Example 1: Continuous addition of diagonal braces to a 40-story 3-bay framework

In this example, cross diagonals are assumed to hide from view behind the central lift core and are added only to the central bay of the 40-story 3-bay planar frame. These diagonal braces are added continuously from the ground level up to a specific story level denoted by a topological variable. The topological variable determines a story level which separates the frameworks along the height into two parts: stories below which are braced in the central bay; and stories above which use an unbraced rigid frame composed of only beams and columns. The optimal stopping level of the central bracing and optimal element sizes are sought by the hybrid OC-GA so as to minimize the total cost of the structure. The optimization involves searching for the most cost effective structural framework ranging from a pure rigid frame to a fully braced frame. Wind loads, as shown in Fig. 3, are derived based on the Hong Kong Wind Code (1983) using a general terrain wind profile. No gravity loading is applied to this framework example. Element strength design constraints and second-order P – Д effects are not considered.

For the example framework having the cross diagonal braces as the topological and sizing variables and cross sectional sizes of the beams and columns as element sizing variables, the minimum cost function can be stated as follows:

where t. – j 1 (4)

j [ = 1 ifstj< T

in which T defines the stopping level of the diagonal braces. The variable t. denotes the on/off status of the jth pair of diagonal cross braces at jth story. For the story levels above story T, the diagonal cross braces of these levels are removed from the central bay. However, for the story levels below or equal to story T, pairs of diagonal braces are added to the central bay of these levels and their associated costs are included in the cost function of the structure as tj =1. It should be noted that all beams are rigidly connected to columns such that the stability of the framework is always maintained whenever any diagonal brace is removed from the structure.

All columns and diagonal braces are selected from among 36 discrete sections over the range of American AISC W14X22 to W14X730; beams are limited to W18 and W24 shapes selected from among 47 discrete sections over the range of W18X35 to W24X492. To account for symmetry and reversal of wind loads, exterior columns and beams are grouped together over two adjacent stories, as were interior ones. Diagonal braces are also grouped similarly to have the same size once over every adjacent two stories. The numbers of sizing variables for beams, columns and braces are 40, 40 and 20 respectively. Together with the topological variable T, the structure has a total of 101 design variables.

Two major parameters affecting the performance of the hybrid OC-GA, the population size and the probability of OC-operator, are studied in this example. Fixed population sizes of 10, 25, and 50 are used with a value of poc ranging from 0.0 (i. e., a pure GA without any OC operation) to 1.0 (i. e. all child designs undertake the OC operation).

Integer representation is adopted. Uniform crossover is applied with a probability of 80% such that 8 out of 10 of the parent designs are chosen to produce offspring designs. The mutation rates for the topological and element sizing variables are 20% and 5%, respectively. The reason for using a higher mutation rate for the topological variables is to increase the exploration of new creation of different forms of the braced frame. A quadratic penalty function is used for all design constraints. Binary tournament selection with elitism is employed in this example. The OC-GA algorithm is set to stop either when the maximum generation reaches 200 or when the same best-fit design is found for 20 consecutive generations. The optimization is carried out using a Pentium 4 3.0GHz computer with 512 MB memory.

Since the hybrid OC-GA is a stochastic algorithm, five independent runs are conducted with randomly generated initial designs. The average final structure weight produced by the hybrid OC-GA method
for three population sizes is plotted with the different values of poc as shown in Fig. 4. For all runs wherein the probability of OC-operator is non-zero, the average final structure weights are found to be significantly over 20% less than those obtained by the pure GA (where poc=0).

With the use of the local OC operator, most of the final least-weight designs are found to satisfy almost all specified drift constraints, with a slight violation of less than 1% in the drift constraints being found in only a few final designs. For the case of the pure GA runs (where poc =0), over half of the final designs are found to be infeasible with a maximum violation of 10% in lateral drift constraints. Based on the results obtained for this example, the hybrid OC-GA method is able to produce more superior designs than the pure GA method.

Considering the results of all OC-GA runs using a value of poc ranging from 0.05 to 1.0, the average structural weights are fairly uniform among themselves with a small variation within 5% of the smallest value as shown in Figure 4,. This implies that under the current settings of the hybrid OC-GA method for this problem, the performance of the OC-GA algorithm was quite insensitive to the value of poc. When the population size is 10, the variation of the average structure weight against the poc is more fluctuating as compared with that of the population sizes of 25 and 50. It is evident that when a small population size is used, the likelihood of resulting in premature convergence to a local optimum becomes higher.

 Figure 4. Effect of poc on average structure weight Figure 5. Effect of poc on average computer time

Since the values of the best-fit designs generated by the hybrid OC-GA method with different values of poc are similar, the computational time required for convergence becomes a crucial factor to determine the most appropriate value of poc to be applied for this example problem. Figure 5 shows the average computation time used for the runs with different values of poc. Apparently the least computation time required is found when the value of pocis around 0.05-0.1. It is evident that the use of a relatively small value of poc can cause a significant reduction in the computational effort required to produce a reasonable final design using the hybrid OC-GA method. However when pocis larger than 0.1, a gradual increase in the computational time is observed with an increasing poc as shown in Fig. 5. In general, experience indicates that the best value of poc for topological and element sizing optimization of building frameworks is found to be about 0.1, meaning that only 10% of the offspring designs are needed to undertake the local OC sizing optimization in order to achieve the best-fit designs by the OCGA method with the least computational effort.

4.2 Example 2: Design of a 40-story 3-bay framework allowing random addition or removal of diagonal bracings

The effectiveness of the hybrid OC-GA method is further investigated in this more complex example 2 design problem in which pairs of diagonal cross bracing members are randomly added to or removed from the central bay of any two adjacent floor levels of the same 40-story, 3-bay steel framework used in example 1. In addition to 80 element sizing design variables for columns and beams, 40 grouped topological and element sizing variables for braces were introduced to form a design problem having a total of 120 design variables. This design problem involves searching an optimal structure from 220 possible topologies, where 2 represents the on/off topology choices and 20 represents the number of grouped stories. All optimization parameters used for this example are kept the same as that of example 1. Four different population sizes of 10, 25, 50 and 100 are considered with the use of different values of poc ranging from 0.05 to 1.0 for this example.

Fig. 6 shows the effect of the poc on the quality of the final designs obtained by the OC-GA method. The structure weights of all final designs of four different population sizes generated by the hybrid OC-GA method are given as point marks; the average values of these runs of each population size are joined by lines as given in Fig. 6. Similar to the first example, much superior designs are generally found by the hybrid OCGA when compared to that of the pure GA (where poc = 0). Regarding lateral drift performance of these runs, almost all final designs of totally 140 OC-GA runs with various values of poc are found feasible with only a few designs within the smallest population of 10 having slight 2% violation in drift. On the other hand, all the final designs generated by the pure GA (where poc=0) are found to be infeasible with some significant violations exceeding 100% in the specified lateral drift constraints.

As shown in Fig. 6, the larger the population size is used, the better the average results of the optimum designs are generally produced. However, the benefit of using a larger population size is apparently not significant particularly when poc > 0.2. It should be noted that the OC-GA method is able to produce steadily optimum designs with smaller spread of scattered final designs for the larger population sizes of 50 and 100. When the population size is relatively small (10 or 25), a larger spread of scattering of final designs is found as poc < 0.2. To improve the robustness and quality of the optimum designs, a larger value of poc (greater than 0.2) is needed to be applied for smaller population sizes of 10 and 25. When the population size is comparable to the similar order of the number of design variables (i. e. 50 or 100 for this example with 120 design variables), small values of poc, in the range of 0.05 to 0.2, is generally found to produce good quality designs.

The scattering of the final design of all four population sizes diminishes with increasing poc values. This suggests that more frequent application of the OC-operator improves the quality of final designs and produce equally good results in terms of the objective function. Although much heavier designs are generally found by the pure GA method (where poc = 0), the final designs generated by GA are found all infeasible with significant drift violations.

For all the OC-GA runs, the design optimization process is terminated when the maximum generation reaches 200 or when almost the same best-fit design is maintained for 20 consecutive generations. The computational time required for solution convergence of each run is given in Fig. 7. For a given value of poc, the average computer time required is generally found to be linearly proportional to the population size. In other words, the time required for a population size of 100 is more or less two times that of a population size of 50. For all the OC-GA runs of a given population size, the average computer time required increases generally with an increasing value of poc. In general, the larger the population size, the smaller the value of poc should be adopted in terms of computational efforts. For this example, a relatively small value of poc ranging from 0.05 to 0.2 is found adequately sufficient to produce the best-fit designs with the least computational effort.

 Figure 7. Effect of poc on computation time

3. Conclusion

In this paper, a novel hybrid OC-GA method incorporating a local search OC algorithm is presented for simultaneous topological and element sizing design optimization of tall steel building frameworks. The hybridization involves the exploration of topologies by the GA method and the refinement of element sizes by the local search OC algorithm. Two 40-story, 3-bay framework examples were tested to demonstrate the effectiveness of the hybrid OC-GA method on generating more superior optimal designs than conventional simple GAs. The incorporation of the OC operator into the GA process has remarkably improved the quality of the best-fit designs in terms of structural material consumption and computational efficiency. Results of the two examples indicate that robust and rapid solution convergence can be found by the hybrid OC-GA method. The use of a relatively small value of the probability of OC operator ranging from 0.05 to 0.2 is generally found to produce good quality best-fit designs with the least computational effort for a population size with a similar order of the number of design variables. The hybrid OC-GA method promises to become a useful tool for optimizing both the topology and element sizes of large-scale practical tall building structures.

Acknowledgement

The work described in this paper was substantially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKUST6302/04E).

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