Implementation of Finite Element Analysis
Project description
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.
Installation
FEMethods is hosted on PyPi, so installation is simple.
pip install femethods
General Layout
FEMethods
is made up of several subclasses to make it easy to define loads and reaction types.
femethods.loads
There are currently only two different load types that are implemented.
PointLoad
, a normal force acting with a constant magnitude on a single pointMomentLoad
, 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.
femethods.loads.PointLoad
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(10, 5)
PointLoad(value=10, location=5)
The location
must be a positive value, otherwise it will raise a ValueError
.
femethods.loads.MomentLoad
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)
MomentLoad(value=2, location=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 displacementPinnedReaction
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 readonly 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, 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)
PinnedReaction(location=7)
>>> print(PinnedReaction(7))
PinnedReaction
Location: 7
Force: None
Moment: None
The location
must be a positive value, otherwise it will 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
from femethods.loads import PointLoad, MomentLoad
Example: Cantilevered Beam with Fixed Support and End Loading
beam_len = 10
r = [FixedReaction(0)] # define reactions
p = [PointLoad(value=2, location=beam_len)] # define loads
b = Beam(beam_len, loads=p, reactions=r, E=29e6, Ixx=125)
b.solve()
print(b)
b.plot()
The output of the program is
PARAMETERS
Length (length): 10
Young's Modulus (E): 29000000.0
Area moment of inertia (Ixx): 125
LOADING
Type: point load
Location: 10
Value: 2
REACTIONS
Type: fixed
Location: 0
Force: 2.0
Moment: 20.0
Example: Cantilevered Beam with 3 Pinned Supports and End Loading
beam_len = 10
r = [PinnedReaction(0), PinnedReaction(2), PinnedReaction(6)] # define reactions
p = [PointLoad(value=2, location=beam_len)] # define loads
b = Beam(beam_len, loads=p, reactions=r, E=29e6, Ixx=125)
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
LOADING
Type: point load
Location: 10
Value: 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
 Add additional element types, such as the bar element
Acknowledgements
Derivation of stiffness matrix for a beam by Nasser M. Abbasi
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