The idea in this article originated from my past civil engineering project, particularly when I was assigned for determining the foundation design of a tower crane structure used in my project. In order to do that, engineers typically compute the support reaction of the tower crane, then using the reaction forces to proceed with the design and analysis of the foundation. The common approach is; either modelling the structure of the tower crane—which is typically done using commercial software, or alternatively using a manual calculation which can be done with hand-writing or spreadsheet. The former is much more rigorous—providing reliability and high accuracy, faster, but yet quite expensive. The later is oversimplified—causing less-reliability and less accuracy, potentially taking longer time but relatively cheap.

To answer the question above, I propose to develop a Python module so that we can automate this task with a considerable rigor but is also very cheap—requiring only cognitive thinking, mathematics and coding skills. And the main strength of this approach is that the module is reproducible, similar to a commercial software. So, the next time we need to compute the support reaction of another tower crane, we can directly use the module.

I am always fascinated with developing the underpinning mathematical framework of the program that I will later develop. It is based on a few considerations: First, it gives a clear foundation to the program, grounded in mathematics. Second, it eases the algorithm development and coding process as a guide. In addition, the emergence bridge between pure mathematics and the applied problem presented by the mathematical framework adds elegance and a philosophical implication.

This article emphasizes the mathematical structure that emerges from my development in the mathematical framework of the Python module for computing the tower crane support reaction. In particular, I use linear algebra for the model and a little functional analysis. Notably I present a bilinear map $\diamond_q$ which maps the product of $\mathbb{R}^q$ with a $q$-direct sum of 3D Eculidean spaces to a 3D Euclidean space, for any $q \in \mathbb{N}$. The map $\diamond_q$ may be equivalent to another established map, and it may be defined via the classical algebraic operation. It is analogous to an inner product, but is different in the traditional sense since the domain consists of different vector spaces. With $\diamond_q$, I can express the application of a load combination in the system as a bounded linear operator on the $q$-direct sum of 3D Euclidean spaces. I also present another map $\overline{\boxtimes}$ which maps the product of a 3D Euclidean space with a $q$-direct sum of 3D Euclidean spaces to the $q$-direct sum via the classical cross product to integrate the moments in the system. The map $\overline{\boxtimes}$ is also bilinear, and it also possesses the antisymmetric property inherited from the cross product in the form of its counterpart $\underline{\boxtimes}$, which is also a bilinear map. I further investigate properties of this linear algebraic model.

Ultimately, the determination of the forces and moments at the support reaction of a TC factored by some load combinations can be reduced to simple matrix multiplications. This result will significantly ease the computational process of the entire system.