Implementation of Finite Element Analysis

# FEmethods

## Introduction

FEmethods is a python module that uses Finite Element Methods to determine the reactions, and plot the shear, moment, and deflection along the length of a beam.

Using Finite elements has the advantage over using exact solutions because it can be used as a general analysis, and can analyze beams that are statically indeterminate. The downside of this numerical approach is it will be less accurate than the exact approach.

The official documentation is on Read the Docs.

## Installation

FEMethods is hosted on PyPi, so installation is simple.

`pip install femethods`

## General Layout

`FEMethods` is made up of several sub-classes to make it easy to define loads and reaction types.

There are currently only two different load types that are implemented.

• `PointLoad`, a normal force acting with a constant magnitude on a single point
• `MomentLoad`, a rotational moment acting with a constant magnitude acting at a single point

All loads are defined by a `location` along the element, and a `magnitude`. The `location` must be positive, and must lie on the length of the beam, or it will raise a `ValueError`

Future goals are to add a library of standard distributed loads (constant, ramp, etc) as well as functionality that will allow a distributed load function to be the input.

The `PointLoad` class describes a standard point load. A normal load acting at a single point with a constant value. It is defined with a location and a magnitude.

```>>> PointLoad(-10, 5)
```

The `location` must be a positive value, and less than or equal to the length of the beam, otherwise it raise a `ValueError`.

A `MomentLoad` class describes a standard moment load. A moment acting at a single point with a constant value. It is defined with a location and a value.

```>>> MomentLoad(2, 5)
```

The `location` must be a positive value, and less than or equal to the length of the beam, otherwise it raise a `ValueError`.

### femethods.reactions

There are two different reactions that can be used to support an element.

• `FixedReaction` does not allow vertical or rotational displacement
• `PinnedReaction` does not allow vertical displacement but does allow rotational displacement

All reactions have two properties, a `force` and a `moment`. They represent the numerical value for the resistive force or moment acting on the element to support the load(s). These properties are set to `None` when the reaction is instantiated (ie, they are unknown). They are calculated and set when analyzing a element. Note that the `moment` property of a `PinnedReaction` will always be `None` because it does not resist a moment.

The `value` property is a read-only combination of the `force` and `moment` properties, and is in the form `value = (force, moment)`

All reactions have an `invalidate` method that will set the `force` and `moment` back to `None`. This is useful when changing parameters and the calculated reactions are no longer valid.

#### femethods.reactions.FixedReaction

The `FixedReaction` is a reaction class that prevents both vertical and angular (rotational displacement). It has boundary conditions of `bc = (0, 0)`

```>>> FixedReaction(3)
FixedReaction(location=3)

>>> print(FixedReaction(3))
FixedReaction
Location: 3
Force: None
Moment: None
```

The `location` must be a positive value, and less than or equal to the length of the beam, otherwise it raise a `ValueError`.

#### femethods.reactions.PinnedReaction

The `PinnedReaction` is a reaction class that prevents vertical displacement, but allows angular (rotational) displacement. It has boundary conditions of `bc = (0, None)`

```>>> PinnedReaction(7)
PinnedReaction(location=7)
>>> print(PinnedReaction(7))
PinnedReaction
Location: 7
Force: None
Moment: None
```

The `location` must be a positive value, and less than or equal to the length of the beam, otherwise it raise a `ValueError`.

### femethods.elements.Beam

Defines a beam as a finite element. This class will handle the bulk of the analysis, populating properties (such as meshing and values for the reactions).

To create a `Beam` object, write the following:

```b = Beam(length, loads, reactions, E=1, Ixx=1)
```

Where the loads and reactions are a list of `loads` and `reactions` respectively.

Note Loads and reactions must be a list, even when there is only one.

The `E` and `Ixx` parameters are Young's modulus and the polar moment of inertia about the bending axis. They both default to `1`.

## Examples

This section contains several different examples of how to use the beam element, and their results.

For all examples, the following have been imported:

```from femethods.elements import Beam
from femethods.reactions import FixedReaction, PinnedReaction
```

```beam_len = 10
# Note that both the reaction and load are both lists. They must always be
# given to Beam as a list,
r = [FixedReaction(0)]                            # define reactions as list

b = Beam(beam_len, loads=p, reactions=r, E=29e6, Ixx=125)

# an explicit solve is required to calculate the reaction values
b.solve()
print(b)
```

The output of the program is

``````PARAMETERS
Length (length): 10
Young's Modulus (E): 29000000.0
Area moment of inertia (Ixx): 125
Location: 10
Magnitude: -2

REACTIONS
Type: fixed
Location: 0
Force: 2.0
Moment: 20.0
``````

### Example 2: Cantilevered Beam with 3 Pinned Supports and End Loading

```beam_len = 10

# Note that both the reaction and load are both lists. They must always be
# given to Beam as a list,
r = [PinnedReaction(0), PinnedReaction(2), PinnedReaction(6)]  # define reactions

b = Beam(beam_len, loads=p, reactions=r, E=29e6, Ixx=125)

# an explicit solve is required to calculate the reaction values
b.solve()
print(b)
```

The output of the program is

``````PARAMETERS
Length (length): 10
Young's Modulus (E): 29000000.0
Area moment of inertia (Ixx): 125
Location: 10
Magnitude: -2

REACTIONS
Type: pinned
Location: 0
Force: 1.3333333333333346
Moment: 0.0
Type: pinned
Location: 2
Force: -4.000000000000004
Moment: 0.0
Type: pinned
Location: 6
Force: 4.666666666666671
Moment: 0.0
``````

## TODO

• Add a more thorough documentation for all the features, limitations and FE fundamentals for each section

## Acknowledgements

Derivation of stiffness matrix for a beam by Nasser M. Abbasi An idiot’s guide to Python documentation with Sphinx and ReadTheDocs by Sam Nicholls for a very helpful guide on how to get sphinx set up

## Project details

This version 0.1.7a2 pre-release 0.1.6.dev0 pre-release 0.1.5.dev0 pre-release 0.1.4.dev0 pre-release 0.1.3.dev0 pre-release 0.1.2.dev0 pre-release 0.1.dev0 pre-release