible loads and supports is also significantly restricted.
Before we move on to the subject of determining support reactions in a truss, let us take a moment to discuss trusses from the point of view of their kinematic analysis.
Determining rigid panels and free rods in a truss – how to make the calculations more difficult
From the previous articles, you already know that every rigid element in a plane has 3 degrees of freedom. On this basis, the condition of static determinacy is formulated for beams or frames.
On the other hand, you also know that the condition of static determinacy for trusses takes a completely different form, namely:
2w-k-r=0
or for single-panel trusses:
2w-k-3 =0
This condition approaches the issue of truss determinacy not directly from the level of the number of panels, kinematic connections in the system, etc. (in general terms—degrees of freedom), but from the number of equilibrium equations that can be written for the truss nodes, as well as the unknown reactions and rod forces. Why we can solve trusses using node equilibrium I will explain in a separate article. For now, however, I would like to draw your attention to why trusses do not use the same conditions as beams and frames, and also—what on earth single-panel trusses actually are…
So, let us look at a truss, starting with a single pin-jointed rod. Such a rod, if unsupported, has 3 free degrees of freedom. If we attach another rod to it using a hinge, then following simple mathematics—we add another 3 degrees of freedom and subtract 2 (due to the hinge connection—blocking relative movement in the horizontal and vertical directions).
n=3t-2p=3⋅2-2⋅1=4
Such a system of two rods already has 4 degrees of freedom. Now let us add another rod, which will be connected to the previous ones by two additional hinges. Altogether we have 3 rods and 3 hinges:
n=3t-2p=3⋅3-2⋅3=3
It turns out that such a system of three interconnected rods has 3 degrees of freedom, exactly the same as a rigid panel in the plane. Moreover, this system of rods is geometrically stable. As a result, a system of three interconnected rods will be called the elementary cell of a truss, and we will further interpret it as a single rigid panel (see the animation below).
Let us now consider how such an elementary cell can be expanded.
To increase the size of a truss, we can add two more rods to the elementary cell, joined together with a hinge… And at this point, something “magical” happens—something that often causes problems, especially when writing the analyzed system in the form of a degrees-of-freedom equation. The error in reasoning usually lies in the fact that we have just added two rods (each with 3 degrees of freedom) and one hinge (removing 2 degrees of freedom), which would lead to the following equation:
n=3t-2p=3⋅5-2⋅4=7
Such a result would suggest that a system of 5 rods in a truss would not form a rigid panel. However, we are forgetting something very important—namely:
when attaching new rods to the elementary cell, we immediately remove two degrees of freedom from each of them!
If those new rods were not connected with a hinge, then relative to the elementary cell they would only be able to rotate around their connections (see the animation below).
It turns out that each newly attached rod at a given hinge increases the number of degreesof freedom removed by that hinge. In other words—every rod added to an existing hinge connection should be considered as if it introduces one more hinge into the system. Taking this fact into account, the number of hinges in the analyzed system (including the “virtual” ones) is equal to 6:
n=3t-2p=3⋅5-2⋅6=3
To summarize—after adding two rods connected with a hinge to the elementary cell—we still obtain a single rigid panel. And this will always be the case whenever such a set of rods is added to a rigid panel. In the end, we may have a truss consisting of a large number of rods connected in many different ways, but we will still interpret it as one panel—such trusses are called single-panel trusses. From the standpoint of planar motion, they have only 3 degrees of freedom, and only these 3 degrees of freedom must be restrained by the support reactions. This type of truss appears in 95% of truss problems encountered in university courses!
And what about the situation when trusses have more complicated connections? I would say—not a big deal—we can analyze them just like multi-panel beams or frames. Such systems may have more support reactions or additional connections between panels in the form of hinges and free rods (those that do not belong to any rigid panel—see the animation below).
In conclusion—the analysis of the degrees of freedom of a truss, considering its connections and rods, can be very unpleasant and often misleading. For this very reason, the static determinacy of trusses is analyzed using simpler formulas related to the equilibrium equations of the nodes. Unfortunately, in order to perform a kinematic analysis—to verify geometric stability—it is still necessary to examine the connections between the rods. Without this, we can never be sure whether our truss can be analyzed using the methods of statics.
Let us now move on to the calculation of the support reactions themselves.
Support reaction calculations – or how to calculate without ending up “behind rods”
Since we now know what single-panel trusses are, how to identify rigid panels in rod systems, and how to deal with determining the static determinacy of a truss, the actual calculation of support reactions should not be a major challenge. Such calculations come down to identifying the unknown reactions (also in kinematic connections), writing the equations of static equilibrium, and then applying pure mathematics.
In statically determinate single-panel trusses, we can expect only 3 support reactions, and we can also write the 3 fundamental equilibrium equations:
∑F_{ix} =0,∑F_{iy} =0,∑M_i =0,
In multi-panel systems, we can divide the truss into subsystems (panels), replacing the hinges and free rods with reaction forces. Then, for each subsystem, we can write 3 equilibrium equations. Alternatively, we can try to write moment equations to the left or right of a hinge, just as we do in beams or frames. In this way, we obtain additional equations without having to split the hinge into reactions.
The last issue worth clarifying is the often problematic interpretation of a connection in which only a single pin-jointed rod and a fixed support are present. At the outset, it is often assumed that such a support provides two independent reactions, which leads to too many unknowns in the equilibrium equations. In practice, however, such a rod can transfer only a force acting along its own axis—therefore, despite the presence of a fixed support, there is only one unknown reaction, and it must necessarily be directed along the axis of the rod.
If you’ve made it this far, thank you for reading the article, and I invite you to try out the EquiTruss module, which allows you to calculate the support reactions (and much more!) of any truss—no matter how many panels it consists of! 😊
I also encourage you to check out the articles on calculating internal forces in trusses!
