The paper discusses effective application of advanced analysis and simulation software in engineering practice. It summarises and classifies procedures for design of structural dimensions. In a series of small examples, it shows features of the convergence of design for allowable stresses.

Nonlinear analysis and simulation software is progressively used to compute the performance of structures. User-friendly structural analysis programs, finite element programs and discrete element programs are commercially available for both general application and special purposes. To date, these tools are not only used to validate code formulas and check exceptional designs, but also directly to design structural dimensions. Clearly, this involves simulations of structural behaviour for several load combinations and construction stages in a number of design cycles. However, it is not always clear how the newly available software tools can be used effectively in engineering practice.

Using advanced software in the dimensioning process is often referred to as performance-based design. Examples in building and civil engineering include damage of a high-rise due to a strong earthquake [Powell 1998] and durability of concrete in a marine environment [Maekawa, Chaube and Kishi 1999]. In mechanical and aerospace engineering similar design methods are already widely adopted and referred to as simulation-based design or virtual prototyping.

Performance-based design has the potential of creating high quality structures for less costs than a traditional deemed-to-satisfy design. In addition it makes it possible to design even more challenging structures of exceptional shapes and dimensions. However, performance-based design needs software that supports all steps in the process. Without user-friendly software it is unlikely to be adopted because it would take more time than a traditional design process. Table 1 shows the different activities in a design process and whether these can be performed best solely by a computer or by an engineer supported by a computer.

Table 1: Activities in a structural design process. Adapted from [Kim, Lee and Hong 1999].

Design procedures are not often published because many experienced engineers incorrectly assume that their methods are obvious. Nevertheless, some design procedures involving nonlinear structural behaviour are described in literature.

Traditional Procedure: Most structures are first designed for the ultimate limit state. The engineer estimates dimensions and uses linear-elastic finite element analyses or structural analyses to determine the flow of forces. The envelopes of section forces for all load combinations are used to select and improve dimensions. This often involves a nonlinear section model specified in a code of practice. If need be, in a second design cycle, linear-elastic analyses are performed to check other performance requirements like deflections and vibrations.

Sometimes, the force flow is estimated manually using equilibrium. For example, the strut-and-tie method for reinforced concrete, which is based on the lower bound theorem of plasticity theory.

Automatic Procedure: Automated design has been applied for reinforced concrete floors [Kolleger et al. 1998]. The procedure includes manly design cycles, which are performed by the software without human intervention. The first cycle starts with minimum reinforcement. The response is simulated to determine the floor’s strength. If it is insufficient, the reinforcement is increased at the location of the largest strain. Simulations and improvements are repeated until the performance is satisfactory.

The algorithm does not arrive at the minimum amount of reinforcement needed for strength but examples show that it comes close [Kolleger et al. 1998]. It can be extended to include a practical reinforcement layout and multiple limit states.

Plastic Procedure: In Switzerland at the ETH Zürich, programs have been developed for reinforcement design in concrete walls [Despot 1995], floors [Steffen 1996], and folded plates [Tabatabai 1996]. The software applies a two-cycle design procedure. In the first cycle a linear-elastic finite element model is used to compute the force flow. Subsequently, the engineer draws reinforcement fields for which the software computes the required reinforcement quantities. In the second design cycle, the software optimises the reinforcement quantities applying plastic optimisation.

A disadvantage of this procedure is that it only includes design for strength since a plastic model does not provide information on deflections or crack widths in the concrete. So, the procedure needs to be supplemented with a model for serviceability limit states.

Eurocode Procedure: The Eurocode [ENV 1992] allows both linear and nonlinear analysis to compute the force flow in a structure. This has been used in a two-cycle design procedure where first a linear model and subsequently a nonlinear model is applied [Wittek and Meiswinkel 1998]. For the nonlinear analysis the code specifies that average material properties shall be used instead of design properties (factored properties). The section forces are used to select dimensions using nonlinear section models and factored material strength.

The procedure includes redistributions due to cracking and creep of concrete. However, due to the average material strengths, components of the nonlinear model will rarely yield. So, the capacity of the structure to redistribute the force flow in the ultimate limit state is not taken into account. For some structures this can result in an over-conservative design.

German Code Procedure: The German code [DIN 1997] allows simulations of the structural performance with factored material properties. This has been used in two ways [Wittek and Meiswinkel 1998]. 1) In the first design cycle a linear-elastic model is used and in subsequent design cycles the reinforcement is reduced until the simulation shows that the performance is as required. 2) The process starts with minimum reinforcement and in subsequent design cycles the reinforcement is increased until the simulation shows that the performance is sufficient.

An important disadvantage of these procedures is that they may need a large number of design cycles, which is not feasible in engineering practice.

Static Wall Procedure: This design procedure [Blaauwendraad 1999] is developed for reinforced concrete walls that are statically loaded. It consists of three cycles and employs a discrete model, which can be upgraded in subsequent cycles. In the first design cycle the model is linear-elastic. The flow of forces is computed, which is used to select initial reinforcement. In the second design cycle the model is partly non-linear. Concrete in tension can crack and distributed net reinforcement can yield. However, concrete in compression and individual reinforcing bars still behave linearly. In the third and final cycle all components of the model behave nonlinearly and the response is a check of the structural performance.

To leave components linear in the second design cycle gives the advantage that the model shows how much strength is needed instead of just failing. Linear components are typically those that can be changed most easily.

Seismic Shear Wall Procedure: A design procedure has been developed for reinforced concrete shear walls that are coupled with beams and exposed to earthquake loading [Fintel and Ghosh 1982]. The procedure uses nonlinear dynamic analysis and accepts damage to the structure in case of a severe earthquake. The first design cycle includes preliminary dimensioning for gravity, wind and earthquake loads. In the second design cycle, earthquake loads are selected to critically excite natural undamped frequencies. Section models include cyclic behaviour, however, they do not break and yielding continues unlimitedly. The software analyses the structural behaviour and the engineer improves the strength of coupling beams such that its plastic deformation can be covered by the available ductility. Usually three to five design cycles are needed to satisfy all performance criteria.

Steel Frame Procedure: A procedure has been proposed [Kim and Chen 1999] to design steel frames for static loading with a geometrical and physical nonlinear simulation program. Material behaviour is modelled realistically. Geometric imperfections and residual stresses are taken into account. The engineer selects initial dimensions by experience and approximation rules. Already in the first design cycle, simulations are performed to check strength and serviceability for the load combinations. If the carrying capacity is smaller than the load, the engineer replaces the member that yields first by a larger size. If the capacity is larger than the load, the engineer reduces the size of the members that do not yield. Simulations and improvements are repeated in subsequent cycles.

Table 2 gives an overview of the summarised design procedures. It shows for each procedure the computations that are adopted in subsequent cycles. Obviously, the word simulation refers to accurate nonlinear computation of structural performance including factored material properties and factored loading (design values). Nonlinear analysis refers to a computation in which the realistic material behaviour has been modified. This can be to improve speed or robustness of a computation or to show the engineer how a structure can be improved. For example, in a computation it can be convenient that when the ultimate strain is reached the reinforcing bars do not break but strengthen instead. This might be referred to as smart material behaviour. It is noted that smart behaviour in this context does not represent a material accurately but rather is a trick to reduce the design effort and the number of design cycles. Linear-elastic and rigid-plastic behaviour may be seen as a special kind of smart behaviour.

Table 2: Computations used in subsequent design cycles of different procedures

Clearly, when structural performance is insufficient we need to make improvements to the structural system or to the dimensions. It can be convenient if the software displays the change of performance per dollar cost for selected design parameters. This is often referred to as sensitivity analysis. It can be applied for any performance criterion and can also be used to optimise a design manually.

In addition to sensitivity analysis other improvement approaches can be used for specific performance criteria. Often applied is allowable stress design. In this approach we select dimensions which only just can carry safely the forces that are computed in a linear or nonlinear analysis. The approach is successful because in many structures the flow of forces varies little with variations in structural dimensions. (Of course, in a statically determinate structure the force flow is actually independent of the dimensions.) In this section the features of allowable stress design are demonstrated in four examples of small structures.

Figure 1 depicts the subsequent steps in design of an initially straight cable. The horizontal axis plots the section area A of the cable. The vertical axis plots the force N in the cable. Both quantities are factored to remove the units. (F is the cable load, E is the elasticity modulus of the cable material and is the allowable stress.) The thick line in the figure represents the structural behaviour as can be obtained by an analysis program. The thin line represents the allowable stress in the cable material. The design process starts with an estimated section area. The thick line in the figure shows the corresponding cable force. Subsequently, the thin line shows which section area is needed to carry this cable force. These steps are repeated until the process is sufficiently converged.

Allowable stress design does not always converge. As an example, Figure 2 shows the design process of a spring that supports a column. The horizontal axis plots the spring stiffness k. The vertical axis plots the spring force N. Again the quantities are factored to remove the units. (F is the column load and l is the column length.) The design process starts with an estimated spring stiffness. In subsequent design cycles the stiffness diverges from the correct solution. It is noted that this example should be analysed with some geometrical imperfection of the column.

Figure 1: Convergence of the design process of an initially straight cable	Figure 2: Divergence of the design process of a spring that supports a column

The process of allowable stress design can be described mathematically as Picard iteration. For trusses it can be shown that the process converges if the following condition is fulfilled in the vicinity of the solution

	(1)

where, is the allowable stress, A_i is the area of the cross section of member i in the truss and N_i is the normal force in this section. In statically determinate trusses condition (1) is always fulfilled because in this case

	(2)

Experience shows that the converged solution of allowable stress design has a minimum amount of material. This is illustrated in Figure 3 for a small truss that we designed. Initially all truss members have equal cross section areas as shown in Figure 3a. After one cycle we define the material volume as 100 %. After five cycles the design is converged, two members have zero area and the volume is reduced to 95 % (Figure 3b). In comparison, Figures 3c and 3d show determinate designs, which each need 114 % material.

Allowable stress design can convergence slowly. Figure 4 shows a column loaded by a force somewhere along its axis. The volume after one design cycle is defined as 100 % (Figure 4a). Clearly, the converged design (Figure 4b) is much better but it takes ten cycles to arrive at this result. Multiple load combinations can reduce the convergence speed even more. This is not practical for manual design improvements. So, In this case we need to accept some over-dimensioning and occurrence of the limit state. However, in automated design the procedure can be maintained until full convergence.

Figure 3. A small truss designed with allowable stress improvements	Figure 4. A column loaded by a force

Many agree that computers will increase the productivity of workers in construction industry. Impressive programs have been developed, which can perform advanced analyses and simulations. It can be expected that this trend will continue and that performance-based design will supplement and perhaps even partly replace codified design in the near future.

It is clearly not feasible to generalise structural design into one unified design procedure. Specific structures and limit states require different procedures. Nevertheless, procedures for design of structural dimensions can be classified as to the subsequent computations that are involved. Hopefully this will encourage exchange of information on this important subject.

Traditional allowable stress design is still a powerful procedure to design and optimise structural dimensions. It can be applied to the full extent in modern structural analysis software.

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