Automated minimum-weight sizing design framework for tall self-standing modular buildings subjected to multiple performance constraints under static and dynamic wind loads

Abstract

In recent decades, the shortage of affordable housing has become an endemic issue in many cities worldwide due to the ongoing urban population growth. Against this backdrop, volumetric steel modular building systems (MBSs) are becoming an increasingly compelling solution to the above challenge owing to their rapid construction speed and reduced upfront costs. Notwithstanding their success in low- to mid-rise projects, these assembled structures generally rely on a separate lateral load-resisting system (LLRS) for lateral stiffness and resistance to increased wind loads as the building altitude increases. However, additional LLRSs require on-site construction, thereby compromising the productivity boost offered by MBSs. To this end, this paper proposes a novel optimisation-driven sizing design framework for tall self-standing modular buildings subjected to concurrent drift, floor acceleration, and member strength constraints under static and dynamic wind loads. A computationally efficient solution strategy is devised to facilitate a meaningful sizing solution by decomposing the constrained discrete sizing problem into a convex serviceability limit stage (SLS) and a non-convex ultimate limit stage (ULS), which can be conveniently solved using preferred local and global optimisation methods, separately. The framework is implemented by integrating SAP2000 (for structural analysis) and MATLAB (for optimisation) through SAP2000’s open Application Programming Interface (API), and demonstrated using a 15-storey self-standing steel modular building exposed to three different levels of wind intensity. A comprehensive performance assessment is conducted on the optimally sized case-study building to investigate its elastic instability behaviour, geometric nonlinear effects on wind-induced response, and impacts of global sway imperfections on member utilisation ratios under different wind intensities. It is concluded that tall self-standing modular buildings can be achieved economically using ordinary corner-supported modules without ad hoc structural provisions, while consuming steel at similar rates to conventional building structures. Furthermore, the proposed sizing framework and solution strategy have proven to be useful design tools for reconciling the structural resilience and material efficiency in wind-sensitive self-standing modular buildings.

Full text

Automated minimum-weight sizing design framework for tall self-standing modular
buildings subjected to multiple performance constraints under static and dynamic wind
loads
Zixiao Wang1, Komal Rajana2, Dan-Adrian Corfar3, and Konstantinos Daniel Tsavdaridis4*
1,2,3,4 Department of Engineering, School of Science & Technology, City, University of London, UK
* Corresponding author, email address: konstantinos.tsavdaridis@city.ac.uk
Abstract
In recent decades, the shortage of affordable housing has become an endemic issue in many cities worldwide due to
the ongoing urban population growth. Against this backdrop, volumetric steel modular building systems (MBSs) are
becoming an increasingly compelling solution to the above challenge owing to their rapid construction speed and
reduced upfront costs. Notwithstanding their success in low- to mid-rise projects, these assembled structures generally
rely on a separate lateral load-resisting system (LLRS) for lateral stability and resistance to increased wind loads as
the building altitude increases. However, additional LLRSs require on-site construction, thereby compromising the
productivity boost offered by the MBSs. To this end, this paper proposes a novel optimisation-driven sizing design
framework for tall self-standing modular buildings subjected to concurrent drift, floor acceleration, and member
strength constraints under static and dynamic wind loads. A computationally efficient solution strategy is devised to
facilitate a meaningful sizing solution by decomposing the constrained discrete sizing problem into the convex
serviceability limit stage (SLS) and non-convex ultimate limit stage (ULS), which can be conveniently solved using
preferred local and global optimisation methods, separately. The framework is implemented by integrating SAP2000
(for structural analysis) and MATLAB (for optimisation) through SAP2000’s open Application Programming Interface
(API), and demonstrated using a 15-storey self-standing steel modular building exposed to three different levels of
wind intensity. A comprehensive performance assessment is conducted on the optimally designed case-study building
to investigate the elastic instability behaviour, geometric nonlinear effects on wind-induced response, and impacts of
global sway imperfections on member utilisation ratios. It is concluded that tall self-standing modular buildings can be
achieved economically using ordinary corner-supported modules without ad hoc structural provisions, while
consuming steel at similar rates to conventional building structural systems. Furthermore, the proposed sizing
framework and solution strategy have proven to be useful design tools for reconciling the structural resilience and
material efficiency in wind-sensitive self-standing modular buildings.
Keywords: structural sizing optimisation, modular building systems, wind effects, SAP2000 Open Application
Programming Interface
1 Introduction
In recent decades, global population growth and increasing urbanisation trends have created endemic housing
shortages and sharp upturns in compact high-rise developments in many major cities worldwide. With the projection
that 68% of the world’s population will live in urban areas by 2050 [1], a partial solution to the growing pressure on
urban housing is to build more rapidly and make them more affordable. Against this backdrop, steel modular building
systems (MBSs), underpinned by modern methods of construction (MMC), design for manufacturing and assembly
(DfMA) methodology, and design for disassembly/deconstruction (DfD) considerations, are becoming an increasingly
compelling solution to the construction industry owing to the excellent strength-to-weight ratio of structural steel [2]
and well-established benefits brought by this construction method, such as halved construction times [3], lower
upfront costs [4], reduced waste generation [5], demountability and reusability [6], reduced on-site labour [7], safer
work sites [8], and better quality control [9]. As shown in Fig. 1, in volumetric modular construction, prefinished
building units are prefabricated offsite in a controlled factory environment and then delivered to site for final assembly
through bolted (or other types of) inter-module connections (IMCs) to form a fully finished building.

To achieve economies of scale, steel MBSs are ideal for tall building applications with repeated units, such as
apartments, student residences, and hotels [9]. Nevertheless, their application to date mostly concerns multistorey
buildings [10], with the optimal number of floors being usually around six. Arguably, this predominance is attributed
to the fact that the structural behaviour of tall MBSs in terms of serviceability and integrity under extreme
environmental loads from high winds (as well as earthquakes) is generally more complex and less well understood
than conventional building structural systems. In particular, the column discontinuity introduced by the IMCs between
stacked modules (see Fig. 1(b)) reduces the overall lateral stiffness of MBSs and increases the effective (buckling)
length of the corner posts. The former effect renders these structures more susceptible to large static and dynamic
structural responses under wind effects, causing serviceability issues (e.g., damage to non-structural components and
wind-induced occupant discomfort), whereas the latter makes the structures more vulnerable to member and/or global
instability under combined gravitational and extreme wind loads. Still, there are exceptional examples of MBSs
reaching over 130m, such as the recently completed 150m tall College Road project in Croydon, the UK (Fig. 1. (d)).
In these cases, however, the structural integrity and stability of the buildings are achieved by means of hybrid
structural systems comprising stacked modules arranged around a reinforced concrete core [11,12], with the
surrounding modules designed to carry gravitational loads only. Nevertheless, the concept of tall self-standing MBSs
is deemed worthy of investigation because of its even shorter construction cycle, as in this case on-site erection of a
separate/additional LLRS is not required. Despite the growing momentum of MBSs in the literature, the existing body
of work concerning with this notion is still limited, with studies mostly focusing on the effect of static and cyclonic
wind loading on the lateral drift performance of steel MBSs between 8 and 11 stories [13–16].
Fig. 1. (a) illustration of a typical tall self-standing MBS comprising corner, edge, and internal IMCs, (b) illustration of two
completed modular units [17], (c) assembly view of corner, edge, and internal IMCs for the sizing design investigation in this
study, and (d) College Road project during construction [18].
On the other hand, another challenge that may have hindered the development of taller self-standing modular
buildings is the increase in structural material consumption required for stiffening and strengthening the individual
modules to sustain larger gravitational and wind actions as the buildings get higher. Although this is generally true for
all tall building structures [19], the increasing effect of building altitude on the structural material usage/costs and
member sizes may be more severe for tall MBSs because of doubled beams and bundled columns from adjacent
modules (see Fig. 1 (c)). Moreover, due to column discontinuity at IMCs, the structural members of individual
modules need to be adequately stiff and large to prevent column instability under combined gravitational and lateral
loads, which can further increase the overall steel tonnage. In this regard, the impact of satisfying performance-based
wind design criteria on the LLRS self-weight also remains unclear for tall self-standing MBSs. Note that this topic
may also be critical for sustainability reasons, as the upfront cost and environmental impacts of tall buildings are

closely related to their structural material usage. Currently, the manufacture of building materials accounts for about
11% of total greenhouse gas emissions [20]. For tall buildings in particular, a large proportion of their embodied
carbon emissions (during material production, construction, maintenance, and demolition) is almost always produced
by their LLRS [21]. As a result, the need to investigate the structural efficiency (broadly defined here as the ratio of
certain structural utility to material consumption) of tall self-standing MBS is genuine in order to foster an economical
and sustainable solution to the ongoing housing crisis. By designing for minimum weight in a performance-based
context, the structural material consumption of steel MBS can be reduced, leading to an equivalent embodied carbon
reduction [22] and a further decrease in upfront costs (note that currently MBSs are already 10% to 20% cheaper, see
[4]). For conventional building structures, their minimum-weight designs are typically achieved through experiential
trial-and-error design cycles [23] or performance-based design optimisation approaches [24–28]. When designing for
wind loads, the sizing of their LLRS is generally governed by serviceability considerations rather than member
resistance [29], which may explain why most sizing frameworks for conventional building structures (such as those
listed above) primarily concern serviceability issues. However, this may not be the case for tall self-standing modular
buildings due to their sensitivity to geometric nonlinear effects. Under combined gravitational and wind loads, their
member forces can be significantly magnified by structural and member deformations as demonstrated later in this
study. Consequently, this nonlinearity must be properly accounted for in the structural analysis/optimisation to
accurately evaluate strength constraint functions. To this end, the direct implementation of conventional sizing
frameworks to self-standing modular structures may result in various issues, such as difficulty in tackling concurrent
serviceability and member strength criteria, slow convergence due to more structural members and design variables
involved, and premature termination of the sizing process during geometrically nonlinear analysis (see detailed
discussion of these issues in Section 3.2).
To this end, this study proposes a performance-based sizing design framework (Section 3.1) for LLRS weight
reduction of tall self-standing MBSs subjected to multiple SLS and ULS design constraints under static and dynamic
wind loads. The framework relies on a sizing optimisation problem formulation that considers codified drift- and
floor-acceleration-based serviceability criteria, together with member strength requirements for structural integrity. A
numerical solution strategy is devised (Section 3.2) to facilitate meaningful and computationally efficient solutions to
the proposed weight minimisation problem by decomposing the sizing workflow into the SLS and ULS stages. The
latter can then be solved sequentially and independently using any combination of well-established local and global
optimisation algorithms. For demonstration, an adapted interior-point method (IPM) for continuous sizing design
under static and dynamic serviceability constraints is provided in Section 4.1, while a genetic algorithm (GA) for
discrete sizing design under member strength constraints is presented in Section 4.2. A numerical application of the
sizing framework, entailing the IPM and GA in Section 4, to a representative 15-storey self-standing steel modular
building under three levels of wind intensity is provided in Section 5 to demonstrate the applicability and potential
gains. The continuous and discrete sizing are achieved by a custom engineering application developed using the Open
Application Programming Interface (OAPI) of SAP2000, enabling the integration of SAP2000 for structural analysis
and MATLAB for constrained structural optimisations. The optimal (sizing) designs of the adopted building are
comprehensively assessed in Section 6 in accordance with Eurocode 3 [30], shedding light on the elastic stability
behaviour, effects of geometric nonlinearity on wind-induced responses, and impacts of global geometric
imperfections on member utilisation ratios of the case-study building under combined gravitational and wind loads.
Finally, Section 7 summarises the main conclusions. In the next section, the limit state design considerations for steel
tall MBSs under wind effects are discussed.
2 Limit State Design Considerations for Tall Self-Standing MBSs under Wind Effects
At present, there is a lack of design standards for MBSs, so the current design practice is based on the regulations
for conventional building structural systems. However, since modular buildings are still building structures, the
common limit state design considerations for conventional buildings should still be applicable to MBSs, which must
satisfy the same, if not more stringent, standards of serviceability and safety [31].
To this end, tall buildings, loosely defined as those exceeding 50m or 14 stories [32], are generally susceptible to
two types of serviceability issues caused by excessive wind-induced deformations and oscillations [33,34] owing to

their increased lateral flexibility and low inherent damping [29]. The first one is related to non-structural damage (to
partition walls, windows and door frames, ceilings, and external cladding/facades) due to large inter-storey drifts
and/or overall deflection in the along-wind (drag) direction, which are induced by the mean/static wind pressures and
thus often referred to as static serviceability criteria. For instance, the limits of H/600 for the total building drift and
h/500 for the inter-storey drift were specified by Griffis [35], where H is the total building height and h is the storey
height. The second problem concerns building occupant discomfort caused by excessive floor acceleration in the
across-wind (lift) direction, which is induced by structural resonance with the vortex shedding phenomenon [36,37],
and is often referred to as the dynamic serviceability criterion. In this regard, many design protocols (e.g.,
ISO6897:1984 [38], AIJ-GEH-2004 [39]) have prescriptive provisions for regulating occupant comfort/habitability
performance under wind excitation. This is usually achieved by requiring the maximum wind-induced floor
acceleration, either peak or root mean square (RMS), associated with a given recurrence interval to be below a
codified threshold value depending on the building’s fundamental frequency of the dominant translational mode
[29,40]. According to a rough criterion by the ASCE 7-95 Standard [41], a building can be considered laterally
flexible and thus prone to the above serviceability issues when the ratio of building height to the least horizontal
dimension exceeds 4, or when the fundamental natural frequency is less than 1 Hz. However, for tall rectangular
buildings with aspect ratios (i.e., height over breath) greater than 3, the dynamic serviceability issue in the across-wind
direction is generally more critical than the static drift issue in the along-wind direction, thereby governing the wind-
related serviceability design [42,43].
Finally, while the above serviceability issues concern wind-induced responses at the global level, tall building
designs under strong winds may also be governed at the local/member level by strength requirements on member
resistance. Specifically, verification of buckling resistance is crucial for slender structural members under combined
axial compression and biaxial bending to ensure member stability under combined gravitational and wind loads. In
this regard, geometric nonlinear effects may need to be considered in the analysis to determine the member design
forces, with appropriate allowances also to account for the effects of global and local/member imperfections (see
Eurocode 3 [30]).
3 Performance-Based Optimal Design Problem Formulation and Efficient Numerical Solution Strategy
3.1 Sizing optimisation problem formulation for minimum structural weight
The general form of the weight minimisation problem of building structures comprising n (or n groups of) frame
members subjected to m performance constraints can be stated as follows
(1)
where is the design variables (DVs) taken as the cross-sectional areas of frame members and bounded by side
constraint amin≤a≤amax, is the objective function taken as the self-weight of the building’s LLRS,
is the j-th performance constraint to be satisfied by the structure, and ρi and li denote the material
density and length of frame member i, respectively. In the above formulation, the DVs a can be continuous, discrete,
or mixed. Importantly, only cross-sectional areas are selected as the DVs here. This consideration facilitates the
weight-minimal design formulation in Eq. (1) and can be supported by expressing all other sectional properties
(contributing to the overall building stiffness) in terms of the cross-sectional area only (see Section 5).
To support the performance-based sizing design for tall self-standing MBSs under wind effects, this study
considers wind-induced drift and floor acceleration constraints for SLS, and member strength constraints for ULS.

Specifically, for a k-storey building (with floor k being the roof), the along-wind inter-storey drift and roof
displacement constraints in Eq. (1) take the forms of
(2)
respectively, where Δj(a) and are the inter-storey drift ratio for storey j and the lateral displacement at the
building roof, respectively, and Δlim and ulim are the corresponding codified permissible limits. Here, the inter-storey
drift ratios are defined as , where and are the wind-induced lateral displacements at the
ceiling and floor levels of storey j, respectively, and hj is the corresponding storey height.
For across-wind floor accelerations, the constraint functions are specified by
(3)
in which is the wind-induced lateral acceleration at the floor level of storey j, and is the codified
acceleration threshold commonly defined as a function of the building’s natural frequency of the first (or dominant)
lateral vibration mode, f1 [24]. Note that the unoccupied roof (or floor k) is excluded from the last equation, as its
acceleration level does not affect occupant comfort under wind excitation.
Finally, for verification of member resistances under along-wind static loads, the strength constraints are given by
(4)
where rj is the maximum utilisation ratio for member (or member group) j from all wind-related ULS load
combinations. In this setting, there are 2k SLS constraints in Eq. (1) in total, comprising k+1 drift and k-1 acceleration
constraints, plus n strength constraints, coming from n structural DVs.
3.2 Efficient numerical solution strategy
From the outset, the optimisation problem in Eq. (1) is nontrivial and computationally challenging to solve for
three reasons.
(1) All three types of performance constraints (on wind-induced drifts, floor accelerations, and member strength) in
Section 3.1 are generally non-convex [44], making the numerically more efficient local optimisation methods
unsuitable for the task because they can be trapped in local optima.
(2) For large structural systems with many DVs (such as tall MBSs), population-based global optimisation methods
(e.g., GA) can be prohibitively slow because their optimisation performance is crucially dependent on population
size [45], which in turn depends on the number of DVs. Although these heuristic methods do not require gradient
information, expensive evaluations of nonlinear performance constraints for a large number of candidate sizing
designs are required to arrive at the final sizing design.
(3) Compared to conventional building structures, tall self-standing MBSs are more prone to global and member
instabilities when subjected to combined gravitational and lateral loads from high winds and earthquakes [46].
This is because the buckling lengths of the corner posts are increased by the discontinuity introduced by the

IMCs (see Fig. 1(b)). Consequently, the effect of geometric nonlinearity (associated with structural and member
deformations) on magnifying wind-induced member forces is more profound, and large-displacement analysis
based on load incrementation strategies may be required to accurately evaluate member design forces for strength
verification. However, this may lead to slow convergence or even failure when a candidate sizing design being
analysed becomes structurally unstable before a specified ULS combination of loads is reached, causing the
sizing optimisation process to terminate prematurely.
In view of the above challenges, a computationally efficient numerical strategy, as shown in Fig. 2, is developed
herein to solve the weight-minimisation problem in Eq. (1). Specifically, the strategy addresses the above three
challenges through implementing the following three techniques, separately:
(i) The two non-convex SLS constraints in Eqs. (2) and (3) are first converted to a convex structural compliance
constraint on the wind-induced total strain energy and a convex natural frequency constraint on the building’s
first translational vibration mode as
(5)
and
(6)
respectively. In Eq. (5), and denote the static along-wind loads lumped at the ceiling and floor levels
of storey j, respectively, while Elim is the upper-bound elastic strain energy selected as the measure of static lateral
stiffness of the building. A good estimate of Elim can be found by , where and
are the lateral displacement limits at the ceiling and floor levels of storey j, respectively, converted from
the code-specific inter-storey drift requirements. Moreover, ftarget in Eq. (6) is the lower-bound (target) frequency
selected as the measure of dynamic lateral stiffness of the building, which can be found either analytically [24] or
by trial and error [19]. By implementing technique (i), the two serviceability constraints become strictly convex
[47], thus addressing challenge (1) partially (due to the strength constraints in Eq. (4) still being nonconvex).
(ii) After converting the two SLS constraints, the optimisation problem in Eq. (1) is decoupled into two sequential
stages: a convex SLS stage and a non-convex ULS stage. This decomposition allows for convenient and
sequential solutions to the two staged subproblems using preferred local and global sizing algorithms, separately.
In this arrangement, the MBS is first optimised for lateral stiffness to meet the converted drift and floor
acceleration constraints, and then for member resistance to satisfy the strength requirements. Upon optimally
stiffening a modular building for serviceability criteria, a large number of its structural members (or member
groups) are no longer governed by strength requirements. Consequently, the number of active DVs in the ULS
stage of the sizing workflow significantly decreases, which, in turn, reduces the population size used by GA and
hence improves computational efficiency, thus addressing challenge (2).
(iii) In the ULS stage of the proposed sizing workflow, the optimal design from the SLS stage, which satisfies the two
serviceability constraints, is used as the lower bound of the side constraint on the DVs a. This manipulation
reduces the likelihood of member instability issues during the geometrically nonlinear structural analysis in the
ULS stage, thus addressing challenge (3).

At this point, the proposed solution strategy for the optimisation problem in Eq. (1) is outlined as follows. The
workflow begins with initialisation, during which the building is first designed to meet all SLS and ULS criteria for
gravitational loads only using standard structural design methods. For wind-sensitive modular buildings, this design
will most likely not meet the codified serviceability and strength criteria, necessitating further stiffening and
strengthening through resizing. The member sizes of this design, indicated as in Fig. 2, are used as the initial
lower bound of the side constraint for the sizing optimisation under drift constraints. The upper bound of the DVs,
denoted as amax in Fig. 2, can be established based on architectural, functional, and other buildability considerations.
After initialisation, the workflow proceeds to the sizing optimisation under the drift constraints in Eq. (2), which
are converted to a single compliance constraint defined in Eq. (5). This optimisation aims to find the optimal set of
DVs, , within the initial side constraint, , by experientially adjusting Elim value in Eq. (5) until the
two sets of drift criteria in Eq. (2) are simultaneously met. Once is found, it is used as the new starting point and
lower bound, i.e., , for the subsequent sizing optimisation under floor acceleration constraints in Eq.
(3), which are now converted to a natural frequency constraint in Eq. (6). This optimisation aims to find the optimal
set of DVs, , within the updated side constraint, , to satisfy the acceleration constraints.
Importantly, the two sequential sizing processes in the SLS stage are both continuous-valued; the discrete optimal
sections, , are only specified/selected (per member design groups) after is determined. This is achieved
by mapping the obtained continuous cross-sectional properties onto different catalogues of commercially available
steel sections (see Section 5 for demonstration).
Finally, the sizing workflow progresses to the ULS stage, in which is used as the new starting point and
lower bound in the discrete sizing process under member strength constraints for ULS load combinations associated
with static along-wind forces. The optimisation process aims to determine the optimal set of standard sections, ,
that minimises M(a) while satisfying the member strength constraints in Eq. (4). The determined sections are adopted
as the final optimal solution to the weight-minimisation problem in Eq. (1) subjected to concurrent performance
constraints in Eqs. (2), (3), and (4).
The applicability of the above solution strategy is illustrated in Section 5 using a 15-storey self-standing modular
building exposed to different wind intensities, for which a numerical implementation of the sizing workflow is
required. To this end, the following section details one local and one global optimisation algorithms for the SLS and
ULS stages of the sizing workflow, separately. First, an IPM is presented in Section 4.1 for continuous sizing
optimisation under convex serviceability constraints. Then, a GA is outlined in Section 4.2 for discrete sizing
optimisation under nonconvex strength constraints. It is worth noting that the two algorithms presented next are just
two possible choices and can be replaced by other local and global optimisation methods deemed appropriate.

Fig. 2. Flowchart of solution strategy for minimal-weight design of tall MBSs subject to wind-related serviceability and ultimate
limit state design constraints.
4 Local and Global Optimisation Algorithms for Minimum-Weight Design of Frame Structures Under
Convex and Nonconvex Performance Constraints
4.1 Interior point method-based continuous sizing under convex structural compliance and natural frequency
constraints
When subject to the convex structural compliance and natural frequency constraint, the sizing optimisation
problem in Eq. (1) is strictly convex. To solve this problem, one can convert the original optimisation problem in Eq.
(1), with reformulated serviceability constraints in Eqs. (5) and (6), into an equivalent, unconstrained form by
replacing the constraint functions with barrier terms. This results in an unconstrained optimisation problem that can be
solved using deterministic local optimisation algorithms, such as the interior-point method [47] detailed in this section.
Specifically, the unconstrained approximate problem for Eq. (1) subject to the compliance and/or frequency constraint
can be formulated as
(7)
where B(a,μ) is the barrier function, μ is a small positive scalar, and ln(cl(a)) is the l-th barrier term, which is not
defined for cl(a) ≤ 0 and is restricted to be positive during the sizing iteration. In case only one constraint (either
compliance or frequency) is considered at a time (which is the case in the current study), the number m in the last
equation reduces to 1 and the summation on l drops. Evidently, as μ decreases to zero, the barrier function B(a,μ)
degenerates to the original objective function M(a) such that the minimum of B(a,μ) should approach the solution of
Eq.(1). Accordingly, instead of solving Eq. (1) directly, it is equivalent to solve the approximate problem in Eq. (7) for

given μ. Mathematically, this is equivalent to finding the stationary point of B(a,μ) at which the gradient of B(a,μ)
equals to zero, i.e.,
(8)
In order to solve Eq. (8), Lagrange multiplier-like dual variables, stored in vector λ={λ1,…,λm}T (where “T”
denotes transpose operation), are next introduced in the following equality
(9)
through which Eq. (8) can be rewritten in the vector-matrix form as
(10)
where is the Jacobian matrix of the constraint function c={c1(a),…,cm(a)}T. To solve for (a,λ), the Newton-
Raphson method is applied to Eqs. (9) and (10) to give
(11)
in which is the Hessian matrix of B(a,μ), and are the diagonal matrices of λ and c,
respectively, is a unit vector, and and are the search directions for the primal and
dual variables, a and λ, respectively. Because μ and cl(a) are both positive, the condition λ ≥ 0 must be enforced at
each iteration (because of Eq. (9)). Finally, for a given value of μ, the sizing DVs and the Lagrangian multipliers, a
and λ, can be found through iteration using the following formula
(12)
where α is the step size at the iteration step p, which can be determined by, e.g., the merit function method [48]. The
above iteration is then executed until the following convergence criteria are satisfied within a prespecified tolerance,
εtol,
(13)
4.2 Genetic algorithm-based discrete sizing under nonconvex member strength constraints
Structural sizing problems under elemental stress constraints are generally non-convex, with several local optima

having different objective function values [44]. For volumetric modular buildings, their modular units generally use
standard hollow sections such that the sizing DVs in Eq. (1) can only attain discrete values. To this end, a global,
discrete sizing algorithm is needed for the minimum-weight design of tall MBSs under code-prescribed member
strength constraints in Eq. (4). This discrete, nonconvex sizing problem can be solved by stochastic global
optimisation methods, such as GAs based on the biological concept of evolution [49]. Specifically, GAs apply the
principle of survival of the fittest to a population of potential solutions/individuals to produce a successively improved
population of solutions to a given optimisation problem. At each generation, a new set of solutions (known as
offspring) is generated by selecting better-fit individuals in the current population (known as parents) based on their
fitness levels and then allowing them to reproduce via genetics-inspired operations, such as crossover and mutation.
GAs are also applicable to integer optimisation problems, making them suitable for the automatic selection of cross-
sections to meet member strength requirements. In this context, an integer GA adapted from Deep et al. [50] is
presented herein for the discrete sizing design of tall MBSs subject to strength constraints on the buckling resistance
of members according to Eurocode 3 [30]. For this task, the DVs in Eq. (1) can only be selected from sets of discrete
values, based on the cross-sectional areas of the standard sections available for the discrete sizing optimisation. This is
achieved conveniently by introducing the integer DVs (where denotes the set of positive integers), which
are linked to the sizing DVs, a, through a one-to-one correspondence. The former DVs contains n location indices of
the cross-sections (for n groups of modular members) in their corresponding lists of standard candidate sections. For
volumetric modular buildings, hot-finished rectangular hollow sections (RHS) are generally used for modular beams,
while hot-finished square hollow sections (SHS) are used for corner posts/bracings. The adapted GA is described in
the following steps.
(1) Generate a sufficiently large initial set of random designs with population size m (based on the number of DVs)
within the design domain confined by the side constraint, xmin≤x≤xmax, where xmin and xmax are the lower and upper
bounds of the location indices.
(2) Check if the maximum number of generations is reached or if the relative change in the best fitness function value
over a prespecified number of generations is less than a pre-specified tolerance. If any of the two conditions is
met, stop the GA; else proceed to step (3). The fitness function of individual/candidate design i in a generation,
, is defined based on the feasibility approach by Deb [49] as
(14)
where Mworst is the objective function value of the worst feasible solution in the current population, and the
summation term, Σcj(xi), reflects the total degree of performance constraint violations. If there is no feasible
solution in the current population, Mworst is set to zero.
(3) Apply the k-way deterministic tournament selection [51] (with a tournament size k, note k < m) to the current
population to create a mating pool. To this aim, k individuals are selected randomly from the current population
for a tournament, with the “champion” of the tournament (the one with the lowest f(xi) value) placed in the mating
pool. This procedure is executed systematically and repeatedly until each individual in the current population
participates in k tournaments exactly. Evidently, the fittest individual/design in the current generation wins all k
tournaments and makes k copies of itself in the mating pool, whereas the least-fit individual loses all k
tournaments and is thus eliminated. In this way, the mating pool always remains the same size as the current
population (i.e., m), with better-fit individuals having a higher chance of being included multiple times.
(4) To create a new set of populations, a probabilistic crossover operation is next applied, with probability Pc∈[0,1],
to m/2 pairs of parents in the mating pool from step (3) to generate m offspring, forming the next generation. For

Pc ≠ 0, the above operation recombines the parents’ genes to reproduce two new solutions from each parenting
pair, whereas for Pc = 0, no crossover is applied and two parents are directly cloned into the offspring generation.
Herein, the extended Laplace crossover operator proposed by Deep et al. [50] is adopted to generate two offspring,
xo1 and xo2, from two randomly selected parents, xp1 and xp2, as follows. First, a random parameter, βi, satisfying the
Laplace distribution, is generated for the i-th “chromosomes” (i.e., the i-th entry of x) of two offspring (where i=1,
…,n) as
(15)
where a and b>0 are two pre-defined parameters, and ui and ri are two random variables with a uniform
distribution between 0 and 1. Then, the i-th chromosomes of two offspring, and , are generated using the
equation below
(16)
After the offspring are generated, a mutation operation, with probability Pm∈[0,1], is next applied to all offspring
following
(17)
where denotes the muted chromosome i of offspring j (j = 1,…,m), and are the i-th entries of xmin
and xmax, respectively, is a random variable between 0 and 1, and p > 1 is the mutation parameter controlling
the strength of the mutation.
(5) Apply integer restrictions to the offspring with non-integer chromosomes, and then evaluate the fitness values of
all offspring (with rounded chromosomes) according to Eq. (14). Specifically, the non-integer chromosomes,
(i = 1,…,n), are rounded to either or , where denotes the integer part of . This ensures
greater randomness in the set of offspring being generated and avoids the same integer values being generated
repeatedly for different offspring.
(6) Increase the current generation number by 1, then go back to step (2).
As a remark, in the numerical part of this work, the continuous and discrete sizing optimisations of the case-study
building are achieved by a custom optimisation application developed using the OAPI of SAP2000, which allows for
the integration of SAP2000 for structural analysis and MATLAB for structural optimisation. The IPM- and GA-based
sizing optimisations presented in this section are implemented using MATLAB’s built-in functions/solvers “fmincon”
and “ga”, respectively.
5 Minimum-Weight Design of a Typical Tall Steel MBS under Wind Loads
5.1 Description of case-study building and design wind actions

To illustrate the applicability of the proposed sizing framework in Section 3, a 15-storey steel modular building
shown in Fig. 3(a) is adopted as a representative structure of wind-sensitive self-standing modular buildings. The
building is 53.5m tall with a 26.1m-by-17.2m rectangular floor plan and comprises 180 modules in total (or 12
modules per storey) interconnected by corner, edge, and central IMCs. All modules share a constant width of 4.20m
and a height of 3.38m but have two different lengths of 9.45m and 7.60m. For each module, floor and ceiling beams
are welded directly to corner posts to form rigid intra-module connections, which is a common practice for
prefabricated volumetric modules [52]. To increase the building’s lateral and torsional stiffness, bracings are arranged
in a limited number of modules at selected locations as shown in Fig. 3 (b) without affecting the usage of the internal
space. In terms of cross-sections, corner posts and bracings are made of hot-finished SHS, while ceiling and floor
beams are made of hot-finished RHS. A linear finite element model of the building is developed using SAP2000® v18
software, comprising 6,854 (1D) frame elements, 1,110 dummy area elements (for modelling façade and internal wall
panels), and 900 rigid membrane elements (for modelling floor, ceiling, and roof panels). As shown in Fig. 3 (c), the
corner, edge, and internal IMCs are modelled using short vertical frame elements with a hinge in the middle, which is
released for biaxial bending. In addition, adjacent modules are horizontally interconnected at the edge and internal
IMCs using one and six 2-joint link elements with six degrees of freedom, respectively, as shown in Fig. 3(c). All link
elements are rigid in their local U1 and U2 directions, with their shear stiffness in local U3 direction, bending stiffness
about the local U2 and U3 axes, and torsional stiffness about the local U1 axis all set to 0. For dynamic analysis, all six
damping coefficients of the link elements are set to 0 for conservatism. Apart from the structural self-weight,
additional gravitational design loads are estimated based on actual MBSs [53] and applied as uniformly distributed
loads, as summarised in Table 1.
Table 1 Selected load combinations based on Eurocode 1 and design loads for sizing optimisation of the case-study building under
SLS and ULS design constraints according to Eurocode 3.
Design loads Floor Ceiling Roof Facade Wall Panels
Superimposed load [kPa] 0.75 1.00 2.50 1.00 0.50
Live load [kPa] 2.00 / 1.00 / /
Design load combinations SLS ULS
Gravitational loads 1.0G + 1.0Q 1.35 G + 1.5Q
1.0G + 1.5Q
Gravitational & wind loads 1.0G + 1.0Q ± 0.6 Wy
1.0G + 0.7Q ± 1.0Wy
1.35G + 1.5Q ± 0.9Wy
1.35G + 1.05Q ±1.5Wy

Fig. 3. (a) 15-storey case-study MBS and member design groups for the sizing optimisation investigation, (b) illustration of a
typical floor, (c) simplified finite element models of corner, edge, and internal IMCs, and (d) the first three vibration modes of the
case-study building.
To support practicality and cost-efficiency, twenty member/design groups are considered for the building, with
common cross-sections changing every three stories, yielding 20 sizing DVs in total as shown in Fig. 3 (a). The side
dimensions of corner posts and bracings are restricted to be within the ranges of 180 to 350 mm and 100 to 260 mm,
respectively, while the sectional depths of modular beams are restricted to be within the range of 120 to 400 mm. To
prevent local buckling of modular members before attaining the yield strength, Class 4 sections are excluded from the
ensuing sizing optimisations. This grouping of structural members is largely driven by fabrication considerations for
volumetric modular buildings and partially by structural rationality. In general, it is preferable to standardise modular
units by requiring the corner posts, floor and ceiling beams, and bracings in adjacent floors to use the same steel
sections to ensure economies in manufacturing and material procurement processes [54].
Prior to sizing the building for wind loads, the building is first designed to satisfy all SLS and ULS requirements
of Eurocode 3 [30] for gravitational load combinations in Table 1, prescribed by Eurocode 1 [55]. The cross-sections
of this design (termed “initial design” from hereafter) with a self-weight of 280.3 metric tons are reported in Fig. 3(b),
which are used as the starting point and the initial lower bound for the side constraints on the sizing DVs in the
subsequent displacement-based optimisation. The first three mode shapes of this design are obtained using standard
linear modal analysis and plotted in Fig. 3(d), with the fundamental vibration mode being translational in the global y
direction. The first three natural frequencies and the corresponding modal participating mass ratios in parentheses are
0.48 Hz (67.0%), 0.49 Hz (60.0%), and 0.53 Hz (73.0%), respectively. For later dynamic analysis, the structural
damping for the fundamental mode is taken as 0.8%, which closely follows the recommended value by Eurocode 1
[55] for steel buildings.

Fig. 4. (a) Static along-wind design forces, (b) power spectral density functions of across-wind design force, (c) along-wind drifts,
(d) along-wind inter-storey drift ratios, and (e) across-wind floor accelerations of the case-study building in Fig. 3 (a) for basic
wind speeds, 25.0, 27.5, and 30.0 m/s.
To assess the impact of wind intensity on the minimal sizing design of the case-study building, three basic
reference wind velocities are considered in the numerical part of this work: vb = 25.0, 27.5, and 30.0 m/s. These
reference velocities represent the 10-minute mean wind velocity at 10m above open flat country terrain with a return
period of 50 years. It is noteworthy that the adopted lower-bound velocity of 25.0 m/s is almost the minimum wind
speed required for the case-study building in Fig. 3 (a) to be governed by wind effects. In contrast, the upper-bound
velocity of 30.0 m/s is already among the highest basic wind speeds on the mainland of Europe according to Eurocode
1 [55]. This allows for the investigation of various structural behaviours of the building under high wind conditions.
Since the fundamental vibration mode of the adopted building is in the global y direction, along which the building's
tributary width is also broader, the case-study structure is more critical in this direction for both wind-induced drift
and floor acceleration. Consequently, the ensuing sizing optimisation only concerns this direction. Nevertheless, the
same sizing process can also be performed in the global x direction if so desired. To this end, the static along-wind
forces in the global y direction are calculated according to Eurocode 1 [55] assuming urban terrain, and plotted along
the building height in Fig. 4 (a). The power spectral density (PSD) functions of the VS-induced across-wind loads in
the global y direction (with the wind coming in the global x direction) are estimated using the across-wind excitation
model by Liang et al. [37] and plotted as continuous surfaces of the excitation frequency and elevation in Fig. 4(b).
Under the static along-wind loads in Fig. 4(a), the initial design reported in Fig. 3(b) is found to be deficient in
meeting the total building drift limit of H/600 and the inter-storey drift limit of h/500 [35] for vb = 30 m/s only, as
shown in Figs. 4(c) and (d), respectively. Furthermore, under the dynamic across-wind loads in Fig. 4(b), the initial
design with f1 = 0.48Hz is deficient in satisfying the occupant comfort stipulation by ISO6897:1984 [38] even for vb =
25 m/s, as shown in Fig. 4(e). The above comfort threshold is specified by the wind-induced maximum floor
acceleration given in terms of the RMS value as
(18)
which is plotted in Fig. 4(e). As seen, the threshold value remains unchanged for different wind speeds as it is only
affected by the building’s fundamental frequency. In view of Figs. 4 (c), (d), and (e), the case-study building in Fig.
3(a) is indeed wind-sensitive, thereby requiring further stiffening (through resizing) to meet the above static and
dynamic serviceability criteria.

5.2 Sizing optimisation under wind-induced drift and floor acceleration constraints
Specifically, for vb = 25.0 and 27.5 m/s, the (continuous) sizing optimisation process under wind serviceability
constraints starts directly with the floor acceleration constraints as the initial design in Fig. 3(b) is not governed by the
wind drift limits. For vb = 30 m/s, however, the sizing process starts with the drift limits first and then proceeds to the
acceleration constraints. Still, all three sizing processes begin with the same initial design, with the cross-sectional
areas (of the structural members) of this design being the lower bound of the initial side constraint, (see Fig. 2).
The variation in the structural self-weight of the case-study building throughout the sizing process is plotted in Fig.
5(a) for three basic wind speeds considered. As seen, the sizing iterations converge after 31, 53, and 79 steps for vb
=25.0, 27.5, and 30,0 m/s, respectively. The relative convergence tolerances for the objective function value (defined
as |M(a(p+1))-M(a(p))|/M(a(p+1))) and step size (defined as |a(p+1)-a(p)|/| a(p+1)|) are both set to 1e-4, where p is the iteration
number. Practically feasible steel sections are specified after convergence by mapping the obtained continuous-valued
cross-sectional properties of different member groups (defined in Fig. 3(a)) onto the corresponding catalogues of
commercially available hot-finished sections. For demonstration, the mapping operation for vb = 30.0 m/s is
graphically demonstrated in Fig. 5(b), (c), (d), and (e) for corner posts, floor beams, ceiling beams, and bracings,
respectively. In these plots, each coloured data point represents a unique hot-finished section, whose cross-sectional
area, A, and second moment of inertia, I, (or the major second moment of inertia, Iy, for RHS) are given as horizontal
and vertical coordinates, respectively, whereas the obtained optimal sectional properties are denoted by black markers.
In this setting, for corner posts (groups 1 to 5) and modular beams (groups 6 to 15), the mapping operation involves
selecting the lightest standard SHS (for each design group) whose I or Iy is not smaller than the optimal value; for
bracing members (groups 16 to 20), the mapping operation considers the cross-sectional area only. To this end, the
selected steel sections for satisfying the floor acceleration constraints in Eq. (18) for three basic wind speeds are
summarised in Table 2, which correspond to the discrete sizing designs represented by the unfilled triangular, square,
and circular markers in Fig. 5(a). The cross-sectional areas of these three sizing designs are collected in (see
Fig. 2). Notably, for corner posts only, one additional buildability constraint is introduced when selecting
commercially available sections, which makes sure that all posts share the same outer dimension along the building
height. This consideration stems from the fact that changing the corner post size along the building height can,
arguably, make the connection between adjacent modules more difficult [56].
Table 2. Optimised steel sections for the case-study building in Fig. 3(a) to satisfy drift and acceleration constraints under three
different wind intensities.
Basic wind
speed [m/s] Floor Corner posts Floor beams Ceiling beams Bracings
25.0
1 – 3 SHS300/14.2
RHS120×60/4.0 RHS120×60/4.0
SHS160/6.3
4 – 6 SHS300/10.0
7 – 9
SHS300/8.0
SHS160/5.0
10 – 12 SHS120/5.0
13 - 15 SHS100/4.0
27.5
1 – 3 SHS350/16.0 RHS120×60/4.0 RHS120×60/4.0
SHS180/12.5
4 – 6 SHS250/8.0
7 – 9
SHS350/10.0
SHS180/10.0
10 – 12 RHS180×100/5.0 RHS160×80/5.0 SHS140/8.0
13 - 15 RHS120×60/4.0 RHS120×60/4.0 SHS100/5.0
30.0 1 – 3
SHS350/16.0 RHS120×60/4.0 RHS120×60/4.0 SHS260/16.0
4 – 6
7 – 9 RHS400×200/12.5 RHS400×200/12.5 SHS 250/10.0
10 – 12 SHS350/10.0 RHS120×60/4.0 RHS120×60/4.0 SHS 180/12.5

13 - 15 SHS150/5.0
Fig. 5. (a) Sizing iterations of the case-study building under drift and floor acceleration constraints for basic wind speeds 25.0,
27.5, and 30.0 m/s, and mapping operation of continuous optimal cross-sectional properties onto different types of hot-finished
steel sections for (b) corner posts, (c) floor beams, (d) ceiling beams, and (e) bracing members.
The wind drift and inter-storey drift ratios of the optimally sized case-study building under the wind speed of
30.0 m/s are plotted in Figs. 6(a) and (b), respectively, together with the corresponding drift limits. The structural self-
weight of the resized building increases slightly to 289.0 t from that of the initial design, i.e., 280.3 t, which is only
designed against gravitational loads. In addition, the wind-borne RMS floor accelerations of the optimally sized case-
study building for the comfort stipulation under three different wind speeds are plotted in Fig. 7, together with the
corresponding ISO6897 comfort thresholds [38]. Notably, as vb increases, the latter threshold becomes increasingly
stringent because the resized building for an increased wind speed is stiffer and thus has a higher fundamental
frequency. In terms of compliance, Figs. 6 and 7 show that the code-prescribed drift and floor acceleration limits are
satisfied on all occupied floors of the optimal sizing designs . However, this is achieved at the expense of
increased structural steel consumption. Specifically, for vb = 25.0, 27.5, and 30.0 m/s, the required steel tonnages of
the discrete optimal sizing designs are found to be 485.2 t, 683.3 t, and 1078.8 t, respectively, which correspond to
73.1%, 143.8%, and 284.9% increments in the structural self-weight compared to the initial design reported in Fig.
3(b). Collectively, the above results suggest that typical tall self-standing MBSs can be governed by wind

habitability/comfort requirements even under moderate wind excitation, whereas the wind drift limits only affect their
(structural) sizing design under high wind speeds.
Fig. 6. Static along-wind load-induced (a) drifts and (b) inter-storey drift ratios of the optimally sized case-study building for
displacement and drift ratio constraints under basic wind speed 30.0 m/s.
Fig. 7. Across wind-induced RMS floor accelerations of continuous and discrete optimal sizing designs of the case-study building
for basic wind speeds (a) 25.0, (b) 27.5, and (c) 30.0 m/s with ISO6897 occupant comfort thresholds.
To shed light on the relative importance of different member groups (see Fig. 3(a)) in contributing to the lateral
stiffness of the adopted building, the cross-sectional areas and major second moments of inertia of the three
continuous optimal sizing designs (shown as the unfilled triangular, square, and circular markers in Fig. 5 (a)) are
plotted in Figs. 8(a) and (b), separately. As seen, for three wind speeds considered, the continuous sizing algorithm
(see Section 4.1) distributes most of the structural steel to the corner posts and then bracings, with their cross-sectional
areas and major second moment of inertia decreasing gradually with floor height and increasing monotonically with
the wind speed. For floor and ceiling beams, however, their cross-sectional areas are retained at the corresponding
lower bounds, i.e., , except for member groups 9 and 14 under vb = 27.5 m/s (corresponding to modular beams in
stories 7, 8, and 9) and groups 8 and 13 under vb = 30.0 m/s (corresponding to modular beams in stories 10, 11, and
12). This finding suggests that the lateral stiffness of braced tall self-standing MBSs is mostly contributed by corner
posts and bracings but not by modular beams. However, when the focal point of the design optimisation moves from
serviceability to structural integrity, those modular beams, which are not stiffened during the SLS stage of the resizing
workflow (see Fig. 2), are inadequate for passing the member strength verification under wind-related ULS load
combinations. In the next section, modular beams are subjected to discrete sizing optimisation under strength

constraints using the GA-based sizing algorithm described in Section 4.2.
Fig. 8. Cross-sectional areas and second moments of inertia (about major bending axis) of three continuous optimal sizing designs
of the case-study building in Fig. 3(a) to satisfy drift and acceleration constraints under three different levels of static along-wind
loads in Fig. 3(d) and across-wind excitations in Fig. 3(e).
5.3 Sizing optimisation under member utilisation constraints
For modular frame members under combined axial compression and biaxial bending, they must satisfy the
following buckling resistance requirements according to Eurocode 3 [30]
(19)
where NEd, My,Ed, and Mz,Ed are the design values of the compression force and maximum moments about the y-y and z-
z axes of the cross-section based on second- or third-order analysis, respectively, χy, χz, and χLT are the reduction
factors due to flexural and lateral torsional buckling, respectively, and kyy, kyz, kzy, and kzz are the interaction factors
(see Section 6.3.3 of Eurocode 3 [30] for more details). In this setting, for all 20 member groups shown in Fig. 3(a),
their maximum utilisation ratios calculated using the above formulas under wind-related ULS load combinations must
be kept below 1.0 to avoid member overstress. However, as stated in the last section, only the modular beam groups
are considered for discrete sizing optimisation under strength constraints using GA, as the utilisation ratios of corner
posts and bracings of the three serviceability-compliant designs in Table 2 are found to be well below 1.0 under the
corresponding levels of wind action. For demonstration, the member utilisation ratios of the serviceability-compliant
design for vb = 30.0 m/s are plotted in Fig. 9 (a), showing that only 8 groups of modular beams are overstressed under
wind-related ULS combinations. To further facilitate the discrete sizing process, the floor and ceiling beams in the
same storey are forced to use the same RHS, which halves the total number of active DVs that need to be optimised by
the GA. This consideration is underpinned by the fact that the floor and ceiling beams in the same storey are found to
contribute equally to the overall lateral stiffness of MBSs [57]. Moreover, it is shown later that they tend to be utilised
and stressed to a similar extent under static wind loads.
To initialise the GA, the population size is set to 50 per generation, and the stopping criterion is chosen that the
best penalty function value does not improve for 50 consecutive generations, or the total number of generations
reaches 150. The penalty function value of a member of a population is the fitness function if the member is feasible
or the maximum fitness function among feasible members of the population plus a sum of constraint violations if the
member is infeasible [50]. As in Section 5.1, the sectional depth of the available RHS for modular beam groups is
limited within the range of 120–400 mm to reduce the vertical spacing between the corner intra-module connections
of stacked modules (see Fig. 1(c)). In this setting, there are 58 candidate RHS (as visualised in Figs. 5 (c) and (d))

available for member strength-constrained discrete sizing optimisation. At this juncture, it is worth noting that
although the total number of DVs has been reduced considerably from 20 to 5 (by deactivating the member groups
that do not violate the strength constraint and by requiring the ceiling and floor beams in the same storey to use the
same section), discrete sizing optimisation under member strength constraints using GA is still computationally
expensive, as large displacement analysis of laterally flexible building structures (such as tall self-standing MBSs with
hinged IMCs) is time-consuming. Accordingly, the maximum number of iterations and the number of stall generations
(whose best fitness value does not improve) are set to relatively low values.
For demonstration, the discrete sizing process under member strength constraints for vb = 30 m/s using GA is
plotted in Fig. 9 (b), which starts with the serviceability-compliant design (in Table 2 and Fig. 5(a)) at a self-weight of
1023.1 tons and converges after 70 generations at 1081.3 tons. As shown, the best penalty function value no longer
improves after the first 21 generations, and the sizing “evolution” is terminated after 50 stall generations at step 70. In
total, 3,500 potential sizing designs are evaluated using SAP2000’s large-displacement solver, and the total core hours
required to arrive at the final sizing design on a Xeon E5-1660V4 processor (base frequency 3.20 GHz) is 99.8 hours.
However, it is important to note that although all genetic operations on the individual members of a generation (see
Section 4.2) are sequential, the constraint function evaluation (involving large-displacement analysis and Eurocode 3-
prescribed structural member verification) for different individual designs within a generation can occur in a parallel
fashion, which is the most time-consuming part of the discrete sizing process. The above discrete optimisation is also
performed on the other two serviceability-compliant designs in Table 2 for their corresponding wind speeds. The
optimal cross-sections of the three final sizing designs satisfying the strength constraints are detailed in Table 3.
Finally, the member utilisation ratios of these three designs are plotted in Fig. 10. By cross-comparing Fig. 10 (c) and
Fig. 9 (a), it is seen that by changing the cross-sections of modular beams in Table 2 to those in Table 3 for vb = 30.0
m/s (with the structural self-weight increasing from 1055.7 tons to 1081.3 tons), the utilisation ratios of modular beam
groups are all reduced to below 1.0. Therefore, the three designs in Table 3 are code-compliant not only in terms of the
(static and dynamic) serviceability requirements but also in terms of member strength requirements. Figure 10 shows a
large variation in utilisation ratio among different member groups under wind-related ULS load combinations,
regardless of wind speed. Corner posts and bracings exhibit relatively low utilisation ratios, reflecting their
governance by global static and dynamic serviceability constraints. In contrast, modular beams display higher
utilisation ratios due to their criticality to strength criteria. This disparity arises because tall building design under
wind loads is primarily governed by lateral stiffness-related serviceability issues. As a result, most structural members
of wind-sensitive tall buildings tend to fall short of full utilisation in terms of member resistance [58].
In the next section, a comprehensive performance assessment is performed on these optimal designs, shedding
light on the structural behaviours of optimised tall self-standing MBSs under static and dynamic wind effects.
Fig. 9. (a) utilisation ratios of modular members of the comfort-optimal sizing design, , under wind-related ultimate limit
state load combinations, and (b) variation of the penalty function throughout the discrete sizing process using GA under member
strength constraints for basic wind speed of 30.0 m/s.

Table 3. Optimal sections for the case-study building in Fig. 3(a) to satisfy member strength requirements under three different
wind intensities.
Basic wind
speed [m/s] Floor Corner posts Floor beams Ceiling beams Bracings
25.0
1 – 3 SHS300/14.2
RHS140×80/4.0
SHS160/6.3
4 – 6 SHS300/10.0
7 – 9
SHS300/8.0
SHS160/5.0
10 – 12 SHS120/5.0
13 - 15 RHS180×100/5.0 SHS100/4.0
27.5
1 – 3 SHS350/16.0 RHS140×80/4.0
SHS180/12.5
4 – 6 SHS250/8.0
7 – 9
SHS350/10.0
SHS180/10.0
10 – 12 RHS180×100/5.0 SHS140/8.0
13 - 15 RHS160×80/5.0 SHS100/5.0
30.0
1 – 3
SHS350/16.0 RHS140×80/4.0 SHS260/16.0
4 – 6
7 – 9 RHS400×200/12.5 SHS250/10.0
10 – 12 SHS350/10.0 RHS140×80/4.0 SHS 180/12.5
13 - 15 RHS160×80/5.0 SHS150/5.0
Fig. 10. utilisation ratios of modular members of the optimal sizing designs (satisfying all design constraints) in Table 3 under
wind-related ultimate limit state load combinations for basic wind speeds (a) 25.0 m/s, (b) 27.5 m/s, and (c) 30.0 m/s.
6 Performance Assessment of Optimally Designed Case-Study Building
6.1 Elastic instability based on undeformed geometries
For tall self-standing MBSs with IMCs in Fig. 1(c) and hot-finished hollow sections, the lack of column
continuity (in terms of rotational stiffness) makes their structural stability under combined gravitational and lateral
wind loads more difficult to predict compared to conventional building structural systems [46]. Therefore, it is deemed
important to investigate the theoretical buckling strength of these structures under different wind intensities by
utilising the optimally sized case-study building in Section 5.3 as a typical representation of tall self-standing MBSs.
To this end, eigenvalue buckling analysis is performed on the three optimal sizing designs in Table 3 under wind
related ULS load combinations in Table 1 in line with Eurocode 1 [55]. For all three buckling analyses, the first 20
buckling modes are investigated, with the eigenvalue convergence tolerance set to 1E-09. All modular members are
meshed into four segments of equal length as further discretisation no longer affect the buckling analysis results.
In this setting, the first buckling modes of the three optimal sizing designs in Table 3 are obtained using the
unstressed stiffness matrix and shown in the left column of Table 4, together with the buckling load factor (BLF) and
buckling load case (BLC). As seen, the critical buckling modes are all associated with the bracing members in the top

storey, with the BLFs equal to 2.68, 2.86, and 9.50 for basic wind speeds of 25.0, 27.5, and 30.0 m/s, respectively.
Moreover, the first 20 buckling modes of the three optimal designs are all affiliated with the bracing members in the
upper part of the structure, while no global buckling mode can be found within the first 20 modes. To further examine
the theoretical buckling strength, the above analysis is repeated for the same optimal sizing designs and wind actions
but now without discretising/meshing the bracing members. This manipulation is used to suppress the buckling modes
associated with the bracing elements, allowing the retrieval of other/higher instability modes. The re-calculated
buckling modes for three wind speeds are listed in the right column of Table 4. For vb = 25.0 m/s, the columns of the
two corner modules in storey 4 on the leeward side of the building are seen to buckle in a bow shape within the global
xz plane (see Fig. 3(a)) under the ULS combination of 1.35G+1.05Q±1.5Wy at the BLF of 22.48. For vb = 27.5 and
30.0 m/s, the critical buckling modes are both associated with the floor and ceiling beams of modules (within the yz
plane) on the ground floor. Again, no global buckling mode is observed within the first 20 eigenmodes. Collectively,
these results indicate that the critical BLF of the first global instability mode is well above 10 for the three sizing
designs. The latter value is a critical threshold recommended by Eurocode 3 [30] for justifying the use of first-order
structural analysis to determine member design forces. However, as demonstrated in the following section, the
increase in wind-incurred member forces caused by the geometric nonlinearity of the structure is significant and must
be accounted for in the analysis by considering third order/large-displacement effects. In this context, Section 5.2.1 of
Eurocode 3 [30] may be unconservative and should be consulted with caution when designing tall self-standing
MBSs.
Table 4. Critical eigenvalue buckling modes of the optimal sizing designs in Table 3 under wind-related ultimate limit state load
combinations for basic wind speeds 25.0, 27.5, and 30.0 m/s.
Basic wind
speed [m/s]
Critical buckling mode
All elements meshed into 4 segments Without meshing bracing elements
25.0
Flexural buckling of bracing (top storey)
BLF: 2.68
BLC: 1.35G+1.5Q±0.9Wy
Flexural buckling of corner posts (4th storey)
BLF: 22.48
BLC: 1.35G+1.05Q±1.5Wy
27.5
Flexural buckling of bracing (top storey)
BLF: 2.86
BLC: 1.35G+1.5Q±0.9Wy
Flexural buckling of modular beams (1st storey)
BLF: 24.42
BLC: 1.35G+1.5Q±0.9Wy
30.0
Flexural buckling of bracing (top storey)
BLF: 9.50
BLC: 1.35G+1.5Q±0.9Wy
Flexural buckling of modular beams (1st storey)
BLF: 17.61
BLC: 1.35G+1.5Q±0.9Wy

6.2 Effects of geometric nonlinearity on wind-induced structural response and member utilisation ratios
This section aims to investigate the sensitivity of the optimal sizing designs in Table 3 to geometric nonlinear
effects in terms of structural response and member design forces under wind-related SLS and ULS load combinations
defined in Table 1. To this aim, the attention is first placed on wind-induced roof drift in the along-wind direction and
VS-induced floor acceleration in the across-wind direction. Specifically, the static displacement response of the
optimal sizing designs is assessed for three different basic wind speeds (i.e., vb = 25.0, 27.5, and 30.0 m/s) using static
P-Δ and large-displacement analyses, separately. The RMS floor acceleration is assessed using the standard linear
frequency-domain analysis of random vibrations [59–61], with the P-Δ and large-displacement effects incorporated in
the stiffness matrix. The latter is achieved conveniently by considering the geometric nonlinear effects on the case-
study building under the set of factored gravitational loads, i.e., 1.0G+0.7Q, and then using the “softened” stiffness
matrix developed from this load case for the linear frequency-domain analysis under the across-wind excitation in Fig.
4(b). In general, this technique is adequately accurate for modelling the geometric nonlinear effects due to the sway of
building structures, while enabling rapid frequency-domain analysis for the purpose of design [62]. This is because the
summed geometric stiffness terms of columns associated with the lateral loads is zero, and only the axial forces caused
by the weight of the structure need to be included in the evaluation of the geometric stiffness terms for the complete
building [63]. Finally, the accuracy of all P-Δ and large-displacement analyses are checked by re-running the analysis
using a smaller step size and convergence tolerance. The roof displacement and RMS acceleration on floor 14 (the last
occupied floor), normalised by the corresponding values obtained using linear/first-order static and dynamic analyses,
are plotted in Figs. 11(a) and (b), respectively. For all wind speeds considered, it is seen that the roof displacements
and floor acceleration predicted by P-Δ and large-displacement analyses are only slightly larger than those predicted
by the first-order analysis. Also, there is a consistent trend that the structural responses predicted by large
displacement solver are always marginally larger than those by P-Δ method, and the geometric nonlinear effects on
wind-induced displacement is larger than on floor acceleration. The above findings confirm that the optimal sizing
designs in Table 3 obtained using the proposed sizing workflow have well-conditioned floor-by-floor stiffness-over-
weight ratios, as the increase in displacements and floor accelerations are insignificant and less than 2% in general.
Fig. 11. (a) static along-wind load-induced roof displacements, and (b) dynamic across-wind load-induced floor acceleration on
floor 14 of the optimal sizing designs in Table 3 (satisfying all SLS and ULS design constraints) predicted by P-Δ and large
displacement solvers, values normalized by the corresponding values predicted by linear solver.
Next, the geometric nonlinear effects on member design forces are numerically quantified by cross-comparing
member utilisation ratios obtained by linear, P-Δ, and large-displacement solvers of SAP2000 under two wind-related
ULS load combinations, 1.35G+1.5Q±0.9Wy and 1.35G+1.05Q±1.5Wy. The mean and peak utilisation ratios for 20
groups of modular members (see Fig. 3(a)), evaluated using Eq. (19) based on the internal forces calculated using P-Δ
and large-displacement analyses, are normalised by the corresponding values from the linear analyses and plotted in
the upper and lower rows of Fig. 12, respectively. As shown, using P-Δ solver does not lead to significant increase in
the member design forces, as both normalised mean and peak utilisation ratios for all member groups and wind
intensities are only slightly above 100%. However, member design forces are “amplified” significantly by large-
displacement analysis, which tracks the positions/orientations of structural elements under the external loads using an

updated Lagrangian formulation and considers the equilibrium equations in the deformed configuration of the
structure [64]. For the adopted modular building, the increase in the peak utilisation ratios for corner posts in the upper
stories and floor beams can be as excessive as 400%. The above observation suggests that the effects of large
displacements and rotations upon the modular members of tall self-standing buildings must be considered properly in
the analysis. Failure to do so may significantly underestimate the internal forces for member design, potentially
causing even more excessive geometric nonlinear effects and member overstress.
Fig. 12. Mean (upper row) and peak (lower row) utilisation ratios of modular member groups of the optimal sizing designs in
Table 3 evaluated by P-Δ and large displacement solvers for basic wind speeds 25.0 m/s (left column), 27.5 m/s (middle column),
and 30.0 m/s (right column), values normalized by the corresponding utilisation ratios evaluated by linear solver.
6.3 Effects of global sway imperfections on member utilisation ratios
In this section, the effects of global geometric imperfections on the wind-induced member forces of the optimally
designed case study building are assessed for the three wind speeds considered before. For the adopted building, it is
numerically verified that the lateral static wind loads are less than 15% of the gravity loads at all floors. Accordingly,
the global imperfections need to be considered for wind-related load combinations according to Eurocode 3 [30].
These imperfections can be due to a lack of verticality/straightness of structural members, lack of mechanical fit, and
minor eccentricities in the IMCs of modular buildings. As in Section 6.2, member imperfections are not allowed in the
structural analysis but accounted for when determining member resistances, which is the standard European and UK
practice [65]. According to Eurocode 3 [30], the assumed shape of global imperfections may be derived from the
global elastic buckling mode of the structure in the buckling plane considered. However, this is not easy to implement
for the adopted building, as its global sway (buckling) modes cannot be identified for the three optimal designs in
Table 3 under the two wind-related ULS load combinations in Table 1. To this end, the first translational vibration
modes of the three sizing designs in Table 3 (all in the y-z plane) are adopted separately as the initial global
imperfections, with the imperfection amplitudes at the roof level all set to H/300 following the recommendations by
Eurocode 3 [30]. This allowance takes into the basic value for global out-of-verticality imperfection and the reduction
factor for building height H (see Eurocode 3 [30]). Notably, the above imperfection amplitude is larger than the
normally specified tolerances for tall MBSs [12], and thus conservative for the purpose at hand. At this juncture,
structural analysis of the three optimal designs in Table 3 under the two wind-related ULS combinations with and
without the above global imperfections is performed using the large-displacement solver of SAP2000, as it is
established in Section 6.2 that the effects of deformed positions/orientations of modular members on member forces
are significant under combined gravitational and wind loads. The averaged and peak member utilisation ratios of the

three sizing designs with global imperfections are normalised by the corresponding values of the same designs without
imperfections and plotted in Fig. 13 for all 20 (member) design groups (defined in Fig. 3 (a)) and for three basic wind
speeds.
Fig.13. (a) average and (b) peak normalised utilisation ratios of modular member groups of the optimal sizing designs in Table 3
(satisfying all SLS and ULS design constraints) with global sway imperfections evaluated using large displacement solvers, values
normalised by the corresponding values of the same optimal designs without imperfections.
As shown in Fig. 13 (a), the normalised mean utilisation ratios of corner posts, floor beams, and ceiling beams
are not significantly affected by the global sway imperfections, as the differences in the ratios are found to be less than
1% across all wind speeds considered. For bracing members, the increases are slightly higher but still within 5%.
Nevertheless, as shown in Fig. 13 (b), the increase in the peak utilisation ratios due to the sway imperfections are
much larger, especially for modular beams and bracing members. Specifically, for corner posts, floor beams, ceiling
beams, and bracings, the maximum increases in the peak utilisation ratios across the three wind speeds are 9.1%,
26.6%, 36.3%, and 37.2%, respectively. This finding suggests that tall self-standing MBSs, even after optimal sizing
for serviceability and strength requirements, can be still sensitive to global sway imperfections, which must be
considered properly in structural analysis such that its amplification effects on member design forces/moments are not
underestimated in design. Nevertheless, it is important to note that the building quality of prefabricated modular units
is closely monitored and controlled in a factory environment through precision manufacturing, and a high level of
precision can be achieved at construction sites by practicing accurate assembly techniques. Accordingly, the global
sway imperfections in realistic tall modular buildings may be much smaller than the assumed value of 1/300H.
7 Summary and concluding remarks
In this study, a performance-based sizing optimisation framework was developed to reduce the structural self-
weight of wind-sensitive tall self-standing modular buildings subjected to multiple performance constraints. This was
achieved by formulating a constrained sizing optimisation problem for tall modular buildings to meet the along-wind
drift and across-wind occupant comfort criteria, which govern the global structural design in the serviceability limit
state (SLS), and to satisfy member strength requirements, which govern the member design in the ultimate limit state
(ULS). An efficient numerical solution strategy was devised to solve the constrained discrete weight minimisation
problem by first reformulating the drift and acceleration criteria as a structural compliance and natural frequency
constraints, separately, and then by decoupling the original optimisation problem into two sequential stages: a convex
SLS stage and nonconvex ULS stage. The latter can be solved using any combination of local and global optimisation
methods, respectively. For numerical application of the framework, an interior point method was used for minimal
sizing under either a structural compliance or natural frequency constraint, whereas a genetic algorithm was used for
minimal sizing under member strength constraints while accounting for geometric nonlinear effects. The applicability
of the proposed sizing framework was demonstrated using a 15-storey self-standing modular building exposed to three
different basic wind speeds ranging from 25.0 to 30.0 m/s. The adopted building has a very common structural
configuration and module arrangement, which arguably makes it a representative benchmark structure for studying the
structural behaviours of tall self-standing modular buildings under different wind conditions. For each wind velocity,

the optimal discrete sizing designs satisfying the drift, floor acceleration, and member strength constraints were
obtained using a structural optimisation application developed using SAP2000's Open Application Programming
Interface, which allows for the integration of SAP2000 (for structural analysis) and MATLAB (for constrained
optimisation). The structural performance of the three optimal designs under combined gravitational and wind loads
was comprehensively assessed in line with Eurocode 3 [30]. Insights were provided into the case-study building’s
elastic instability behaviour, geometric nonlinear effects on wind-induced responses, and impacts of global sway
imperfections on member design forces, with the main findings summarised as follows.
1. By comparing the sectional properties of different member groups of the three optimal designs, it was found that
increasing the size of corner posts and bracings was more effective in enhancing the lateral stiffness of the case-
study building than enlarging the ceiling and floor beams. This observation emphasises the relative importance of
corner posts and bracings in the static and dynamic serviceability design of tall self-standing MBSs with similar
structural layouts to that of the case-study building.
2. For the three wind speeds considered, it was found that the serviceability design of the case-study building was
governed by the across-wind floor acceleration constraints but not by the along-wind drift requirements. In this
regard, inter-module connections (IMCs) may play a critical role in affecting the dynamic acceleration response
of tall self-standing MBSs under wind excitations; these bolted joints can offer additional energy dissipation to
the building through friction and material damping when, for instance, high damping material(s) is used.
Alternatively, supplementary damping devices [60,61,66,67,68] can be utilised to enhance the habitability of tall
self-standing modular buildings under wind excitations. This, in turn, reduces the amount of structural material
needed to increase the lateral stiffness of the building (see [19]). Furthermore, at the member level, the size of
corner posts and bracings of the case-study building was governed by overall dynamic stiffness requirements to
satisfy the comfort stipulation, whereas modular beams were governed by strength criteria to satisfy the buckling
resistance requirement.
3. Regarding the elastic buckling behaviour, the optimal sizing designs of the case-study building with hinged IMCs
demonstrated no global buckling modes under wind-related ULS load combinations for all wind intensities
considered. This finding suggests that the resilience of tall self-standing MBSs against global instability under
combined gravitational and high winds (e.g., vb = 30.0 m/s) may be achieved by structural sizing for
serviceability and integrity requirements. More importantly, the critical elastic buckling factor of 10.0 by
Eurocode 3 [30], which may be used to justify the first-order structural analysis when satisfied, was found
unconservative for the case-study building. This finding warrants further research to determine a more
appropriate value for tall self-standing MBSs.
4. In terms of geometric nonlinearity, its effects on amplifying the wind-induced (static) displacement and
acceleration responses of the optimally sized case-study building were insignificant. Therefore, the global SLS
design of tall self-standing MBSs, with similar structural layouts to that of the case-study building, may be
performed using first-order analysis. However, for structural member design, the effect of large displacements
and rotations (i.e., third-order effects) on modular members must be accounted for in the analysis. Using second-
order analysis may still underestimate the internal forces for member design, potentially causing even more
excessive geometric nonlinear effects and member overstress.
5. It was found that global sway imperfections had significant impacts on the member utilisation ratios of the case-
study building calculated using Eurocode 3 [30], particularly for the modular beams and bracing members. This
finding suggests that tall self-standing MBSs may be highly sensitive to global sway imperfections, which must
be adequately considered in structural analysis to avoid underestimating their amplification effects on member
design forces.
6. For the basic wind speeds of 25.0, 27.5, and 30.0 m/s, the corresponding optimal sizing designs of the case-study
building was found to have a structural self-weight of 78.3, 112.7, and 160.6 kg/m2, respectively. At the moderate
wind speed of 25.0 m/s, the self-weight of the optimal sizing design was comparable to the average weight of
steel frameworks for typical 10- to 15-storey high-rise buildings (usually around 75 – 90 kg/m 2 [69]) that use
conventional structural systems.

As a final remark, the current study has confirmed that tall, self-standing modular buildings are structurally
feasible under high wind conditions with reasonable steel tonnages. With the proposed optimisation-driven design
framework, such buildings can achieve even greater cost reduction, beyond the current rate of 10% to 20% [4] (which
is based on non-optimally sized modular buildings), as well as faster construction times (as all building components
can be prefabricated offsite with no onsite construction required). From a structural optimisation standpoint, the
proposed framework offers an efficient and flexible sizing approach for minimum-weight design of self-standing
modular structures subjected to concurrent, wind-related serviceability and strength constraints. The proposed
framework is envisioned to serve as a useful tool for creating wind-resilient and material-efficient tall self-standing
modular buildings.
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