In this paper, a connection between reciprocal diagrams and typical features in membrane structures is drawn . Although numerous form -finding techniques have been developed in the field of tensile membrane structures, the introduction of reciprocal diagram s allows for an intuitive understanding and can therefore facilitates the development in early design stages . To begin with, we introduce relevant basic concepts , including planar reciprocal diagrams and force density method . The s econd part of the paper focuses on the construction of planar reciprocal diagrams for typical membrane features including membrane patches, ridge/valley cables (cables that go through a patch of membrane ) and edge cables . The con cept of anchoring force polygon is introduced. The t hird part of the paper discusses form -finding in 3D with force densities found in the planar reciprocal construction , followed by progressive steps of updating force densities based on current geometry . Several benchmark studies were undertaken . Comparison s to analytical solutions or finite -element results were done , showing good agreement . Keywords : membrane structures, reciprocal diagrams, graphical methods, force density method, form finding , isotropic stress 1. Introduction In the field of graphic statics, Maxwell’s construction of 2D reciprocal diagrams (1) is a powerful tool for its clear presentation of form and force of an equilibrium system. The idea has been implemented in recent work in funicular vault structures (2), tensile membrane under vertical loading (3) and optimal trusses (4), among many others . This paper focuses on the reciprocal diagram construction of tensile membrane s including typical features such as edge cables and ridge/valley cables , followed by the form finding process using the thereby informed force densities . The proposed method is fast and intuitive , rather straightforward to implement due to the use of 1D element only . The current paper is organised as the following. First part of the paper reviews basic concepts of reciprocal diagrams (5), Frei Otto ’s edge cable technique (6) and the force density method developed by Schek (7). The s econd part extends Otto’s technique to other features, namely ridge/valley cables and membrane patches . A comp lete construction of planar reciprocal diagram is therefore feasible . Here, Singer’s extension of the classical force density method to membrane structures was inspirational (8). In the third and last part of the paper , lifting of the planar equilibrium was discussed, including a This preprint research paper was submitted on 27-Mar-2024 and has not been peer reviewed. 2 progressive procedure of updating force densities. Several case studies are done and compared with analytical or finite -element results. In the drawing of form and force diagrams , blue is used for compression and red for tension. Green denotes boundary conditions like anchoring forces or constrain ts. Black lines are used for constructional purposes, unless otherwise stated . 1.1. Reciprocal diagrams The application of reciprocal figures in graphic statics is a powerful tool for its clear presentation of form and force in equilibrium. The 2D form and force diagrams share the same number of edges and can be constructed to any chosen angle regarding the corresponding pair of edges (5). Typically used are the parallel (Cremo na convention) and orthogonal (Maxwell convention) constructions. This paper adopts the latter, as it remains the relative position of corresponding edges, enabling an intuitive reading of diagrams. Figure 1. Reciprocal diagrams of an explanatory self-stressed truss following Maxwell convention . (a) Form diagram ; (b) force diagram contain pairs of edges that are reciprocal to each other . The close polygon s found in force diagram denote equilibrium in corresponding nodes in form diagram , e.g. triangle ② in (b) for node ② in (a). 1.2. Membrane edges of a planar isotropic stress field Considering a planar membrane of isotropic stress es, namely same principal stress es without shear, the bending free edge s, say materialised as edge cable s, are of arc profiles . Such a planar stress field is also termed hydrostatic, like soup film (9). The analogy was widely used in Frei Otto’s research of retractable membrane roofs (6), where the authors have drawn inspiration for the correlation to reciprocal diagrams, see section 2.1. In the case of anisotropic stress with one dominating direction e.g., in warp pointing perpendicular to the span of an edge , the cable profile on the other hand follows a parabola profile . The difference of the two is diminishing as the sag to span ratio f/L (or rise) decreases. For a sensible rise around 10%, this can well be ignored. Mathematically , the Taylor series for the lengths of both arc and parabola profiles 𝑠𝑠𝑎𝑎 and 𝑠𝑠𝑝𝑝 tells no difference within the second order of sag f. In practise therefore , L +8𝑓𝑓2 3𝐿𝐿 is often considered a good approximation for the cable length , Eq. (1) and (2). In this paper, we consider primarily form- finding for isotropic stress fields. This preprint research paper was submitted on 27-Mar-2024 and has not been peer reviewed. 3 𝑠𝑠𝑎𝑎=𝐿𝐿+8𝑓𝑓2 3𝐿𝐿−32𝑓𝑓4 15𝐿𝐿3+𝑂𝑂(𝑓𝑓5) (1) 𝑠𝑠𝑝𝑝=𝐿𝐿+8𝑓𝑓2 3𝐿𝐿−32𝑓𝑓4 5𝐿𝐿3+𝑂𝑂(𝑓𝑓5) (2) Figure 2. Edge representation of arc and parabola (a) geometries of different rises ( sag to span ratio of 10%, 20% and 30% ), (b) isotropic vs. directional membrane stress conditions. Black dot represents the centre and focal point respectively. 1.3. Force Density Method Force density method (FDM ) is a form -finding method developed initially for cable net structures such as the Olympic stadium roof in Munich (7). The form i.e., 3D coordinates of free nodes can be linearly solved with equilibrium condition only, given anchor point positions, external loads, and force densities in a net. In the absence of external loads, the linear equation of FDM reads: 𝐶𝐶𝑇𝑇𝑄𝑄𝐶𝐶𝑖𝑖𝑥𝑥𝑖𝑖=−𝐶𝐶𝑇𝑇𝑄𝑄𝐶𝐶𝑓𝑓𝑥𝑥𝑓𝑓 (3) with 𝐶𝐶=�𝐶𝐶𝑖𝑖 𝐶𝐶𝑓𝑓� being the branch -node matrix of the network, 𝑄𝑄 the diagonal matrix of force densities, and 𝑥𝑥𝑇𝑇=�𝑥𝑥𝑖𝑖𝑇𝑇 𝑥𝑥𝑓𝑓𝑇𝑇� the coordinates, where subscript 𝑖𝑖 designates free nodes and 𝑓𝑓 for fixed nodes. FDM has been extended for surface structures of isotropic stress field since late 20th , (8), (10) which had inspired the connection between a triangular membrane element and its reciprocal diagram in this paper. This is further discussed in section 2.3. 2. Planar reciprocal diagrams for membrane structures Three type s of building elements , namely edge cables, ridge/valley cables, and the membrane patches are considered. The reciprocal construction shown here demonstra tes their form and force clearly which leads to an intuitive design procedure . 2.1. Edge cable s and the anchoring force polygon Considering a membrane corner anchoring two edge cables as the following Figure 3. As discussed, when the stress field is isotropic, the edges are of arc profiles. The well -known formular for This preprint research paper was submitted on 27-Mar-2024 and has not been peer reviewed. 4 hydrostatic pressure 𝐹𝐹=𝑝𝑝 ⋅ 𝑟𝑟, gives the constant cable force 𝐹𝐹 as the product of pressure 𝑝𝑝 and radius 𝑟𝑟. Assuming a membrane stress of unity , the edge cable force s 𝐴𝐴𝐴𝐴�����⃗ and 𝐴𝐴𝐴𝐴�����⃗ has the amplitude of its radius 𝑟𝑟1 and 𝑟𝑟2, pointing tangential ly to their respective arc profile s. The anchoring force 𝐴𝐴𝐴𝐴������⃗ as the summation of the se two cable forces , is in the direction of the share d chord 𝐴𝐴𝐶𝐶����, and has the amplitude of |𝑂𝑂1𝑂𝑂2|. This can be proven by checking the congruence of △𝐴𝐴𝑂𝑂1𝑂𝑂2 and △DAB (11). One shall notice, t he form edge 𝐴𝐴𝐴𝐴���� and the force edge 𝑂𝑂1𝑂𝑂2������� constitute a reciprocal pair, sitting orthogonally to each other . Figure 3. Graphical solution of anchoring force of t wo edge cables of arc profile at an anchor point A , with a n isotropic stress field of unity . With such , an anchoring force polygon (green dashed) can be constructed given a series of edge cables spanning a n isotropic stress field of unity by simply connecting the centres consecutively . The amplitude and direction of planar anchoring forces at these anchor point s are thereby determined graphically . This preprint research paper was submitted on 27-Mar-2024 and has not been peer reviewed. 5 Figure 4. Anchoring force polygon (green dashed) of an isotropic stress field of unity in plane . 2.2. Ridge/valley cables In addition to two edge cable s at an anchor point as discussed in previous case, oftentimes a ridge/valley cable is present. That is, a cable spanning inside a membrane patch. Such cables can be used to adjust the membrane shape, deviating it from soup -film like geometries , which extends the design space of membrane structures. The construction of reciprocal diagrams of ridge/valley cables can be done by “shifting” the diagrams apart . This correlate s physically to the fact that the additional force s introduced by the ridge/valley cable s need to be anchored , see the increase in amplitude of anchoring force edge (green dashed) between (a) and (b) in following Figure 5. The equilibrium is now denoted by the quadrilateral force polygon in Figure 5 (b), as opposed to the force triangle in Figure 5 (a). Note that the added ridge/valley cable do es not have to be colinear with the existing anchoring cabl e, in other words it may alter the previous anchoring cable direction. This preprint research paper was submitted on 27-Mar-2024 and has not been peer reviewed. 6 (a) (b) Figure 5. Reciprocal construction of r idge/valley cable in addition to two edge cables at one anchor point. equilibrium is now denoted by the quadrilateral force polygon in Figure 5 (b), as opposed to the force triangle i n Figure 5 (a). 2.3. Membrane patch es After the discuss ion of typical cable elements in membrane structures, let’s now turn to the membrane patch itself. A planar membrane patch subjected to isotropic stress of σ is discretised into a tri-mesh , a part of which shown as Figure 6 (a). Inspired by Singer ’s work on extended FDM for surface structures (8), the following reciprocal construction can be done. That is, the force introduced by a triangular face on one of its three vertices 𝑉𝑉𝑖𝑖, is perpendicular to the opposing edge 𝐴𝐴𝑖𝑖, and its amplitude equates 1 2σ𝑙𝑙𝑖𝑖, with 𝑙𝑙𝑖𝑖 the length of 𝐴𝐴𝑖𝑖, i=1,2,3 . Note that the tri -mesh is neither form nor force diagram. This enables the lumping of stress es to forces on the tr i-mesh nodes , and therefore the use of 1D element only. When the isotropic stress is unity i.e., σ= 1, the force diagram of this triangular face is simply its medial triangle as Figure 6 (c), and the corresponding form diagram is obtained by connecting the orthocentre of this face to its three vertices as Figure 6 (b). This preprint research paper was submitted on 27-Mar-2024 and has not been peer reviewed. 7 Figure 6. Reciprocal construction of tri-mesh under isotropic stress of unity : (a) tri-mesh , (b) form diagram, (c) force diagram. Note that the tri -mesh is neither form nor force diagram. One shall notice the above construction gives three tension edges for an acute triang ular face . In the case of an obtuse face, whereby the orthocentre sits outside , the constructed force edges are one compression edge and two tension edges . In the very special case of a n orthogonal triangle, which practically would not be encountered in a regular discretization, only gives two meaningful tension edges. Numerically, the third edge of zero length and infinite force density , is one that starting from a node and ending at itself . Its corresponding row can be eliminated from the branch -node matrix . Physically, this zero -length edge has infinite geometric stiffness but no length to “exhibit” any forces . Figure 7. For an obtuse triangular face (b) whereby the o rthocentre sites outside, the form edge through the obtuse vertex corresponds to a compression force . 2.4. Compatibility of features Until now the construction for three components , namely edge cables, ridge/valley cables and membrane patches are described. Their form and force edges are compatible (in the sense that they do not overlap ) given a triangular discretization which embeds the cable elements . The form edges of cable elements sit always on the tri -mesh edges, whilst the form edges for membrane patch sit always off the tri-mesh edges (with the exception in an orthogonal triangle as mentioned ). This simplify the computational implementation of the method since no special topological manipulation is needed . In terms of the force diagram , the compatibility ( denoting equilibrium ) is naturally given during the construction of close force polygons . For edge cables, their force edges share a “pole” i.e., the centre of their arc profiles. For patches, the media n triangle s denote the internal equilibrium of the tri-mesh faces. For ridge/valley cables, the choosing of prestressing forces are subjected to the designer, given the summation of these additional forces alone do not violate condition of equilibrium. The introduction of ridge/valley cables causing a “shifting ” effect in the force diagram, is in a similar fashion as adding creases in a funicular vault as described by Rippmann (12). This preprint research paper was submitted on 27-Mar-2024 and has not been peer reviewed. 8 Figure 8. Compatibility of membrane features in the reciprocal diagram construction : coloured solid line s for form edges and coloured dashed lines for force edges. Introdu ction of ridge/valley cables “shift” the force diagram apart . 3. 3D form- finding and comparison of results The planar equilibrium described by the constructed form and force diagram s presents a set of force densities for the given network . A linear FDM can therefore be used to lift the geometry in to 3D when new anchor point heights are prescribed . A lift with only vertical displacement to approach a minimum surface is a practical ly meaningful approach , since the planar equilibrium is preserved when projected back , e.g. i n cases where the force direction in plan is of primary importance . By lifting , the length of a form edge is increased by factor 1/𝑐𝑐𝑐𝑐𝑠𝑠(θ𝑏𝑏), where θ𝑏𝑏 is the angle between that bar 𝑏𝑏 and the plane for reciprocal construction . The corresponding force has to increase by the same factor for the invariant force densit y. In more general cases where displacement is not restricted to vertical, this factor is also known as deformation gradient F. In both cases , the lifted geometry does not correspond to an isotropic stress field anymore , see formulation by Bletzinger (13). The resulted non-isotropic stress field S, referring the deformed /actual geometry, is related to the imposed isotropic stress σ by F through 𝑆𝑆=𝑑𝑑𝑑𝑑𝑑𝑑𝐹𝐹𝐹𝐹−1⋅σ⋅𝐹𝐹−𝑇𝑇. For 1D- elements (14), we have 𝐹𝐹= l/L , hence actual stress 𝑆𝑆=𝐿𝐿 𝑙𝑙σ, where l and 𝐿𝐿 are the actual and reference bar lengths respectively . To summarize these relations within a tri-mesh face : 1 2σ=𝑞𝑞𝑖𝑖=𝑁𝑁𝑖𝑖 𝐿𝐿𝑖𝑖=𝑛𝑛𝑖𝑖 𝑙𝑙𝑖𝑖=1 2li 𝐿𝐿𝑖𝑖Si, i = 1,2,3 [𝑁𝑁/𝑚𝑚] (4) This preprint research paper was submitted on 27-Mar-2024 and has not been peer reviewed. 9 Figure 9. A visual hint: the lifted net (ao-bo-co) does not correspond to a n isotropic stress field anymore (not perpendicular to median trian gle of actual geometry △abc). Shown here is the case with vertical displacement only. This can be resolved via a progressive procedure namely updating the force densities according to actual geometry . In other words, the planar equilibrium construction through reciprocals provides an intuitive entry point to find starting values for the following FDM steps , with many design intensions like the determination of anchoring forces, the introduction of ridge/valley cables already implemented . This two -fold procedure is implemented in Python in Rhino3D ®. For handling matrices , NumPy 1.24.2 (15) was used via “remote procedure call” mechanism provided by the COMPAS framework 1.17.5 (16). 3.1. Catenoid Catenoid is a minimum surface of revolved catenary and can be analytically defined in cylindrical coordinates as the following . The code is tested with the geometry setup with two circular rings of 6 meter s diameter, various distance s apart . Solving constant 𝑐𝑐 numerically with geometrical boundary conditions gives corresponding