form finding of shells by structural optimization

Engineering with Computers (1993) 9:27-35 9 1993 Springer-Verlag London Limited Engineering

C~nputers

Form Finding of Shells by Structural Optimization

K.-U. Bletzinger and E. Ramm University of Stuttgart, Institut fiir Baustatik, D-7000 Stuttgart 80, Germany

Abstract. Shell structures are known to be extremely parameter sensitive; even small changes of the initial design, e.g., to the shape of the shell, may drastically change the internal stress state. The ideal case for concrete shells is a pure membrane stress state in compression for all loading conditions. Since in many realistic situations the solution for an 'optimal' shape is not obvious, the need for form finding methods is evident. This paper presents computational methods of structural optimization as a general tool for the form finding of shells. The procedure as a synthesis of design modelling, structural analysis and mathematical optimization is discussed with special emphasis on the modelling stage. Several examples show the power of the approach and the similarities to experimental solutions.

Keywords. Computer aided geometrical design; Form finding; Shells; Structural optimization

1. Introduction

1.1. Shells as Structures of Optimal Behaviour

Shells are the most efficient structural alternative for a number of extreme situations such as structures of long spans, minimum mass, or high resistance [13. At the same time, they appear very light and graceful, and meet aesthetical demands in a natural manner. There is no doubt that shells are the epitome of structural elegance. The extraordinary behavior of shell structures is caused by the 'double arch effect' which, in contrast to one-dimensional curved structures, allows them to carry several different load constellations by a pure membrane action. That means that shells designed to act as membranes are already optimal structures. In some cases they may also show typical characteristics of 'over-optimized' specializa- tions with high sensitivity with respect to small

Correspondence and offprint requests to: K.-U. Bletzinger, Univer- sitfit Stuttgart, Institut f/Jr Baustatik, Pfaffenwaldring 7, Postfach 801140, D-7000 Stuttgart 80, Germany.

changes of certain parameters, e.g., the reduction of the buckling load due to only small initial imperfec- tions. Since realistic situations of design, such as single loads, support conditions, free edges, or shape incompatibilities, make it very difficult to fulfil the basic membrane-oriented design rules, a modification of the original design could substantially improve the structural behaviour. Additionally, the specific properties of the chosen material have to be considered. It is obvious that the shape of a concrete shell should be designed to avoid tension to the greatest possible extent. The ideal situation would be a pure membrane state in compression, hopefully avoiding any buckling. The problem is to find shapes of that kind.

1.2. Traditional Form Finding Methods

It can be recognized that most shells are of regular shapes which are often analytically defined. This reflects interactions of the usual design practice and classical shell theory which gives closed solutions for analytical defined geometries only. Typical design practice is to experiment with standard geometries like spheres, cylinders, toil, cones and HP-surfaces. To adjust the final shape to the prescribed plan, segments are cut out or different shell types are put together, neglecting the basic conditions of membrane theory with respect to both equilibrium and compatibility. This leads to high bending stresses or large displacements, which are usually avoided by additional stiffening elements. Typical examples are the cylindrical roof shell with edge beams and diaphragm walls at the ends, or the spherical shell over a polygonal plan with heavy beams at the free edges.

An experimental principle long used to find the optimal shape of structures in compression is the hanging model and its inverse. The inverted catenary was used, for example, in 1748 by Giovanni Poleni to compare the shape with Michaelangelo's design of St. Peter's in Rome, and extensively by Gaudi at the beginning of this century for many structures around

28 K.-U. Bletzinger and E. Ramm

Barcelona (chapel in Colonia Gfiell, Sagrada Familia, etc.). Extended to two dimensions, the hanging fabric can be used to find shapes of shells over almost arbitrary plans. In many cases, heavy edge beams can be avoided by this method, to yield pleasing, naturally shaped shells with free edges. This principle was used very successfully and economically in many practical applications by H. Isler [-2, 3].

The basis of the hanging model experiment is that one characteristic load case is used to generate the final shape by large deflections of a given membrane. So far, the method can easily be simulated by modern analysis codes which are able to consider geometrically non-linear structural behaviour. The procedure is powerful [1] but has some drawbacks: for example, it is not possible to find a compromise when different load cases are dominant, or to consider criteria for the genesis of shapes not based on elastic deformations.

2. Structural Optimization - A General Tool of Shape Design

Computational methods which offer a more general approach to shell design than the principle of inverting hanging models are the procedures of structural optimization. Their foundation is a strict separation of structural geometry, mechanical behaviour and the design objective which is responsible for the generation procedure. The idea of form finding by these methods is therefore very much related to the obvious engineering approach:

(1) choose a shape; (2) evaluate the structural behaviour according to the

given load cases and support conditions, by finite element methods, for example;

(3) check stresses, displacements, buckling load and other safety requirements;

(4) compare the quality of the design with the chosen optimality criteria;

(5) if necessary, propose a new and better design by means of sensitivity analysis, and repeat the process.

This procedure is absolutely different from the hanging model and other related principles where the generating rule itself (i.e., mechanical reactions to given loads) is already the criterion for optimality. Nevertheless, the same results can be achieved if equivalent objectives and load conditions are chosen. Because of their general formulation, methods of structural optimization can tackle problems with many load conditions, arbitrary design objectives

which are not necessarily related to mechanical behaviour, and loads such as body forces and support conditions which change with every modification of shape and which can only be solved with difficulty by experiments, and sometimes not even then.

The bulk of investigations is still devoted towards optimization of cross-sections. In shape optimization, most work addresses two-dimensional systems [4]. Shell problems are usually restricted to the axi- symmetric case; relatively little has been described for general shells so far [5]. Usually, as in this paper, non-linear structural response is not taken into account because of the exponential increase in the complexity of the problem. Considerable research is currently underway to include these phenomena.

The methods of structural optimization have reached a remarkable level [6], and they have been used as design tools to improve structural quality in many industrial applications, especially in aircraft [7] and automotive as well as in other mechanical industries. Their potential for alternative design in civil engineering has not yet been fully exploited. This lack of practical acceptance may be explained by some frustrating experiences in the 1960s and early 1970s. But progress in computational methods and hardware since those times is also considerable, and this has rekindled interest in structural optimization.

A typical problem of structural optimization is characterized by an objective f(x) and constraints 9(x) and h(x) which are non-linear functions of the optimization variables x. It can be stated as:

minimize:

subject to:

f(x) hj(x) = 0; j = 1 , . . . , m e gj(x) ~ 0; j -= m e -t- l . . . . , m

x L~

Form Finding of Shells 29

mainly in compression, stress levelling fs can be used E8]:

fs = fv (~ - a")2 dV (3)

where cr, is a prescribed goal of stress. There are other objective functions, such as construction cost, weight, or natural frequencies, which are interesting in the shell structure design and often dominate the two objectives of natural significance already mentioned above. Applying the methods of multicriterion optimization, several, even conflicting, objectives can be considered simultaneously to obtain an 'optimal' compromise of structural design. This is done, for example, when strain energies of more than one load case have to be minimized (see the example in subsection 4.2).

Inequality constraints g(x) are taken into account to impose safety and reliability requirements. Typical constraints of this type are stress and displacement limits. If the stiffness of a structure is to be maximized for a prescribed structural mass, an equality constraint h(x) is introduced. Otherwise, unrealistically massy solutions could be obtained, as is usually the case if external loads dominate the self-weight of the structure.

Form finding implies optimization of geometry. Characteristic optimization variables are therefore geometric parameters defining the structural shape. The number of variables can be reduced dramatically without loss of generality if CAGD concepts are used [-9, 10]. By these methods, shapes of free form shells can be described by the coordinates of a few so-called 'design-nodes' which can be chosen as variables. In addition, the thickness variation can be optimized where discrete thicknesses at design-nodes are taken as variables. The use of these methods in modelling and modifying surfaces will be described in more detail in the next section.

Form finding of shells results in non-linear optimization problems which exhibit many different properties of mathematical optimization. Depending on the objective (strain energy, weight, etc.), the constraints (equality, non-equality) and their combinations, the optimization problems vary from totally uncon- strained (often stress levelling as objective) via little constrained but with equality constraints (strain energy minimization with fixed structural mass) to highly constrained problems like weight minimization, which tends to reduce mass until the limit of material resistance is reached. Therefore, only robust and sophisticated methods can be recommended, such as SQP techniques (sequential quadratic programming)

[11], which are able to handle all these problems, or approximation methods such as the method of moving asymptotes [-12] (MMA), which are superior in special cases like weight minimization. Both methods have been successfully used in shape optimal design of free form shells [-8, 9, 13].

As already mentioned, structural optimization is understood to be a synthesis of various individual disciplines [10]: (a) design modelling (CAGD), (b) structural analysis (e.g., FEM), (c) behaviour- sensitivity analysis, (d) mathematical programming and (e) interactive computer graphics as an important additional aid. Expertise in all these fields is necessary to get satisfactory results. The authors developed their own program system CARAT (Computer Aided Research Analysis Tool) [14, 15] which provides the designing engineer with coordinated program modules. Design modeller, FE-analysis and optimization algorithms are coordinated from the root of development and are integrated in a general design procedure based on a unique in-core database which allows fast data exchanges without any loss of accuracy.

3. Modelling and Modifying Structural Shapes

Design modelling, as an important part of structural optimization, is the backbone of the whole procedure. The general methods of Computer Aided Geometrical Design (CAGD) [16, 17] are the basis of modern pre-processors to design structural geometries in two and three dimensions. Shapes are approximated piecewise by 'design patches'. Within each design path, the resulting shape r is parametrized in terms of shape functions ~i, patch parameters u, v, w, and design nodes rdi which describe the location of the patch in space:

r(u, v, w) = ~ ~i(u, v, w)r~i (4) i=1

Many different shape functions are available, e.g. Lagrangian interpolation, Coons' transfinite interpolation, B6zier and B-spline approximations. De- pending on their formulation, a huge variety of shapes can be described without severe restrictions on the manifold solutions. In shape design of free form shells, one-dimensional cubic B4zier and B-splines, and two-dimensional bi-cubic B6zier patches (Fig. 1) appear to be superior to others. This is because only the corner nodes of those design patches interpolate the resulting shape. The remaining inner nodes only approximate the shape, which yields no differences in


roo

Fig. 1. B6zier patch.

continuity patches

the results compared with equivalent Lagrange interpolation schemes, but allows the construction of continuity conditions between adjacent design patches if composite surfaces are to be defined. This is demonstrated by an example which also reflects the interactive capabilities of CARAT. A plate is defined by four 16-noded B6zier patches as shown in Fig. 2. These elements are connected in such a way that the shape generated is continuous in slopes across the common edges of adjacent patches. To obtain this, corresponding design nodes of the involved patches have to remain on a common line during all subsequent shape modifications. The same rule holds for the second dimension which leads to linear dependencies between, at most, nine design nodes. These topological relations are formulated in super- imposed 'continuity patches' [9]. They are generated automatically and preserved during manual user interactions and shape optimization (Fig. 2(a), (b), (c)). Figure 2 shows different types of continuity patches depending on whether they are connecting two or four design patches, or if they are defined at an isolated corner. In all cases, four nodes are independent and control the locations of the remaining nodes, leading to a reduction of geometrical degree of freedom which is very welcome in structural optimization to stabilize the procedure.

The idea of continuity patches is very helpful in interactive design of free form shells which can serve as initial shapes for subsequent optimization runs or as valuable interactive pre-processor tools for input preparation of complex shapes. Figure 3 shows the

continuity des ign nodes

a) four B6zier patches defining a plate

b) shift design nodes

c) generated continous shape Fig. 2. Interactive surface modification; continuity patches connecting four B6zier patches.

plan of a free form shell described by a total of 16 B6zier patches, and the generated shape modelled by 8-noded isoparametric shell elements. The generated model of a sea urchin shell is shown in Fig. 4, and was the subject of a biomechanical study together with biologists [18]. The stiffening effects of the wrinkles were the main objective of this investigation. The shape was interactively adjusted to measured data using B-splines which were linearly blended in cylindrical coordinates (Coons' interpolation).

Another important fact which must be considered when 'membrane' shapes are to be determined is the generation and modification of corresponding support conditions. CARAT supplies rules to generate support conditions which are tangential and normal to surfaces or edges, respectively, and remain so during the whole form finding process.


Fig. 3. Free form shell using 16 B~zier patches.

M

ground plan generated shape

Fig. 4. Finite element model of a sea urchin shell.

4. Examples

4.1. Bi-parabolie Roof Shell

This example is used to demonstrate the effects of different objective functions and the variety of shapes which can be generated by using only two variables. The structural situation is shown in Fig. 5. A shell of rectangular plan (b = 6m, l = 12m) and uniform constant thickness (t = 0.05m) is supported by diaphragms at the smaller edges. The shape is generated by four B6zier patches. The design nodes are linked (a) to preserve double symmetry and (b) to describe a bi-parabolic surface which can be controlled by two vertical coordinates as indicated. In the initial design, both coordinates are set to sl/2 = 3m, describing a cylindrical shell. The structure is loaded

parabolic shape functions

variable

linked design node \ diaphragm

Fig. 5. Parabolic roof shell: problem statement.

~-- - ' - -b = 6m- - - - -~ "~t ~- - - - -~1 = 12mr

material properties: E = 30,000 MPa, v = 0.2 (concrete)

32 K,-U. Bletzinger and E. Ramm

optimizer: SQP

intial values: Sl = 3m s2 = 3m

a) initial shape

1200 optimal values:

Z~ s2* = 3.12m 1080

~ 960

840

e~

720

600 0 1 2 3 4 5

iterations :

360, optimal values: Sl* = 0.90m

~ 340 i s2 * = 1-97m

b) optimal shape, minimization of strain energy

280 k ~ 260 , ~ .

240: 012345678

iterations c) optimal shape, stress leveling

36 i t optimal values: sl* = 1.64m

34 s2* = 1.34m

~ 32 .~ 30

28

26

24 0123456

iterations d) optimal shape, weight minimi~tion Fig. 6. Parabolic roof shell; initial design and optimal solutions.


load:

snow:

p=5~ nl 2

and dead load

load cases: ~!:!i!ii~:.i!i!i~:.~!~!iiiiiii~i:i~iiiiiiii!~i!iiiiii!:i:~:~:!!i!i~:i!i!iii!i~i~i!!i!!%i| P

I::iiiii]i:i~iiiii~i~i~iiiii!ii]!iiii::iiii!i!::~i:i] g [i!!i!!!i!i!!ii!i!i!ii~-i!i!::iii!iii~iiiiiii!iil]

oround olan: ,i:iiiiiiiiili~ii:i~:~i:i:iiii~il, '~ ::::::::::::::::::::::::

~iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii~,, lOm

Ii:i:!:i:i:!:~:i:!:~:!:!:i:i:!:]:i:] i

1,9-----------I~,~-------I~ lOm lOm

Fig. 7. Free form shell subjected to two load cases.

t o = 8.1cm

,~ lOm =, . . . lOm =,

t* = 8.0cm

initial shape optimal shape

by uniform vertical load p = 5 kN/m 2 (snow). Support conditions are fixed hinges. Because of the symmetry of loads and the structure, only one-quarter of the shell has to be analyzed. This was done by 72 8-noded isoparametric shell elements which are 2 x 2 reduced integrated. The material properties of concrete are chosen as E = 3.0 x 107 kN/m 2, v = 0.2.

In a first optimization run, strain energy was chosen as the objective function without a stress constraint, assuming the structure is sufficiently reinforced to resist high tension forces. The resulting shape (Fig. 6(b)) is an anticlastic surface (HP), very similar to a minimal surface which acts almost like a membrane in tension and compression. Since the structural thickness is fixed, the result is alternatively restricted by an upper bound (6 m) on variable s 1.

To get a more suitable design for concrete, the objective 'stress levelling' was used to reduce tension stresses in the lower fibres of the structure, which are caused by interactions of normal forces and bending moments. A goal stress of o a ~--- - -100 kN/m 2 was prescribed. The optimal structure (Fig. 6(c)) is a synclastic shape (EP) where the area of tension in the lower fibres is reduced to a minimum. Tension cannot

be avoided totally because of the simple shape function and the rectangular plan of the structure. It is remarkable that the diaphragm - although possible - does not disappear. If it vanishes, the resulting shape has a horizontal tangent plane at the corner, leading to negative curvature and increased bending.

By using 'weight' as ~he objective function, any shape between the 'minimal surface' and a plate can be determined, which is forced by the additional constraints on stresses and displacements. Figure 6(d) shows a result obtained with constraints on v. Mises effective stresses are not allowed to exceed an arbitrarily chosen value of a m = 400 kN/m 2.

4.2. Non-regular Shell with Two Loading Cases

A concrete shell of quadratic plan and uniform thickness t (E= 3.4 x 107kN/m 2, v=0.2 , 7 = 25 kN/m 3) has been investigated with respect to strain energy minimization (Fig. 7). The total material volume is kept constant during shape optimization; no further constraints are introduced. Two combinations of dead load and uniform live load p = 5 kN/m 2, which acts either on the entire shell or on one-half of


side views: A-A . B-B

~r. i'

7.3m 18rn I ,d l L ~

I~ r~ 25.3m r . , , i

elan with 23 desien variables s,:

$1 '11

$23 ~'22 - -$21 -$20

lore

10m " 141 ~-4 El v l

continuity patches IS sl

'" ....... " Ss l 0 variable height

~ : ~~ " fixed height l

~ls | X linked height

Is Fig. 8. Geometric model of free form shell.

it, are considered as different load cases. The sum of the non-weighted individual strain energies defines a compromise objective. The initial geometry with a thickness of t = 8.1 cm was evaluated in a preliminary design process. The shape is defined by four Brzier patches. Design variables are linked and continuity patches are introduced to preserve symmetry and continuity of the structure. The optimal shape exhibits a maximal stiffness. Since a pure membrane state is otherwise not possible, the shell needs a boundary stiffened by a distinct negative curvature.

4.3. Tennis Hall

The shell in Fig. 8 is related to the reinforced concrete tennis hall designed by H. Isler [3]. The material data are E = 3.0 x 107 kN/m 2, v = 0.2, ~ = 25 kN/m 3. One load case of dead load plus uniform live load, 25 kN/m 2, is considered. One-quarter of the shell is idealized by 4 Brzier elements linked by continuity patches and by 23 vertical nodal coordinates S~ as design variables (Fig. 8). A total of 126 reduced integrated 8-node shell elements are used for a linear structural analysis. Strain energy is minimized. It can be recognized that the shape near the free edges is sensitive and might even result in a sharp local curvature representing a kind of edge beam. If this is not tolerated by prescribed geometrical constraints,

the optimal shape again shows a clear negative curvature near the free boundary (Fig. 9) as was demonstrated by Isler in hanging model experiments and the related beautiful shell structures.

5. Conc lus ions

Shape-sensitive structures like shells require high quality design, analysis and manufacturing. Therefore, the main objective is a membrane-oriented design, avoiding as far as possible bending, and also buckling phenomena.

The present paper presents the methods of structural optimization as general computational tools to find the shape of shells subjected to different load cases and certain boundary conditions. The key to the approach is a flexible design modeller which allows the generation and modification of even complex shapes by only a few design parameters. Different design objectives can be applied. It was shown that useful information on optimal shells of preferable stress state can be achieved by the objectives of 'minimal strain energy' and 'stress levelling'. Although the static analysis is assumed to be linear in this investigation, the entire optimization procedure is highly non-linear, demanding sophisticated algorithms and experienced personnel. Together with progress in computational sciences and hardware,


Fig. 9. (a) Initial and (b) optimal shape of free form shell.

[3 = .

structural optimization can become a valuable design aid for shell structures, which will reduce the planning time and the experimental expense.

Acknowledgements

This work is part of the research project SFB 230'Natural Structures - Light Weight Structures in Architecture and Nature' supported by the German Research Foundation (DFG) at the University of Stuttgart. The support is gratefully acknowledged. The authors also would like to thank their former research associate and colleague Stefan Kimmich.

References

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2. Isler, H. (1990) Elegante Modelle - die moderne Form des Schalenbaus, Deutsche Banzeitung, db 7, 62-65

3. Ramm, E.; Schunck, E. (1986) Heinz Ister - Sehaten, Kr~mer, Stuttgart

4. Haftka, R.T.; Grandhi, R.V. (1986) Structural shape optimization - a survey, Comp. Meth. Appl. Mech. Engng, 57, 91-106

5. Kruzelecki, J.; Zyczkowski, M. (1985) Optimal structural design of shells - a survey, SM Archives, 10, 101-170

6. Schmit, L.A. (1981) Structural synthesis - its genesis and development, AIAA Journal, 19, 1249-1263

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analyse mit finiten Elementen, PhD Dissertation, Institut f/Jr Baustatik, Universit/it Stuttgart

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10. Braibant, V.; Fleury, C. (1986) Shape optimal design and CAD oriented formulation, Engineering with Computers, 1,193-204

11. Schittkowski, K. (1981) The nonlinear programming method of Wilson, Han and Powell with an augmented Lagrangian type line search function, Numerische Mathematik, 38, 83-114

12. Svanberg, K. (1987) The method of moving asymptotes - a new method for structural optimization, Int. J. Num. Meth. Engng, 24, 359-373

13. Ramm, E.; Bletzinger, K.-U.; Kimmich, S. (1991) Strategies in shape optimization of free form shells, Nonlinear Computa- tional Mechanics - State of the Art (Wriggers, P. and Wagner, W., Editors), Springer-Verlag, Berlin, 163-192

14. Btetzinger, K.-U.; Kimmieh, S.; Ramm, E. (1991) Efficient modeling in shape optimal design, Computing Systems in Engineering, 2, 483-495

15. Kimmich, S.; Ramm, E. (1989) Structural optimization and analysis with program system CARAT, Proc. GAMM - Seminar, October 5-7, 1988, Siegen, on Discretization Methods and Structural Optimization - Procedures and Applications (Eschenauer, H.A. and Thierauf, G., Editors), Lecture Notes in Engineering, Springer-Verlag, Berlin, 186-193

16. Brhm, W.; Farin, G.; Kahmann, J. (1984) A survey of curve and surface methods in CAGD, Comp. Aided Geom. Des., 1, 1-60

17. Faux, I.D.; Pratt, M.J. (1979) Computational Geometry for Design and Manufacture, Ellis Horwood, Chichester

18. Philippi, U.; Nachtigall, W. (1991) Constructional morphology of sea urchin tests, to be presented at the 2nd International Symposium on Natural Structures - Principles, Strategies, and Models in Architecture and Nature, SFB 230, Stuttgart, October 1-4

form finding of shells by structural optimization

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