In 3D-raamwerkprogramma’s wordt wringing vaak gemodelleerd volgens de theorie van Saint-Venant, waarbij wordt uitgegaan van vrije welving. Welving is het fenomeen waarbij de dwarsdoorsnede van een profiel door torsie uit zijn vlak vervormt. In de praktijk wordt welving echter vaak verhinderd door constructieve verbindingen of opleggingen, wat leidt tot extra interne spanningen in het element. Dit rapport behandelt de afleiding en de validatie van de 3D element-stijfheidsmatrix voor dubbel symmetrische profielen, waarbij verhinderde welving correct wordt gemodelleerd volgens de theorie van Vlasov.
De resulterende 14 × 14 matrix is analytisch afgeleid en succesvol geïmplementeerd in een symbolisch Python-script met SymPy. Validatie met diverse standaardbelastinggevallen bevestigt dat het model het elementgedrag correct beschrijft. De resultaten tonen aan dat het meenemen van verhinderde welving leidt tot een hogere torsiestijfheid in vergelijking met de theorie van Saint-Venant. Omdat berekende vervormingen hierdoor kleiner uitvallen, kan het toepassen van dit model overdimensionering van constructies voorkomen. Het ontwikkelde script biedt bovendien een waardevol controlemiddel om de output van commerciële software onafhankelijk te kunnen verifiëren.
Voor vervolgonderzoek wordt aanbevolen de afleiding uit te breiden naar asymmetrische profielen, waarbij axiale rek, buiging en wringing niet langer onafhankelijk van elkaar zijn. ... Read more

The method of Macaulay makes use of singularity functions to describe the integrations in the Euler-Bernoulli beam equation in a single equation, instead of splitting the structure into parts with equal load situations. In this report, an expansion for this method is derived, which makes it possible to solve structures with multiple stiffnesses and varying stiffness slopes using this method as well. ... Read more

This report has been written as part of the final assignment for the Bachelor’s program in Civil Engi-neering. I chose this project to combine and build upon my two favorite subjects from the program: structural mechanics and object-oriented programming in Python, which I explored further through a minor in computer science. I was also excited to gain experience with contributing to open-source Python projects via GitHub.

First and foremost, I would like to thank Tom van Woudenberg for supervising my project, particularly on the GitHub side. His expertise in GitHub workflows gave me the structure and confidence needed to contribute efficiently to SymPy. Furthermore, I am also grateful to Fransesco Messali for his thourough feedback on the report itself; his guidance on improving clarity for readers less familiar with the subject matter was especially valuable. Lastly, I would like to thank Sai Udayagiri from Google Summer of Code for his feedback on both the code and GitHub practices.

Lucas Verlaan
Delft, June 2025 ... Read more

The design of 2D rectangular steel trusses demands a critical balance between structural performance and constructability, a core challenge in civil engineering. Using distinct cross-sectional profiles minimizes material use but elevates structural complexity, whereas standardized profiles facilitate construction simplicity at the cost of efficiency. This thesis develops a multi-objective optimization framework to navigate these trade-offs, targeting four essential objectives: mass minimization to reduce material requirements, connection degree to simplify joint configurations, symmetry to enhance aesthetics and standardization, and beam continuity to streamline assembly processes. By controlling the number of unique HEA profiles, the study delivers tailored solutions for preliminary structural design, aligning with engineering priorities and stakeholder preferences to optimize truss performance and practicality.
A computational framework employs the Tree-structured Parzen Estimator (TPE), a sample-efficient Bayesian optimization method, to efficiently explore the complex, discrete design space of truss configurations. TPE performance is rigorously validated against exhaustive search (EXS) to ensure accuracy in identifying optimal designs. Stakeholder-defined weights, implemented through weighted scalarization, enable customized trade-off analyses, though without direct stakeholder engagement. This approach supports the exploration of diverse configurations, effectively balancing performance and standardization while addressing the computational demands of large search spaces, thus providing a robust tool for 2D truss optimization.
The findings indicate that intermediate profile grouping often produces designs that balance structural performance and constructability. The multi-parallel plot, a dynamic visualization tool, potentially empowers stakeholders, including engineers and project managers to transparently explore trade-offs, pending practical validation. Despite limitations, such as untuned TPE hyperparameters and a focus on 2D trusses, this promising framework enhances transparency and adaptability in preliminary structural design. By integrating efficient optimization with intuitive visualization, the study establishes a foundation for future advancements in steel truss optimization, offering a versatile methodology with potential to inform broader structural engineering applications. ... Read more

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...
... Read more

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. ... Read more

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 ... Read more

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. ... Read more

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. ... Read more

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. ... Read more