Oto druga cześć artykułu związanego z podstawami statyki. Głównym tematem omawianym w tej części będzie geometryczna niezmienność układu, która jest związana z jego podparciem.
W ramach artykułu omówimy tematy:
Supports – how to support something so it doesn’t fall…
Typically, to immobilize a body, so-called kinematic constraints are used – I know, it sounds intimidating, but it represents something very simple – supports. To paraphrase – to immobilize a body, you simply have to grab it and attach it to something. In most mechanical analyses done in academic settings, our body will be attached to a mythical ground that, not only is perfectly rigid (it doesn’t move), but also, along with a perfectly rigid support, provides almost divine-quality kinematic constraints. In reality, every support has its own stiffness, and every attachment to the ground needs to be properly designed (foundations, bonds, etc.), but for now, we can ignore that.
Now – attention – we will introduce an important concept. If supports serve to block a body’s movement, then in mechanics, we will say that they restrict the degrees of freedom of the system. So, how many types of support can we have…? – Essentially, as many combinations of degrees of freedom as we can create.
For example:
- If we want to block all three degrees of freedom of a plate with one support, we will call it a fixed support.
- If we want to block horizontal and vertical movement but allow rotational movement around the support, we call it a pinned support.
- If we want to block linear movement in a specific direction (it doesn’t necessarily have to be horizontal or vertical), we call it a roller support.
There are still at least a few combinations of degrees of freedom we can calculate (all of which are available in our EquiBeam application).
Geometric immutability of the system – so why can’t I move it…
To summarize the topic of restricting degrees of freedom – to immobilize a body, we need to block all its degrees of freedom. Such a system – attention – will be called geometrically rigid or stable.
Does that mean that to immobilize a plate, we simply need to support it with supports whose total blocked degrees of freedom equal 3? Unfortunately, not quite… Of course, if we fix the plate in place, taking away three degrees of freedom will fully immobilize it, but there may be cases of support where, despite blocking three degrees of freedom, we block the wrong ones…
Imagine a beam supported by three roller supports:
Without any further analysis, we can see that it cannot move in the y-axis direction. After a moment’s thought, we also conclude that this arrangement of supports prevents rotation. But… we haven’t restricted its ability to move along the x-axis at all. Such a system, despite attempting to block three degrees of freedom, will still be able to move – such a system will not be geometrically stable.
The analysis of systems in terms of their geometric rigidity is called kinematic analysis, which I will briefly discuss in a moment. This is a topic that isn’t very simple because we are not limited to analyzing a single plate. What’s more – we often analyze systems composed of many different plates, properly connected (e.g., at hinges) and supported in specific ways. EquiBeam allows for the analysis of geometric immutability of such beam systems.
Kinematic analysis – can it move, or not?
As long as we are analyzing a single plate, performing kinematic analysis is relatively simple – as you know, we only need to block “just 3 degrees” of freedom. Blocking degrees of freedom related to linear movement along an axis (sometimes called translational degrees of freedom) always occurs by applying two non-parallel reaction forces. Automatically – if all reaction force directions are parallel – the system is definitely not geometrically stable. Additionally, if only two non-parallel reaction forces exist, the system can rotate around the point where their directions intersect. To summarize
a single plate is immobile if three reaction forces act on it, where at least one is non-parallel to the others, and the directions of all three do not intersect at one point.
All the above cases are illustrated in the graphic below
Kinematic analysis of multiple plates connected by kinematic constraints is much more complex. In such cases, none of the plates need to be fully fixed by supports because sometimes it’s enough that they “support each other.” Let’s look at the graphic below.
Both of the plates shown have two degrees of freedom restricted by supports – the first by a fixed support, the second by two roller supports. Let’s temporarily ignore the hinged connection between them. In this situation, as you know from the previous analysis, each of these plates has only a rotational degree of freedom left – the first plate can rotate around the fixed support, the second around the intersection point of the roller support reactions. Knowing the instantaneous center of rotation of each plate, we can easily determine the theoretical direction of motion for any of its points – including the hinge connection point.
Now – attention – if the directions of motion of the hinge point, analyzed for the first and second plates, are not parallel, this means that the plates block each other’s rotational movement through this connection. Such a system is geometrically stable. If the directions were parallel – the system of plates would not block its own rotation!
The final example presented in this article is the expansion of the above system with additional plates connected by hinges. In such a case, if a plate is attached by a hinge to an immobile system (which we have already determined), it only has a rotational degree of freedom – it can rotate around this hinge. To immobilize it, we need to apply a reaction force that doesn’t pass through this hinge – as shown in the graphic:
The above considerations should close the topic of kinematic analysis for beam-like systems. In trusses and frames, more complicated situations can arise, for example, due to the existence of hinged rods, but regardless of the case, you should always keep in mind the relationships discussed above. I will expand the analysis for frame and truss systems in a separate article.
In further discussions, we will assume that our systems are supported in such a way that they are geometrically immutable. In the next part of the article, we will discuss the calculation of support reactions and learn what statically determinate systems are.
