Graphic statics. In a broad sense, the term graphic statics is used to describe the technique of solving particular practical problems of statics using graphical means. Actively used in the architecture of the 19th century, the methods of graphic statics were largely abandoned in the second half of the 20th century, primarily due to widespread use of frame structures of steel and reinforced concrete that facilitated analysis based on linear algebra. The beginning of the 21st century was marked by a "renaissance" of the technique driven by its addition to the computer-aided design tools thus enabling engineers to instantly visualize form and forces.

History 

Markou and Ruan trace the origins of the graphic statics to da Vinci and Galileo who used the graphical means to calculate the sum of forces, Simon Stevin's  parallelogram of forces and the 1725 introduction of the force polygon and funicular polygon by Pierre Varignon. Giovanni Poleni used the graphical calculations (and Robert Hooke's analogy between the hanging chain and standing structure) while studying the dome of the Saint Peter's Basilica in Rome (1748). Gabriel Lamé and Émile Clapeyron studied of the dome of the Saint Isaac's Cathedral with the help of the force and funicular polygons (1823).

Finally, Carl Culmann had established the new discipline (and gave it a name) in his 1864 work Die Graphische Statik. Culmann was inspired by preceding work by Jean-Victor Poncelet on earth pressure and Lehrbuch der Statik by August Möbius. The next twenty years saw rapid development of methods that involved, among others, major physicists like James Clerk Maxwell and William Rankine. In 1872 Luigi Cremona introduced the Cremona diagram to calculate trusses, in 1873 Robert H. Bow established the "Bow's notation" that is still in use. It fell out of use, especially since construction methods, such as concrete post and beam, allowed for familiar numerical calculations. Access to powerful computation gave structural engineers new tools to compute stresses for shell structures such as Finite element method.

While the method is not commonly used for construction today, graphic statics was proposed as an educational tool to teach intuition in engineering education. It is employed in classes at MIT and ETH. for architecture and structural engineering students.

Concepts

Polygon of forces 

To graphically determine the resultant force of multiple forces, the acting forces can be arranged as edges of a polygon by attaching the beginning of one force vector to the end of another in an arbitrary order. Then  the vector value of the resultant force would be determined by the missing edge of the polygon. In the diagram, the forces P1 to P6 are applied to the point O. The polygon is constructed starting with P1 and P2 using the parallelogram of forces (vertex a). The process is repeated (adding P3 yields the vertex b, etc.). The remaining edge of the polygon O-e represents the resultant force R.

In the case of two applied forces, their sum (resultant force) can be found graphically using a parallelogram of forces.

Digital Adaptations of Graphic Statics 
With the advent of computational tools and parametric design, graphic statics has undergone significant evolution, transitioning from manual drawing techniques to digital workflows. These adaptations have enhanced its precision, accessibility, and integration into modern architectural and engineering practices.

Software Tools 
Several software platforms have integrated graphic statics principles, enabling designers to explore equilibrium-based forms and optimize structures efficiently. A few examples include:

 RhinoVAULT: A plug-in for Rhinoceros 3D developed by the Block Research Group at ETH Zurich. RhinoVAULT uses Thrust Network Analysis (TNA) to apply graphic statics for the design of compression-only structures, including shell vaults and domes.
 Grasshopper Add-ons: Extensions such as Kangaroo Physics and Millipede in Grasshopper 3D have incorporated elements of graphic statics to facilitate form-finding and structural analysis within parametric design frameworks.
 3D Graphic Statics Tools: Emerging tools like Compass and eQuilibrium allow for the visualization and manipulation of 3D graphic statics diagrams, broadening its applications in three-dimensional design.

Applications 
Digital adaptations have expanded the scope of graphic statics, making it a valuable tool for:

 Material Optimization: By visualizing force flows, designers can reduce material usage while maintaining structural efficiency.
 Complex Geometries: Parametric tools enable the exploration of intricate geometries that would be challenging to model manually.
 Interactive Design: Real-time manipulation of force diagrams in software provides immediate feedback on structural behavior, fostering intuitive decision-making during the design process.

Educational Impact 
The digitization of graphic statics has also influenced its role in education. Many universities such as MIT  now teach graphic statics through interactive software, enabling students to experiment with equilibrium concepts in a hands-on manner.

Limitations 
Despite its advantages, digital graphic statics faces challenges such as scalability for highly complex systems and integration with advanced analytical tools like Finite Element Method (FEM). However, ongoing research continues to address these limitations.

References

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Structural analysis