Truss Analysis

Truss Analysis (structures/fem/truss-analysis)

Use when the task is a 2D pin-jointed (plane) truss solve with the direct stiffness method: building the element stiffness matrix of each bar from E, A, L and its orientation, assembling the global stiffness matrix, applying the support conditions, solving the nodal displacements by Gaussian elimination, and recovering member axial forces and support reactions. The logic module is pure Python standard library (no numpy); the dense linear solve is implemented here. Units are SI: coordinates in m, E in Pa, A in m^2, forces in N, displacements in m.

Domain quick reference

• A pin-jointed truss bar carries only axial load. With direction cosines c = cos(theta), s = sin(theta) of the bar axis (measured counterclockwise from global +x), the 4x4 element stiffness matrix in global coordinates, over degrees of freedom [u_xi, u_yi, u_xj, u_yj], is

  k = (E A / L) * [[ c^2,  c s, -c^2, -c s],
                   [ c s,  s^2, -c s, -s^2],
                   [-c^2, -c s,  c^2,  c s],
                   [-c s, -s^2,  c s,  s^2]]
  

• Global assembly: each element scatters into the global K of size 2N x 2N over dofs [2i, 2i+1, 2j, 2j+1] (0-based node indices), so the global displacement vector is [u_x0, u_y0, u_x1, u_y1, ...].

• Support conditions: remove every constrained dof, solve the reduced system K_red u_free = F_red by Gaussian elimination with partial pivoting, then expand back (constrained entries stay 0).

• Member axial force, tension positive: F = (E A / L) * ((u_xj - u_xi) c + (u_yj - u_yi) s).

• Reaction at a constrained dof d: R_d = sum_c K[d][c] u_c, the force the support exerts on the structure, positive along +x/+y.

Worked example (verified by running scripts/truss_analysis_logic.py): the symmetric three-bar truss with nodes (0,0), (4,3), (8,0) m, bars 0-1, 1-2 and 0-2 of E = 200 GPa, A = 1e-3 m^2, a downward load of 100 kN at the apex node 1, a pin at node 0 (x and y fixed) and a vertical roller at node 2 (y fixed) gives

displacements: [0, 0, 1.3333e-3, -5.25e-3, 2.6667e-3, 0] m
member forces: [-83.333e3, -83.333e3, +66.667e3] N
reactions:     R_0x = 0, R_0y = +50e3 N, R_2y = +50e3 N

The two inclined bars are in compression (83.3 kN each), the bottom chord carries 66.7 kN tension, and the vertical reactions split the 100 kN load evenly. The same two bars without the bottom chord, pin plus vertical roller, is a mechanism: K_red is singular and the solve raises ValueError.

Workflow

1. List the nodes as (x, y) coordinates in m; the list position is the 0-based node index.
2. List the elements as (i, j, E, A): 0-based node indices, Young's modulus in Pa, cross-section area in m^2.
3. List the loads as {(node, axis): force} with axis 'x' or 'y' (positive along +x/+y) and the supports as [(node, axis)] pairs to fix to zero displacement.
4. Check determinacy first: m bars + r support reactions >= 2 j nodes is necessary for stability; a singular solve later means a mechanism.
5. Solve the whole model with truss_analysis(nodes, elements, loads, constraints), which returns displacements, member_forces and reactions in one dict.
6. Or step through the method: element_stiffness_matrix(E, A, L, theta_deg) per bar, assemble_global_stiffness(nodes, elements), solve_displacements(K, F, constraints), member_forces(nodes, elements, displacements), reaction_forces(nodes, elements, displacements, constraints).
7. Sanity check: the sum of all reactions plus all applied loads must be zero in both directions (global equilibrium).

Pitfalls

• Confusing truss-analysis with fem/modal-analysis: modal-analysis computes natural frequencies and mode shapes of 2-DOF mass-spring systems (eigenvalue problem); truss-analysis is static, has no masses and never solves an eigenvalue problem.
• Confusing truss-analysis with fem/calculix-linear: calculix-linear drives continuum FEA in CalculiX (ccx) with element-basis stress checks and margins of safety; truss-analysis is a hand-scale 2D bar solve with its own Gaussian elimination and no FEA software.
• Confusing truss-analysis with the vehicle-design structures-integration wing-box-sizing leaf: wing-box-sizing sizes spars, stringers and covers of a transport wing box under bending and torsion; truss-analysis only solves pin-jointed bar models.
• Picking an unstable support set: a two-bar V arch with a vertical roller is a mechanism (m + r = 5 < 2 j = 6); the reduced system is singular and solve_displacements raises ValueError. Add a third bar or a second horizontal reaction to stabilize it.
• Mixing up the sign conventions: member force positive means tension and negative means compression; reactions are the force the support exerts on the structure, positive along the +x/+y axes.
• Using 1-based node numbers or a different dof ordering: indices are 0-based and the dof of (node, axis) is 2*node + (0 for 'x', 1 for 'y').
• Mixing units: everything is SI (m, Pa, m^2, N, m of displacement); feeding mm^2 for A with Pa for E scales every stiffness by 1e6 and corrupts the displacements.
• Forgetting that adding a bar changes the load path: with the bottom chord present the apex drops 5.25 mm and the chord takes 66.7 kN tension; without it the structure is a mechanism. Always recheck determinacy when the model changes.

Behavior contract (gate 3)

The truss logic is exercised by the gate 3 contract test: scripts/test_truss_analysis.py against scripts/truss_analysis_logic.py (stdlib unittest, offline). Run: python3 scripts/test_truss_analysis.py

Compliance

• FAR-25 and MMPDS are referenced, not reproduced: standards-map.yaml marks them gated: false and reference-only: true; only the summary paraphrase above is used, never standard text.
• compliance: STANDARDS-REF, gated: false.