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Cost-based optimization of steel frame member sizing and connection type using dimension increasing search

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Abstract

This paper describes a novel method to select the optimal member sizes and connection types for steel frame structures to minimize total installed cost. The proposed dimension increasing search (DIS) method addresses geometric constraints associated with connection, safety, and serviceability constraints. Solving this optimization problem requires addressing a non-convex objective function of total installed cost, a large number of constraints, the time-consuming structural analysis for evaluation of the constraints, and intractability resulting from high-dimensional variables. The DIS method involves initial grouping of design variables to reduce the search dimension from the original dimension. Random search coupled with forward checking and branch-and-bound techniques are then employed to optimize the configuration of the steel frame. This approach eliminates designs that are more expensive or do not satisfy geometric constraints without performing structural analysis. The variable groups are then split into subgroups to increase the search dimension, and the increased dimensional design variables are optimized until the original search dimension is achieved. We benchmarked the DIS method against leading methods using three numerical examples with increasing problem scales. The DIS method was more computationally efficient compared to other leading methods by an order of magnitude, and the solutions found by the method have a 9% lower total installed cost on average. The DIS method also showed scalability; the computational time increased only slightly as the problem scale increased. The scalability of the DIS method demonstrates the potential for its successful industrial application to large-scale building projects.

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Abbreviations

\({\varvec{x}}\) :

Sizing variables

\(t\) :

Connection type variable

\(f\) :

Total installed cost function

\(c\) :

Safety and serviceability constraints

\(g\) :

Geometric constraints

\({f}^{i}\) :

Optimal solution in the \(i\) th stage

\({d}^{i}\) :

The search dimension in the \(i\) th stage

\({f}_{m}\) :

Member sizing cost

\({f}_{c}\) :

Connection cost

\({\xi }_{j}^{k}\) :

A stochastic movement size for \(j\) th element in variable at \(k\) th step in random walk

\({s}_{max}\) :

Maximum step size in random search

\(p{^{\prime}}\) :

A random number between 0 and 1 to determine the step size

\({x}_{j}^{k}\) :

The \(j\)th element in variable \(x\) , at \(k\) th step in random search

\(dt\) :

Decreasing threshold

\(p\) :

A random number between 0 and 1 to determine the moving direction in random research

\(it\) :

Increasing threshold

\({n}_{max\_width}\) :

The maximum search width

\({n}_{max\_depth}\) :

The maximum search depth

\({G}_{i,s}\) :

Variable group \(s\) at stage \(i\)

\({\sigma }_{i}\) :

The stress at steel member \(i\)

\({\sigma }_{i}^{a}\) :

The allowable stress at steel member \(i\)

\({P}_{u}\) :

The required axial strength

\({M}_{uy}\) :

The normal flexural strength in y direction

\({\phi }_{b}\) :

The flexural reduction factor

\({M}_{ny}\) :

The required flexural strength in the y direction

\({\phi }_{c}\) :

The resistance factor

\({P}_{n}\) :

The normal axial strength

\({M}_{ux}\) :

The normal flexural strength in x direction

\({M}_{nx}\) :

Required flexural strength in the x direction

\({d}_{k}\) :

The inner-story displacement

\({d}_{k}^{a}\) :

The allowable inner-story displacement

\({D}^{k,l}\) :

The lower bound for the connection’s dimension property

\({D}_{j}^{k}\) :

The dimension property for connection \(j\)

\({D}^{k,u}\) :

The upper bound for the connection’s dimension property

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Acknowledgements

The authors would like to thank CIFE (Center for Integrated Facility Engineering at Stanford University) for the funding provided and the access to computational resources. We also appreciate the software support provided by Autodesk, especially Patrick Tierney.

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  1. Stanford University, Stanford, USA

    Bo Peng, Forest Flager, Steve Barg & Martin Fischer

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  1. Bo Peng
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Correspondence to Bo Peng.

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This research was funded by Autodesk and CIFE. Besides, there is no conflict of interests with other people and institute.

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Appendix A: Detailing parameters

Appendix A: Detailing parameters

See Table 7.

Table 7 The unit cost for the cost estimation
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1.1 Replication of results

We attached the code for the experiments mentioned in the Sect. 4 (Table 8). Three folders are corresponding to the numerical experiments of three steel frames. The cost data applied in this paper is explained in the work of Barg et al. (2017).

Table 8 Code for the experiments
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Peng, B., Flager, F., Barg, S. et al. Cost-based optimization of steel frame member sizing and connection type using dimension increasing search. Optim Eng 23, 1525–1558 (2022). https://doi.org/10.1007/s11081-021-09665-5

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