🏗️ Pure Python 2D Frame Analysis Engine (Zero Dependencies)

A modular, object-oriented 2D Frame Analysis software developed strictly from scratch in Python. The core objective of this project is to implement the Finite Element Method (FEM) and global matrix assembly processes without relying on any external numerical or data manipulation libraries (such as NumPy or Pandas).

🚀 Engineering Objective

Commercial structural software often acts as a "black box." This engine was built to demonstrate a fundamental, under-the-hood understanding of computational matrix mathematics, memory allocation (banded storage), and structural mechanics principles required to analyze 2D frame structures.

✨ Core Features & Software Architecture

🧮 1. Custom Matrix & Math Library

Built entirely from standard Python lists to handle all linear algebra operations.

• Matrix Class: Handles standard dense operations ($[R]^T [k] [R]$), vector multiplications, and transpositions.
• SymmetricBandedMatrix Class: A highly memory-efficient storage scheme that compresses the massive $N \times N$ Global Stiffness Matrix into an $N \times m$ array (where $m$ is the semi-bandwidth), drastically reducing RAM usage.
• Custom Solver: Implements a localized Banded Cholesky Factorization algorithm to solve the global linear system ${F} = [K]{D}$.

🧱 2. Object-Oriented Assembly (Direct Stiffness Method)

The structural analysis workflow is divided into highly decoupled, specialized modules:

• Structural Elements Module: Generates the $6 \times 6$ local stiffness matrix ($[k]$) and transformation/rotation matrices ($[T]$) based on material properties ($E, A, I$).
• Frame Analyzer Module: Automates equation numbering (masking restrained DOFs), calculates the structural semi-bandwidth dynamically, and assembles the global $[K]$ and ${F}$ matrices.

📊 3. Zero-Dependency Data Structure

To maintain a lightweight and purely algorithmic footprint, structural models (Nodal coordinates, Connectivity, Sections, Nodal Loads, and Boundary Conditions) are defined directly via 2D Python arrays (List[List[float]]) rather than external spreadsheets.

🛠️ Program Output

The execution script solves the 2D frame and outputs highly precise data to the console:

1. Equation Numbering Matrix and Active DOFs.
2. The compressed Banded Storage format of the Global Stiffness Matrix.
3. Nodal Displacements ($U_x, U_y, R_z$).
4. Member End Forces (Axial, Shear, Moment) in local coordinates.

💻 Installation & Usage

Prerequisites

• Python 3.8 or higher.
• No pip install required. (Zero external dependencies).

Running the Engine

1. Define your structural topology directly inside the execution script (Assignment2_Q2_Main.py).
2. Run the program:

   python Assignment2_Q2_Main.py
   

Proving that robust structural engineering tools can be built purely on fundamental logic and efficient algorithms.