# What are degrees of freedom in mechanics and civil engineering?

When solving plane structures, such as trusses, frames and beams, the first step is always to determine the total number of degrees of freedom.

This is extremely important, because it allows us to understand whether our structure is statically determinate or indeterminate.

Indeterminate structures, such as continuous beams, are much more difficult to solve because the system of equilibrium equations has more unknowns than equations. In such cases, boundary conditions must be taken into account to find more equations to add to the system.

## What are degrees of freedom?

A 2d member laying on a plane can move in three distinct ways:

- It can translate along the X axis;
- It can translate along the Y axis;
- It can rotate.

So, for plane structures, every member (beams, trusses, etc) has three distinct ways to move, i.e. three
**degrees of freedom**.

Members are connected together by **joints**, which restrain or take away some of the degrees of freedom of the
connected members.

Joints (often called nodes) fall into two categories:
**supports** and **hinges**.

A **support** works by constraining the beams connected to it with respect to the plane. A pin, for example, is a
type of support that allows members connected to it to rotate, but prevents them from translating.

In other words, the vertical and horizontal degrees of freedom are constrained, while the rotational degree of freedom is unconstrained. Because they constrain the structure with the overall plane, supports are also called external constraints.

A **hinge** is usually a joint that forces the beams connected to it to have the same translational displacement, but
allows them to rotate freely.

You can think of it like a sort of “internal pin”. This is the type of joint that truss structures use. Because hinges define how beams interact with one another, they are also called internal constraints.

Tip

An easy way to remember the distinction between supports and hinges is to think of supports as defining how the beams are attached to the ground, and hinges as defining how the beams are attached to one another.

## What are the types of support?

A support can “lock” the beams connected to it by blocking one or more of the three degrees of freedom in the plane. This means that there are 5 possible support types:

icon | name | r_{x} |
r_{y} |
r_{r} |
---|---|---|---|---|

Fixed | ||||

Pin | ||||

Skater | ||||

Roller | ||||

Fixed rotation |

## What are the types of internal constraint?

By far, the most common type of internal constraint is the hinge. This is the type of joint used in truss structures. But it doesn’t end there: you could have a structure where a beam is supported by another beam with an internal roller. When you have a lot of members connected to a joint, it can get pretty complicated.

## How can I calculate the number of degrees of freedom of a structure?

You can easily calculate the number of degrees of freedom of any structure by multiplying the number of members by 3. So, if \(n_b\) is the number of members, then \[n_{dof}=3\cdot n_b\]

You can think of a member as a continuous “chunk” of a structure. Here are some examples:

## How can I calculate if a structure is statically determinate, indeterminate or if it is a mechanism?

Once you know the number of degrees of freedom \(n_{dof}\), then you can calculate the degree of indeterminacy by counting the number of restraints \(n_r^{tot}\). The degree of indeterminacy is given by the following formula: \[f=n_{dof}-n_r^{tot}\]

- If \(f=0\) the structure is statically determinate and stable;
- If \(f < 0\) the structure is indeterminate to the degree of \(f\);
- If \(f > 0\) the structure is a mechanism with f degrees of freedom.

But how do we calculate the number of restraints \(n_r^{tot}\)? It’s a two-step process.

### Step 1: calculate the number of external restraints

That is, the restraints given by the supports. The next table shows how many restraints each support type has:

icon | name | \(n_r\) |
---|---|---|

Fixed | 3 | |

Pin | 2 | |

Skater | 2 | |

Roller | 1 | |

Fixed rotation | 1 |

You simply have to consider every support your structure has and add the numbers together. This will become much clearer once we go through some practical examples.

### Step 2. Calculate the number of internal restraints

The joints where two or more beams connect together also contribute to the total number of restraints of a structure. The total number of members attached to a joint also plays a role in this case, as you can see in the next table

icon | name | \(n_r\) |
---|---|---|

Hinge | \(2\cdot (n_b-1)\) | |

Int. skater | \(2\cdot (n_b-1)\) | |

Int. roller | \(1\cdot (n_b-1)\) | |

Continuous moment | \(1\cdot (n_b-1)\) |

So, for every joint where two or more members connect, we have to count the number of restraints and add it to the total number of restraints \(n_r\).

Let’s go through some examples.

## Example 1

Calculate the determinacy of the following structure:

The structure has one member, so the number of degrees of freedom is: \[n_{dof}=3\cdot 1=3\] First, we calculate the number of external restraints:

Joint | \(n_r\) |
---|---|

1 | 2 |

2 | 1 |

3 | 1 |

So \[n_r^{ext}=4\] Since there are no hinges where more than one member meet, the contribution of the internal restarints is 0: \[n_r^{int}=0\]

The total number of restraints therefore is \[n_r^{tot}=n_r^{ext}+n_r^{int}=4\] The structure is indeterminate to the first degree.

## Example 2

Calculate the determinacy of the following structure:

The structure has two members, so the number of degrees of freedom is: \[n_{dof}=3\cdot 2=6\] First, we calculate the number of external restraints:

Joint | \(n_r\) |
---|---|

1 | 2 |

2 | 1 |

3 | 1 |

So \[n_r^{ext}=4\] Next we need to calculate the contribution given by the hinge in joint 2: \[n_{r,2}^{int}=n_r^{int}=2\cdot(2-1)=2\]

The total number of restraints therefore is \[n_r^{tot}=n_r^{ext}+n_r^{int}=4+2=6\] The structure is statically determinate.

## Example 3

Calculate the determinacy of the following structure:

The structure has two members, so the number of degrees of freedom is: \[n_{dof}=3\cdot 2=6\] First, we calculate the number of external restraints:

Joint | \(n_r\) |
---|---|

1 | 3 |

3 | 2 |

So \[n_r^{ext}=5\] Next we need to calculate the contribution given by the internal roller in joint 2: \[n_{r,2}^{int}=n_r^{int}=1\cdot(2-1)=1\]

The total number of restraints therefore is \[n_r^{tot}=n_r^{ext}+n_r^{int}=5+1=6\] The structure is statically determinate.

## Example 4

Calculate the determinacy of the following structure:

The structure has two members, so the number of degrees of freedom is: \[n_{dof}=3\cdot 2=6\] First, we calculate the number of external restraints:

Joint | \(n_r\) |
---|---|

1 | 3 |

3 | 1 |

So \[n_r^{ext}=4\] Next we need to calculate the contribution given by the hinge in joint 2: \[n_{r,2}^{int}=n_r^{int}=2\cdot(2-1)=2\]

The total number of restraints therefore is \[n_r^{tot}=n_r^{ext}+n_r^{int}=4+2=6\] The structure is statically determinate.