Implementation of Finite Element Analysis

# FEMethods

## Introduction

FEMethods is a 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.

## 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 force acting with a constant magnitude acting at a single point

All loads are defined by a location along the element, and a value (magnitude). The location must be positive 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 value (magnitude).

>> PointLoad(5, -10)


The location must be a positive value, otherwise it will 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, otherwise it will 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 resisit 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)
Fixed Reaction
Location: 3
Force: None
Moment: None


The location must be a positive value, otherwise it will 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)
Pinned Reaction
Location: 7
Force: None


The location must be a positive value, otherwise it will raise a ValueError.

### femethods.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.beam import Beam
from femethods.reactions import FixedReaction, PinnedReaction


beam_len = 10

reactions = [FixedReaction(0)]     # define reactions

b = Beam(beam_len, loads, reactions, E=1, Ixx=1)
b.get_reaction_values()
print(b)
b.plot()


The output of the program is

PARAMETERS
Length (length): 10
Young's Modulus (E): 1
Area moment of inertia (Ixx): 1
Location: 10
Value: -2
REACTIONS
Type: fixed
Location: 0
Force: 2.0
Moment: 19.999999999999986


beam_len = 10

reactions = [PinnedReaction(0), PinnedReaction(2), PinnedReaction(6)]     # define reactions

b = Beam(beam_len, loads, reactions, E=1, Ixx=1)
b.get_reaction_values()
print(b)
b.plot()

PARAMETERS
Length (length): 10
Young's Modulus (E): 1
Area moment of inertia (Ixx): 1
Location: 10
Value: -2

REACTIONS
Type: pinned
Location: 0
Force: 1.3333333333333321
Moment: 0.0
Type: pinned
Location: 2
Force: -3.9999999999999973
Moment: -8.881784197001252e-16
Type: pinned
Location: 6
Force: 4.666666666666664
Moment: -3.552713678800501e-15


## TODO

• Add a more thorough documentation for all the features, limitations and FE fundamentals for each section
• Add a general solve function for elements that will define all unknowns (nodal displacements, reaction forces)

## Acknowledgements

Derivation of stiffness matrix for a beam

## Project details

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