In this post you will learn what is the degree of static indeterminacy? How to determine whether a system (beam, frame, truss) is statically determinate. In addition, you will learn about the possible cases of supports that we encounter in mechanics tasks.
A static system is called such a system that is stationary. That is, it cannot move under the action of external forces. If it moves then it will be a mechanism.
The branch of mechanics in which we will deal with the motion of bodies is kinematics and dynamics. Statics is that part of mechanics in which the elements remain at rest, for this to be the case several conditions must be met.
In this entry:
- Supports. Types and symbols
- Degree of static indeterminacy
- Statically determinable systems
- Statically indeterminate systems
- Mechanisms
Supports. Types and symbols
To make the system stationary we need to support it. In mechanics assignments, you will most often encounter supports like the following. In the drawings they are marked support reactions For individual supports:
| You can find all of the aforementioned supports in the app Beam calculator Where you can test ways to support beam systems. |
We can divide static systems into two types:
- Statically determinate systems - the degree of static non-determinism of such a system is zero.
- Statically indeterminate systems - the degree of static indeterminacy of such a system is greater than zero.
Degree of static indeterminacy
To calculate the degree of static nonconvexity of a given system, you can use the formula below. This formula is best used when calculating beams i ram.
| N=R-J-3 where: N - degree of static indeterminacy R - the number of support reactions. That is, the sum of all reactions for our supports J - number of internal joints - if not present P=0 3 - the number of equilibrium equations. In static systems it is 3 |
For truss We will use the formula as follows:
| 2n = m + r where: n - number of truss nodes (nodes) m - number of truss bars (members) r - number of support reactions (reactions) |
A truss is determinable if the above relation is satisfied. That is, the doubled number of nodes must be equal to the sum of the bars and the number of reactions.
Statically determinable system
So what is this static determinability? A statically determinable system is one for which we can calculate the support reactions. For example, our beam or will be statically determinable if, using the three equilibrium equations, we can calculate all reactions. The degree of static indeterminacy in such systems is zero.
N=0
Using the formula above
N=R-J-3 => N=3 - 0 - 3 = 0
where:
R=3 - sum of all reactions for our supports
J=0 number of internal joints, not present
3 - number of equilibrium equations
And for a simple truss, the calculation will look like this:
2n = m + r => 2*5 = 7 + 3 => 10 = 10
where:
n = 5 - number of nodes
m = 7 - number of truss bars
r = 3 - number of support reactions
Statically indeterminate systems
A statically equivocal system is one for which we are unable to calculate the support reactions. The degree of static indeterminacy in such systems is greater than zero. The number of unknowns of the support reactions is greater than the number of equilibrium equations.
N > 0
where:
R=5 - sum of all reactions for our supports
J=0 - number of internal joints, not present
3 - number of equilibrium equations
The beam is statically non-determinable twice - (result N=2)
Such a system is stiff and has no possibility of working. The beam under load or temperature change has no possibility of displacement, in other words, the possibility of working.
Mechanisms
The last type of systems I will discuss are systems that have the possibility of movement - mechanisms. The degree of static indeterminacy for such systems is negative.
N < 0
N=R-J-3 => N=2-0- 3 = -1
That's the end of the static non-convexity topic, thank you and feel free to browse other posts 😊