Beam Calculator

DEMONSTRATION SOFTWARE ONLY

This Beam Calculator tool is provided solely for educational and demonstration purposes. It is NOT intended for use in actual structural engineering design, analysis, or decision-making processes. Under no circumstances should the results from this tool be used to design, evaluate, or approve any real-world structural component, connection, or system.

NO WARRANTY OR GUARANTEE

This software is provided "as is" without any warranties of any kind, either express or implied, including but not limited to warranties of merchantability, fitness for a particular purpose, accuracy, completeness, or non-infringement.

NOT FOR PROFESSIONAL USE

This tool has not been validated, verified, or certified for use in professional engineering practice.

LIMITATION OF LIABILITY

In no event shall the authors, developers, contributors, or any affiliated parties be liable for any direct, indirect, incidental, special, exemplary, or consequential damages arising in any way out of the use of this software.

USER RESPONSIBILITY

By using this tool, you acknowledge that: (1) this is demonstration software only; (2) you will not use the results for any actual engineering design; (3) you assume all risk associated with any use of this tool.

Overview

This tool performs 3D finite element analysis of beam structures using the Rhapson Core engine. Unlike simple hand calculations with tabulated formulas, this implementation uses the matrix displacement method to solve the full system of equilibrium equations, making it suitable for complex loading patterns, multiple spans, and various support conditions.

Finite Element Model

The beam is discretized into 3D Euler-Bernoulli beam elements with 12 degrees of freedom per element (6 per node):

• 3 translations (u_x, u_y, u_z)
• 3 rotations (θ_x, θ_y, θ_z)

Each element captures axial deformation, bending in two planes, and torsion. The element stiffness matrices are assembled from section properties (EA, EI_y, EI_z, GJ) and transformed into the global coordinate system using direction cosines.

Load Handling

The engine supports multiple load types and converts them to consistent nodal loads:

• Distributed loads: Converted to fixed-end forces and moments using exact integration. For a uniform load w on a span of length L, this generates shear forces wL/2 and moments wL²/12 at each end.
• Partial UDLs: Loads over a portion of the span are handled by computing the equivalent fixed-end forces for that segment.
• Point loads: Applied as concentrated forces at specific locations, with shape functions determining the distribution to nodal DOFs.
• Point moments: Applied directly to rotational DOFs.

Solution Method

For beam analysis, the solver operates in linear mode (single load step, no geometric stiffness) since most beams undergo small deformations. The equilibrium equation is:

K × u = f

Where K is the global stiffness matrix, u is the displacement vector, and f is the applied load vector. The system is solved using sparse LU factorization via the CSparse library, which efficiently handles the banded structure of frame stiffness matrices.

Boundary Conditions

Supports are enforced through DOF partitioning—the stiffness matrix and load vector are partitioned into free and prescribed DOFs. Only the free DOFs are solved, making the system smaller and ensuring exact satisfaction of support constraints. Support types include:

• Pin: Translations fixed, rotations free (restrains u_x, u_y, u_z)
• Roller: Only vertical translation fixed (restrains u_y)
• Fixed: All DOFs restrained (restrains all 6 DOFs)

Load Cases and Combinations

The engine supports multiple load cases analyzed independently, then combined using load factors:

• Load cases: Each case (Dead, Live, Wind, etc.) is analyzed separately with its own loads
• Combinations: Results are superposed using user-defined factors (e.g., 1.2D + 1.6L for LRFD)
• Envelopes: Maximum and minimum values across all combinations are tracked for design

Post-Processing

After solving for displacements, the engine computes design quantities at stations along each span:

• Reactions: Computed by equilibrium: R = K_constrained × u - f_fixed-end
• Shear force V(x): Recovered from element forces, including contributions from distributed loads
• Bending moment M(x): Computed from end moments and load effects, ensuring equilibrium at every station
• Deflection δ(x): Interpolated from nodal displacements using cubic Hermite shape functions

Diagrams are sampled at 100 stations per span to capture smooth curves and accurately locate maximums.

The Rhapson Core Engine

While this beam calculator uses linear analysis, it's powered by the Rhapson Core library—a comprehensive nonlinear frame analysis engine. Rhapson Core implements:

• Newton-Raphson iteration with incremental load control for nonlinear problems
• P-Delta (geometric stiffness) for second-order effects in slender members
• Line search backtracking for robustness near instability
• Member end releases via static condensation for internal hinges
• Modified Newton methods with tangent stiffness caching

The engine has been validated against OpenSees for nonlinear frame analysis, achieving machine-precision agreement (~10^-10 cumulative error). Methods follow McGuire's Matrix Structural Analysis, Cook's Finite Element Analysis, and Crisfield's Non-linear FEA of Solids and Structures.

Why Finite Element Method?

Traditional beam formulas from textbooks only work for simple cases (single span, uniform load, specific supports). The FEM approach handles:

• Multiple spans with varying properties
• Complex loading patterns (multiple point loads, partial UDLs, moments)
• Any combination of support types
• Exact solutions (within numerical precision) rather than approximate formulas
• Easy extension to 3D frames, nonlinear analysis, and dynamic problems

Technical Details

Element theory: Euler-Bernoulli (no shear deformation—valid for slender beams where L/h > 10)
Shape functions: Cubic Hermite polynomials for transverse displacement, linear for axial
Solver: Sparse direct (CSparse LU factorization)
Convergence: Single-step linear solve (typical runtime < 10ms for typical beams)
Coordinate system: Right-handed global XYZ, Y-up convention