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. Assembly & Solution

Each element is assigned a $6 \times 6$ local stiffness matrix $[\mathbf{k}_e]$ comprising axial ($EA$) and flexural ($EI$) terms, rotated into global coordinates via the transformation matrix $[\mathbf{T}]$: $$[\mathbf{K}_e] = [\mathbf{T}]^T [\mathbf{k}_e] [\mathbf{T}]$$ and assembled into the global stiffness matrix $[\mathbf{K}]$. Point forces and moments enter the nodal load vector $\{\mathbf{F}\}$ directly; distributed loads are converted to equivalent nodal forces from the fixed-end shear and moment solutions of the loaded spans.

After imposing support conditions on the constrained degrees of freedom, the linear system $$[\mathbf{K}]\{\mathbf{U}\} = \{\mathbf{F}\}$$ is solved for the nodal displacements $\{\mathbf{U}\}$. Element end forces (axial $N$, shear $V$, bending moment $M$) follow from back-transforming the displacements to local coordinates: $\{\mathbf{f}\} = [\mathbf{k}_e]\{\mathbf{u}\} - \{\mathbf{f}_{eq}\}$, where $\{\mathbf{f}_{eq}\}$ are the equivalent nodal forces of the element.

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. 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.

3. Hints & Tips

3.1. Pasting and Scaling Images

You can trace over images by pasting them directly into the sketcher on the canvas. Simply use Ctrl+V (or Cmd+V on Mac) to paste an image from your clipboard straight onto the canvas. Once placed, click the image to select it, and use the grab corners to dynamically scale it or the scale function where you can pin point two points on the image and set the known length. The image will rescale accordingly on the canvas!

3.2. Precisely Placing Loads

In order to place a load at an exact distance along a beam simply hover on the start or end of that beam and a grey dashed line will appear when moving the cursor. The drag the cursor in the direction of the distance and type in your distance and press enter.