Requisitos específicos de estruturas de membranas

Artigo técnico

Para o dimensionamento de estruturas de membranas é necessário um procedimento específico, que tem de ser respeitado ao contrário da maioria das estruturas convencionais. A determinação de formas pré-carregadas adequadas e a criação de respetivos padrões de corte têm-se tornado uma parte indispensável no planeamento de estruturas de membranas. O artigo que se segue abordará de forma breve os dois processos principais no planeamento de estruturas de membranas. O objetivo é elucidar as propriedades físicas das estruturas de membranas, recorrendo a exemplos para reforçar as teses.

Introdução

As estruturas de membranas fazem parte das tendências atuais da Engenharia Civil, que se caracterizam pela sua beleza, leveza, eficácia estática e expressão de forma. Devido à rigidez à flexão nula, a forma de uma estrutura de membrana não pode ser separada do seu pré-esforço. 

As formas não podem ser escolhidas livremente, elas têm de ser encontradas. Estas estruturas com formas variadas são feitas com rolos de tecido e folhas. Os padrões de corte são elaborados a partir de faixas de material planas. Os padrões de corte são depois unidos e esticados para estarem na posição final, atingindo assim a estrutura prevista. A determinação dos padrões de corte é uma parte muito sensível durante o processo de planeamento. A sua qualidade tem um forte impacto na qualidade da estrutura no seu todo. O artigo debruçará-se precisamente com estes dois processos, com a determinação da forma (form-finding) de estruturas de membranas e com a determinação de padrões de corte, destacando a aplicabilidade na prática.

Form-Finding

Como mencionado mais acima, a forma das membranas e dos cabos não pode ser escolhida livremente, ela tem de ser calculada. Esta tarefa é uma ligação essencial entre o dimensionamento e as leis físicas [1]. A forma é o resultado das condições de fronteira dadas e o equilíbrio de forças no espaço. Estes forças são o somatório do pré-esforço necessário, da pressão para estruturas pneumáticas e eventualmente outras cargas como o peso próprio, que normalmente têm pouca influência. Ao prescrever as condições de fronteira e os pré-esforços necessários, é possível criar uma variedade enorme de formas [2, 3].

A definição de condições de fronteira pode normalmente ser efetuada na sua totalidade, ao contrário que, geralmente, tal não é possível obter para o pré-esforço prescrito. Quando é utilizado o software para form-finding, é necessário o pré-esforço para a urdidura e a trama. No entanto, o pré-esforço na membrana é muitas vezes mais diversificado que ambos os valores de entrada. Surgem questões como o porquê do pré-esforço ser diferente do prescrito e mais à frente sobre qual o pré-esforço que pode ser atingido fisicamente e qual não pode. Além disso, se forem utilizadas ferramentas de form-finding diferentes, normalmente são gerados valores de pré-esforço diferentes para os mesmos valores de entrada. Neste ponte surge a questão sobre qual a solução que é mais correta.

Primeiro, consideraremos a possibilidade de existência do pré-esforço definido. As estruturas de membranas têm curvatura dupla, de maneira que a curvatura de Gauss não é zero. Isto significa que não pode existir na estrutura completa somente um valor de pré-esforço para a urdidura e um valor de pré-esforço para a trama. A única exceção é o pré-esforço isotrópico, o qual pode existir se for estável nas fronteiras definidas. Quando são utilizados na urdidura e na trama dois valores diferentes para o pré-esforço, a forma resultante terá tensões em grandezas que podem estar perto dos valores de entrada, mas não podem ser iguais, visto ser teoricamente impossível.

Como foi afirmado, é possível ter pré-esforço isotrópico com precisão na membrana, se tal forma for fisicamente realizável. Esta solução existe e pode ser obtida para a maioria das formas, tais como parabolóides, abóbadas cilíndricas, cones ou membranas pneumáticas (Figura 01). Para formas cónicas, o pré-esforço isotrópico, de um ponto de vista físico, não é estável. O pré-esforço isotrópico também é possível para formas mais complicadas, onde não há zonas cónicas.

Figura 01 - Formas básicas de estruturas de membranas [4]

The first example of form-finding will be shown on the hypar structure (Figure 02) for both, isotropic and orthotropic prestress respectively. Different results for form-finding with isotropic prestress requirement will be shown and further discussed.

Figura 02 - Estrutura de membrana na forma de um parabolóide hiperbólico

Figura 03 - Malha de EF e orientação de urdidura (vermelho) e trama (verde)

The orientation of warp fibers runs from one high point to the next high point (Figure 03). The required prestress for the first form-finding calculation is nwarp = nweft = 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 form-finding 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).

Figura 04 - Vetores dos esforços internos principais 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 nwarp = 4.00 kN/m and nweft = 2.00 kN/m (Figure 05). Again, concentrations should be avoided for such an orthotropic definition and the resulting prestress should be smooth.

Figura 05 - Vetores dos esforços internos principais 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).

Figura 06 - Vetores dos esforços internos principais n1, n2 e esforço normal 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 form-finding 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).

Figura 07 - Estrutura de membrana de abóbada cilíndrica

Figura 08 - Vetores dos esforços internos principais n1, n2 para representação do pré-esforço isotrópico uniforme

Figura 09 - Estrutura de membrana

Figura 10 - Vetores dos esforços internos principais n1, n2

Since form-finding is process of tensioning the structure, the more accurate results will be obtained if whole statical system is incorporated in this form-finding 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 form-finding 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, form-finding 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).

Figura 11 - Parabolóide hiperbólico dividido por cortes geodésicos (esquerda) e cortes planos (direita)

Figura 12 - Padrões de corte criados por cortes geodésicos (esquerda) e cortes planares (direita)

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:
Ewarp = 1.600 kN/m,
Eweft = 1.200 kN/m,
vwarp/weft = 0.05,
G = 400 kN/m.

The second material is an orthotropic fabric mesh without coating:
Ewarp = 1.600 kN/m,
Eweft = 1.200 kN/m,
vwarp/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.

Figura 13 - Padrão de corte da textura com tratamento de superfície (em cima) e para a malha parcial sem tratamento de superfície (em baixo)

Figura 14 - Forma diferente dos padrões de corte para a utilização de diferentes materiais

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 form-finding 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 form-finding 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

[1]  Otto, F. & Rasch, B. (1996). Finding Form: Towards an Architecture of the Minimal. Fellbach: Edition Axel Menges.
[2]  Forster, B. & Mollaert, M. (2004). European Design Guide for Tensile Surface Structures. Brüssel: TensiNet.
[3]  Veenendaal, D. & Block, P. (2012). An Overview and Comparison of Structural Form Finding Methods for General Networks. International Journal of Solids and Space Structures 49, pages 3741 - 3753. Amsterdam: Elsevier.
[4]  Architen Landrell: Basic Theories of Tensile Fabric Architecture.
[5]  Bletzinger, K.-U. & Ramm, E. (1999). A General Finite Element Approach to the Form Finding of Tensile Structures by the Updated Reference Strategy. International Journal of Solids and Space Structures 14, pages 131 - 146. Amsterdam: Elsevier.
[6]  Wüchner, R. & Bletzinger, K.-U. (2005). Stress‐Adapted Numerical Form Finding of Pre‐Stressed Surfaces by the Updated Reference Strategy. International Journal for Numerical Methods in Engineering 64, pages 143 - 166. Amsterdam: Elsevier.
[7]  Němec, I. et al. (2010). Finite Element Analysis of Structures: Principles and Praxis. Aachen: Shaker.
[8]  Moncrieff, E. & Topping, B.-H.-V. (1990). Computer Methods for the Generation of Membrane Cutting Patterns. Computers and Structures 37, pages 441 - 450. Amsterdam: Elsevier.
[9]  Bletzinger, K.-U. & Linhard, J. & Wüchner, R. (2010). Advanced Numerical Methods for the Form Finding and Patterning of Membrane Structures. CISM International Centre for Mechanical Sciences 519, pages 133 - 154. Berlin: Springer.
[10]  Dlubal Software: Analysis & Design Software for Tensile Membrane Structures.

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)

Ligações

Contacto

Contacto da Dlubal

Tem alguma questão ou necessita de ajuda? Então entre em contacto connosco ou consulte as perguntas mais frequentes (FAQ).

+49 9673 9203 0

(falamos português)

info@dlubal.com

RFEM Programa principal
RFEM 5.xx

Programa principal

Software de engenharia estrutural para análises de elementos finitos (AEF) de estruturas planas e espaciais constituídas por lajes, paredes, vigas, sólidos e elementos de contacto

Preço de primeira licença
3.540,00 USD
RFEM Estruturas de Membranas
RF-FORM-FINDING 5.xx

Módulo adicional

Determinação da forma (form-finding) de estruturas de membranas e de cabos

Preço de primeira licença
1.750,00 USD