2.
2.3. Pre-stress of Bending and Torsion
For a rigid joint its corresponding orientations of the beam-ends, orientations of the beam-ends connected to that joint, maintain their relative difference in the transient phase. For a straight joint, the two corresponding orientations of the beam-ends coincide with each other throughout the process. Therefore, by assigning orientations as in Fig. 3a, we can assign the pre-stress of an initially straight joint. The assignment of an initially angled joint is shown in Fig. 3b.
Fig. 2 (a) Definitions of included angles, θa,z and θb,z ; (b) Definitions of included angles, θa,y and θb,y ; and (c) Definitions of the beam axial direction, .
2.3. Pre-stress of Bending and Torsion
For a rigid joint its corresponding orientations of the beam-ends, orientations of the beam-ends connected to that joint, maintain their relative difference in the transient phase. For a straight joint, the two corresponding orientations of the beam-ends coincide with each other throughout the process. Therefore, by assigning orientations as in Fig. 3a, we can assign the pre-stress of an initially straight joint. The assignment of an initially angled joint is shown in Fig. 3b.
Fig. 3 (a) In the area enclosed by the dashed line, the orientations of the consecutive beam ends are equivalent. This builds up the pre-stress of moments of an initially straight joint; (b) In the area enclosed by the dashed line, the orientations of the two consecutive beam ends are the same as its own beam orientation. Hence no pre-stress of moments and torsion is built.
2.4 Weighted Stiffness
In our scheme, the profile stiffness can be used as an active factor in the form-finding process. The real stiffness - a profile stiffness of a practical profile that can be used in the bearing structure - is useful to explore the structural behaviour of elastic grids under geometric constraints. Besides, if the grid lengths are pre-determined, the real stiffness is helpful to solve a grid pattern in accordance with these pre-determined grid lengths.
If minimising the bending strain energy is a more critical issue than maintaining grid lengths, the fictitious stiffness - a profile stiffness that is designed on purpose and is only used in the form-finding stage - is used, which is defined by multiplying the bending and torsional stiffness by a weighted factor lager than one and keeping the axial stiffness unchanged. The fictitious stiffness enables the grids to have a larger range of strained lengths, which is helpful to reduce curvatures of the bending elements. Besides, this character can be utilised to generate smoothly curved grid patterns with various grid lengths.
3 Geometric Constraint
In contrast with the implicit integration method the explicit method does not require a global stiffness matrix. The motion update is treated locally for each node. The nodal motion is only affected by the physical quantities around the node. As a result, DR is an ideal method to deal with local perturbations that are caused by physical contacts or geometric constraints, because the local instability will not cause the entire system to crash. The geometric constraint discussed in this paper is a curve or surface defined by the non-uniform rational basis spline (NURBS). If a structural node is constrained, its movement will be restricted within the constraint domain. The static state derived in this manner is a state that exhibits the lowest strain energy while satisfying all geometric constraints, which is crucial for combining material aspects with geometric design requirements.
3.1 Projection Method
By counting only tangential nodal residuals, a smooth tangential movement on the constraint surface/curve is achieved. The slight deviation caused by the tangential movement is adjusted by projecting the constrained node to the constraint at the end of each time step.
3.2 Constraint Force Method
We can also manipulate the form by applying constraint forces. A nodal constraint force may be defined by a vector which points toward the closest point on the geometric constraint and is proportional to the distance between the node and the point.
The magnitude of the constraint force is used to control the fitness of the obtained form to the target geometry. Smaller constraint forces result in a form that exhibits less strain energy but with the price of a larger deviation from the target geometry. An agreeable result may be attained by tuning the magnitude of the constraint forces.
4 Examples
Four examples are presented in this section. The first example shows a beam that is relaxed from a strained geometry. The second example demonstrates the form-finding process of a single-layer grid shell. The third example and the fourth example show the form-finding processes of double-layer grid structures.
4.1 Relaxed Beam
This example is used to verify the pre-stress assignment as proposed in Sec. 2. If the pre-stress is correct, the strained curved beam will resume a straight line as in Fig. 4. The diameter of the barrel and the length of the barrel are 20m and 40m respectively. The initial geometry of the bended beam is defined as a helix on the barrel. The beam is composed of 36 elements with a square profile that has a side length of 5cm. Each element has a length of 141cm. The elasticity and shear modulus are defined by E=107KN/m2 and G=E/2, respectively.
Once the beam is released, the motion is triggered by the residuals and the system is eventually damped to a static state. The computation terminates once the residual of each node is less than 10-4KN. The variation in strain energy throughout the process is shown in Fig. 5.
4.2. Weald and Downland Gridshell
The Downland Gridshell, which is charcaterised for its unique triple-bulb geometry, is our first form-finding example. The task is to find a single-layer grid pattern that complies with the material aspects and satisfies the geometric constraints.
Fig. 4 (a) Pre-stressed beam on a barrel surface; (b) Beam relaxed from a strained geometry
Fig. 5 Strain energy versus time step
The grid, which is composed of 102 initially straight planks, has a uniform square profile and a width of 5cm (Fig.6). Each beam element has an unstrained length of 1m. The connections between the crossing rods are revolute joints, which enable free rotation along the local z-direction. The elasticity and shear modulus of the material are defined as E=107KN/m2 and G=E/2, respectively.
Two geometrical constraints are given. First, the grid nodes on the two longer edges must stay on the curved boundaries. Second, the remaining grid nodes must remain on the triple-bulb surface.