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Performance-based control co-design of building structures with controlled rocking steel braced frames via neural dynamic model

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Controlled rocking steel braced frames (CRSBFs) are modern systems for resilient building structures known for their effective energy dissipation and self-centering features. In a CRSBF, post-tensioned (PT) strands and shear fuses are the distinct features providing the self-centering and energy dissipation mechanisms. As a relatively new structural system, there is a need to study this system’s optimal design. Control co-design has gained interest in recent years as a class of integrated engineering system design methods as an alternative to the traditional approach of optimizing the structural design and sequentially optimizing the control system’s design. It considers the direct relationship between physical and control system design decisions to discover non-obvious design solutions that enable new performance and functionality levels. This paper proposes a performance-based control co-design methodology for building structures integrating CRSBFs that concurrently minimizes the mainframe weight and determines the CRSBF design parameters subject to design code requirements as the optimization constraints. The patented neural dynamic model of Adeli and Park is used in this research to solve the nonlinear optimization problem. The seismic performance evaluation of the proposed methodology includes a shear frame and a nonlinear 6-story 3D building structure. The 6-story building model consists of 282 nodes with 1,692 degrees of freedom. A total of 610 frame elements, 8 PT strands with nonlinear material, and 32 nonlinear links are determined without requiring high-performance computing. Results show that the energy dissipation mechanism effectively reduces the seismic demand in structural members. The proposed performance-based control co-design methodology can lead to 21% reduction in the total weight of the 6-story frame structure.

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A m a x :

maximum overall cross-sectional area of the strands

A s t r :

cross-sectional area of the strand

A w :

web area of the W section

C v :

shear post-buckling strength reduction factor


objective function

F y :

steel yield strength

G :

total number of inequality constraints

K :

total number of equality constraints

L :

Lagrange function

M c x :

available flexural strength

M r x :

demand flexural strength

P c :

available axial strength

P r :

demand axial strength


power spectral density function of the acceleration

S 0 :

scale factor for acceleration spectrum

W i :

weight of the ith frame element

W m a x :

maximum overall weight of the frame

g j(x):

jth inequality constraint

h k(x):

kth equality constraint

j :

number of inequality constraint

k :

number of equality constraint

M :

size of design variable vector

n :

iteration number in the optimization algorithm

r n :

exterior penalty parameter

r 0 :

initial penalty parameter

w j :

square root of absolute value of gj(x)

x :

design variable vector

x :

design variable vector at equilibrium point

α :

Lagrange multiplier for inequality constraint term

γ :

Lagrange multiplier for equality constraint term

δ f :

fuse shear deformation

𝜖 :

strictly positive number used to obtain penalty parameter

𝜖 PT :

strain in PT strand

λ :

Lagrange multiplier for objective function term

ξ g :

ground damping ratio

ϕ v :

safety factor for shear

ω g :

ground natural frequency; and


the gradient of a function


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The authors gratefully acknowledge the permission from Professor Hojjat Adeli and Professor Hyo Seon Park’s to use the patented ND model (US patent 5,815,394 issued date on September 29, 1998).

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Correspondence to Mariantonieta Gutierrez Soto.

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The data, models, and codes used during the study are available in the DesignSafe-CI online repository located at (structural model) and (optimization model).. The section of the code generated and used during the study to implement the patented ND model for optimization (US patent 5,815,394 issued date on September 29, 1998) is proprietary.

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The authors declare no competing interests.

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Javadinasab Hormozabad, S., Gutierrez Soto, M. Performance-based control co-design of building structures with controlled rocking steel braced frames via neural dynamic model. Struct Multidisc Optim 64, 1111–1125 (2021).

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