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Designing Regular and Irregular Elastic Gridshells by Six DOF Dynamic Relaxation
Jian-Min Li and Jan Knippers
1 Introduction
An elastic gridshell defined in this paper is a single-layer or double-layer shell structure that consists of initially straight and continuous members, has equal/regular or various/irregular grid lengths and allows scissoring movements in the joints during the erection process (Fig.1). Designing a grid shell of this type needs to fulfil many geometrical and mechanical constraints. The grid pattern needs to be as close to the target surface as possible. Meanwhile, important from material aspects, an ideal grid pattern should exhibit the lowest strain energy compared with other patterns.
Most of form-finding methods for elastic grid shells are based on simplified structural models, which consist of only three degrees of freedom (DOF) per node [1][2][3] such that bending and torsion effects cannot be accurately considered. We use dynamic relaxation method (DR) with six DOF per node and Bernoulli beam elements, which have 12 DOF each, to solve the optimised grid pattern, which fulfils the geometric constraint and exhibits the lowest strain energy [4]. One advantage of our scheme is that it can handle both the form-finding and loading analyses. The obtained form can be directly transformed to a bearing structure. There is no need to transfer the form-finding result to another finite-element solver.
To address the feature of bending-active elastic gridshells [5], we demonstrate a simple method to create pre-stress in grid shells such that we can start the form-finding or the loading analysis from a strained geometry. With this method, we are no longer restricted from beginning in an unstrained state and, thus, the tedious and tricky pre-stressing procedure for bending-active structures is prevented.
The projection method and constraint force method are integrated into our scheme. The projection method is applied to find grid patterns that exactly fulfil the geometric constraints, while the constraint force method allows the obtained forms to deviate from the target surfaces. The fitness of the obtained forms to the target surfaces is controlled by the magnitude of constraint forces. Smaller constraint forces, which result in less fitness, generate grid patterns that exhibit lower strain energy.
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, the real stiffness facilitates the form-finding of a grid pattern in accordance with the pre-determined grid lengths. Meanwhile, the fictitious stiffness (a profile stiffness that is designed on purpose and is only used in the form-finding stage) enables the grids to have a larger range of strained lengths, which facilitates the reduction of bending stresses and can be utilised to generate smoothly curved grid patterns with various grid lengths.
Jian-Min Li, Jan Knippers
Institute of Building Structures and Structural Design (ITKE), University of Stuttgart, Germany
Fig. 1 Structural model of a single-layer grid shell with equal grid lengths
The rest of the article is structured as follows: In Sec. 2, we describe the structural modelling of elastic beams. In Sec. 3, we illustrate how to apply geometric constraints. In Sec. 4, four examples are given. In the final section, our conclusions are provided.
2. Beam Modelling
Our beam model is visualised as two separated nodes/particles with an interval as the beam length and each node is assigned a mass term and an inertia term. The nodes will shift and rotate freely if no force or torque is exerted. Once the internal forces are applied, the nodal movements become coupled and the individual nodes will behave as an integral structure.
2.1. Dynamic Relaxation with Six DOF per Node
DR is an explicit time integration method [6], which is applied in our scheme to simulate the above-mentioned system. With the information of the state of the previous time step and the forces currently exerted on the system, we can calculate the state of the subsequent time step.
Translational motion is described by nodal positions and translational velocities, while rotational motion is described by nodal orientations and angular velocities. The rates of change of translational velocities and rotational velocities are proportional to the residuals of translational forces and torques exerted on nodes. The update formulations for translation motion and rotational motion are listed in Tab. 1. The derivation and the application of these formulations are illustrated in our previous work.
| Table 1: Translation and rotation formulations of DR | |
| Translational part | Rotational part |
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With proper damping, the kinetic energy is reduced to zero and a static state can be derived. We use two different damping methods: For the translational degrees of freedom we use kinetic damping, setting every translational velocity to zero once the translational kinetic energy of the system begins to decrease. For the rotational degrees of freedom we use velocity damping, multiplying every angular velocity by a factor less than one at each time step [7].
2.2. Beam Mechanism
To calculate the internal forces in beam elements, we need to define additional orientations of the beam-ends. In this paper, we consider only rigid joints. That means each beam end will rotate simultaneously with the corresponding node and maintain their relative difference in orientations. The Euler-Bernoulli beam element, which has 12 DOF each, is used to calculate internal forces in beams. The axial force, bending moments and torque are determined by the corresponding positions and orientations of the two beam-ends of the beam element. The formulations for calculating internal forces are listed as follows:
Where θ is the included angle that is defined by the orientation of the beam end and the axial direction of the beam. Its geometric definition is shown in