Here’s the third part of the article about the basics of statics. In this article, you will learn how and why we calculate support reactions and what statically determinate systems are. I will also show how we analyze such systems using EquiBeam.
In this article, we will cover the following topics:
Support Reactions – How to Separate the System from the Ground
Using the knowledge from the previous parts of the article, we know that we can analyze a body that is supported in some way. But how does this support relate to the previously discussed conditions of static equilibrium?
The matter is actually quite simple to grasp – if a support blocks a certain type of movement, any attempt to apply a force in that direction will result in a reaction force from the support (or a moment). This brings us to an important concept – support reactions.
Support reactions are all the forces and moments generated by a support to prevent the movement of the body restricted by that support.
From this, we can draw one conclusion – in a support, we can distinguish as many support reactions as the number of degrees of freedom the support restricts.
For example:
- In a fixed support (which restricts 3 degrees of freedom), we can distinguish reactions in the form of a horizontal force, a vertical force, and a moment.
- In a pinned support (which restricts 2 degrees of freedom), we can distinguish reactions in the form of a horizontal force and a vertical force – technically, this is one reaction, but we can split it into two components perpendicular to the axes of the coordinate system.
- In a roller support (which restricts 1 degree of freedom), we can distinguish one reaction in the form of a force perpendicular to the support’s surface.
In problem-solving, the step where we replace the support with its corresponding set of support reactions is called detaching from constraints.
What determines the values of support reactions? We can simplify it by saying that it depends on the load.
More specifically, in statically determinate systems (which I’ll discuss shortly), it depends on the type of load, its value, and its location. In statically indeterminate systems, the stiffness of the system also matters, but that’s a topic for another article.
To simplify what I just wrote – if no forces act on the system, the values of the reactions will also be zero. However, if forces are applied, the values of the reactions (not necessarily all of them) will be non-zero.
How can we calculate the values of support reactions? This brings us back to the initial question
we calculate the support reactions for statically determinate systems using the static equilibrium equations
If the system is to remain stationary, the forces acting on it must be balanced – this directly relates to the fact that the values of the support reactions must appropriately counterbalance the loads acting on the system! Writing out the equilibrium equations essentially gives us a system of equations, the solution to which provides the values of the support reactions.
At this point, we’ve tackled one of the common difficulties students face – calculating support reactions (at least in theory) – so now you know why supports are used in the analysis of beams, frames, and trusses, why equilibrium equations are written, and what they can be used for – calculating the values of reactions.
To close this topic, we should, quoting a famous line: “ask ourselves one important question”: why do we even need to know the values of reactions…? Here’s the answer – knowing the support reactions allows us to calculate the values of internal forces, which is essential for strength calculations, as well as displacements in the system, which are a crucial part of any design work.
Given that the analysis of reactions and equilibrium equations in this article was presented only theoretically, for an unlimited number of examples of such analysis, I invite you to our EquiBeam application!
Finally, I’d like to address the topic of statically determinate systems – as these are what you’ll definitely encounter early in your studies…
Statically Determinate Systems – What Can and Cannot Be Quickly Calculated
In principle, the topic of statically determinate systems isn’t entirely a mechanical problem – it’s more related to basic mathematical laws. As you already know from earlier parts of the article, one of the first steps in the mechanical analysis of a system is determining the values of the support reactions. You also know that their values can be calculated by writing out equilibrium equations and solving the resulting system of equations.
The question is – can we always calculate the values of reactions from the system of equations? Unfortunately, no… and the obstacle here is simple, brutal mathematics.
Equilibrium equations typically form a system of linear equations. In such a case, mathematics is clear – to calculate the specific values of unknowns in a system of equations, the number of unknowns must equal the number of equations.
Of course, this is not a sufficient condition because we can encounter inconsistent or undetermined systems, but these generally don’t occur in geometrically immutable mechanical systems. This means that when analyzing a single body (for which we can write 3 equilibrium equations), we can only have three unknowns in the form of support reactions.
In the case of more bodies, we can write three equilibrium equations for each of them. However, in such systems, it is also necessary to add additional supports and connections between the bodies, which, as constraints, also generate additional unknowns in the form of reactions at the connections.
Typically, the most common connection between bodies is a hinge, which restricts their vertical and horizontal displacements – thereby taking away 2 degrees of freedom – automatically generating two reactions.
Based on all the above information, we can formulate a simple mathematical rule related to the ability to calculate reaction values based on equilibrium equations for geometrically immutable systems. For flat bodies loaded in their plane (beams and frames), it can be formulated as follows:
3\cdot t-2\cdot p-r=0
gdzie: t – is the number of bodies in the system, p – is the number of hinged connections, r – is the number of support reactions.
For trusses, where in addition to reactions, the unknowns are the values of forces in the truss members, and for each node, we can write two equations, the formula looks like this:
2\cdot w-k-r=0
gdzie: w – is the number of nodes in the truss, k – is the number of truss members, r -is the number of support reactions.
Pomimo tego, że samo prawo jest proste, niestety nie jest ono pełnym wyznacznikiem tego czy reakcje da się policzyć (jest to warunek konieczny, ale nie wystarczający – jak w matematyce).
Although the rule itself is simple, unfortunately, it’s not a full determinant of whether reactions can be calculated (it’s a necessary condition but not sufficient – as in mathematics). To be sure, we can apply a modification of the above formulas and a short algorithm (outlined below). The modification of the formulas involves creating a coefficient that defines the excess number of unknowns – ksn
k_{sn}=2\cdot p+r-3\cdot tlub
k_{sn}=k+r-2\cdot wProposed algorithm:
- calculate the value of the ksn coefficient based on the number of bodies, types, and number of supports and connections
- if ksn < 0 – we know that too few degrees of freedom have been restricted – the system is not geometrically immutable – we solve it using kinematics, not statics,
- if ksn > 0 – the system may (but doesn’t necessarily have to be) geometrically immutable – perform a kinematic analysis,
- if the kinematic analysis shows that the system is geometrically immutable, return to analyzing the ksn coefficient.
All systems that meet the condition of geometric immutability and for which ksn = 0 are called statically determinate systems – we can calculate all reaction values based on the equilibrium equations.
If the system is geometrically immutable and ksn > 0 the system is statically indeterminate – we cannot calculate the reactions using only equilibrium equations, but that doesn’t mean we can’t calculate them at all.
We can now also introduce a nice term for the ksn coefficient – it is the degree of static indeterminacy of the system.
To calculate statically indeterminate systems, we use additional methods that consider the system’s stiffness (its displacements) such as:
- the Clebsch method,
- the force method,
- the displacement method,
- the Menabrea-Castigliano method.
These and other methods are available in our EquiBeam application.
Statically Determinate Tasks in EquiBeam – Why Did We Do This…
You can learn more about EquiBeam as a whole from the app section on our website. Here, I just want to emphasize that in EquiBeam, you’ll find the “Statically Determinate Beams” method, which is designed specifically for calculating reactions, internal forces, and drawing their diagrams.
When solving a statically determinate problem in EquiBeam, you will get the support reactions broken down, along with the equilibrium equations and their full solution, up to the values of the support reactions. The program will also determine whether the beam is geometrically immutable.
In the attached video, I present examples of solving such a beam in EquiBeam. I encourage you to use EquiBeam to practice solving beam system problems!
If you’ve made it to this point – thank you for reading the article. I’m sure it will clarify the basics of mechanical calculations, assessing whether a system is statically determinate, analyzing support reactions, and calculating their values.
