The method of Macaulay is a method to determine force and deflection properties of structures. The method makes it possible to describe discontinuous beams with a single equation. The fact that the entire beam can be described in a single equation makes this method well-suited for use in programming. In past Bachelor End Projects students from the TU Delft have made advancements in the application of the method of
Macaulay, making it able do be used for more complicated structures.

SymPy is a Python library that has a Beam module that specialises in calculations of beams. In this
module the method of Macaulay is used for some of these calculations. The goal of this project is to use the advancements made by previous Bachelor End Projects to extend the implementation of acaulay’s method in SymPy.

This leads to the following research question: How can the current implementation of Macaulay’s method be extended in Python to be able to calculate and analyze more complicated beams and structures, based on the advancement made by previous Bachelor End Projects at the TU Delft?

The advancements made by previous Bachelor End Projects can be divided up into different subjects.
These subjects can be implemented one by one. During the course of the project there has been focus on implementing two of these subjects. The first subject is rotation and sliding hinges. The second subject is calculations of influence line diagrams. Both these subjects are completely implemented into the SymPy code.

In the implementation of rotation and sliding hinges, the mechanics calculation done by the SymPy code are changed. These hinges are now directly added to the load equation of the beam using singularity functions. The main advantage of this is that calculations on beams with hinges can be done on the beam as a whole, instead of having to cut the beam up into different parts. The fact that the beam can be calculated as a whole also makes this method able to be scaled to calculate beams with multiple hinges.

There was already an existing implementation to calculate influence line diagrams, but this implementation could be improved. In the new implementation the moving load is added to the load equation using a singularity function. The ability to apply the boundary conditions that come with hinges is also added in the new implementation. This makes the new
implementation able to calculate correct influence lines for beams with and without hinges.

There are more advancements made by previous Bachelor End Projects than implemented during this project. The advancements on some other subjects, for example normal forces or spring connections, can still be implemented into the current Beam module. For other advancements, those regarding 2D structures, it will be better if they are implemented in a new module. This new module could specialize in 2D structures, while the current module stays focused on calculations on single beams...

The Macaulay method is an integration method used to calculate the force distribution and deformations of a structure similar to the conventional integration method. However, where the conventional integration method requires a new equation for each discontinuity along the beam, the Macaulay method only requires one equation. Because of this less solution conditions are needed. Up until now, the Macaulay method was only applicable for one-dimensional structures. In this report the Macaulay method will be extended such that it is also applicable on kinked, branched and enclosed two-dimensional structures, including statically indetermined structures and trusses.

This report describes the development of a Python based tool for 2D structural analysis in civil engineering, using the SymPy library (Meurer A, 2017) and applying the Macaulay method (Macaulay, 1919) to enable symbolic computation for structural analysis. The tool serves as a proof of concept demonstrating the feasibility of symbolic computation in analyzing internal forces. Central to the project is the extension of existing SymPy modules, specifically the Beam module, to support 2D analysis by introducing new modules: "Column" and "Structure2D." The existing "Beam" module in SymPy, which computes vertical load effects, was used to handle shear forces and bending moments by creating a load equation for the vertical direction. To handle normal forces a "Column" module was proposed (although not implemented in this iteration), designed to compute horizontal load effects based on the material’s elasticity modulus, length, and cross-sectional area. The "Structure2D" module integrates both vertical and horizontal load computations by transforming a 2D structure into two corresponding 1D representations. This method enables separate computations for each direction, converting 2D problems into simpler 1D computations while preserving load and support behavior. The tool’s workflow involves users defining structure components (members, loads, supports) on a 2D grid. Once defined, the structure is "unwrapped" to map each member along a continuous line. Each load is split into its horizontal and vertical components and applied to the corresponding axis. Through examples, the report showcases the tool’s ability to calculate shear forces, bending moments, and reaction loads. Although the project successfully demonstrates the potential of symbolic computation for 2D structural analysis, several limitations exist. The tool currently lacks a functional Column module, restricting it to single horizontal reaction loads and limiting its ability to solve for horizontal deflections and axial forces fully. Additionally, the structure must be non-branching and linear, excluding complex configurations like truss networks. Material properties must be uniform across members, a constraint that simplifies the model but limits its real-world applicability. Addressing these limitations would involve developing a more advanced unwrapping algorithm for intersecting members and introducing a complete Column module. In summary, this prototype validates the possibility of a symbolic 2D structural analysis tool, providing civil engineering students and educators with an analytical framework that combines Macaulay’s method and modern symbolic computation. This tool lays the groundwork for future developments that could expand its functionality, to move away from proof of concept to a fully featured open-source product

Macaulay's method, which makes use of singularity functions to describe external forces on beam structures, is extended for two-dimensional structures. Furthermore an extension on the method of Macaulay is described for singularities in the rotation and displacement field of the structures, based on the article of Falsone, G. (2002), ‘’The Use of Generalised Functions in the Discontinuous Beam Bending Differential Equations’’ out of the International Journal of Engineering Education. Additionally the modeling of springs is added to the extension of the Macaulay's method.

The method of Macaulay utilizes singularity functions to describe discontinuous forces acting on beam structures. This method has been extended to determine internal forces and deformation properties, where the beam is inclined at a certain angle or curved.

Macaulay's method utilizes singularity functions to describe discontinue forces acting on beam structures. This method has been extended to determine internal forces and influence lines, making use of a single expression. The method can be applied to analyze all possible scenarios for both one-dimensional and two-dimensional constructions.