Knowledge-Based Global Optimization of Cold-Formed Steel Columns

� Corresponding author.

E-mail address: schafer

0263-8231/$ - see front ma

doi:10.1016/j.tws.2004.01.0

Tel.: +1-410-516-7801; fax: +1-410-516-7473.

@jhu.edu (B.W. Schafer).

tter # 2004 Elsevier Ltd. All rights reserved.

01

Thin-Walled Structures 42 (2004) 785–801

www.elsevier.com/locate/tws

Knowledge-based global optimizationof cold-formed steel columns

H. Liu, T. Igusa, B.W. Schafer �

Department of Civil Engineering, Johns Hopkins University, Baltimore, MD 21218, USA

Received 26 August 2003; received in revised form 31 October 2003; accepted 5 January 2004

Abstract

Cold-formed steel member cross-section shapes are difficult to optimize because of thenonlinear behavior of such members under buckling loads. Traditional gradient-based opti-mization schemes, employing deterministic design specifications for the objective function,are inefficient and severely limited in their ability to search the full solution space of membercross-sections. Herein, a new global optimization approach that is well suited for optimiza-tion of such cross-sections is introduced. There are two distinguishing characteristics of thisapproach: (1) it operates within a low-dimensional expert-based feature space rather thanthe high-dimensional design space of cross-section parameters; and (2) it uses a numericalimplementation of the direct strength method (DSM) for the objective function. Throughthe use of Bayesian classification trees, the most significant coordinates of the expert-basedfeature space are defined; these coordinates are of low dimension and are in terms offeatures which provide insight into structural behavior. The classification trees are then usedto efficiently generate candidate member cross-section prototypes for subsequent refinedlocal optimization.Optimization results are presented for three structurally distinguishable length regimes to

provide proof-of-concept of the proposed scheme. It is demonstrated that an expert-basedfeature space and its associated classification tree can effectively encapsulate the knowledgegained in the design optimization process and can be subsequently used as a startingframework for related design optimization problems. This is, in essence, a highly efficientknowledge transfer mechanism that is absent in most optimization schemes. Optimization ofthin-walled members stands to benefit greatly from the combination of more flexible andgeneral design methodologies (e.g., the DSM) and novel, emerging, optimization schemessuch as the one presented herein.# 2004 Elsevier Ltd. All rights reserved.

H. Liu et al. / Thin-Walled Structures 42 (2004) 785–801786

Keywords: Cold-formed steel; Global optimization; Knowledge; Classifier model

1. Introduction

One of the advantages of cold-formed steel is its flexibility in forming differentcross-section shapes. Ironically, in practice, only limited cross-sections are adopted.Among them, in the US, the C- and Z-shapes are the most widely used. However,these cross-section shapes have never been proven superior to alternatives. In fact,using the same amount of steel, it is not difficult to find cross-section designs withhigher load capacity than the traditional C-, Z- and R-shapes (Fig. 1). Optimizingthe cross-section shape of a cold-formed steel member is interesting from a struc-tural mechanics viewpoint, and due to the vast geometric possibilities in the design,the problem is also challenging from an optimization viewpoint.There has been interesting work in cold-formed steel member optimization

reported in the literature. In this past work, the prescriptive rules of the governingUS design specification (e.g., [1]) have been used for evaluating the objective func-tion. Seaburg and Salmon [2] investigated the optimization of hat-shaped membersusing gradient-based search techniques; Adeli and Karim [3] applied a neuraldynamics model to optimize hat-, I- and Z-shapes; Karim and Adeli [4] also usedthis model to perform a comprehensive parametric study for the global optimum ofhat-shaped beams. Recently, Lu [5] conducted a genetic algorithm (GA) optimiza-tion of Z- and R-shape purlins and used finite-strip analysis within the objectivefunction evaluation.To advance the state-of-the-art of cold-formed steel member optimization, two

major issues must be addressed: the nonlinearities of the strength-based objectivefunction and the inclusion of the set of all feasible cross-sections. The highly non-linear nature of the strength of thin-walled cold-formed steel members is due to thefact that member strength is controlled by a complex combination of overall, dis-tortional and local buckling modes and material strength. Common gradient-basedoptimization methods tend to be unreliable for such highly nonlinear objectivefunctions.Previous optimization searches [2–5] were conducted only within a predefined

scope of prototype shapes. The restriction of cold-formed steel member design to

Fig. 1. C-, Z-and R-shape cross-sections.

787H. Liu et al. / Thin-Walled Structures 42 (2004) 785–801

one or a few fixed cross-section prototypes may lead to suboptimal results as

illustrated by the following example. For a typical C-shape column under pureaxial compression, the local buckling mode is the dominant mode. However, asmall change from this prototype, e.g., the addition of lip stiffeners and web stiffen-

ers, can markedly increase the local buckling stress and make the distortionalbuckling mode dominant, as indicated by Schafer [6]. As a result, load capacity isenhanced. Fig. 1 shows that the number of design variables for typical (a) C-shape,

(b) lipped C-shape and (c) lipped C-shape with one web stiffener are 2, 3 and 5,respectively. Most optimization methods can only handle design variable vectors(X) of fixed dimension. Thus, the three shapes have to be considered as three differ-

ent classes of prototype shapes and be treated as three independent optimizationproblems, which can be inefficient. If we use our expertise [6,7], we know thatshape (c) is usually superior to shape (a) or (b). Some natural questions are: If we

are looking for the optimum design among all three shapes, does this expertise suf-fice, excluding (a) and (b) and optimizing shape (c) only? If we do so, what is the

probability that we would miss the real optimum design? How do we know if thisexpertise is reliable or not? To address these questions, we use, in this paper, aninnovative global optimization method introduced by Liu and Igusa [8–10]. In this

method, knowledge functions are constructed in a manner such that all three cross-section shapes can be included simultaneously in the search. This method hasthe following characteristics that make it well suited for the shape optimization

problem:

1. minor restrictions on prototype shapes;2. no convergence problems;3. direct inclusion of expertise to provide guidelines in optimization formulation

and to improve efficiency; and4. transferable knowledge, where the result of one optimization problem is used to

efficiently solve other similar problems.

2. Optimal design of cold-formed steel columns

In the most common formulation for structural optimization, the design goal isto minimize the weight of the structure to resist a given load. In this paper, a

closely related optimal design formulation, maximum load capacity, is used. Thisformulation is from the perspective of a manufacturer, who, starting from a givenrectangular-sized sheet of steel, would like to design a cross-section with the high-

est possible load capacity. The two formulations are interchangeable. In this paper,we study optimal column design under pure axial compression, so the loadcapacity refers to the axial compression load capacity. The optimization problem is

of the form:

maxX

PnðXÞ ð1Þ


subject to constraints

giðXÞ � 0 i ¼ 1; . . . ; k; hjðXÞ ¼ 0 j ¼ 1; . . . ; l ð2Þ

Here, the design variable vector X specifies the locations and angles of the folds

that define the cross-section; the objective function Pn(X) is the predicted axial

compression load capacity of the column; and the constraints gi(X) and hj(X) are

geometric in nature, ensuring that the folds are within the steel sheet dimensions

and the cross-section contains no intersections.The calculation of the load capacity Pn(X) follows the direct strength method

(DSM) [11], a new design method recently approved as an alternative design pro-

cedure in the AISI specification. This method has two stages: elastic buckling

analysis and prediction of ultimate strength. A variety of applicable rational

analysis methods can be used for elastic buckling prediction, and we choose the

finite-strip method. In particular, the open source code CUFSM [12] is used for

this purpose. The advantage of the DSM analysis is that any cross-section may be

analyzed, thereby permitting a wider search in the design space than traditional

cold-formed steel analysis methods which are tied to conventional cross-section

shapes. The elastic buckling analysis returns the elastic buckling loads of three

modes: local (Pcrl), distortional (Pcrd) and overall (Pcre). The overall buckling mode

includes flexural, torsional and flexural–torsional buckling, and Pcre is the mini-

mum of the critical elastic column buckling loads for these modes. In the strength

prediction of the local buckling load, local–overall interaction is explicitly included.

A typical elastic buckling curve calculated by CUFSM is shown in Fig. 2. The

elastic buckling load Pn, normalized by the compressive yield load Py ¼ AgFy, is

plotted with respect to the half-wave length lhalf-wave normalized by the column

Fig. 2. Elastic buckling curve for the illustrated cross-section.


length l. Note that for a given column, the three characteristic buckling loads arenot necessarily all present.The equation for calculating nominal axial strength Pne for overall buckling is in

terms of the nondimensional parameter kc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPy=Pcre

p:

Pne ¼ 0:658k2c� �

Py for kc � 1:5 ð3aÞ

Pne ¼0:877

k2c

!Py otherwise ð3bÞ

The nominal axial strength for local buckling is in terms of kl ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPne=Pcrl

p:

Pnl ¼ Pne for kl � 0:776 ð4aÞ

Pnl ¼ 1� 0:15Pcrl

Pne

� �0:4 !

Pcrl

Pne

� �0:4

Pne otherwise ð4bÞ

Finally, the nominal axial strength for distortional buckling is in terms of

kd ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPy=Pcrd

p:

Pnd ¼ Py for kd � 0:561 ð5aÞ

Pnd ¼ 1� 0:25Pcrd

Py

� �0:6 !

Pcrd

Py

� �0:6

Py otherwise ð5bÞ

Herein, only the nominal load capacity is considered. We do not include a safetyfactor X (ASD) nor a resistance factor / (LRFD). Thus, our objective function is:

PnðXÞ ¼ min Pne;Pnl;Pndf g ð6Þ

3. Knowledge-based global optimization

In this section, the basic concepts and analysis steps underlying the proposedglobal optimization process are described. For illustration, the first part of theexample, the preliminary design of a relatively long cold-formed steel column, ispresented here; the remainder of the example is presented in the next section.

3.1. Introduction to the example

The design example begins with a plane steel sheet with thickness t ¼ 1 mm andwidth w ¼ 280 mm. The objective is to fold the steel sheet into a shape with uni-form cross-section geometry so that the column can withstand the highest possibleaxial compression load. Considering manufacturing feasibility and costs, werequire (a) either symmetric or antisymmetric cross-section and (b) at most fourfolds on each side of the central longitudinal axis.The Young modulus is E ¼ 210; 000 MPa, the yield strength is Fy ¼ 227 MPa

(33 ksi), and the column length for the long-column design is l ¼ 2:9 m (9.5 ft).


Since there are up to fourfold locations and angles to be considered, the designspace X is composed of two eight-dimensional subspaces. In the preliminarydesign of this column, a lattice is used to discretize this relatively large and com-plex design space. The lattice is constructed by assuming that a fold can occuronly on one of 20 uniformly spaced points on the cross-section and that the angle

at a fold is a multiple of 45v. As shown in Fig. 3, the design vector can be defined

as X ¼ fsym; h1; h2; . . . ; h10g, where sym ¼ true or false indicates whether thecross-section is symmetric and hi is the fold angle at point i, in which at least 5 are

equal to 180v. To reduce the number of duplicate cross-sections, the value of h1 is

limited to 90v

or 135v. Cross-sections which cross themselves are considered

invalid. With this discretization, the number of all possible designs is approxi-mately 1.2 million.

3.2. Formulation of the feature space

The global optimization process begins with the identification of a set of featureswhich, through expert judgment, is believed to be relevant to the optimizationproblem. A feature of a system is a continuous or discrete quantity that providesinformation that may be useful in designing the system. This information must gobeyond the raw information given by the design vector X. Herein, the features ofinterest are those that can be formulated by a deterministic function on X. Thus, afeature fj would be given as a function

fj ¼ FjðXÞ for X 2 X ð7Þ

If m features are identified, then the index would range over j ¼ 1; . . . ;m. While theBayesian classification tree, which we will introduce in the next subsection, canidentify unimportant and duplicate features, it cannot generate new features. Thus,all potentially relevant features must be included at the start of the analysis.

variables in design space X for the initial global optimization: (a) al
Fig. 3. Discrete lowable folding
positions, (b) design variables.


For the column example, the following features were enumerated:

f1 B
oolean variable indicating cross-section symmetry type (symmetric,antisymmetric)
f2 s
trong-axis moment of inertia (Istrong) f3 w eak-axis moment of inertia (Iweak) f4 a spect ratio, for the cross-section with width b and depth h in the strong
and weak principal axes directions (b/h)
f5 d istance between the shear center and centroid (ls) f6 n ormalized length of the longest segment, excluding the lip (last segment)
(dmax/t)
f7 n ormalized length of the lip (c/t) f8 m aximum of f6 and f7 (max{dmax,c}/t) f9 w arping constant (Cw) f10 t he largest ratio of lengths of consecutive segments (maxfdi=diþ1g) f11 r atio of lengths of the lip and its neighboring segment (c=dn�1)
The feature set is the set of all features and the feature vector is f ¼ ff1; . . . ; fmg,where for the column example m ¼ 11. The space of all possible values for the fea-ture vector is the feature space D. In the column example, the feature set wasdefined using expertise in cold-formed steel member analysis and design. Thus,from the knowledge science perspective, the feature vector f contains informationon the structural behavior of the column beyond what is immediately availablefrom the raw design vector X.

3.3. Construction of the knowledge function

In the following, it is shown how a knowledge function can be defined and con-structed to effectively use the information in feature vector f to perform efficientoptimal design. The type of knowledge function that is of interest herein is theclassifier. The most basic classifier is binary, where a design specified by X is eithergood or not. A good design can be defined in terms of the objective function asPnðXÞ > P0, where P0 is a predetermined threshold value. For the column example,P0 would be the lower limit for the strength of a good column.In general, the objective function Pn(X) requires considerable computational

effort, making it ill-suited for direct use in design over large design spaces. Thus, itis useful to define a knowledge function ~PPnðfÞ that sacrifices classifying accuracyfor computational efficiency. While the original objective function Pn(X) can deter-mine, with 100% certainty, whether a given design X is good or not, the knowledgefunction can only provide a likelihood that the designs corresponding to featurevector f is good. The knowledge function is hence defined as follows

~PPnðfÞ � Pr PnðXÞ > P0ð Þ ð8Þin which X can be any design corresponding to feature vector f. When this prob-ability is close to either 100% or 0%, the function is an accurate classifier; onthe other hand, a probability of 50% provides no classification information


whatsoever. Ideally, the knowledge function should be relatively easy to computeand give values close to 100% or 0% for as wide a range of values for f as possible.The first step in constructing such a knowledge function is the identification of aninformation-laden set of features as described earlier. The remaining steps canfollow one of several approaches. Herein, it is shown how Bayesian classificationtrees can be used to define knowledge functions. The details of the learning algor-ithm underlying the evolutionary construction of Bayesian classification trees infeature spaces are described in Refs. [10,13]. In this paper, these details are omittedand only the final trees for the cold-formed steel column design problem are pre-sented.

4. Optimization of cold-formed steel columns

In what follows we will construct knowledge functions for long- and short-col-umn optimal designs. Then we will show how the knowledge embedded in thesefunctions can be transferred to the optimal design of intermediate-length columnsthrough a highly efficient knowledge transfer process facilitated by the underlyingBayesian classification trees. Finally, we will show how the knowledge functionscan be combined with local search algorithms, resulting in a multi-start global opti-mization strategy.

4.1. Knowledge function for long columns

To begin, a training data set is needed to: (1) find an appropriate value for thethreshold P0 for defining good designs, as in Eq. (8), and (2) provide input for theBayesian classification tree learning algorithm. For the long-column design prob-lem, 500 sample designs {X(i)} with valid cross-sections are generated randomly indesign space. For each sample design, the corresponding feature vector fðiÞ ¼ FðXðiÞÞand axial compression load capacity P

ðiÞn ¼ PnðXðiÞÞ are computed. The training data

set is then given by the set of ordered pairs (fðiÞ;PðiÞn ). A histogram of the capacities

normalized by the compressive yield load Py is shown in Fig. 4a. A threshold valueof P0 ¼ 0:3Py is chosen to define good column designs; as indicated in Fig. 4a, only

8% of the training data set fall in the good design category. The training data set isthen used in the Bayesian classification tree learning algorithm. While the initialtrees are large and include all 11 features, the final tree, shown in Fig. 5, is small andis only in terms of features f1 (symmetry), f3 (weak-axis moment of inertia) andf7 (lip length).The tree is interpreted as follows: the rounded rectangles are the nodes, where

the top node is the root of the tree. The binary relation within each node defines afeature criterion. The labeled paths immediately below each node indicate whetherthe feature criterion is satisfied (Y) or not satisfied (N). The remaining rectanglesare the leaves which store the values for the knowledge function ~PPnðfÞ. Thus, theknowledge function value for any feature vector f is determined by starting at theroot and following the appropriate paths leading to a leaf. For example, a columndesign given by a C-section with width b ¼ 0:29w and depth h ¼ 0:42w will have


feature values f 1 ¼ symmetric, f3 ¼ Iweak ¼ 0:0036tw3 and f 7 ¼ 0:29w=t. The

corresponding leaf is the second from the top, with knowledge function value of

3%, indicating a 3% chance that the column design is good, with axial compression

load Pn > 0:3Py.

From the optimization point of view, the value of the classification tree is in

identifying a small subregion of feature space that corresponds to a high likelihood

of good design. For the long-column example, this subregion is defined by the path

leading to the right-most leaf. This path indicates that, out of all antisymmetric

cross-sections with weak-axis moment of inertia of at least 0.0003tw3 and lip length

of at most 0.2w, approximately 93% will be good designs. All other cross-sections

would fall into one of the remaining three leaves, which, as indicated by Fig. 5, all

Fig. 5. Classification tree for long columns (~PPnðfÞ � PrðPnðXÞ > 0:3PyÞ).

Fig. 4. Histogram of Pn/Py for compressive load capacity for long columns: (a) training data set, (b)

Type I and Type II feature subregions.


have low likelihood of good design. Hence, the tree can be used as a classifier,

classifying any cross-section design as either Type I or Type II, as indicated in

Fig. 5, where approximately 93% of Type I cross-sections and less than 8% of Type

II cross-sections are good designs.The accuracy of the tree is checked next. The idea is to check how well the Types

I and II classifications truly reflect high and low likelihood of good design. A

second set of sample designs D ¼ fXðjÞg with valid cross-sections are generated

randomly in the original design space. Using the classification tree in Fig. 5, the set

D is rapidly divided into sets D(I) and D(II) of Types I and II designs. Then, the

load capacity Pn is computed for 500 designs randomly selected from D(I) and

another 200 designs randomly selected from D(II). The results, shown by the histo-

grams in Fig. 4b, do indeed demonstrate the accuracy of the classification tree,

with 94.0% of Type I and 8.3% Type II designs satisfying the criterion Pn � P0 for

good designs. It is noted that 8% of the designs in D fall in D(I), indicating that

only 8% of all valid cross-sections are Type I designs. Furthermore, while Type I

designs are defined by a single, simply defined region in feature space, these desir-

able designs are in numerous subregions scattered throughout the original design

space X. With the classification tree, however, Type I designs can be rapidly ident-

ified in these scattered subregions.After the check of tree accuracy, as presented in Fig. 4b, the best of the Type I

designs, which correspond to the highest load capacities Pn in the figure, are

chosen as candidate near-optimal cross-sections for the final step in global design

optimization. Details of this final step are shown for the intermediate-length col-

umns in Section 5.

4.2. Knowledge function for short columns

For the short-column design problem, the column length was chosen to be the

same as the width of the steel sheet, l ¼ w ¼ 280 mm; the other steel properties are

Fig. 6. Histogram of Pn/Py for compressive load capacity for short columns: (a) training data set, (b)

Type I and Type II feature subregions.


the same as before. The development of the knowledge function follows the sameprocedure as that introduced in the preceding subsection for long-column design.After generating the training data set, the following were found: (a) with the

threshold value of P0 ¼ 0:9Py, 8% of the training data set fall into the good designcategory, as indicated by Fig. 6a; (b) in the final Bayesian classification tree, onlytwo features are relevant, as indicated by Fig. 7; (c) Type I designs are specified bycross-sections with lip length smaller than 0.1w and the longest segment less than0.25w; and (d) the classification accuracy of the tree is very high, as indicated byFig. 6b.It is worthwhile to note that the features in the classification trees in Figs. 5 and

7 have been used in current code design specifications. This demonstrates that themachine-learned classification tree is consistent with the expert knowledge embo-died in design specifications.

4.3. Knowledge transfer for intermediate-length column design

In this subsection, we obtain a knowledge function for intermediate-lengthcolumns without a training data set. This is performed by a knowledge transferprocess whereby the information that is embedded in the knowledge functions forshort and long columns is combined under the supervision of an expert. It is notedthat expert opinion and guidance is critical—thus, the knowledge transfer processis by no means a black-box procedure.To be sure that the previously obtained training data sets would not be closely

related to the intermediate-length column design problem, we choose a length of1600 mm, which is nearly the average of the long and short column lengths. Tofurther complicate the design optimization problem, we change the steel yieldstrength from Fy ¼ 227 MPa (33 ksi) to Fy ¼ 378 MPa (50 ksi).We begin with an expert opinion (with knowledge and insight on the structural

mechanics of cold-formed steel columns) on the column design problem. In thetraining data set for short columns, it was found that column strength is domi-nated by local buckling; hence the classification tree contains the two features (lipand longest segment lengths) most relevant to local buckling. In the training dataset for long columns, overall buckling is dominant and features directly related to

Fig. 7. Classification tree for short columns (~PPnðfÞ � PrðPnðXÞ > 0:9PyÞ).


the overall buckling load capacity such as the weak-axis moment of inertia areused in the classification tree. In intermediate-length columns, both local and over-all buckling modes are expected to be important. Thus, the subregion of featurespace that correspond to cross-sections that can resist both buckling modes wouldbe the most logical place to look for good designs. This subregion would be givenby the intersection of the two Type I feature regions designated by the classi-fication trees in Figs. 5 and 7 for long and short columns. It is noted that the con-trolling features of the distortional buckling mode are not obvious.The classification tree shown in Fig. 8 is the direct combination of the two pre-

vious trees, and the right-most leaf is the desired intersection of the two Type I fea-ture regions. For simplicity, the complement of this intersection, which would becomposed of all remaining leaves, are lumped into a single Type II leaf, as shownin the figure. Since a training data set was not used to construct this tree, theknowledge function values ~PPnðfÞ of each leaf are not immediately known.As before, the accuracy of the new classification tree can be evaluated. Two sets

of sample designs and their associated load capacities are evaluated for Type I andType II feature subspaces. From these sets of sample designs, the following infor-mation is obtained: (1) Type I designs constitute only 0.8% of the entire space. (2)The distributions of load capacities for Type I and Type II designs subspaces,shown in Fig. 9, clearly show that Type I designs have significantly higher loadcapacities. (3) The best designs are characterized in terms of four features: anti-symmetry, weak-axis moment of inertia of at least 0.003tw3, lip length smaller than0.1w, and the longest segment less than 0.25w. With the classification tree, it waspossible to rapidly identify 200 Type I designs. It is noted that if a training data setwith randomly chosen designs was used without the benefit of the classification treein Fig. 8, only one out of 125 samples would fall in the desirable Type I category.Thus, to obtain 200 Type I designs, 25,000 sample designs would have to be

Fig. 8. Classification tree for intermediate-length columns.


evaluated, as compared with only 200 that were evaluated with the help of theclassification tree. While the classification accuracy of the tree, which can be visua-lized by the degree of overlap of the Types I and II histograms in Fig. 9, is not ashigh as those of the preceding two trees, it is noted that the tree in Fig. 8 was con-structed without the benefit of a training data set for intermediate-length columns.

5. Multi-start global optimization

The knowledge-function-based optimization of the preceding section was per-formed in the information-laden feature space. Through the use of classificationtrees, it is found that the most promising designs lie within a well-defined sub-region in feature space. It is found, however, that these designs tend to be widelyscattered in the original design space X. From the optimization viewpoint, suchwidely scattered designs with significantly above-average performance values areideal for the final multi-start, gradient-based local optimization process. This isillustrated in this section for intermediate-length columns, which are the most diffi-cult columns to optimize. We begin with the seed set of the best five designs ident-ified by the tree of Fig. 8; the five cross-sections are shown in Fig. 10.

Pn/Py for compressive load capacity for intermediate-length
Fig. 9. Histogram of columns: Type I and
Type II feature subregions.

Fig. 10. Best designs among the 200 samples at intermediate length.


Since local searches are gradient based, the discrete variables that were sufficientfor the feature-space-based optimization of the preceding section, defined inFig. 3b, must be replaced by continuous variables. Since Type I cross-sections areall antisymmetric, the variable h1 in Fig. 3b is fixed at 90

v. Keeping the maximum

number of folds at eight, there are eight continuous variables to completelydescribe the cross-section geometry; these variables are defined in Fig. 11. The localoptimization problem is formulated as

maxPnðfx1; . . . ; x4;/4; . . . ;/4gÞ ð9Þsubject to the constraints that all segments have nonnegative lengths, i.e., 0 � xi �w=2 for i ¼ 1; 2; 3; 4 and

Pi¼4i¼1 xi � w=2, and the cross-section does not intersect

itself.

. Continuous design variables for local optimiz
Fig. 11 ation.
Fig. 12. Nonsmooth behavior of the objective function and the predominant buckling mode: (a) over a

180vrange for design angle /1, (b) over a 10

vrange for /1.


The objective function Pn is governed by the three buckling modes described inSection 2. As the mode switches, the gradient of the function and sometimes thefunction itself will change abruptly. This is illustrated in Fig. 12a,b, where suddenchanges in the objective function and its derivative can be observed when the angleh1 for the first design in Fig. 10 varies within a 10

vrange. With such high non-

linearity, standard gradient-based searches for the local optimum cannot be used.It is found that a surrogate-based optimization method [14], which uses asmoothed approximation for the objective function that increases in accuracy asthe search converges, is appropriate for this problem. The final local optimizationresults are presented in Table 1 and Fig. 13. It is observed that the local optimiza-tion process provides only a minor (less than 9%) improvement in the compressionload capacity and an even smaller change in the cross-section shapes.Our optimization result is compared with a currently used C-shape. In the

SSMA product list [15], product 600S162-33 has a similar w/t ratio. After pro-portionally enlarging it to make it equivalent to the size of the steel sheet usedhere, we find that the load capacity for this C-shape intermediate-length columnwith 50 ksi steel is 24.2 kN. This is only 38% of the strength of the optimal designdetermined herein. It can be seen that considerable savings could be achieved bysystematically investigating optimal cold-formed steel shapes.

6. Conclusions and summary

It has been demonstrated that knowledge-based global optimization such as thefeature-space-based method introduced herein is well suited to cold-formed steelmember design. The results of the study are summarized in the following.

Table 1

Local optimization results for intermediate-length columns

Seed
Local optimum (kN) Percentage improvement
over initial seed (%)

Number of objective

function evaluations

1
63.7 8.3 1233
2
63.5 5.2 1009
3
61.3 5.3 900
4
62.8 8.1 1065
5
60.5 4.3 896
Fig. 13. Final optimized shapes for intermediated-length columns.


The DSM, which is an entirely new design method for cold-formed steel, is an

enabling tool for advanced optimization schemes and allows for a new exploration

of optimal cold-formed steel cross-sections. Optimized cold-formed steel shapes

have much higher load capacities than commonly used shapes, one example

demonstrates 300% improvement over the common C-shape.Efficiency of knowledge-based global optimization without gradient information

is remarkably high. For the example of an intermediate-length column, by effec-

tively using the knowledge encapsulated in the classification trees for short- and

long-length columns, only 200 objective function evaluations were needed to find

near-optimal cross-sections. The subsequent gradient-based local search required

approximately 1000 objective function evaluations for an improvement in load

capacity of only 8%.Expert knowledge for the optimization process can be quantitatively added and

evaluated using the feature-based classification trees. For long columns, the classi-

fication tree shows that load capacity is primarily governed by the weak-axis

moment of inertias, as expected. The tree also indicates the need to have reason-

able lip lengths, showing that local and possibly distortional buckling are signifi-

cant for thin-walled long columns. The requirement of antisymmetry that is found

in the classification tree is initially unexpected; however, further study shows that

antisymmetric shapes usually have higher weak-axis moments of inertia than

symmetric shapes. For short columns, the classification tree indicates that the

width-to-thickness ratios of the lip and the largest segment must be sufficiently

small to develop reasonable local buckling capacity. This is in agreement with the

expected thin-walled behavior in short columns.Proof-of-concept of this novel global optimization method is provided by the

examples presented. In the future, the 20 allowable folding positions must be

increased to describe potentially important details such as small stiffeners. The

width-to-thickness ratio should also be included as a design variable. Constraints

on the geometry that may be related to the important issue of manufacturability

should also be added. Furthermore, a knowledge function specifically dealing with

distortional buckling mode would be useful in constructing a classification tree that

could identify designs resistant to all three buckling modes. Our final intent is to

provide a universally effective classification tree that may identify cross-section

designs that are nearly optimal for a broad range of cold-formed steel members.

Acknowledgements

This material is based upon work supported by the National Science Foundation

at Johns Hopkins University under Grant Numbers DMI-0087032 and CMS-

0084590. This research support is gratefully acknowledged.


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Knowledge-Based Global Optimization of Cold-Formed Steel Columns