One of the basic rules used in optimizing structural elements, is to eliminate bending moment. Bending moment produces an uneven distribution of stresses in cross sections which broadly speaking, must be designed according to the maximum design stress. This results in a lot of material ending up underutilized.
Eliminating Bending moment means basically that beams are to be connected with spherical joints, so that beams function as struts and tie. Members in tension can be substituted with ropes and if the whole structure is in tension the result is a funicular shape. Then, if we take the opposite of each load vector, the structure will then be in pure compression and thus requiring struts.
Mathematician Luigi Cremona has probably been one of the people who investigated the most the equilibrium of such shapes, along with Carl Culmann. The Cremona diagram is still used today as a visual representation of force equilibrium in a given point. On the left you can see a picture from Cremona’s main work Le strutture reciproche nella statica grafica, 1879, where the static equilibrium of a cantilevered roof is found.
One of the most interesting results of this application is the study for the Sagrada Familia by Gaudi, depicted in the left picture. One can see how the cable structure is completely in tension under vertical loads, here sand sacks. Therefore, the reversed model made of struts is in compression.
This field has not been investigated for some time until recently. Philippe Block, professor at ETH and founder of the BLOCK research group, has been studying this subject and his PhD thesis represents a good reference.
In his work, Block derived a new methodology for the form finding of a 3D funicular shape within a given envelope, exploiting the basic principles of graphic statics. He and his team indeed produced some nice tools you can find on the research group website.
New explorations in this field are related to the revamping interest in reciprocal frame structures. They are so called because composed of beams laying on each other, therefore transferring only shear and normal forces through connections. On the left you can see a special type of reciprocal frame where connections can transfer also traction load, thus making possible changes in curvature, whilst traditional ones (also known as nexorade) were used to span planar distances such as roofs. They were used because of their easiness to build since the connection were mono-directional not preventing the two connected frames to distance one from the other. This image is from a workshop held together by Gennaro Senatore and Daniel Piker, the latter is doing some investigation on this matter.