Fig. 6 (a) Initial grid and geometric constraints; (b) Transient state; (c) Equilibrium state under constraint; (d) Bearing structure after constraints are removed and bracing and bearing conditions are added; the enlarged part shows the nodal orientations.
4.3. Effect of the Weighted Stiffness (Irregular Gridshell)
In this example, we apply the weighted stiffness in the form-finding stage to the above grid structure to see its effect on the grid patterns: A factor of 10E4 is applied to the bending and torsional stiffness to enable the beam element to have a larger range of the strained length. This fictitious stiffness setting is only used in the form-finding stage. Once the form is found, the material stiffness is recovered and the strained lengths of the found form are taken as the new unstrained grid lengths.
The comparison of the strain energy between the two forms, which are found with the real stiffness setting and the fictitious stiffness setting, is showed in Tab. 2, and the comparison of their geometries is showed in Fig. 7. The in-plane bending strain energy of the irregular grid shell is largely reduced, which is only 25% of the in-plane bending strain energy of the regular gridshell.
| Table 2: Comparisons of the grid length and the strain energy between the regular grid and the irregular grid | |||||||
| max. length[m] | min. length[m] | strain energy [kNm] | axial strain energy [kNm] | torsional strain energy [kNm] | out-of-plane bending strain energy [kNm] | in-plane be ding strain energy [kNm] | |
| regular grid irregular grid | 1.0003 1.0602 | 0.9996 0.9308 | 9.5E+01 6.7E+01 | 5.2E-01 3.2E-03 | 2.5E+01 2.6E+01 | 3.6E+01 3.3E+01 | 3.2E+01 8.2E+00 |
4.4 2D Hybgrid
Hybgrid, which was proposed by Truco and Felipe [8], is an innovative double-layer structural type that is composed of uniform flexible chord members and can generate various geometries by controlling the strut lengths between the chords (Fig. 8). Currently, the only available form-finding for Hybgrid is based on physical models. Thus, the testing of our method constitutes a significant benchmark.
The grid is composed of three chords: the upper chord, the middle chord and the lower chord. Every chord is composed of 40 beam elements; each contains an unstressed length of 12.88cm. The connections between the chords consist of revolute joints. Each chord has a uniform rectangular profile that exhibits a width of 80mm and a thickness of 4mm. The elasticity and shear modulus are defined as E=107KN/m2 and G=E/2, respectively.
Fig.7 Comparison of the grid patterns of the irregular grid and the regular grid, which are presented by the orange colour and the purple colour respectively. Both grid patterns are smooth, but the irregular grid has less in-plane bending curvatures.
Fig. 8 (a) Initial geometry; (b) Transient state; (c) Transient state; (d) Equilibrium state under constraint; (e) Bearing structure after boundaries are removed and struts and supporting conditions are added.
Fig.9. (a) Initial grid; (b) Transient state; (c) Transient state; (d) Equilibrium state under constraint; (e) Bearing structure after constraints are removed and struts and supporting conditions are added; (f) The enlarged part shows the nodal orientations.
The upper chord nodes that will be subsequently connected with struts after form-finding are constrained by the upper curve, only movable on the curve. Similarly, the lower chord nodes that will be subsequently connected with struts are constrained by the lower curve. The two constraint curves, which contain an interval of 0.375m between the curves, are equivalent. The constraint curve is defined by two arcs with a curvature of 4.4m.
4.5 3D Hybgrid
We also applied our scheme in the form-finding of the 3D-Hybgrid (Fig. 9). The plane grid is composed of 18 composite beams. Each composite beam has a structure that is similar to the 2D Hybgrid in the last example. In plain view, two crossing beams have a 60-degree included angle. Two types of constraint are applied: the surface constraint and the curve constraint.
For the surface constraint, the upper chord nodes that will be subsequently connected with struts after form-finding are constrained by the upper surface. Similarly, the lower chord nodes that will be subsequently connected with struts are constrained by the lower surface. The two surfaces are equivalent with an interval of 0.375m between the surfaces. The surface is defined as a surface of revolution that is generated by an arch of radius of 2.9m with a revolution of radius of 22.9 m.
For the curve constraint, we constrain four nodes, which are located on the top layer around the central crossing point, by two curves that are produced by projecting two straight crossing lines, which have a 60-degree included angle, on the upper constraint surface. This constraint helps grids to maintain the included angles in general.
5 Conclusions
We present a general scheme to solve grid patterns for elastic grid shells according to material properties and geometric constraints. Our form-finding scheme is based on the dynamic relaxation method with six DOF per node and the Euler-Bernoulli beam theory.
We can build the pre-stress of bending moments and torsions directly into structures by assigning specific initial orientations of beam-ends. This method enables us to begin the structural simulation from a strained state such that the tricky pre-stressing process of bending-active structures is prevented.
Using the projection method and the constraint force method, we can apply geometric constraints to grids. The projection method can generate grid patterns that exactly fit the geometric constraints, while the constraint force method produces grid patterns that are close to the target geometry and have less strain energy compared with the results derived from the projection method.
The profile stiffness plays as an active role in the form-finding process in our scheme. The real stiffness facilitates the form-finding of a grid pattern in accordance with the pre-determined grid lengths. The fictitious stiffness 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