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.
FEMethods is hosted on PyPi, so installation is simple.
pip install femethods
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 pointMomentLoad, 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)
PointLoad(magnitude=-10, location=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)
MomentLoad(magnitude=2, location=5)The location must be a positive value, and less than or equal to the length
of the beam, otherwise it raise a ValueError.
There are two different reactions that can be used to support an element.
FixedReactiondoes not allow vertical or rotational displacementPinnedReactiondoes 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.
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))
FixedReactionLocation: 3Force: NoneMoment: NoneThe location must be a positive value, and less than or equal to the length
of the beam, otherwise it raise a ValueError.
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))
PinnedReactionLocation: 7Force: NoneMoment: NoneThe location must be a positive value, and less than or equal to the length
of the beam, otherwise it raise a ValueError.
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.
This section contains several different examples of how to use the beam element, and their results.
For all examples, the following have been imported:
fromfemethods.elementsimportBeamfromfemethods.reactionsimportFixedReaction, PinnedReactionfromfemethods.loadsimportPointLoad, MomentLoadbeam_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 listp= [PointLoad(magnitude=-2, location=beam_len)] # define loads as listb=Beam(beam_len, loads=p, reactions=r, E=29e6, Ixx=125)
# an explicit solve is required to calculate the reaction valuesb.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
Magnitude: -2
REACTIONS
Type: fixed
Location: 0
Force: 2.0
Moment: 20.0
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 reactionsp= [PointLoad(magnitude=-2, location=beam_len)] # define loadsb=Beam(beam_len, loads=p, reactions=r, E=29e6, Ixx=125)
# an explicit solve is required to calculate the reaction valuesb.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
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
- 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
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