## How to determine whether a structure (beam, frame, truss) is statically determinate. Additionally, you will learn about possible cases of supports that we encounter in mechanics tasks.

A static structure is a structure that is stationary. This means that it cannot move under the influence of external forces. If it moves, it will be a mechanism.

The branch of mechanics in which we deal with the motion of bodies is kinematics and dynamics. **Statics** is the part of mechanics in which the elements remain at rest, and for this to be the case, several conditions must be met.

In this entry:

**Supports. Types and symbols**

For the structure to be stationary, we must support it. In mechanics tasks, you will most often encounter the following supports. The drawings show **support reactions** and symbols for individual supports:

All the supports mentioned above can be found in the application |

Static arrangements can be divided into two types:

Statically determinate structure - the degree of static indeterminacy of such a structure is zero.

Statically indeterminate structure - the degree of static indeterminacy of such a structure is greater than zero.

**Degree of static indeterminacy **

To calculate the degree of static indeterminacy of a given structure, you can use the formula below. This formula is best used when calculating **beams** and **frames**.

*N = R - J - 3*

where:

N - degree of static indeterminacy

R - number of support reactions. That is, the sum of all reactions for our supports

J - number of internal joints - if there are none P=0

3 - number of equilibrium equations. In static structure it is 3

For **truss** we will use the formula below:

*2w = p + r*

where:

w - number of truss nodes

p - number of truss bars

r - number of support reactions

A truss is determinate if the above relationship is satisfied. So twice the number of nodes must be equal to the sum of the bars and the number of reactions.

**Statically determinate structures **

A statically determinate structure is a structure for which we can calculate support reactions. For example, our beam or will be statically determinate if we are able to calculate all reactions using three equilibrium equations. The degree of static indeterminacy in such structure 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, none

3 - number of equilibrium equations

And for a simple truss, the calculation will look like this:

*2w = p + r => 2*5 = 7 + 3 => 10 = 10*

where:

w = 5 - number of nodes

p = 7 - number of links

r = 3 - number of support reactions

**Statically indeterminate structure**

A statically indeterminate structure is a structure for which we are unable to calculate the support reactions. The degree of static indeterminacy in such structure is greater than zero. The number of unknown 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, none

3 - number of equilibrium equations

The beam is twice statically indeterminate - ( result=2)

Such a structure is too rigid and cannot work. Under the influence of load or temperature change, the beam cannot move, or in other words, cannot work.

**Mechanisms**

The last type of structure that I will discuss are structures that can move - mechanisms. The degree of static indeterminacy for such structure is negative.

N < 0

N=R-J-3 => N=2 - 0 - 3 = -1

That's the end of the topic of static indeterminacy, thank you and I invite you to browse other entries 😊

PS. All beam images used in the post were generated in beam solver.

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