Smaller cable nets

To construct a cable net one must know how the cables are best arranged. Generally, two arrangements are worth considering:

• Geodesic mesh, in which the cables run along the geodesic lines in the surface.

A geodesic line is the shortest way between two points on a surface. This approach minimises the use of material but the manufacturing can be quite complicated.

• Uniform mesh in the unstrained state. From a constructional point of view this approach is the best one. As, an error in length of 0.1 % can give rise to an error in force of about 50 %, accurate placing of the connections is crucial [63].

An equidistant mesh enables the mounting to be done in a factory. At the building site, the net can be assembled on the ground and hoisted into position.

Figure 3.7 shows an example from [72], where certain elements are assigned with a constant unstrained length.

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

1 2

3

4 5

6

Figure 3.7: The uniform mesh approach. Elements on the sides of non-shaded areas should have equal unstrained lengths. Redrawn from [72].

Table 3.3: Coordinates in metres for the fixed nodes of the structure in Figure 3.7.

Node Coordinates

x y z

1 0.0000 0.0000 0.0000 2 −6.4897 2.2285 3.0000 3 −6.9932 9.3859 0.0000 4 −2.4917 11.7810 4.0000 5 3.8300 11.5901 0.0000 6 7.0874 4.2153 5.0000

Considering the magnitude of the prestressing force, the usual rule of thumb is that the magnitude should be such that no element goes slack under any load condition.

However, according to Leonhardt and Schlaich [63] this rule “causes unacceptably high forces.” They further conclude that “it will never be possible to establish general rules for finding the shape of cable net structures.” This is due to the many factors which affect the shape and load-bearing characteristics of a cable net.

Among the factors are: prestress magnitude and distribution, mesh geometry of the net, edge rigidity (concrete ring or edge cables), angle between net cables and edge cables and stiffness ratios of the members [63].

In general, the magnitude of the prestressing force is determined by the allowable deformations and fatigue strength of the cables. However, an increase in prestressing force is not as effective in reducing the deformations as an increase in curvature of the net [63]. Of course, the distribution of the prestressing force must be quite uniform.

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3.4. EXAMPLES

Several configurations have to checked before finding one which best satisfies all requirements put on the structure.

A common approach for prestressed cable nets is given in reference 72. In this approach, the interior cables are assigned with unstrained length constraints and the elements in contact with the edges are assigned with force constraints. In this way a good compromise between load-bearing behaviour and construction ease is reached. A special problem which may arise when using the outlined technique is that the elements connected to the edge cables may have too large angle changes, Figure 3.8. Since the real cable is continuous and has a finite bending stiffness, such angles cannot occur in practice. A technique to avoid these angle changes is to assign force constraints also to the interior cables. For a three-dimensional structure a configuration that satisfies all the constraints exactly may not be found. But if the modified damped version of the non-linear force density method is used, the iterations stop at a satisfactory shape.

To find the shape of the structure in Figure 3.7, the following procedure was used:

1. All interior cable were assigned with a force density equal to 200 kN/m. The force density for the edge cables was 1200 kN/m. The interior cables had the following constraints: cable force equal to 200 kN and unstrained length of 1 m. No constraints were assigned to the edge cables. The heights of all the fixed nodes are changed to z = 0. Thus, the cable net lies in the x–y plane.

The stiffness of the cables was AE = 100 MN.

2. With the prescribed force densities, the linear force density method gave the shape shown in Figure 3.8. The net has a nearly square mesh, but some large distortions occur near the edges.

3. With all the fixed nodes still in the same plane, 20 iterations with the non-linear force density method gave a fairly smooth layout without large angle changes near the edges. Note that this configuration does not satisfy the constraints. This step is an intermediate step to get rid of the irregularities in the edge areas and get a nearly equidistant interior net with a uniform force distribution.

4. In this, the last step, all the fixed nodes have their original positions given by Table 3.3. The force constraints for all interior cables assigned with unstrained length constraints are removed. Only interior cables connected to the edge ca-bles still have force constraints. The reason for this modification is that for a three-dimensional cable net both force and unstrained length constraints can-not generally be satisfied within the net. However, for the plane configuration in step 3 it is possible to satisfy both constraints. To obtain the final shape shown in Figure 3.9 required 19 iterations.

The reason for the somewhat slow convergence of the non-linear force density method, if one compares with the Newton-Raphson technique used in finite element analy-ses, is probably due to the highly distorted meshes in some parts of the net. If

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

large changes in the force densities of only a few elements are needed to satisfy the constraints, slow convergence follows.

Figure 3.8: Too large angles of the cables occur near the edges if only unstrained length constraints are used for interior cables. Dashed line = desired position of the cable. Redrawn from [72].

Figure 3.9: Three dimensional view of the equidistant cable net with smooth force distribution.

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3.4. EXAMPLES

The next example will show that different final configurations will be obtained de-pending on the initial force density values of the edge cables. In the first example the target shape is an interior mesh width of 1 metre and a prestressing force of 200 kN for each of the cables in the edge area. The stiffness of the cables was AE = 1000 MN. For both nets the starting value of the force density for each interior cable was qinterior = 200 kN/m. For the first net, Figure 3.10, the force density in each of the edge cables was qedge = 5qinterior and for the second net, Figure 3.11, it was qedge = 50qinterior. The same procedure as for the previous example was used. To obtain the plane net, with force constraints for all interior cables, 8 iterations were required for net 1 and 7 iterations for net 2. This difference is probably due to the fact that the starting shape of net 2 is closer to the final shape, see Figure 3.11(b).

As above, the final three-dimensional configuration is obtained by removing the force constraints from interior cables not connected to the edge and fixing the support nodes in their original positions. For both nets the final shape was obtained with only 4 iterations. The solutions show that, depending on the starting values of the force densities in the edge cables, which are unconstrained, different configurations and force values are obtained.

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

(a) Initial configuration. qinterior = 200 kN/m and qedge= 1000 kN/m.

(c) Final configuration. Coordinates of fixed points (m) and edge cable forces (MN).

Figure 3.10: Hyperbolic paraboloid net 1.

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3.4. EXAMPLES

(a) Initial configuration. qinterior = 200 kN/m and qedge= 10000 kN/m.

(c) Final configuration. Coordinates of fixed points (m) and edge cable forces (MN).

Figure 3.11: Hyperbolic paraboloid net 2.

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

One of Frei Otto’s most famous membrane structures is the star-shaped pavilion over the dance fountain in Cologne. It was built for the 1957 Federal Garden Exhibition.

The tent is still standing, although it was planned for a single summer [87]. This structure inspired the author in the next example—a star-shaped cable net. The net vertices lie on two circles with radii 14.421 m and 8.165 m, respectively. The height difference between the vertices is 4 m. All cables have the axial stiffness AE = 100 kN. The starting values of the force densities are: 200 N/m for the interior cables, 2000 N/m for the valley cables, 6000 N/m for the ridge cables and 1000 N/m for the edge cables. An unstrained length of 1.8 m are assigned to all net cables that are perpendicular to the ridge cables. Force constraints are assigned to all interior cables (not ridge or valley cables) and ridge cables. The force values are 200 N and 8000 N, respectively. To obtain the final shape, shown in Figure 3.13, 10 iterations were required.

Figure 3.12: The pavilion over the Cologne Dance Fountain. Reproduced from Ar-chitectural Design, No. 117, “Tensile Structures”, 1995.

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3.4. EXAMPLES

(a)

(b)

Figure 3.13: A star-shaped cable net structure inspired by Frei Otto’s pavilion over the Cologne Dance Fountain.