Free-form thrust lines

Structures with a non-funicular shape can be transformed into a free-form thrust line by adding a tension stiffening system shaped according to the bending moment in the free-form curve. The reaction forces of the stiffened free-form structure will be equivalent to those of the funicular solution (dash-dotted line). Explore possible configurations of the free-form thrust line. Note that the stiffening system shown is the visualization of the bending moment in the free-form curve, not the actual geometry of the structure.

Design of structures with a free-form shape

This case study explores the use of thrust lines as a strategy to design structures with a free-form non-funicular shape. Such structures can be transformed into a free-form thrust line by adding a tension stiffening system shaped according to the bending moment in the free-form curve.

Waterloo International station

Waterloo International station was the London terminus of the Eurostar international rail service from its opening in November 1994 until November 2007. It is located on the Western side of Waterloo railway station, London. It was designed by Grimshaw Architects and structural engineer Anthony Hunt and completed in May 1993. It won numerous awards, including the Royal Institute of British Architects’ Building of the Year award (1994) and the AIA London UK Chapter Design Excellence Award (1995).

The tapering span of the roof structure reflects the constraints of the site, the asymmetry of the trusses is derived from the required clearance for the trains across the tracks. The inversion of structure and skin reflects the inversion of tension and compression in the roof structure. The arches are made up of curved trusses which are triangular in section. Both the compression and tension members are curved and tapering, they are created from telescoping circular sections, thus responding to the distribution of forces in each truss.

Berlin Central station

Berlin Central station is the main railway station in Berlin, Germany, located on the site of the historic Lehrter Bahnhof. It was designed by Gerkan, Marg and Partners and structural engineer Hani Azer and opened in May 2006. The curved roof structure spans about 85 metres and is clad with a glass panels with an integrated photovoltaic system.

Bergisel ski jump

The Bergisel ski jump, situated on the Bergisel mountain overlooking Innsbruck was designed by Zaha Hadid Architects following an international competition in 1999 and opened in 2002. It was part of a larger refurbishment project and has replaced the first Bergisel ski jump which was built for the 1964 Winter Olympics. Structurally the ski jump consists of a 50 m tall vertical concrete tower, a green-metal bridge integrating the ramp and the cafe, and the foundation dug into the Bergisel Mountain. Conceptually, however, the structural elements are not different systems, but parts of the overall structure.

Construction of the bending moment in the free-form curve

The bending moment in the free-form curve is proportional to its deviation from the funicular solution, M=H*y. By choosing the tension force T in the stiffening system, the shape can be approximated, since M=T*d. Use the slider to go through the construction of the bending moment in the free-form curve:

  • Constructing a Bezier curve through the supports and placing the hinge on the control curve (Step 1).
  • Constructing a funicular through 3 points for the given loading, i.e. the self-weight of the structure (Step 2).
  • Finding the reaction forces for the funicular solution (Steps 3-4).
  • Determining the bending moment in the control curve: M=H*y, with H the horizontal thrust of the structure and y the vertical intercept between the control curve and the funicular (Step 5).
  • Choosing the tension force in the stiffening cable, T, to determine d, the perpendicular distance to the control curve since M=T*d (Step 6).
  • Repeating the procedure (Step 7).
  • Connecting the struts results in the visualization of the bending moment in the free-form curve (Step 8).

Click here to open the interactive drawing that shows the graphical solution. Use the slider in the bottom right corner to browse through the different steps.

Move the handles to manipulate the control curve. Use the slider to vary the tension force T. Switch between different configurations inspired by Waterloo station, Berlin station and Bergisel ski jump.