Moving Roof


# variable geometry truss(VGT)    # kinetic architecture    # adaptive structure  # structural design competition

Architect:
Hayashi Kazuki

Location:
Daiba park, Tokyo, Japan

Project Area:
29000㎡

Project Term:
2018.6-2018.8

Description:
What will you behave if you are under a roof which moves up and down corresponding to the people underneath? We will not only stare at the movement, but also interact with the roof by capturing, sprawling, chasing, or inflating. In that sense, I believe this architecture has potential to stimulate our hidden sensitivity. I propose the structure using adaptive trusses with actuators, called Variable Geometry Truss(VGT).


Site plan.


Elevation(afternoon).Elevation(night).


Main structure.


Structure diagram.


Presentation boards.


 

Mathematical approaches:
To achieve this structure, we need the following information:

1. What is the target surface?
2. Are the axial forces within the range that the actuator can handle?

[1. What is the target surface?]
I introduced a tensor product Bezier surface to define the target surface, whose control points go upwards and downwards depending on the number of people around each point. Usually, Bezier surface is defined as a function of parameters \(u\) and \(v\) as
$${\bf p}(u,v)=\sum_{i=0}^{m}{\sum_{j=0}^{n}{{B_{i}^{m}(u)B_{j}^{n}(v){\bar{\bf p}}_{ij}}}}$$

Under condition that the control points are evenly spaced and their locations are all within the range of [0,1] in x and y direction, the height of any arbitrary point on the surface can be derived as a explicit function of its \(x\) and \(y\);
$$z(x,y)=\sum_{i=0}^{m}{\sum_{j=0}^{n}{{B_{i}^{m}(x)B_{j}^{n}(y){\bar z}_{ij}}}}$$

\( {\bar z}_{ij} \) is set corresponding to the number of people around the fixed coordinates (\({\bar x}_{ij}\), \({\bar y}_{ij}\)) of control point \( {\bar{\bf p}}_{ij} \), obtained with a monitoring system.

To visualize the sequence of the target surface movement, grasshopper component “RoofWave” is developed within the framework of Grasshopper Software Development Kit(SDK). The original source codes are also provided below.

Developed component.

 

[2. Are the axial forces within the range that the actuator can handle?]

Linear structural analysis is conducted for various geometry of VGT to confirm that every axial stresses are within the range of allowable stress. Let \(\bf u\) denote a vector of displacement about all degrees of freedom. \(\bf u\) can be obtained by solving the following stiffness equation:

$${\bf Ku}={\bf p}$$

where \({\bf K}\) is a global stiffness matrix and \(\bf p\) is a load vector. After obtaining elongation of \(i\) th member \(d_i\) from \(\bf u\), axial stress of \(i\) th member \(\sigma_i\) is given by

$$\sigma_i = \frac{Ed_i}{L_i}$$

where \(L_i\) is member length of member \(i\) and \(E\) is Young’s modulus, respectively.

Result of structural analysis for various geometries.