Geometry Optimization in Structural Design.pdf

SEAOC 2014 83rd Annual Convention Proceedings

Geometry Optimization in Structural Design

Alessandro Beghini, PhD, SE, Associate Mark Sarkisian, SE, Partner

Skidmore, Owings & Merrill, LLP, San Francisco, CA

Abstract Several practical tools for the characterization of the optimal layout of material in a structure have been developed in recent years based on both numerical and analytical approaches. This paper revisits such methods and demonstrates their relevance in the conceptual design of high-rise buildings, long span structures and smaller specialty structures. Introduction Structural optimization has been attracting increasing interest in the building industry to identify optimal geometric layouts. Designs based on an optimal material distribution for the structural system are not only efficient, lightweight and minimize the embedded carbon, but are also often aesthetically pleasant from an architectural point of view. Once the geometric layout of a structure and its components are defined during the conceptual phase of design, only minor adjustments are usually performed in later design stages, making the conception of the layout of upmost importance. The optimization tools available to engineers are various and their utilization depends on the specific project or application considered. Commercial software employing gradient based optimization, for example, has been successfully applied for topology and shape optimization. Custom made optimization tools have also been developed by accessing the advanced programming interface of commercial software and utilizing several of their built-in functions. In addition, engineers have been

rediscovering a variety of theorems including Maxwell's theorem and Rankine’s theorem for funicular frames. Maxwell's theorem enables the engineer to quickly calculate the optimal volume of a frame for given loads and boundary conditions while Rankine’s theorem can be applied to identify optimal cable layouts for given loads. This paper provides an overview of the optimization techniques and tools currently used by engineers to conceive optimal and innovative architectural layouts. Particular attention will be given to specific practical applications for a variety of projects. Principal Stresses and High-rise Building Mechanics Principal stress trajectories represent an analytical approach that informs optimal placement of material in a structure. For the case of high-rise buildings, the principal stress trajectories can be traced analytically by numerically solving the governing differential equation as indicated in what follows. In the design of a high rise building, the structure can initially be analyzed as a vertical cantilever beam fixed at the top of the foundation. The fundamental laws of mechanics controlling such problem are rather simple, since the problem is statically determinate. For the sake of brevity, we only underline a few fundamental properties characterizing the behavior of this structure, in particular in relation to the concept of principal stress trajectories. The problem under consideration is described in Fig. 1 where a high-rise building of aspect ratio H/B and unitary thickness is loaded under a uniform wind load,


w. Wind load profiles are technically represented by power laws with exponent depending on the exposure (i.e.: the building location in relation to the surroundings: urban areas, coastal regions, open country, etc.); however a uniform load distribution captures the fundamental aspects of the problem without introducing unnecessary complexity.

Figure 1: Schematic representation of a high-rise building as a cantilever beam of rectangular cross section. Using simple statics, the moment and shear diagram with respect to the elevation x can be calculated as follows:

1 (1)

1 (2)

The cantilever beam problem is treated according to the Euler−Bernoulli beam theory, which hinges upon two important assumptions: the cross section stays plane during the deformation process and remains orthogonal to the beam centerline. Both these assumptions are not accurate for high-rise buildings due to the presence of shear lag and shear deformation; however, as mentioned already, the scope of this paper is to establish a theoretical benchmark against which to compare the numerical results. To this end, we deem the Euler−Bernoulli theory as providing sufficient accuracy. The simple expressions in (1) and (2) are employed to calculate the flexural σx and shear τxy stresses in the cantilever beam under consideration. Notice that in a

beam the stresses σy (stresses transverse to the beam axis) are assumed to be negligible. The formulas to calculate σx and τxy from the centerline moment and shear according to Saint Venant’s Principle for various cross sectional shapes can be found in a variety of solid mechanics textbooks (Sokolnikoff 1951; Timoshenko and Goodier 1987; Love 1944). The plane stress state described by σx and τxy can be rotated in the principal coordinate system to derive the principal stress directions for the principal stresses σ1 and σ2 by solving the following equations derived from the Mohr’s circle:

tan 2 ,

tan 1 (3)

Where is the angle between the principal directions and the Cartesian coordinate system in Fig. 1.

Figure 2. Elevation of the principal stress trajectories for a tubular high-rise building. The solution of the above equations leads to two sets of characteristic lines (see Fig. 2); along these lines there is no shear stress and the normal stress is acting at each location along the tangent to the line. Equation (3) has been solved by finite difference to trace a discrete number of principal stress trajectories. One set of lines


represent compression lines while the other set represents tension lines.

Figure 3. Application of the principal stress trajectories to the design of a high-rise building. The trajectories are acting as streamlines such that the lateral wind force ‘‘enters’’ the continuum at a certain location along the height and flows through the trajectories to the foundation (this is due to the non-shear condition along these lines). Since the principal stress trajectories represent the natural flow of forces in the structure, they offer an analytical method to identify the optimal layout of structural material in a high-rise. The optimality comes from the idea of understanding how the forces are ‘‘moving’’ through the structure to the foundation and embrace this flow with the structural members as indicated in the high-rise design in Fig. 3. The principal stress trajectories in Fig. 2 show the following important characteristics in relation to the behavior of high-rise buildings:

• The tension and compression lines meet at a 45◦ angle at the centerline since in beam theory there is a state of pure shear stress at the centerline. • The stresses at the beam edges are vertical because the problem is purely axial. It can be noted how the lines become very dense toward the edges, emphasizing how in a high rise the most efficient way to carry the overturing moment is to put material as far away as possible from the neutral axis. • The trajectories tend to be more vertical toward the base of the cantilever and closer to 45◦ bracing toward the top. This is caused by the fact that at the top there is mainly shear-type loading while the bottom of the cantilever is controlled by the overturning moment. The principal stress analysis conducted in two dimensions can be extended to three dimensions assuming a cantilever beam with the cross section of a hollow tube (see Fig. 2b). In this case, when the wind is blowing orthogonally to one of the tube faces, the side of the tube parallel to the wind directions are behaving similarly to the two dimensional problem while, in the sides orthogonal to the wind direction, the stress trajectories are mainly vertical. This result emphasizes the typical behavior of a tubular high rise structure which behaves similarly to a simple I-beam section. The faces of the tube orthogonal to the wind direction are acting as flanges and mainly carry the overturning moment, while the faces of the tube along the wind direction are carrying the shear force. It should be noted that the analysis presented in this section is based on uniform material distribution. As the material is redistributed the paths may change. More details on high-rise building mechanics and applications of principal stress to building design can be found in Stromberg et al (2011). Graphic Statics and Rankine’s Theorem Graphic Statics is a graphical method of solving for the forces in a structural frame using two reciprocal diagrams, which can be created using simple drafting tools. Each of the reciprocal diagrams in Graphic Statics consists of a set of points, straight lines interconnecting all the points, and polygons defined by those lines. These polygons can be closed (i.e. defined by areas with


closed loops of finite lines) or open (i.e. defined by chains of finite lines between two nodes with external loads). The lines in the first diagram, called the form diagram, represent structural members, or rather lines of action of the structural members. The lines in the second diagram, known as the force diagram, represent forces carried by the members from the form diagram. For every line of action in the form diagram, there is a reciprocal line in the force diagram. That reciprocal line is drawn parallel to the original line and its length is proportional to the force in the original member. Thus, for a given structure, graphic statics can be a very powerful tool to gain insight to the total load path of the structure, as it provides all of the information about the load and the path using these two reciprocal diagrams. The development of the theory of reciprocal diagrams dates back to the work of Maxwell (1890). More recently Zalewski and Allen (1998) showed possible applications of the methodology for truss design. Several examples of application of Graphic Statics to structural design are also shown in Beghini et al. (2013, 2014a, 2014b). The information about the internal forces can be interpreted from the form and force diagrams as follows: The capital letters, A,B,C...,G are sequentially placed clockwise in the intervals between external forces (open polygons) in the form diagram and the numbers, 1,2,3,..., are placed in the internal spaces (closed polygons) between members. Each line in the form diagram is bordered by two polygons. Thus, a member may be called using the corresponding letter or number of the adjacent polygons, e.g. B-3 or 6-7, and a joint called with a series of letters and numbers, e.g. A-B-3-2-1-A. Similarly, the external forces are called using the adjacent open polygons, for example FAB. The open polygons denoted by capital letters in the form diagram correspond to points (nodes) on the load line of the force diagram, denoted by the lowercase letters, a,b,c,…g.. The numbers denoting the closed polygons in the form diagram also have corresponding nodes in the force diagram. The axial force in a truss member can then be determined by measuring the length of the reciprocal line in the force diagram. The relative scale of the force diagram is set by drawing the load line representing the external forces to a scale. For example, the force in

member A-1 in the form diagram of Fig. 4 is proportional to the length of the line between points a and 1 in the corresponding force diagram. Similarly, the force in the member between polygons 2 and 3 is proportional to the length of the line between points 2 and 3 of the force diagram. The remaining forces in the other members can be computed likewise. Thus, the forces acting on a node in the form diagram correspond to a polygon in the force diagram, where each force is a side of the polygon. For example, at node A-B-3-2-1-A, the force polygon is given by points a-b-3-2-1-a. Reading clockwise around joint A-B-3-2-1-A in the form diagram, we can determine if members A-1 and 2-3 are in tension or compression. Since we go from 1 to A clockwise around the joint, we read from 1 to a on polygon a-b-3-2-1-a, i.e.: the force in A-1 is from left to right, towards the joint A-B-3-2-1-A of the form diagram. Thus, member A-1 is in compression. Likewise, moving from 3 to 2 on the force polygon goes from the upper left to the lower right, or away from the joint in the form diagram, so member 3-2 is in tension. The remaining forces can be interpreted likewise.

Figure 4. From diagram (top) and force diagram (bottom) for a simply supported truss.

Moreover, in Graphic Statics, the member lengths and forces can be determined solely using geometry. These diagrams can be constructed with simple drafting tools


(straight edges, triangles, and a scale) or by the use of simple equations for lines and the intersection of lines required to solve a truss. Within graphical methods, Rankine’s Theorem (Rankine, 1858) was formulated to generalize the solution of the equilibrium of a funicular. An example of the application of Rankine’s Theorem for a cable with a series of loads is shown in Figure 5. Unfortunately, Rankine’s Theorem (and graphical methods in general) progressively lost popularity with the development of mathematical analytical tools at the beginning of the 20th century and the advances in numerical methods and computational power towards the end of the century. However, recently engineers have re-discovered this theorem, which provide powerful insight to the design problem, and have been using it for practical design application as shown in what follows.

Figure 5. Generation of a funicular polygon using Rankine’s Theorem.

Cable Supported Structures with Assigned Load Path The concepts of graphic statics and Rankine’s theorem described in the previous section can be used in a variety of design application as shown by Fivet and Zastavni (2012), Zastavni (2008), and Beghini et al. (2014a,b). In particular, the authors focus here on the use of Rankine’s theorem for the design of cable-nets with a primary cable and secondary cables as shown in Figure 5 supporting an assigned path with prescribed loads. Such path could be the deck of a pedestrian bridge or it could be the edge of a canopy cantilevering off a high-rise building. The red line in the figure indicates the main cable from which secondary cables are strung. The secondary cables support the assigned path. Notice that the force density method (Schek, 1974) has historically been used to identify the layout of cable-nets. However, its original formulation would not be suitable for this application because it is not possible to simultaneously assign the location of a point load and the value of it. Extensions have been introduced in the literature to overcome such issue with non-linear formulations. Beghini et al. (2013) presents a linear analytical approach to the solution of the problem, which gives the designer a significant degree of insight in the equilibrium solution. Such insight would be hidden in the numerical calculations using the force density approach. The first step for the application of Rankine’s theorem is to extend the formulation of the equilibrium of a funicular in a three dimensional space assuming that the three dimensional solution is the composition of two two-dimensional funiculars (Beghini et al., 2013). Therefore, the main cable equations (red line in figure 4) in the coordinate system indicated in the figure are:

∑

(4)

∑

(5)

In the above equations, and are the vertical reactions at one end of the main cable in the z and y direction respectively; H is the level of prestress in the


main cable; are the assigned design loads; are the unknown horizontal thrusts; and . The locations of the nodes of the main cable are the unknown variables of the problem considered and they are defined by the intersection of equations (4) and (5) with the equation for the secondary cables given by:

(6)

, ̅ in the above equation indicates the assigned

coordinates of the path. Notice that the secondary cables are assumed to be in a plane parallel to the y-z plane and that the value of the coordinate is given. Therefore, the unknowns in the above equations are only the coordinates , . Equation (4) is independent of equation (5) and (6), hence the coordinates are defined by the level of prestress considered and the gravity design loads. The

coordinates and the thrust are calculated solving equations (5) and (6) simultaneously. The mathematical details of the solution can be found in Beghini et al. (2013), where the solution for parallel secondary cables described above is also generalized to the case where the cables are non-parallel.

Figure 5. Problem statement. Canopy Design Application The methodology outlined in the previous section has been used to derive the geometry of the currently under-

construction canopy for a high-rise building in China. Figure 5 shows a rendering of the canopy. The final design incorporate two sets of cables: upper and lower. The former resists gravity loads due to the structure self-weight and superimposed loads (glass and other architectural components) while the latter is mainly required to resists the vertical uplift forces due to wind and seismic events. The lateral forces on the canopy are resisted by a grid of in-plane moment connected steel members. The initial cable geometry was adjusted using finite element analysis to account for the cables self-weight. The analysis was also required to check for any potential structural instability since lightweight structures are typically susceptible to it. Other possible cable layouts with non-parallel secondary cables and their impact on the structural performance of the canopy are described in Beghini et al. (2013).

Figure 6. Rendering of the canopy.

Upper cables resist gravity loads and downward forces from wind and seismic events

Lower cables resist uplift loads


Density Methods Topology optimization with density methods is a numerical method that enables the identification of optimal geometries for bracing systems without the need to specify a priori the number of bracing members, their locations and their connectivity. For application in structural optimization for architecture, the design domain is taken to be the outer skin or shell of the building so that the resulting structural system is expressed in the exterior as an integral part of the architecture itself. Thus, the optimal layout problem in terms of an objective function can be stated using the design variables, , and the displacements, , as follows: min ,

. . , 0, 0

(7)

where the design field, , and the structural response, , are related through the equality and inequality constraint functions, . A common optimization problem solved in structural engineering applications is the minimum compliance problem: min ,

. . ,

,

(8)

where represents the equilibrium equation constraint, while is the constraint on the available volume of material for the design, . The global stiffness matrix is given by which depends on the design variables,

; and are the vectors of nodal displacements and forces, respectively. The minimum compliance problem corresponds to the maximization of the system stiffness and it is employed to calculate the optimal material layout in the following examples. In density methods, a void is signified by a null material density ( 0), while 1 represents solid material. For regions of gray material, or intermediate densities, the commonly used Solid Isotropic Material with Penalization (SIMP) model is employed (Zhou and Rozvany (1991), Rozvany et al. (1992), Bendsoe (1989), Bendsoe and Sigmund (1999)):

(9)

This power-law relationship uses the Young’s Modulus of solid material and the penalization power 1 to force the material to tend towards 0 or 1 (void or solid respectively) where the element density assumes a value somewhere in this range. The optimization process can also include continuation on the penalization power from 1 to 4 in steps of 0.5 until convergence.

Figure 7. Picture of the model for the conceptual design of the upper and lower “bridges” spanning several towers in the Zendai competition. An example of the aforementioned topology optimization framework for an architectural design application integrated with structural engineering principles is the conceptual design for the Zendai competition (China) (see Figure 7). The aim in this project was to create a unique and innovative design for the upper “bridge” structure spanning several towers. The design space was approximated as a beam that was discretized with a mesh of polygonal elements to eliminate any mesh bias (Talischi et. al 2010). The gravity load on the mesh was applied as a series of equal point loads at nodal locations. The mesh was constrained with pin supports at the nodes corresponding to the locations where the towers would support the “bridge”. Using topology optimization techniques, the resulting design solution is unique and innovative with an organic look as illustrated by the picture of the model in Figure 7. The results show member lines intersecting at 90 degree angles reminiscent of the geometries of Michell frames (Michell, 1904). In particular, the left-most and


right-most areas of the design domain give way to the development of a bounded Michell-like truss. Additional examples of application of topology optimization in Architectural design are indicated in Beghini et al. (2014).

Figure 8. Rendering of the high-rise building used in the optimization example.

Figure 9. Topology optimization of the outer surface of a high-rise building in China. Resulting contour plot (left) and interpretation of the results in a structural system (right). Another example of application of topology optimization in Architectural design is illustrated in Fig. 8 and 9 for the design of a high-rise tower in China. In this application the outer surface of the building was discretized using shell elements and the topology optimization was conducted using commercial software. The resulting contour plot in Fig. 9 (left) indicates in red the areas requiring larger amount of material and in blue the regions with very low material requirements. The contour plot is later interpreted into an actual structural system for the building as indicated in Fig. 9 on the right.


Combining Continuum and Beam Elements One of the main challenges for the engineer using topology optimization to identify optimal layouts of material for the design is the interpretation of the results. In the prototypical two-dimensional high-rise building problem illustrated in Stromberg et al. (2012) and reproduced in Figure 10 (left), for example, large bands of material tend to form at the ends of the domain to resists the overturning moments induced by the lateral (wind) loading. As a consequence, there is little material available to form the diagonals. In addition, the thick bands of material cause unrealistic flexural stiffness giving spurious results in the layout geometry of the structure and makes the identification of the bracing workpoints very difficult. To circumvent these issues, a methodology has been introduced in Stromberg et. al (2012) to combine continuum and discrete (beam/column) elements. The use of beam elements at the edge of the domain prevents the formation of the thick bands of material and gives a realistic representation of the flexural properties of the column as indicated in Fig. 10 on the right. Figure 11 shows an example of a conceptual design using the combined element technique for a 288m tall high-rise building in Australia, where the behavior is modeled using both 2D continuum (Q4) elements and beam/column elements. The final results show a bracing system in which the densities increase as the load increases throughout the height of the structure, indicating an increase in member sizes of the diagonals. Here, several patterns emerge naturally (i.e. no layout constraints were applied in this study), which provide an aesthetic value to the design as well. The image shows another important result illustrated in Stromberg et al. (2012): the optimal location of the bracing workpoint is not in the middle of the module (i.e.: 45 degrees bracing) but at three quarter of the height. Conclusions This paper describes some the methodologies applied to explore optimal structural systems using both numerical and analytical approaches. Several examples of applications of such methods are discussed. The resulting designs are not only structurally efficient, but also show interesting architectural features.

Figure 10. Issues associated with topology optimization using continuum elements only (left) and advantages of combining continuum and discrete elements (right).


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Geometry Optimization in Structural Design.pdf