Topology optimisation of braced frames for high rise buildings

1. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
30 – 31, December 2014, Ernakulam, India
84
TOPOLOGY OPTIMISATION OF BRACED FRAMES FOR
HIGH-RISE BUILDINGS
Joshua Mathew1
, Nisha Babu2
1
M.Tech Computer Aided Structural Engineering, SreeNarayanaGurukulam College of Engineering, Kadayiruppu P.O.,
Kolenchery-682311, Ernakulam, Kerala
2
Assistant Professor, Department of Civil Engineering, SreeNarayanaGurukulam College of Engineering, Kadayiruppu
P.O., Kolenchery-682311, Ernakulam, Kerala
ABSTRACT
This paper describes a topology optimisation technique using continuum quadrilateral finite elements to design
structural braced frames that are part of the lateral system of a high-rise building. Continuum topology optimisation is
treated as the problem of improving the performance of a continuum design domain in terms of the efficiency of material
usage and overall stiffness. This work explores the analytical aspects of optimal geometry for braced frames to
understand the underlying behaviour. Conventional bracing layouts and the optimised layout were analysed for deflection
and subsequently compared.
Keywords: Braced, Bracing, Frame, Layout, Topology Optimisation.
1. INTRODUCTION
1.1 General
Topology optimisation has been around for several years in various engineering fields and recently it has gained
popularity in structural engineering with the advent of structural expressionism. Using topology optimisation, engineers
can find the best concept design that meets the design requirements. The structure can be altered to suit a particular
architectural style without compromising its structural integrity.
Even though, this technique can be applied to various segments in civil engineering, this paper focuses on the
topology optimisation of the lateral bracing system in high-rise buildings in order to find an optimal bracing layout to
minimise deflection and material usage which in turn reduces cost and the impact it has on the environment.
The goal of topological optimisation is to find the best use of material for a body such that an objective criterion
(i.e., global stiffness, natural frequency, etc.) takes out a maximum or minimum value subject to given constraints (i.e.,
volume reduction). This method is based on the concept of gradually removing unnecessary or inefficient material from a
structure to achieve an optimal design.
Topology optimisation is implemented through the use of finite element method for analysis.To develop the
lateral system, topology optimisation is carried out on an isotropic, homogeneous material.
Braced frames are commonly employed in high rise buildings to resist lateral loads in the form of wind or
seismic forces. They have been used in several existing buildings like the John Hancock Center (Chicago, IL), Broadgate
Tower (London, UK)and Bank of China Tower (Hong Kong).The diagonal braces of such systems are conventionally
arranged to range between 45° to 60°.
INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND
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2. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
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The optimisation technique reveals the most efficient layout of the available material that also minimises the
displacement. Thus, the optimised lateral system would save material cost as well as minimise the impact on natural
resources.
1.2 Optimisation of lateral load resisting systems
Multi-storey buildings are designed not only for gravity loads but also for lateral loads. In the design for lateral
loads, the structural system must be adequate stiff to resist wind and seismic forces. The effect of lateral load on
buildings increase rapidly with the increase in height of the building.[9]
2. OBJECTIVE AND SCOPE
2.1 Objective
To study a technique for the design of an optimal braced frame system.
To verify the constant state of stress in an optimised frame under certain conditions.
To analytically derive the optimal geometry for a braced frame and then confirm it using Altair®
OptiStruct®
.
To analyse and compare conventional bracing layouts with the optimised layout for displacement in SAP2000.
2.2 Scope
This paper explores an optimal bracing layout to maximise structural performance whilst minimising the
material required. Structural performance could include tip displacement, frequency, compliance, critical buckling load,
etc.The focus of this paper is to minimise the compliance whereby the stiffness is maximised and to relate this feature to
the structure.Adopting such a technique, the end result would be an innovative, lightweight and structurally efficient
design.
3. METHODOLOGY
3.1 Topology optimisation formulation
1)Problem statement: Topology optimisation consists of searching for the optimal layout of material in a given design
domain in terms of an objective function. Throughout this work, the aim is to maximise the stiffness of the structure. The
minimum compliance problem can be stated in terms of the density, ρ, and the displacements, u, stated as follows:
The compliance of the structure is denoted by c, K(ρ) represents the global stiffness matrix which depends on
the material densities, while u and f are the vectors of nodal displacements and forces, respectively. The volume
constraint, VS, represents the maximum volume permitted for the design of the structure. The design, or topology of the
solution, is determined by the material density, ρ. A zero density value signifies a void whereas one represents solid
material.
The weight of a structure is used as the objective function and the constraint is imposed on the mean compliance
of the structure. Continuum topology optimisation is treated as the problem of improving the performance of a
continuum design domain in terms of the efficiency of material usage and overall stiffness.The weight of a structure is
used as the objective function and the mean compliance is treated as the constraint. In other words, the performance
objective is to minimise the weight of a continuum design domain while maintaining its overall stiffness within an
acceptable performance level. [9]
3.2 Overview of the softwares used
3.2.1 Altair®
OptiStruct®
: A structural analysis solver for linear and non-linear structural problems under static and
dynamic loadings. Based on finite-element technology, and through advanced analysis and optimisation algorithms,
OptiStruct helps designers and engineers rapidly develop innovative, lightweight and structurally efficient

3. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
30 – 31, December 2014, Ernakulam, India
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designs.Topology optimisation generates an optimised material distribution for a set of loads and constraints within a
given design space. The design space can be defined using shell or solid elements, or both.
3.2.2 SAP2000 v14: SAP2000 is general-purpose civil-engineering software ideal for the analysis and design of any type
of structural system. Basic and advanced systems, ranging from 2D to 3D, of simple geometry to complex, may be
modelled, analysed, designed, and optimized using a practical and intuitive object-based modelling environment that
simplifies and streamlines the engineering process.
3.3 Optimal braced frames – analytical aspects
A benchmark is established and presented in Stromberg et al[6]with the help of Principle of Virtual Work,
deriving the analytical aspects of an optimal braced frame to compare with the numerical results illustrated later on.
Optimal design result in a state of constant stressconfirms that, minimum compliance leads to constant stresses. In
general, for the compliance minimisation problem, a state of constant strain energy density represents the condition of
optimality.Since the strain energy density is related to the Von-Mises stress, the effective stresses in optimal structures
are constant. [6]The constant stress condition is verified later for the continuum with the help of
Altair®
HyperWorks®
software.
Considering the problem in Fig.2 (top), a point load representing the wind (lateral) force acting on the frame is
applied at the top left corner and symmetry is enforced. The topology optimization of the continuum mesh does not lead
to a simple 45° bracing angle due to the interaction of shear and axial forces. The 45° bracing angle would be the
outcome of a pure shear problem as shown in Fig. 2 (bottom). However, the cantilever problem (used to model a high-
rise) is never pure shear because the overturning moment PH does not appear in a pure shear problem. Therefore, the
topology optimisation results in a ‘‘high-waisted’’ cross bracing. The actual location of the intersection point of the
braces at 75% of the height H is confirmed in Fig. 2(top right).[6]
Fig.1: Illustration of the differences between the case of a cantilever structure (top) and the pure shear problem
(bottom) [6]
The braced frame central work point is always located at 75% of the module height.
3.4 Optimisation Parameters
Even though, there are several objectives that can be taken into consideration (frequency, compliance, buckling,
tip displacement etc.), for this work we maximise the overall stiffness of the building; therefore; minimum weighted
compliance is used as the objective function for topology optimisation. The peripheral skin or shell of the building is
considered as the design domain.
Objective: Minimise weighted compliance.
Design Constraints: The volume fraction should not exceed 20%.
Design variables: The density of each element in the design space.
Wind load which has a windward and a leeward component was considered as the only lateral load.
The dimensions of the module were assumed to be 48 m by 41.5 m. An anti-symmetric point load of P = 2 MNwas
applied to the top corners of the shell componentwith a thickness of t = 0.200 m and Young’s Modulus E = 200,000
MPa(steel).
For the topology optimisation, a volume fraction of 20% is used.

4. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
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Fig.2: Problem statement
Fig.3: Behaviour of wind load
An anti-symmetric point load of P = 2 MNwas applied to the top corners of the shell component on account of
the behaviour of wind. As naturally occurring wind blows across a building, the wind hits the windward wall causing a
direct positive pressure. The wind moves around the building and leaves the leeward wall with a negative pressure.
Topology optimisation was run on quadrilateral elements with an element size of one for the lateral load. The
process is iterative and concludes once both the objective and the constraint are satisfied. An optimal bracing layout was
obtained in the 21st
step.
It is required to design a bracing system for this steel frame in order to control the lateral drifts.
Lateral bracing systems in multi-storey steel buildings are mainly designed to resist lateral loads hence it was
assumed that the effect of floor loads that were carried by beams and columns on the layout of bracing systems could be
neglected. All topologies obtained are symmetrical about the vertical axis of the frame as expected under the reversible
wind loading conditions. It should be noted that the exact dimensions of bracing members as well as columns that need to
be resized are not shown in the result, which only illustrates the basic layout of the bracing system for the frame
structure. Bracing members are rigidly connected to the frame members. [9]
3.5Modelling of braced frames in SAP2000
Various braced frame models were studied in SAP2000, each illustrating a specific aspect of braced frame
structural behaviour. A sixteen storey model measuring 48m by 41.5m was used to study the effects of different bracing
schemes. The intersection of the bracing was constrained to move along the centreline of the module due to symmetry,
the height ratio, z/H, was varied from 0.5 to 1 (z being the distance of the brace work point from the base) and the
corresponding tip displacement was plotted (see Fig. 13). The optimal z/H ratio (i.e. the one that minimises the deflection
at the top of the frame) is shown to be 0.75H in Fig.13.

5. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
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Rolled steel beam ISWB 200 was the steel section that was adopted for modelling the bracing layouts.
Fig.4: Dimensions of rolled steel beam ISWB 200
Fig.5: Properties of rolled steel beam ISWB 200
Fig.6: Undeformed and deformed shape of bracing layout with a z/H ratio of 0.75 (Optimised brace)
The efficiency of the optimal bracing layout was then compared with other conventional bracing layouts in
terms of displacement in SAP2000.
Inferences made on the models are detailed in the succeeding chapter.

6. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
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4. RESULTS AND DISCUSSION
4.1 Altair®
OptiStruct®
Results
After 21 iterations, all the conditions in the problem statement were satisfied. The final topology of the shell
element is as shown in Fig.9 and Fig.10. Density of the element ranges from a value of one to zero. The colour red
represents the most critical element and colour blue represents the least efficient element. All the other colours in-
between stand for values in between one and zero.
Fig.7: Final topology of the shell component with both efficient and inefficient elements
Fig.8: Final topology of the shell component after the exclusion of inefficient elements
1) Optimal Braced Frame - Constancy of Stress:Constant state of stress in optimised frame is verified in the continuum
using Altair®
HyperWorks®
- the Von-Mises stresses are nearly constant within each optimised member.
Fig.9: Topology optimisation of the frame using quadrilateral elements and corresponding plot of Von-Mises stress

7. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
30 – 31, December 2014, Ernakulam, India
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Fig.10: Optimal brace work point
Since the shell was meshed with quadrilateral elements each with an element size of one, there are 48 elements
in the Z-direction. It was observed that the brace work point was formed 36 elements above the base. Hence, the height
ratio, z/H was found out to be 0.75 confirming the analytical result. The optimal z/H ratio (i.e. the one that minimises the
deflection at the top of the frame) was found to be 0.75H.
This result has been further confirmed by modelling various bracing layouts using discrete members in SAP2000.
4.2 SAP2000 Results
The joint displacement for each bracing setup was obtained from SAP2000 and plotted for comparison.
Fig.11: Plot of deflection versus brace height intersection ratio
Comparing z/H ratios varying from 0.5H to 1H, the tip displacement is the least when the z/H ratio is 0.75H
taking both the translation in X and -Z direction into consideration.

8. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
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Fig.12: Plot of deflection for various bracing layouts
Comparing the optimised bracing layout with various other conventional bracing layouts, the tip displacement is
the least in the optimised bracing taking both the translation in X and -Z direction into consideration.
5. CONCLUSION AND RECOMMENDATIONS
5.1 Conclusion
The development of a lateral braced frame system in a high-rise building presented in this work enables the
structural engineer to swiftly identify an efficient bracing layout.
The analysis conducted for a single module braced frame can be extended to a frame with multiple modules
along the height and a single load applied at the top by observing the relationships between the geometry of the frames
and the forces in its members. The cumulative displacement at the top would be significantly reduced as the modules are
increased.
The constant state of stress in an optimised frame under certain conditions was verified. The optimised layout
was analytically derived and then verified using Altair®
OptiStruct®
, an optimisation software.
Conventional bracing layouts and the optimised layout were analysed for deflection in SAP2000 and subsequently
compared. After the development and analysis of an optimal bracing layout, it was found out that it performed better than
the conventional bracing layouts considering the displacement factor.
This work can further be extended to three dimensional structures to optimise its structural performance such as
the tip displacement, frequency, compliance, critical buckling load, etc.
6. ACKNOWLEDGEMENT
The first author acknowledges the support from Altair®
HyperWorks®
, India for providing the student edition
license of the software.
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