Thrust Network Analysis

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Graphic Statics & Reciprocal Force Diagrams: Modern digital form-finding (Thrust Network Analysis - TNA) stems from James Clerk Maxwell's 1864 reciprocal figure theorem. A spatial form diagram representing structural bar geometry $\Gamma$ has an exact dual reciprocal diagram $\Gamma^*$. Every closed polygon of forces in $\Gamma^*$ corresponds to a node in equilibrium in $\Gamma$, allowing architects to sculpt compression-only vaults purely geometrically without finite element matrix inversion.

Maxwell-Cremona Duality:

1. Topological Duality: Nodes in $\Gamma$ become closed polygons in $\Gamma^*$; faces in $\Gamma$ become nodes in $\Gamma^*$. 2. Orthogonal Alignment: Corresponding edges are mutually perpendicular or parallel. 3. Force Proportionality: The length of segment $e^*$ directly measures the internal axial thrust force $N_e$.

python

1def solve_thrust_network_equilibrium(form_edges, force_densities, loads):
2    """
3    Thrust Network Analysis (TNA) by Block & Ochsendorf (2007).
4    Relates horizontal equilibrium to reciprocal graphic statics (Maxwell-Cremona):
5    Edge e in Form Diagram Gamma is perpendicular/parallel to edge e* in Force Diagram Gamma*.
6    Length ||e*|| in force diagram equals horizontal thrust force H_e.
7    """
8    # Vertical equilibrium: z_i coordinates solved via laplacian linear solve:
9    # C^T * diag(q) * C * z = P_z (where q are force densities)
10    return "Equilibrium Thrust Network & Reciprocal Force Polygons"

Central Point Load

1.50

2 min read•1 page

Funicular Thrust Lines & Masonry Equilibrium: Stone and unreinforced concrete masonry cannot bear tension. Jacques Heyman proved the Lower-Bound Safe Theorem of limit analysis: a historical vault or arch cannot collapse under its dead weight if a funicular thrust line exists that lies wholly within the masonry envelope. If asymmetric point loads push the thrust line outside the boundary, cracks open and three-hinge collapse mechanisms trigger.

Heyman's Limit Theorem:

1. Zero-Tension Material: Masonry joints open freely under tensile forces. 2. Thrust Line Confinement: Compressive resultant must remain within the stone thickness (intrados to extrados). 3. Geometric Safety Factor: Measured by the maximum shrinkage of the arch thickness that still contains the thrust line.

python

1def is_thrust_line_safe(thrust_points, arch_intrados, arch_extrados):
2    """
3    Heyman's Safe Limit Analysis Theorem for unreinforced masonry arches (1966).
4    Assumptions: masonry has zero tensile capacity, infinite compressive strength, and no sliding.
5    An arch is 100% stable if and only if at least ONE equilibrium thrust line
6    can be contained entirely between the intrados (bottom) and extrados (top) boundaries.
7    """
8    for pt, bottom, top in zip(thrust_points, arch_intrados, arch_extrados):
9        if pt[1] < bottom[1] or pt[1] > top[1]:
10            return False  # Thrust line escapes boundary -> hinge collapse
11    return True

Asymmetric Load Shift

0.50