Static determination and geometric stability

Now we will acquire some knowledge of the subject of static determinability and geometrical constancy of structural elements, at this stage in fact it is a very important topic, because after checking these things you know if you can go to the next stage of calculating your project.

These are the two basic conditions that should be met and must always be checked before calculating.

First, let’s get acquainted with the symbols that correspond to particular supports.

Pin Support

Roller Support





Fixed End Support

Fixed Support – Example 1

Fixed Support – Example 2






By checking these conditions, we first start with static determination.
Static determination is a quantitative condition that checks if the “shield” is immobilized. For this to happen, all the points of freedom must be taken away from the construction elements.
There is a formula for it, here it is.

n – the number of constraints
t – number of discs (“shields”)
3 – constant multiplication

The formula shows that the condition is fulfilled if each shield is accompanied by three bonds.
Time for example.

We have a beam consisting of one disc, supported on one side with an articulated joint and on the other hand articulated sliding. In total, we have one shield and three support reactions.

From the formula.n=3*t

We have three bonds (Pin = 2, Roller= 1), therefore n = 3.
The entire beam consists of one shield, so t = 1.
We get.


The condition is met!


The element that divides the beam into separate discs is only a joint.

In the above example, we can see what the wrong number of shields looks like. The fact that we see a beam as if it were a “split” by articulated joint does not mean that we have two shields here because it is missing joint in the beam.
Let’s check the condition of static determination.


The condition is unfulfilled! The system is stiffened.


Another example. We see that there is a joint in the middle of the beam, that means it separates the beam into two disks. Let’s check the condition of static determination.


The condition is met!

However, in the case of trusses, it looks a bit different, and formula looks different.
We have such a truss.

The formula looks like this.

p + r = 2*w

Number of nodes –  w
Number of bars – p
Number of reactions – r

Number of nodes – “w” = 7
Number of bars – “p” = 11
Number of reactions – “r” = 3 (I believe you know why 3)
Let’s see.

11 + 3 = 7 * 2

The above truss is statically determinable.


Geometric stability.
In simple terms, what is the difference between static determination and geometric constancy? Well, static determination is not a sufficient condition to determine whether the building will not collapse. A lot depends on how the supports are spread over the beam, frame, truss or arch. It may turn out that we calculated the given patterns and agree, but here comes geometric immutability, which checks whether the placement of supports is good.
Here is an example.

We have
– one shield t = 1,
– number of constraints n = 3,

3 = 3 * 1
3 = 3

The condition of static determination is met!

However, considering the placement of supports, we see that there is something wrong.
Loading our beam with vertical forces, nothing will happen, because the supports will balance the forces acting. However, by charging the above system with horizontal forces, the entire beam will be moved immediately, and thus the construction disaster.


Let’s check the example below.
Is the beam statically determinate and geometrically stable?

Static determination.


The condition is unfulfilled! The system is stiffened.

Geometric stability.
The location and the type of supports will take away the possibility of moving the beam, regardless of the forces we use.
The condition is met!

For correct verification of geometrical constancy, we need knowledge in the field of construction.
We must feel whether the structure with the adopted support structure will be safe.