Truss Analysis (`trusses`)

Check	IND	US	EU	ISO
Member axial utilization	implemented	—	—	—

Purpose

Determine axial forces in two-dimensional pin-jointed truss members under nodal loading. The module identifies tension and compression members, flags zero-force links, and reports axial stress utilization against allowable values for preliminary truss sizing.

Physics & theory

Truss members are assumed two-force elements carrying only axial force $N$ along the member axis. At each pin joint, equilibrium $\sum F_{x} = 0$ and $\sum F_{y} = 0$ must hold. Because moments cannot be transferred at pins, the structure stiffness matrix involves only translational DOFs (two per node in 2D).

Axial stress is $σ = N / A$ . Tension members are limited by yield or net-section rupture in detailed design; compression members require buckling checks handled separately in the Column Buckling module. The solver uses the direct stiffness method with bar elements of stiffness $E A / L$ .

Indeterminate trusses are solved by the same matrix approach; degree of indeterminacy must be matched by sufficient supports and member connectivity.

External supports restrain selected nodal translations; internal joints are ideal pins with no moment capacity. Loads apply only at nodes as concentrated forces — member self-weight and distributed loads must be converted to equivalent nodal forces by the user.

The solver validates connectivity, positive member areas, and sufficient boundary restraints before assembling the stiffness matrix.

Governing equations

\frac{E A}{L} [1 - 1 - 1 1] {u} = {F}

σ = \frac{N}{A}, U = \frac{∣ σ ∣}{σ _{allow}}

Numerical method

Bar-element FEM: each member contributes axial stiffness in global coordinates after direction-cosine transformation. The assembled system is solved for nodal displacements; member forces $N = (E A / L) (u_{j} - u_{i})$ are recovered. Zero-area or disconnected members produce singular systems and are rejected at validation.

Inputs

Parameter	Description
Nodes	$x, y$ coordinates, support conditions
Members	End nodes, cross-sectional area $A$ , elastic modulus $E$
Loads	Nodal force components
Allowable stress	For utilization screening

Outputs

Member axial force (signed tension/compression), axial stress, utilization ratio, reaction forces at supports, deformed shape (optional visualization).

Design codes & checks

Indicative: Member axial utilization
US: AISC 360 tension/compression member context (screening only)
EU: EN 1993-1-1 member rules (screening)

Assumptions & limitations

Pin joints, members connected at centroidal axes.
No joint eccentricity, secondary bending, or buckling in this module.
2D planar truss; no 3D spatial truss.
Linear elastic; no cable slack or compression-only release logic unless configured.

References

Hibbeler, R. C. Structural Analysis, 10th ed. Pearson.
AISC. Steel Construction Manual, 16th ed.
EN 1993-1-1:2005. Eurocode 3 — Tension and compression members.
ISO 10721:1997. Steel structures — Static analysis and design.
PhyCalcPro verification benchmarks in src/data/verification/ where available for this module.
Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.

Truss Analysis

Assumptions

Limitations

Engineering checks