Lucas: Axial Force Analysis in SymPy

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Lucas: Axial Force Analysis in SymPy#

In this report, the development and implementation of the new column module in SymPy is described.
This module models and solves for axial force problems in one-dimensional structural elements. This
implementation is based on Macaulay’s method, which uses singularity functions to express the Euler-
Bernoulli differential equations as a single piecewise expression. These functions are well-suited to
symbolic manipulation and integration within SymPy, a Python library for symbolic mathematics and
physics.

The column module addresses a limitation in the existing Structure2D module, which is also part of
SymPy’s continuum mechanics framework. Structure2D enables modeling of two-dimensional structures
with members in both the x and y directions. Internally, these structures are ”unwrapped” into a
one-dimensional system. While a ’beam’ module is already used to solve for shear forces and bending
moments, axial force support in Structure2D is currently limited. Only one horizontal reaction force can
be solved, making use of a simple sum of horizontal loads. The column module provides the foundation
for extending this functionality to include symbolic treatment of multiple axial loads and reactions,
similar to how the beam module supports transversal loads.

The column module can be used by initializing a column object with its length, elastic modulus and crosssectional
areas. Users can apply supports and loads, including distributed loads, as well as define
telescope hinges which allow discontinuities in displacement. The singularity functions of the supports,
loads and hinges are added to a continuous load equation. The unknowns in the load equation are
solved with force and deflection boundary conditions. At the supports, the deflection is zero and at the
hinges the axial force is zero. Due to the load expression needing to be integrated twice to solve for
displacements, two integration constants also have to be solved. The remaining equations are provided
by boundary conditions for absence of axial forces just outside of the element.

The system of equations is solved symbolically using a linear solving function. The unknown forces
and displacements are automatically replaced into the load equation. From this solved load equation,
the singularity expressions for the axial force and deflection along the column can be set up and plotted.
The column module has been tested for multiple different use cases with different load types and
telescope hinges. The results have been compared with those from Matrixframe, showing that the they
are correct.

While the current implementation offers core functionality for analyzing axially loaded elements, further
improvements can be made by developers in the future. These include enhancements such as
improved visualization, more flexible boundary conditions and inclusion of variable cross-sectional properties.
While the column module is a standalone feature, it should also be adjusted to comply with future
integration into the Structure2D module.

Verlaan (2025)

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