An integrated design system for cable reinforced membranes using interactive computer graphics

004W949/81/03026l-20$02.W0 @ 1981 Pergamon Press Ltd.

Computers & Strncluns. Vol. 14, No. 3-4. pp. 261-280. 1981 Printed in Great Britain.

AN INTEGRATED DESIGN SYSTEM FOR CABLE REINFORCED MEMBRANES USING INTERACTIVE COMPUTER GRAPHICS ROBERTB. HABERt University of Illinois, Urbana-Champaign, IL 61801,U.S.A. and JOHN F. ABELS and DONALDP. GREENBERG~ Cornell University, Ithaca, NY 14850,U.S.A. (Received 30 October 1980;received

for publicationI3 January 1981)

Abstract-A computer-aided design program for cable reinforced membrane structures is described. The program makes extensive use of interactive computer graphics techniques for data entry and visual feedback on the current status of the design. Nonlinear finite element analysis is used as the basis of design. Routines are provided for automated generation of finite element meshesusing discrete transfinite mappings; solution of the initial equilibrium problem through least squares force adjustment methods, iterative shape smoothing, or nonlinear displacement analysis; and nonlinear stiffness analysis including frictional slip behavior between membrane material and reinforcing cables. The program is capable of handling both pretensioned and air-supported structures. The overall design approach and structure of the program are emphasized. INTRODUCTION There has been a rapid growth in the use of membrane

tension structures in the past several years. These structures possess significant advantages for enclosing large volumes, For example, the new haj terminal, currently under construction in Jeddah, Saudi Arabia, will cover an area of 105 acres using 210 modules each 150 feet square U,2]. Two major classes of membrane structures can be identified: prestressed membranes and air-supported membranes. Prestressed membranes consist of flexible material stretched between a supporting structure. The supporting structure may be a continuous rigid frame, as in the Florida Pavillion in Orlando, Florida[3], or the membrane may be supported only at selected points with free edges in between, as in the ice rink roof for Edgar M. Queeny Park in St. Louis[4]. Steel reinforcing cables must be used to support free edges of the fabric. Prestressed membranes derive a large amount of their stiffness from a large initial prestress that is applied by cutting the membrane undersize and stretching it into shape. A pressure difference between the interior and exterior provides support for air-supported membranes. This requires that the building be relatively air-tight, with facilities for controlling and maintaining the pressure. The internal pressure does not have to be large since membrane weights are relatively small. The Pontiac Silverdome, the professional sports stadium for the Detroit area, has an air-supported roof which encloses 10 acres[5]. Tension membrane structures have a number of advantages: (1) Membrane roofs are extremely /igIrt weight. (2) Membrane structures are able to cover very large,

tAssistant Professor of Civil Engineering; formerly Research Assistant, Department of Structural Engineering, Cornell University. SAssociate Professor of Civil Engineering. DProfessor of Architecture and Director, Program of Computer Graphics.

c/ear spans. (3) Since there is no need to design the membrane section for flexure or buckling, the membrane section can be fully stressed. (4) Membranes provide a great deal of architectural design flexibility. (5) Membrane structures are well suited to prefabricated construction techniques.

The unique properties of membrane structures present a challenging array of problems to the designer. Solutions to some of these problems have only recently been established and others are still under development. These difficulties can be grouped into three categories. First, membranes typically possess complex, doubly curved shapes. This produces a basic problem of surface representation. Traditional two-dimensional drawings are inadequate for the presentation of these shapes and exact mathematical representations are generally not available for arbitrary membrane geometries. A clear representation of the membrane shape is needed for architectural, engineering and construction purposes. Second, the shapes of membrane structures cannot be chosen arbitrarily. Since membranes possess no flexural stiffness, the membrane shape, the loads on the structure and the internal stresses must interact to satisfy equilibrium at all times. The preliminary design of membrane structures involves the determination of an initial configuration that satisfies this requirement for a single set of loads. The problem of selecting this configuration is called the initial equilibrirtm problem. In addition to satisfying the equilibrium condition, the choice of the initial equilibrium configuration must satisfy all other design and construction requirements. Third, it is also necessary to study the behavior of membrane structures under a variety of load conditions to insure that the structure can withstand all of the forces that it will encounter during its useful life. The lack of flexural stiffness leads to large deformations under loads acting normal to the membrane surface. In some cases the loads themselves will be deformation dependent. There may also be slippage between the membrane and its reinforcing cables. A nonlinear analysis is required to include all these effects in the stress analysis of membranes. 261

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All three of the above sets of problems must be solved in the design of a membrane structure. This demands a combined architectural and structural design approach which is difficult to obtain with the prevalent fragmentation of the design professions. Recent advances in mathematical surface representation, nonlinear finite element analysis and interactive computer graphics technology present possibilities for overcoming these membrane structure design problems. This paper presents a computer-aided design system, based on these methods, that was developed at Cornell University. The membrane design system is intended to provide a unified approach to the design and analysis of tension structures which allows the designer to interact with a computer through intuitive, graphical means. There are three main phases in the design process using this system: initial surface representation, solution of the initial equilibrium problem and nonlinear displacement analysis. This paper is intended as an overview of the design system. Three other papers currently in preparation will present the specifics of the analytical techniques used in the system[9,12,16]. MATHEMATICAL SURFACEREPRESENTA’MON

A prerequisite to the analysis of a membrane structure by the finite element method is the establishment of a finite element mesh to describe the geometry of the structure. It is advantageous to generate the finite element grid directly from a mathematical model of the initial or trial shape of the overall structure. This eliminates the need to define the location of each node point individually and provides a more convenient method for defining and modifying the trial form of a structure. The laws of equilibrium require that funicular membrane structures possess curved surface geometries. In most cases these surface forms are doubly curved and may take on either synclastic or anticlastic local curvatures. Complex boundary geometry is also quite common. Slope discontinuities may arise either on the interior of the surface or along the boundaries. These properties make it difficult or impossible to describe a membrane surface using a single simple mathematical model. Finite element modeling provides an attractive approximate method for surface description. Although finite elements are well suited to the representation of surface geometry for purposes of analysis, they are not well suited to the problem of shape design since each node point in the mesh must be located independently. This node-by-node description requires a large amount of data and provides no unified method of manipulating the surface representation on a global level. Thus there is a need for a surface representation at a level between a single finite element and the entire surface of the membrane. This suggests that a hierarchical partition of the membrane surface be used. The primary partition breaks the entire surface into a series of regions. Each region can be described by a simple standardized mathematical approach to surface representation. As many regions are used as are needed to describe the overall surface geometry adequately. A secondary partition is imposed on each region in the form of a conventional finite element mesh. The node point coordinates for the finite element mesh are generated directly from the mathematical model used to describe each region. Standard shape functions within each element provide the local surface model.

Several techniques are currently available for surface representation in a computer environment. Transfinite mappings [6,71 were chosen in the present system for this purpose. These mappings allow the designer to interpolate a surface through a series of control curves. The generated surfaces match the entire length of the curves exactly. A special form of these mappings in which a discrete representation of the interpolation curves is used was chosen for the present design system. This form of the mappings has favorable properties for finite element mesh generation with interactive computer graphicsPI. Details of the mappings used in the membrane design system are presented in[9]. The membrane design system was designed to allow the user to describe and manipulate the surface descriptions interactively using real-time computer graphics techniques. The use of a digitizing tablet for the input and manipulation of geometric data has been emphasized in lieu of conventional keyboard data entry methods. Feedback on the current state of the design is provided by a refresh vector display capable of generating dynamic perspective images in real time. A cursor is displayed on the vector display to represent the current position of the pen on the digitizing tablet. The main steps involved in developing a surface model are: 1. Data space preparation. 2. Interpolation curve definition. 3. Mesh generation. 4. Local mesh modification. The definition of a hypothetical membrane structure will be used to illustrate the surface description process. It should be emphasized that the above list does not represent a rigid order of operations. It is possible to return to any step at any time to modify previously defined data. This capability provides a highly flexible design environment. The user controls the overall flow of the design process by interacting with the graphics display shown in Fig. 1. Such a display is referred to as a page. The main control page, shown in Fig. 1, contains a list, or MIAU,of the major design functions which the user may choose to activate. Pointing to a menu item activates the design programs for the function indicated. A special page for controlling the specific design function is then displayed. This page contains additional menu items as well as one or more graphical displays of the current structural configuration. Program control is returned to the main control page whenever a particular design process is completed. Data space preparation To begin a design the user points to the DATA SPACE item at the top of the main menu. This transfers control to the the data space preparation page shown in Fig. 2. This page is used to describe and scale a threedimensional space in which the membrane structure will be situated. An orthogonal grid in the X-Y plane is also defined which will be used to obtain numerical precision when inputting geometric plan data through the tablet. The user may change the spacing of the grid lines in the X and Y direction (grid lines per inch on the screen) using either the tablet or the keyboard. The SCALE function is used to assign an absolute scale to the grid (number of structural length units per grid interval). When the structural geometry does not coincide with a uniform grid it may be useful to move some of the grid

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lines away from their regular spacings. This may be accomplished by activating the DRAG function and dragging individual lines with the pen as shown in Fig. 2. Interpolation

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The second step in the mesh generation process is to create the discrete curve descriptions that will be used for interpolation in the transfinite mappings. A discrete curve description consists of a list of key node coordinates representing a series of points along the curve. The locations of the key nodes on a curve determines the number and spacing of the elements that will be generated on the curve. Since the tablet can only digitize two coordinates at a time, the space curves are originally defined as planar curves in the X-Y plane. After the curves have been defined in two dimensions, they are manipulated in three dimensions to form space curves. The definition of the two-dimensional curves is accomplished using the PLAN OUTLINES page shown in Fig. 3. Three general techniques are available for generating discrete curve descriptions: point-by-point description, automated curve generation and inking. In the point-by-point description, each point in the discretized curve key node list is specified independently. Each point may be entered either by specifying its coordinates through the keyboard or by pointing directly to its location using the pen. The automated curve generators allow the designer to describe a mathematical curve definition. This mathematical definition is then automatically converted to discrete form according to the instructions of the designer. The designer may specify the number of key nodes that will be used to define the curve. Weightings may be specified to crowd the key nodes toward either end of

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the curve. This will produce local refinement in the resulting element mesh. Figure 3 shows the input of a circular arc segment defined by three data points. The automatic curve generators available in the membrane design system are: straight lines, quadratic and cubic Lagrange polynomial curves, circular arcs, closed circles, ellipse segments, closed ellipses, and hyperbolas. The third method of curve definition is inking. In this method the designer traces a curve on the digitizing tablet. The curve is initially stored in discrete form using a large number of closely spaced key nodes. When the trace is completed, the curve description is filtered using a numerical procedure so that the curve has the correct number of key nodes at the spacing specified by the designer. A number of subsidiary functions can be activated from the PLAN OUTLINES page, Fig. 3. A SYMMETRY function allows the user to reflect a previously defined curve about an arbitrary axis. The CENTER and ZOOM functions control the translation and magnification of the display. This allows the user to inspect detailed views of any area of the structure. The DELETE function is used to erase previously defined curves that are no longer wanted. Once the curves have been defined in two dimensions, they are ready for manipulation in three dimensions using the 3-D OUTLINES page shown in Fig. 4. This page provides three simultaneous views of the structure: a perspective view, an orthographic plan view and an orthographic elevation view. Each of these views can be rotated, translated and zoomed independently in real time. The FULL switch allows the designer to point to any of the individual views to obtain a full screen display of the view. A hardcopy “snapshot” of the display can

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be obtained by activating the SNAP function. These basic conirol functions are available throughout the membrane design system. Manipulation of the curve data can be performed at four levels: changes may be applied to all the curve data as a unit, to individual curves, to single data points used for curve generation and to individual key nodes in a particular curve list. The user specifies which portion of the data is to be changed by pointing first to the appropriate menu item and next to the curve, data point, or key node to be relocated. Several methods for changing the location of curve data are available. Data may be dragged in the X-Y plane or in the 2 direction by activating the appropriate DRAG function. Figure 4 shows the’inner circle being dragged in the Z direction. The main display port is used as a potentiometer to control the direction and rate of drag with the pen. When the KEYBOARD function is activated the user may control any of the curve manipulation functions through the keyboard instead of through the tablet. The COPLANAR function is used to project curve data into an arbitrary plane specified by selecting three points. The MATCH functions allow the designer to match selected coordinates of any point in the structure to a designated reference point. The SEGMENTS and WEIGHTS switches permit the designer to change the number of key nodes and the spacing of the key nodes on any curve previously defined by one of the automatic curve generators. After a key node has been dragged to a new location, if the PROJECT function is activated the key node will be projected to the nearest location on the mathematical curve from which it was generated. The curve definitions in the present example are completed by returning to the PLAN OUTLINES page and

using the symmetry function. Once the curve description is complete, the designer may view the outlines from any angle as shown in Fig. 5. At this point the designer is ready to proceed to the mesh generation process. Mesh generation The discrete cruve descriptions created in the boundary curve definition phase described above are used as interpolation curves for discrete transfinite mappings. The mesh generation process is controlled by the designer using the CREATE MESH menu page shown in Fig. 6. The user may choose whether to mesh a region with cables or membranes, the form of mapping to be used in discretizing the region, as well as which curves are to be used to describe the region. The three displays provide continuous feedback on the current status of the finite element mesh. The user must select a sufficient number of interpolation curves to define one of the transfinite mappings. Each new curve is selected by hitting the SELECT switch and then pointing to one of the curves in the main display. Complex curves are formed as concatenations of two or more curves by pointing to each of the individual curves in turn. When the curves for a region have been selected the user activates the appropriate mapping option to produce the desired grid. The LOFTING, TRANSFIN and TRIMAP switches activate the uni-, bi and tri-directional mappings respectively. The order of the mapping is selected automatically based on the number of curves selected. The CURVES switch is used to generate a cable model along a designated curve. Once a grid has been computed it is displayed for evaluation by the designer. The element numbering and the node numbering can be displayed using the ELEM

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I’S and NODE #‘S switches. If the grid is acceptable, the STORE switch is used to add the new region to the permanent description of the structure. The designer may then proceed to mesh additional regions of the structure. The DELETE switch is used to remove any existing regions from the structure description. Figure 6 shows a linear lofted region that was generated between two closed curves and Fig. 7 shows a region generated with the bilinear transfinite mapping. The mesh is completed using a second linear lofted region similar to the one shown in Fig. 6. Figure 8 shows the completed mesh in a full screen view. It is possible to modify the curve data base even after the finite element mesh has been produced. Figure 9 shows the 3-D OUTLINES page with one of the boundary curves being redefined by dragging a data point in the X-Y plane. The NEWMESH switch instantly remeshes the structure to produce the grid shown in Fig. 10. Local mesh modification The transfinite mappings provide the designer with convenient methods of defining and manipulating the overall form of the membrane surface. In some cases it becomes necessary to make local modifications to the element mesh to meet the requirements of a particular design. These modifications can be effected using the EDIT MESH page shown in Fig. 11. This page permits the designer to reposition or delete existing nodes, to create new nodes and to add and delete individual elements. The upper set of functions are used to manipulate the nodes in the finite element model. New nodes can be positioned by activating the ADD NODE switch. It is also possible to enter three-dimensional coordinates for

new nodes using the keyboard. The locations of new or existing nodes can be altered using the DRAG-XY, DRAG-Z, MATCH-X, MATCH-Y, MATCH-Z, and COPLANAR functions. The operation of these functions is similar to the equivalent functions which were described for the 3-D OUTLINES page. The user can delete elements from the structural model by activating the DELETE switch and pointing to each element in turn. To add elements to the model the user first selects an element type, activates the ADD ELEM switch and then points to the nodes which define each new element. As each new element is defined it is added to the finite element data base and displayed. Figure 11 shows a series of two-noded elements being added to define a radial reinforcing cable. INITIAL EQUILIBRIUMPROBLEM

Membrane structures are usually constructed so that the structure will experience a significant prestress at all times. Thus, there is generally no compatible unstressed configuration for the entire structure, even if no external loads are applied and the self-weight of the structure is neglected. This means that the designer must specify a stressed reference configuration for the structure. To obtain a workable design, it is important that the reference configuration represent a realistic condition for the structure and that the reference configuration satisfy the laws of equilibrium. This problem is complicated by the fact that membrane structures cannot develope flexural stresses to accommodate unbalanced forces acting normal to the membrane surface.. Instead, these forces must be balanced through careful control of the prestress distribution and the membrane surface shape. The problem

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is further complicated by the fact that some of the loads acting on the reference configuration. such as self-weight and inflation pressures, may be dependent on the shape of the reference configuration. The problem of finding a reference configuration that satisfies the laws of equilibrium has been termed shape finding by some researchers. This nomenclature is inappropriate since it does not adequately describe methods in which variables besides the shape are adjusted to satisfy equilibrium. For this reason the selection of an appropriate reference configuration will be referred to here as the initial equilibrium problem. The basic parameters involved in the initial equilibrium

problem are: (I) Surface topology, (2) Body forces, (3) Surface tractions. (4) Surface geometry, (5) Geometry boundary conditions, and (6) Internal stress distribution. A “solution” to the initial equilibrium problem consists of a combination of these parameters describing a configuration of the structure that satisfies the requirements of equilibrium. Since the initial equilibrium problem is a pure statics problem it is not necessary to introduce the equations of kinematics. The selection of material constitutive properties is therefore not a part of the initial equilibrium problem, although they are required for subsequent stiffness analysis. Nonetheless, some methods for solv-

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ing the initial equilibrium problem do require the designation of material properties, although for these methods the properties need not be the same as those of the actual structure. In some cases fictitious material properties may be introduced to control the solution of the reference configuration. A variety of initial equilibrium solution methods have been developed [lo, 1II. The membrane design system makes use of three different solution methods which can be used individually or in combination to maintain control over the various parameters. Detailed formulations for these methods are presented in [12]. Least squares solution methods The membrane design system contains two related least squares methods for solving the initial equilibrium provlem. In these methods all the parameters for the initial equilibrium problem are specified directly by the designer with the exception of the internal stress distribution which becomes the problem unknown. The geometry of the finite element representation is established using the transfinite mapping methods described above or through a previous analysis. The finite element representation is also used to describe the internal stress distribution with a finite number of unknown generalized stresses. This permits equilibrium equations of the form AT=F

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To be written, in which A is a rectangular matrix based on the known structural geometry, 7 is a vector of unknown generalized stresses and F is a vector of nodal forces. If eqn (1) is overdetermined (more equilibrium equations than unknown stresses) then no exact solution exists. An approximate solution can be obtained via the least squares residual method which minimizes the sum of the squares of the equilibrium error at each nodal degree of freedom. This method is a generalization of the least squares methods for cable networks described in[13,14]. It should be emphasized that this method satisfies equilibrium in a least squares sense only. The accuracy of the solution will depend on the feasibility of the initial problem statement. The stresses computed by the overdetermined method often are not smooth from point to point on the structure. This may require a stress smoothing step to obtain a more nearly uniform stress distribution. If eqn (1) is underdetermined (more unknown generalized stresses than equilibrium equations) then the solution is exact but is not unique. A new method has been developed for the membrane design system in which the designer is able to specify an ideal stress distribution. The deviations in the actual stresses from the ideal stresses are then minimized in a least squares sense subject to the equilibrium constraint via a Lagrange multiplier method. This method satisfies equilibrium exactly and also affords the designer some control over the stress distribution. Iterative smoothing method Another logical choice for the problem unknown is the surface geometry as defined by the nodal coordinates of the finite element mesh. The iterative smoothing method is a new method in which the designer may specify all the parameters directly except the nodal coordinates. The internal stresses are considered to be constant and not

dependent on the final equilibrium shape. The loads acting on the structure may be shape dependent but must be specified in a form so that they may be calculated directly for any given shape. The problem is then to determine a set of nodal coordinates that will establish equilibrium under the prescribed stresses and loads. Equilibrium equations for the solution may be weitten as K,x=F

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in which Kc is the updated Lagrangian geometric stiffness matrix[15] for the structure, x is a vector of nodal coordinates and F is a vector of nodal loads acting on the structure. (See [16] for a derivation of eqn (2)) All terms in eqn (2) occur in the solution equilibrium configuration. Note that eqn (2) is nonlinear since KG is a function of both the constant stresses and the unknown nodal coordinates. In the iterative smoothing method a solution to eqn (2) is obtained by Gauss-Seidel or Jacobi iteration. The term “smoothing” arises from the fact that in these methods the coordinates of each node are expressed as a weighted average of the loads acting on the node and the coordinates of the nodes surrrounding the node. It is necessary to establish a trial shape to initiate the equilibrium iterations. The geometry of the trial shape is established using the transfinite mapping methods or through a previous analysis. Nonlinear displacement analysis approach with slip The nonlinear displacement analysis method has been widely used to solve the inital equilibrium problem[17201. In this method both the surface shape and the internal stress distribution are treated as unknowns. The designer begins by establishing a trial configuration in which all the basic parameters are chosen independently. This configuration is generally not in equilibrium. A stress-strain constitutive relationship is also established for the material in the structure. This law may be chosen to control the initial equilibrium solution and need not represent the actual material in the structure. A nonlinear stiffness analysis is then performed to bring the structure into equilibrium. This generally produces a change in both the stress distribution and the structure shape. Appropriate choices for the constitutive law allow the designer to gain control of either the stress distributions or the dimensions of individual elements but not both. A constitutive tensor with zero components will make the equilibrium stress distribution equal to the trial stresses. This case is similar to the iterative smoothing method described above. Very large components will keep the element dimensions constant but will cause large changes in the internal stress distribution. Intermediate values for the components of the constitutive tensor will result in changes in both the shape and the stresses. Membrane structures are often reinforced with cables that can slide along the fabric. This condition is diffiicult to analyze using the conventional displacement approach. The axial force should be constant along these cables if friction is neglected, or should vary directly with the friction forces. Although the designer may specify a proper set of trial stresses, straining during the displacement analysis will produce an irregular equilibrium stress distribution. The membrane design system incorporates a special slip displacement analysis capability to handle this prob-

An integrated design system for cable reinforced membranes using interactive computer graphics

lem. While the stress magnitudes cannot be selected directly with this method, the change in stress from point to point on the structure can be controlled. This method may be mixed with the conventional nonlinear displacement method by allowing slip only in certain parts of a structure. The parts of the membrane design system used for nonlinear displacement analysis are described in the below section on large deformation analysis. Combined approach

Each of the methods described above is useful for solving certain cases of the initial equilibrium problem, but no single method is optimal for all situations. Ideally, a designer should have direct, independent control over each of the design variables. However, in practice, control of one or more variables must be sacrificed to gain complete control of any single variable. It is possible to use the solution methods in combination to create a more flexible design tool. This combined approach lets the designer experiment with various methods to find the optimal solution. Approximate results from one solution method may be used as the input data to another method to obtain an improved solution. For example, it is possible to combine the shape adjustment of the iterative smoothing method with the stress adjustment of one of the least squares methods. A mathematically defined surface shape may be improved with a small number of shape adjustment iterations. This improved shape is input to one of the least squares methods to obtain an appropriate stress distribution for the shape. If necessary, these stresses may be smoothed. The new stresses can in turn be used as input for further

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shape adjustment iterations. This process is repeated until a desired combination of shape and stresses are obtained. The nonlinear displacement method may be employed as a final correction step. The membrane design system employs a common data base for all the methods used. This permits the designer to move from one method to another without having to reformat the data. Also, the designer is able to recover previous design configurations if the current configuration is unacceptable. Computer graphics displays provide an important means for evaluating the current state of the design. These features make it easy to employ a combined design approach with the membrane design system. Although the surface topology, body forces, surface tractions and geometry boundary conditions are prescribed in all the solution methods, they may be altered interactively using computer graphics techniques at any time during the solution process. Thus in the context of an iterative interactive design methodology these parameters may also be treated as unknowns. Computer graphics techniques for the initial equilibrium problem

The solution of the initial equilibrium problem is controlled interactively by the designer using computer graphics techniques in the membrane design system. This allows the designer to evaluate the current status of the design and to control the solution in a natural manner. Boundary conditions are specified with the menu page shown in Fig. 12. The designer creates a list of nodal restraint types which determine which of the three

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translation and three slip degrees of freedom are free or fixed for a given node. The properties of each restraint type are then attached to various nodes in the structure by pointing to individual nodes or to boundary curves used in the mesh generation process. Figure I2 shows a rigid boundary condition being applied to the edge of the double cone structure by pointing to curves. A solution of the initial equilibrium problem by one of the least squares methods can be obtained using the menu page shown in Fig. 13. The designer is able to set up the solution process graphically. The form of the stress distribution may be constrained in a variety of ways. The stresses in different elements may be constrained to be the same. Membrane stresses may be constrained to be isotropic, or to have principal stress directions coincident with the directions of the warp and fill in woven cloth. Also the stresses in selected elements can be specified directly by the designer. If the underdetermined solution method is used the designer can also describe the ideal stress distribution by graphicat means. After the problem has been described the program automatically selects either the overdetermined or underdetermined solution method and solves the appropriate least squares problem. The designer may also request one or more iterations of stress smoothing if the stress distribution is uneven. Figure 14 shows the menu page used to solve the initial equilibrium problem by the iterative smoothing method. The designer specifies the stress distribution by setting up a list of element stress types. The stress description of each type is then assigned to individual elements, cables, or regions by pointing to them in the display. A uniform, isotropic membrane stress resultant of 360 pounds per linear inch is being assigned to the membrane

elements in Fig. 14. A uniform axial cable force of 15,000 pounds is being assigned to the radial reinforcing cables in Fig. 15. After the element stresses have been specified the designer can initiate the iterative smoothing analysis. The results of each iteration are displayed as they are computed. The designer may interupt the analysis at any point to change the prescribed stresses or the applied loads to control the shape of the structure. Figure 16 shows the trial shape generated by transfinite mappings and Fig. 17 shows the converged initial equilibrium shape for the double cone structure. The trial shape and inflation pressure distribution for an air-supported structure are shown in Fig. 18. The final shape and pressure distribution are shown in Fig. 19. The membrane design system also has the capability of generating shaded images of any structural configuration with the hidden surfaces removed on a raster display device. These displays help the designer to evaluate a design. Changes in shading clarify local surface curvatures and emphasize slope discontinuities which occur where reinforcing cables distort the membrane surface. The design can be modified if the curvatures are too small or the reinforcing cables are found to be ineffective. Figure 20 shows a raster display of a design proposed for an airline terminal cover. Shaded displays can also be merged with optically scanned color photographs of proposed construction sites to obtain realistic architectural renderings as in Fig. 21. LARGE DEFORMATIONANALYSISWITH FRICTIONALSLIP

The solution of the initial equilibrium problem determines a single structural configuration which is in equilibrium under one set of loads. The behavior of a struc-

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An integrated design system for cable reinforced membranes using interactive computer graphics

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Fig. 21. Black and white reproduction of color architectural rendering.

ture under a variety loadings must be determined to insure that stresses and deflections can be kept within tolerable limits. The lack of flexural stiffness in membrane structures leads to large deformations in response to changing loads. Therefore, it is necessary to consider geometric nonlinearities when investigating the behavior of these structures. Membrane structures are often detailed in a way that

permits relative slip between reinforcing cables and the membrane. In general, there are friction forces transmitted between the cables and the membrane associated with the slip. The distribution of friction forces along a cable will constrain the distribution of stresses in the cable. This frictional slip must be considered to model the behavior of membrane structures properly. The membrane design system employs a new stiffness

An integrated design system for cable reinforced membranes

formulation based on a mixed Eulerian-Lagrangian displacement model which can account for frictional slip between membranes and reinforcing cables. In this method the node points can move in the structural continuum to become associated with new material points. Details of the slip formulation are presented in [Ml. This slip analysis capability is embedded in a conventional geometrically nonlinear finite element stiffness analysis for membranes and cables. The membrane design system makes use of computer graphics techniques to specify material properties and loads in preparation for a displacement analysis. The menu page shown in Fig. 22 is used to create a list of material property types. The designer may then associate items on this list with elements in the structure by pointing to individual elements, cables, or regions. The designer may specify a reference load case for the initial equilibrium solution and up to five additional load cases for stiffness analysis using the menu page shown in Fig. 23. Each load case may be composed of any combination of concentrated point loads, uniform pressure loads, snow loads, wind loads and self-weight dead loads. Automated load generation routines are available for each of these load types. Figure 23 shows a wind load distribution on the double cone structre. The load cases may be superposed by specifying an incremental load history for each load case with the menu page shown in Fig. 24. Special editors are also included for specifying prescribed displacements and for describing the slip problem. All of the information needed to perform a stiffness analysis can be generated using the graphical editors of the membrane design system. The graphical input serves to speed data preparation and to reduce input errors.

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After an automatic finite element analysis has been performed the designer may view computer graphics displays of displaced shapes using the menu page shown in Fig. 25. The analyst may view the displacements at any load step and may also scale the displacements to emphasize the structural behavior under load. Figure 25 shows the initial shape and Fig. 26 shows the displaced shape for part of a pavillion structure. CONSLUSIONS

A unified approach to the problems encountered in the computer-aided design and analysis of cable reinforced membrane structures has been described. The solution of these problems requires a combination of technologies based on mathematical surface representation, nonlinear finite element analysis and interactive computer graphics. The membrane design system described represents a powerful tool for the designer. An intuitive approach to design is encouraged by continuous visual feedback of the current status of the design. The use of graphical input techniques helps to speed communication of data to the computer, to eliminate typographic input errors and to facilitate control of interactive analysis. New analysis tools for solving the initial equilibrium problem and for modelling cable slip provide the designer with more control in the design phase and a better understanding of a structure’s behavior. The design and analysis of a membrane roof can be completed in only a few hours with this system. This contrasts with a period measured in weeks needed to complete the same task using other design systems. The savings in time and improved feedback allow the designer to improve and refine his design to a greater extent than was previously possible.

Fig. 22. Menu page for assigning material properties.

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Fig. 23. Menu page for specifying loads.

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An integrated design system for cable reinforced membranes using interactive computer graphics

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Acknowledgemenfs-The work presented in this paper was supoorted bv a research nrant from the Birdair Structures division of Chemfad, Inc. The computer programs were implemented at the Cornell Program for Computer Graphics. The authors wish to acknowledne the aid of John Hollvdav.

II. K. Linkwitz, Combined use of computation techniques and models for the process of form findinn for orestressed nets grid shells and -membranes. Sonderdrick dkr Beitriige zum Internationalen Symposium “ Weitgespannte Flachentragwerke,Stuttgart,pp. 17-28 (1976). 12. R. B. Haber and J. F. Abel, Initial equilibrium solution methods for cable reinforced membranes. Submitted for RFSF.RF.NCES publication. 1. Anonymous, Tent structures: are they architecture? Archi13. W. C. Knudson and A. C. Scordelis, Cable forces for desired tectural Record, pp. 127-134(May 1980). shapes in cable-net structures. IASS Pacific Symp. Part II on 2. Anonymous, The era of swoops and billows. Progressive Tension Structuresand Space Frames, Tokyo and Kyoto, pp. Architecture,p. 110(June 1980). 93-102 (1972). 3. A. Morrison, The fabric roof. Civil Engng, pp. 60-65 (Aug. 14. H. Ohyama and S. Kawamata, A problem of surface design 1980). for prestressed cable nets, IASS Pacific Symp. Part II on 4. Anonymous, Tension design: its ups and downs. Engineering Tension Structuresand Space Frames, Tokyo and Kvoto, DD. News Record, p. 15, (12 Oct. 1978). 103-115(1972). 5. Anonymous, Crane, winches, A-frames, unroll 10 acre, airsupported stadium dome. EngineeringNews Record, p. 17 (4 15. K-J. Bathe, E. Ramm and E. L. Wilson, Finite element formulations for large deformation dynamic analysis. Int. I. Sept. 1975). Num. Meth. in Engng 9.353-386 (1975). 6. W. J. Gordon and C. A. Hall, Construction of curvilinear coordinate systems and applications to mesh generation. Int. 16. R. B. Haber and J. F. Abel, Contact slip analysis using mixed disolacements. Submitted for oubhcation. .I. Num. Meth. in Engng 1,461-477 (1973). 7. R. E. Barnhill, G. Birkhoff and W. J. Gordon, Smooth inter- 17. I. H. Argyris, T. Angelopouios and B. Bichat, A general method for the shape- finding of lightweight tension strucpolation in triangles, J. Approximation Theory 8, 114-128 tures. Computer Methods in App!. Mech. and Engng 3, (1973). 135-149(1974). 8. R. B. Haber, M. S. Shephard, J. F. Abel, R. H. Gallagher and D. P. Greenberg, A general two-dimensional, graphical finite 18. E. Haug and G. H. Powell, Analytical shape finding for cable nets. h4SS Pacific Svmo. Part II on Tension Structures and element preprocessor utilizing discrete transfinite mappings. Space Frames, Tokya and Kyoto, pp. 83-92 (1972). Int. .I. Num. Meth. in Engineering(1981). 19. P. G. Smith and E. L. Wilson, Automatic design of shell 9. R. B. Haber and J. F. Abel, Discrete transfinite mappings for structures. 5th Conj. on Electronic Computation, ASCE, pp. the description of three-dimensional surfaces using inter289-317(1970). active computer graphics. Int. I. Num. Meth. Engng (1981). 20. E. Haug and G. H. Powell, Finite element analysis of non10. R. B. Haber, Computer-aided design of cable reinforced linear membrane structures. IASS Pa& Symp. Part II on membrane structures. A thesis presented to the faculty of the Tension Structuresand Space Frames, Tokyo and Kyoto, pp. graduate school of Cornell University in partial fulfillment 83-92 (1972). for the degree of Doctor of Philosophy, Ithaca, N.Y. (Aug. 1980). ~-~r