Calculation Documentation

1. General Finite Element (FE) Calculations

1.1. Modeling Approach

The core analysis engine utilizes the Direct Stiffness Method (DSM) for 2D frame elements based on Euler-Bernoulli beam theory. The structure is discretized into a series of nodes and beam elements. Each node in the 2D plane possesses three degrees of freedom (DOFs):

• Translation in the global X-direction ($u_x$)
• Translation in the global Y-direction ($u_y$)
• Rotation about the Z-axis ($\theta_z$)

1.2. Stiffness Matrices

For every element, a $6 \times 6$ local stiffness matrix $[\mathbf{k}_e]$ is formulated accounting for axial ($A, E$) and flexural ($I$) rigidities. Using a rotation transformation matrix $[\mathbf{T}]$, the local stiffness matrix is transformed into the global coordinate system: $$[\mathbf{K}_e] = [\mathbf{T}]^T [\mathbf{k}_e] [\mathbf{T}]$$ These element matrices are subsequently assembled into the global stiffness matrix $[\mathbf{K}]$ representing the entire structure.

1.3. Loading & Solving

Applied point forces and moments are directly assigned to the nodal force vector $\{\mathbf{F}\}$. For distributed loads along beam spans, they are converted into Equivalent Nodal Forces (ENF) using fixed-end moment and shear reaction formulations, which are applied to the boundary nodes of the affected elements.

Boundary conditions (supports) are enforced using a penalty method or matrix reduction approach. The resulting system of linear equations is solved to obtain the nodal displacements $\{\mathbf{U}\}$: $$[\mathbf{K}]\{\mathbf{U}\} = \{\mathbf{F}\}$$

With the global displacements known, they are transformed back into local element displacements. Finally, the internal forces (Axial $N$, Shear $V$, Bending Moment $M$) at the element ends are extracted by multiplying the local displacement vector by the local stiffness matrix and subtracting the equivalent nodal forces.

2. Influence Lines Generation

2.1. Definition

An Influence Line illustrates the variation of a specific structural response (e.g., a support reaction, internal shear force, or bending moment at a fixed target point) as a moving concentrated unit load traverses across the structure.

2.2. Computational Algorithm

Rather than relying on the kinematic Müller-Breslau principle, this application utilizes a robust Moving Unit Load (Brute Force) evaluation. This guarantees geometric accuracy independent of internal hinge kinematics. The calculation sequence is as follows:

1. Path Extraction: The algorithm identifies all connected horizontal and angled beam elements in the model to formulate a continuous, walkable path length ($L_{total}$).
2. Discretization: The path is subdivided into 1000 evaluation steps ($n = 1000$).
3. Iteration: A virtual downwards vertical load of magnitude $P = 1.0$ is injected into the model at each step position $x_i$.
4. Analysis: The exact same FE Solver described in section 1 is executed independently for each of the 1000 load positions.
5. Data Mining: After each solve, the requested feature (e.g., reaction $R_y$ at Support 'A', or moment $M$ at $2.5m$ on Beam 'B') is precisely extracted holding the surrounding model state constant.
6. Plotting: The 1000 extracted values $\{v_0, v_1, ..., v_{1000}\}$ are mapped to their corresponding geometric coordinates $(x_i, y_i)$ on the canvas, taking into account outward normals for perpendicular drawing on skewed beams.