Page T1.2 . 19 September 2001                     
ArchitectureWeek - Tools Department
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Parametric Propagation of Form


The basic structure of the terminal is that of a flattened three-pin bowstring arch. Because of the asymmetrical geometry of platforms, the center pin was moved to one side, allowing the arch to rise steeply on the west side to clear the structural envelope of the train, with a more gentle incline over the platforms on the east.

The arched roof of the train shed follows the curve of the railway, and increases in span down the length of the station to accommodate the increase in the width of the platforms. The roof is supported by a series of three-pin arches.

Each arch and the related cladding are different as the roof changes width along the curved tracks. The variability of the arches is visible in both plan and isometric views.

Using conventional CAD modeling techniques, a single arch form could be modeled then duplicated 35 times down the length of the track (there are 18 pairs of arches, i.e. 36 in total, curiously numbered 2 to 37), with adjustments for the curvature of the track in plan. A laborious process of resizing individual trusses and arches would then need to be carried out.

Rather than model each arch separately, therefore, a single parametric model of an arch was modeled, such that it encoded the underlying design rules for the whole family of arches. The complete roof model then became a series of instances of this parametric arch, each with a different value for the span parameter.

A complete three-pinned arch is composed of two bowstring trusses. The longer trusses have tension rods on the inside, whereas the shorter trusses have tension rods on the outside. The cladding on the (western) short truss side is all glass, whereas the (eastern) long truss side is clad in stainless steel decking to reduce solar gains.

Applying CAD Techniques

By expressing parametric relationships between graphical objects in a CAD model, it becomes possible to simply describe a whole family of possible outcomes. This fundamental CAD principle is just what was needed in the case of Waterloo, in which a range of structural arch forms, similar, yet variable in terms of scale, dimension, and position, could be described.

The parametric expression of relationships between graphical objects is a way of modeling a complex set of design relationships that would be very difficult to model using conventional CAD techniques.

However, given that most desktop CAD systems now have an associated computer programming environment, it is at least possible for CAD users themselves to set up parametric relationships within these environments, even if the same systems have no direct means of parametric expression.

The two primary elements involved in parametric expression for the Waterloo train shed are the minor and major trusses that form an arch. More specifically, their horizontal spanning distances B and C respectively, as shown at right.

The size of the reduced truss is dictated by the parametric truss scaling factor (hx/H). The position of reduced trusses is determined by a parametric relationship which maintains the vertical cast pin dimension of 9.5 feet (2915 millimeters).

H and hx can then both be derived from a simple Pythagorean equation (the square on the hypotenuse is equal to be the sum of the squares on the other two sides).

In a conventional CAD approach (in which all key dimensions are explicitly modeled), any of the key arch dimensions (hx (or H), B, C) could of course be changed, but only by means of a long and laborious series of delete operations followed by new CAD drawing operations.

A parametric CAD model (in which some dimensions are derived from others), on the other hand, can quickly and more easily be changed by choosing a particular dimension and changing its value. Just one of the graphical arch objects, together with the associated parametric expressions, constitutes a parametric CAD model. Any other arch can be derived by supplying new values to the parametric expressions.

The truss scaling factor is based on the ratio of the pin hypotenuse (hx) to the other sides of the triangle. In other words:

hx = [ (29152 + (B + C)2 ] 1/2

where B = minor truss span; C = major truss span

The parametric model can be extended from just the description of arches, through to the description of the connections between pairs of arches. This model can then in turn be extended to the whole shed form, so that when any dimensional change is made, it is then propagated through the whole model.

Parametric expressions, therefore, allow users to change the values of key parameters, and to observe the propagation of changes on dependent expressions, and hence upon the dependent geometry. This is often referred to as strategic manipulation.

A Technology for Architects?

At present; it is still the case that parametric design software is expensive and runs on powerful computing environments. Parametric design software has to date primarily been developed for mechanical engineers, civil engineers, and industrial designers.

The design strategies adopted in these fields tend to be compositional in nature, thus lending themselves to parametric representation. The extent to which this approach is appropriate for architectural design is an issue for further debate.

In summary, parametric expressions can represent a set of geometrical forms instead of just one. The parametric expression of design relationships allows for the exploration of variation.

The downside of parametric expression is that designers invariably have to think a little differently from conventional CAD system users. CAD based on parametric expression, however, is at the leading edge of CAD technology.

Parametric design is a relatively new form of expression in architectural design, with the potential to support design creativity. It is also a further argument in support of the case for modeling vs. drafting approaches to CAD.

Peter Szalapaj is a lecturer in CAD at the School of Architecture, University of Sheffield, United Kingdom.

This article is excerpted from CAD Principles for Architectural Design, copyright 2001, available from Architectural Press and



ArchWeek Image

An isometric view of the Waterloo Station train shed, its arches varying in span from 107 to 159 feet (32.7 to 48.5 meters).
Image: Nicholas Grimshaw & Partners

ArchWeek Image

Plan view of the train station showing varying roof truss sizes and expansion joints.
Image: Nicholas Grimshaw & Partners

ArchWeek Image

Construction line grid defining the truss geometry. The arch on the west needed a much steeper curve than the arch on the east to allow trains to arrive right underneath it.
Image: Nicholas Grimshaw & Partners

ArchWeek Image

Minor and major trusses spanning lengths B and C respectively. The longer trusses have tension rods on the inside, whereas the shorter trusses have tension rods on the outside.
Image: Nicholas Grimshaw & Partners

ArchWeek Image

Parametric expression of the truss scaling factors.
Image: Nicholas Grimshaw & Partners

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A cross section from the end of the Waterloo Station.
Image: Nicholas Grimshaw & Partners

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Computer rendering of the train shed interior.
Image: Nicholas Grimshaw & Partners

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CAD Principles for Architectural Design by Peter Szalapaj
Image: Architectural Press


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