Specific Requirements of Membrane Structures
Technical Article
This paper is focused on the specific aspects of designing membrane structures. These structures have specific requirements such as formfinding and cutting patterns generation. These topics are the focus of many research works and there are many methods proposed to achieve the desired results. However in this paper these issues will be discussed more from the practical point of view than from the theoretical one. The paper will be followed by examples, which will complement the discussed topic and show the special attention required during the design process.
Introduction
Membrane structures are one of current trends in civil engineering, since they are beautiful, light, statically effective and also challenging. There are many tasks an engineer has to face during their design and construction.
When designing these structures, the shape cannot be chosen freely, but it has to be calculated. For generation of the proper shape the formfinding process has to be used where one of the shaping factors is the prescribed prestress. Although the engineers define the prestress using specific value in warp and weft, the resulting prestress of the formfinding process often differs from the prescribed one. These differences will be discussed in the following text and will be supported by examples.
When the shape is proposed and the membrane structure is evaluated using nonlinear static analysis, the cutting pattern generation process has to be performed in order to manufacture the structure. This step is usually the most sensitive task in the whole design process. Current possibilities of the cutting pattern generation process will be presented. This chapter will be followed by practical examples as well.
FormFinding
As mentioned above, the shape of membranes and cables cannot be chosen freely, but has to be calculated. This task is an essential connection between the design and the physical laws [1]. The shape is the result of given boundary conditions and equilibrium of forces in space. These forces are the sum of the required prestress, pressure for pneumatic structures and eventually other loads such as selfweight, which often have little influence. By prescribing the boundary conditions and the required prestresses, we are able to create an enormous variety of shapes [2, 3].
Defining the boundary conditions can be usually fulfilled absolutely while it is generally not possible to achieve this for the prescribed prestress. When the software for formfinding is used, prestress for warp and weft is required. However, the prestress in the membrane is often much more diverse than both input values. There arises the question why this prestress is different than the one prescribed and further which desired prestress can be physically achieved and which one not. Moreover, if different formfinding tools are used by civil engineer, they usually generate different resulting prestress for the same input values. Here arises another question, which solution is more optimal.
First we will deal with the possibility of existence of defined prestress. Membrane structures have double curvature, therefore the Gaussian curvature is not zero. This leads to the fact that orthotropic prestress with only one value of prestress for warp and one value of prestress for weft cannot exist in the entire structure. The only exception is isotropic prestress which can exist if it is stable within given boundaries. When two different values of prestress in warp and weft are used, the resulting shape will have stresses of values which can be close to the input values but can not be equal since it is theoretically impossible.
As stated, it is possible to have accurate isotropic prestress in the membrane if such shape is physically possible. This solution exists and can be achieved for most shapes, such as hypar, barell, vault and inflatable (Figure 01). For conical shapes isotropic prestress is not physically stable. Isotropic prestress is possible also for much more complicated shapes, where there are no conic regions.
Figure 01  Basic Shapes of Membrane Structures [4]
The first example of formfinding will be shown on the hypar structure (Figure 02) for both, isotropic and orthotropic prestress respectively. Different results for formfinding with isotropic prestress requirement will be shown and further discussed.
Figure 02  Hypar Membrane Structure
Figure 03  Warp (red) and Weft (green) Orientation
The orientation of warp fibers runs from one high point to the next high point (Figure 03). The required prestress for the first formfinding calculation is n_{warp} = n_{weft} = 2.00 kN/m. The results will be shown as vectors of the principal internal forces using the color scale.
Structural engineers often find themselves in the situation where different softwares find different solutions for the same input values in the formfinding process. In practice it is quite common that the resulting shape has concentrations of forces in the corners (Figure 04, right). However, the exact isotropy can also be reached (Figure 04, left).
Figure 04  Vectors of the Principal Internal Forces n1, n2
The question arises, which result is right. From the theoretical point of view, both shapes are in equilibrium, therefore both shapes are realizable. However, the left example shows a more uniform use of material and more uniform longterm effects, such as creep. When further loads are applied, the corners of the left membrane will break later than the corners of the right membrane. Generally, it is advantageous to find a shape with prestress as smooth as possible, without local concentrations, so the entire membrane is well tensioned and simultaneously the load bearing capacity is not reduced by excessive tension in some regions.
As it was mentioned before, isotropy is the only homogenous prestress which can be precisely achieved. The achieved precision is limited practically only by the mesh size. For larger elements there will be higher deviation since these elements cannot approximate the corresponding shape as precisely as in the fine mesh, but this deviation should still remain within a small range and no significant concentration should appear.
When orthotropic prestress is required for the structure, the magnitude of prestress in warp and weft will oscillate around the input values but never reach the exact magnitude of the input values, since it is not theoretically possible. However, a shape with prestress that has result values very similar to the input values can be achieved. In our case the input values are n_{warp} = 4.00 kN/m and n_{weft} = 2.00 kN/m (Figure 05). Again, concentrations should be avoided for such an orthotropic definition and the resulting prestress should be smooth.
Figure 05  Vectors of the Principal Internal Forces n1, n2
For most shapes, such as hypar, barell vault and inflatable, concentrations can be avoided and the prestress can be smoothly distributed in the membrane structure. However, for conic structures with highpoints or lowpoints, the regions of prestress concentration cannot be avoided. But still, the concentration is necessary only in the region of the high point, while in the corners bellow there is no need for any concentration (Figure 06).
Figure 06  Vectors of the Principal Internal Forces n1, n2 and Axial Forces N
Furthermore, there is one more way to recognize if the region in the membrane needs the concentration of forces or not. This can be intuitively derived from a simple formula (1). This formula represents the equilibrium of forces, where n1 and n2 are the principal forces, 1/R1 and 1/R2 are curvatures in the directions of those forces and p is the external load if defined in the formfinding process.
$$\frac{\mathrm n1}{\mathrm R1}\;+\;\frac{\mathrm n2}{\mathrm R2}\;\;\mathrm p\;=\;0\;(1)$$
For tensioned structures, where there is no inner pressure and the selfweight has not a significant influence, the equilibrium is given by perpendicular prestresses and opposite curvatures. Generally, we can evaluate if there is a need for a rapid change of curvatures for the proposed structure. If there is such a need, it implies substantial changes of forces. This fits to the cone shape where the tangential and radial curvatures have to be changed rapidly when reaching the top of the structure (Figure 06, Figure 10 conical regions). On the other hand, there is no need for changing the curvatures in e.g. corners of hypar membranes, so there is no need for significant changing of prestresses in those regions (Figure 04 left, Figure 05, Figure 08, Figure 10 hypar parts).
Figure 07  Barrel Vault Membrane Structure
Figure 08  Uniform Isotropic Prestress Displayed by Vectors of the Principal Internal Forces n1, n2
Figure 09  Membrane Structure Composed of Hypars and Conical Parts with FE Mesh
Figure 10  Vectors of the Principal Internal Forces n1, n2
Since formfinding is process of tensioning the structure, the more accurate results will be obtained if whole statical system is incorporated in this formfinding calculation (Figure 04, Figure 05, Figure 06, Figure 08, Figure 10). This interaction of whole statical system is even more important in further nonlinear static analysis.
At the end of this chapter one last fact should be mentioned. Generally, the formfinding procces is characterized as calculation of the form for the given prestress. This can be described by the following formula (2). This formula states that the shape is in equilibrium if there is no change in the virtual work. This virtual work consist of the internal virtual work, where prescribed prestress σ is multiplied by the changes in the strain δê of the membrane and the external virtual work, where the external load p acting on the structure is multiplied by the changes of the deformation δu of the membrane [5, 6, 7].
$$\;\mathrm{δW}\;=\;\mathrm{δW}^\mathrm{int}\;\;\mathrm{δW}^\mathrm{ext}\;=\;\mathrm t\;\cdot\;\int_\mathrm\Omega\mathrm\sigma:\mathrm{δêdΩ}\;\;\int_\mathrm\Omega\overrightarrow{\mathrm p}\cdot\mathrm{δudΩ}\;=\;0\;(2)$$
Along some theoretical challenges which are necessary to overcome during the implementation into numerical methods, another general problem arises. This formula assumes that the internal prestress σ is known. However, except the isotropic prestress, it is practically impossible to define in advance the spatial prestress in equilibrium. Therefore, two values of prestresses, one for warp and another one for weft, are defined although they cannot be reached exactly. Then, there is the challenge to find the prestress in equilibrium, which will be as close to those input values as possible. Therefore, formfinding should not be considered only as a process of finding unknown shapes, but as a process of finding unknown shapes for a generally unknown prestress approximated by two defined values.
Cutting Patterns Generation
One of the most characteristic features of membrane structures is their double curvature. Since these structures have to be manufactured from roles of textile, the spatial shape has to be converted into patterns in plane. This process consists of two essential steps, dividing the spatial shape by cutting lines and flattening the prestressed spatial patterns into relaxed flat patterns.
In order to cut the structure, theoretically, any line can be used, but for practical reasons the most common line is the geodesic section. Having straight patterns after flattening is a wellknown advantage of the geodesic cuts. When planar cuts are used the patterns are curved. This statement can be proved by the example of two hypars, where geodesic (Figure 11, left) and planar cuts (Figure 11, right) are used. The resulting patterns are shown in figure (Figure 12).
Figure 11  Hypars Divided by Geodesic Cuts (left) and Planar Cuts (right)
Figure 12  Cutting Patterns Created by Geodesic Cuts (left) and Planar Cuts (right)
The second step of cutting pattern generation is a much more complex task since the closest approximation of spatial pattern is calculated in the plane. There are many methods proposed for this analysis [8], some of them are based on a simplified geometrical approach, others on more advanced mathematical mapping and recent advanced methods are based on the nonlinear analysis performed by finite element method (FEM) [9].
This last method is the most general approach when solving the flattening process using nonlinear analysis and there is the possibility of taking into account the material properties. If we do not wish to take into account the orthotropic nature of the fabric and its transverse contraction in the flattening process, isotropic material with Poisson's ratio v = 0 can be used. However, if the intention is to use material data in the flattening process, it is possible to achieve more precise patterns.
During membrane material testing, usually, only axial stiffness in warp and weft directions and the Poisson's ratios are determined. However, the shear stiffness should also be determined. The influence of the shear stiffness will be shown on the following example. There is used one of the middle patterns of the left structure above (Figure 11) with different material inputs for flattening process. The resulting patterns are presented below (Figure 13, Figure 14).
The first material is a coated fabric with orthotropic behaviour:
E_{warp} = 1.600 kN/m,
E_{weft} = 1.200 kN/m,
v_{warp/weft} = 0.05,
G = 400 kN/m.
The second material is an orthotropic fabric mesh without coating:
E_{warp} = 1.600 kN/m,
E_{weft} = 1.200 kN/m,
v_{warp/weft} = 0.05,
G = 10 kN/m.
When observing the resulting shapes of the whole patterns (Figure 13) they look the same, but when they are zoomed, the difference is obvious (Figure 14). From the point of view of the precise material data, the pattern quality can be improved.
Figure 13  Patterns of Coated Fabric (upper) and Fabric Mesh without Coating (lower)
Figure 14  Detail of Patterns of Coated Fabric and Fabric Mesh without Coating
With the flattening process the compensation estimated by biaxial test is also applied to simulate the release of the prestress in the membrane.
Using geometrically nonlinear analysis, with or without taking into consideration material orthotropy, we can calculate flat patterns for the spatial patterns with minimal energetical deviation between them. These calculations using finite element analysis (FEM) are the most natural way and correspond with the way of analysing civil structures.
During calculations in the process of distortion energy minimization, there is the possibility to take into account also other requirements. One of the most common requirements of structural engineers is that the length of the borderlines of the neighbouring patterns has to be same. Another requirement can be setting special compensation for some of the borderlines of the pattern, this is often called decompensation. Using nonlinear analysis the solution with minimum distortion energy is found taking into consideration construction requirements since they are necessary for the manufacturing process itself.
Conclusions
The aim of this paper was to deal with formfinding and cutting pattern generation process from the practical point of view. Since those two processes are crucial for designing membrane structures, the distribution of calculated prestresses by formfinding process was discussed and further the current method for solving of cutting pattern was shown with regard to minimization of distortion energy. The paper was followed by examples calculated in engineering software RFEM [10] for complementation of the presented text.
There is not the intention of the paper to present which solution should be used by the engineer, but there is an intention to present the current possibilities of methods for designing the shape and for calculation of the patterns. The possible existence of prestresses, their distribution within the entire structure and influence of material properties in the cutting pattern generation process was described above.
References
Authors
Ing. Rostislav Lang
doc. Ing. Ivan Němec, CSc.
Ing. Hynek Štekbauser
Institute of Structural Mechanics, FAST VUT v Brně (Faculty of Civil Engineering, Brno University of Technology), FEM consulting Brno
Reviewer
Prof. Ing. Jiří Studnička, DrSc., ČVUT v Praze (Czech Techncal University in Prague)
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Figure 01  Basic Shapes of Membrane Structures [4]

Figure 02  Hypar Membrane Structure

Figure 03  Warp (red) and Weft (green) Orientation

Figure 04  Vectors of the Principal Internal Forces n1, n2

Figure 05  Vectors of the Principal Internal Forces n1, n2

Figure 06  Vectors of the Principal Internal Forces n1, n2 and Axial Forces N

Figure 07  Barrel Vault Membrane Structure

Figure 08  Uniform Isotropic Prestress Displayed by Vectors of the Principal Internal Forces n1, n2

Figure 09  Membrane Structure Composed of Hypars and Conical Parts with FE Mesh

Figure 10  Vectors of the Principal Internal Forces n1, n2

Figure 11  Hypars Divided by Geodesic Cuts (left) and Planar Cuts (right)

Figure 12  Cutting Patterns Created by Geodesic Cuts (left) and Planar Cuts (right)

Figure 13  Patterns of Coated Fabric (upper) and Fabric Mesh without Coating (lower)

Figure 14  Detail of Patterns of Coated Fabric and Fabric Mesh without Coating