The funicular concept can be best described  and visualized with cables or chains,  suspended from two points, that adjust their form for any load in tension.  But funicular  structures may also be  compressed like arches. Yet, although funicular tension  structures adjust their form for pure tension under any load, funicular compression  structures may be subject to bending in addition  to compression since their form is rigid  and not adaptable.  The funicular line for tension and compression are inversely identical;  the form of a cable becomes the form of an arch upside-down.  Thus funicular forms may  be found on tensile elements.

1  Funicular tension triangle under single load
2  Funicular compression triangle under single load
3  Funicular tension trapezoid under twin loads
4  Funicular compression trapezoid under twin loads
5  Funicular tension polygon under point loads
6  Funicular compression polygon under point load
7  Funicular tension parabola under uniform load
8  Funicular compression parabola under uniform load


IBM traveling exhibit by Renzo Piano A series of trussed arches in linear extrusion form a vault space  The trussed arches  consist of wood bars with metal connectors for quick assembly and disassembly as  required for the traveling exhibit.  Plastic  panels form the enclosing skin,  The trussed  arches provide depth and rigidity to accommodate various load conditions

Suspension roof
Exhibit hall Hanover by Thomas Herzog


Trusses support load much like beams, but for longer spans.  As the depth and thus dead weight of beams increases with span they become increasingly inefficient, requiring  most capacity to support their own weight rather than imposed live load.  Trusses replace  bulk by triangulation to reduce dead weight.

1  Unstable square panel deforms under load. Only triangles are intrinsically stable polygons
2  Truss of triangular panels with inward sloping diagonal bars that elongate in tension under load (preferred configuration)
3  Outward sloping diagonal bars compress (disadvantage)
4  Top chords shorten in compression Bottom chords elongate in tension under gravity load
Gable truss with top compression and bottom tension

Warren trusses
Pompidou Center, Paris by Piano and Rogers

Prismatic trusses
IBM Sport Center by Michael Hopkins
(Prismatic trusses of  triangular cross section provide rotational resistance)

Space trusses 
square and triangular plan

Note: Two way space trusses are most effective if  the spans in the principle directions are  about equal, as described for two-way slabs above.  The base modules of trusses should  be compatible with plan configuration (square, triangular, etc.)


Horizontal systems come in two types: one way and two way. Two way systems are only  efficient for spaces with about equal span in both directions; as described below.  The  diagrams here show one way systems at left and two way systems at right

1  Plywood deck on wood joists
2  Concrete slab on metal deck and steel joists
3  One way concrete slab
4  One way beams
5  One way rib slab
6  Two way concrete plate
7  Two way concrete slab on drop panels
8  Two way concrete slab on edge beams
9  Two way beams  
10  Two way waffle slab
11 Deflection ∆ for span length L1
12 Deflection ∆=16 due to double span L2 = 2 L1

Note: Deflection increases with the fourth power of span.  Hence for double span deflection  increase 16-fold.. Therefore two way systems over rectangular plan are ineffective  because elements that span the short way control deflection and consequently have to  resist most load and elements that span the long way are very ineffective.

Rupture Length (material properties, i.e., structural efficiency)

Rupture length is the maximum length a bar of constant cross section area can be  suspended without rupture under its weight in tension (compression for concrete & masonry).

Rapture length defines material efficiency as strength / weight ratio:

R = F / λ

R = rupture length
F = breaking strength
λ = specific gravity (self weight) 

Rupture length, is of particular importance for long-span structures.  The depth of  horizontal span members increases with span.  Consequently the weight also increases  with span.  Therefore the capacity of material to span depends on both its strength and  weight.  This is why lightweight material, such as glass fiber fabrics are good for long- span structures.  For some material, a thin line extends the rupture length to account for  different material grades.

The graph data is partly based on a study of the Light weight Structures Institute, University Stuttgart, German.

Structural design for: Strength, Stiffness, Stability, Synergy

Structures must be designed to satisfy three Ss and should satisfy all four Ss of structural design – as demonstrated on the following examples, illustrated at left.

1  Strength to prevent breaking
Stiffness to prevent excessive deformation
3  Stability to prevent collapse
4  Synergy to reinforce architectural design, described on two examples:

  Pragmatic example: Beam composed of wooden boards
  Philosophical example: Auditorium design

Comparing beams of wooden boards, b = 12” wide and d = 1”deep, each.  Stiffness is  defined by the Moment of Inertia, I = b d^3/12

Note:  The same amount of material is 100 times stiffer and 10 times stronger when glued  together to transfer shear and thereby engage top and bottom fibers in compression and  tension (a system, greater than the sum of its parts).  On a philosophical level, structures  can strengthen architectural design as shown on the example of an auditorium:

•  Architecturally, columns define the circulation
•  Structurally, column location reduces bending in roof beams over 500% !