Truss topology optimization of timber–steel structures for reduced embodied carbon design
Introduction
The building and construction industry is a major source of carbon and greenhouse gas emissions, e.g. accounting for 39% of the annual global carbon emissions in 2017 [1]. Typically, the construction related emissions are categorized as either the operational carbon that is proportional to the energy used for building operations (e.g. lighting, heating/cooling, etc.), and the embodied carbon that relates to material extraction and preparation, transportation to sites, construction processes, maintenance, and demolition. Operational carbon generally constitutes a larger fraction of building carbon footprints than the embodied carbon [2]. However, with the recent emphasis of lowering the operational energy requirements, the embodied carbon of buildings become increasingly important to consider [3], [4].
Several studies have proposed tools to measure and benchmark the embodied carbon content of buildings [5], [6], [7]. To aid designers, Pomponi and Moncaster [8] highlight several mitigation strategies to lower the embodied carbon. Specifically, it is mentioned that significant savings can be obtained e.g. by using (i) more environmentally friendly materials, and (ii) structural optimization. However, at current there is a lack of design methods that aim at reducing the structural embodied carbon footprint [9].
The European Technical Committee TC350 [10] defines four stages of a building’s life cycle in which contributions to the embodied carbon can occur. In a Life Cycle Assessment (LCA), carbon emissions associated with all stages of the life cycle should be included [11]. The structural elements most often accounts for the largest proportion of the entire embodied carbon [12]. Therefore, this work focuses on constraining or minimizing the embodied carbon associated with the product stage of the structural elements of new construction. This includes the carbon associated with the raw material supply, and transport to a factory for manufacturing of the structural components. A full LCA is not performed and transportation of construction materials and components to the specific construction site is not included in the current work. It should here be noted that this work relies on using site specific carbon inventory databases for construction materials and components. At current, some databases might not be able to provide carbon coefficients that cover all processes associated with the product stage. The relative difference in the used values will in some cases effect the design solutions. For some design scenarios, database values might only be available for location specific raw material use. In those cases, the herein presented framework will still allow designers to generate structural design solutions that are carbon efficient in terms of the raw material use. Further, it should be emphasized that the embodied carbon related to restoration or repurposing over the lifetime of the structure is not considered herein.
To estimate the embodied carbon of a whole structure or building, the Global Warming Potential () is convenient to use [13]. The is defined as the sum of the material quantity of each material times the embodied carbon coefficient associated with the material: where and are the volume and density of material and is the material’s embodied carbon coefficient. If a structure consists of a single material, the is directly proportional to the weight of the structure. In this case, structural optimization methods that minimize the weight will therefore also minimize the . However, if a structure consists of multiple materials, it is not sufficient to solely minimize the structural weight.
More environmentally friendly materials tend to be combined with more common structural materials to make up for relative strengths and weaknesses that occur with changing of the loading directions. For example, timber and steel are often combined in trusses, as timber is 30% weaker in tension caused by non-uniformity in the wood grain orientations [14] while steel is more susceptible to compression buckling due to its slenderness. Moreover, timber is a more environmentally friendly material, with an that is 3.5 times smaller than that of steel [15].
This paper focuses on truss design with reduced as there is relatively little existing research on the topic. Brown and Mueller [16] used multi-objective optimization to find planar steel truss geometries for long span building roofs that optimize embodied and operational energy. Stern et al. [17] extended the planar roof trusses to multiple materials (timber and steel) and used shape and sizing optimization on various spans. It was found that compared to a baseline all-steel truss, a steel–timber truss can yield savings of 31%–57% depending on the span length. This work seeks to go further by using truss topology optimization that increases the design freedom, and demonstrate the algorithm on both 2D and 3D timber–steel design examples.
Topology optimization is a freeform design approach where member sizes and connectivity are found through an iterative design process [18]. The design problem is formulated as a formal optimization problem and solved using a rigorous mathematical program. Although both objective and constraint functions can be chosen by the design engineer, most approaches typically consider a minimum compliance problem (equivalent to maximizing the stiffness for elastic, static conditions) that is subject to a weight constraint. While topology optimization has been used extensively in mechanical and aerospace applications where it has been shown to lead to new solutions that typically outperform conventional low-weight design [18], its use in structural design is still relatively limited. The existing examples include using continuum and discrete element topology optimization for tall buildings [19], [20], [21], [22], design strut-and-tie layouts for reinforced concrete structures [23], [24], [25], [26], [27], and more recently to explore design of super-long spanning girder bridges [28].
Although, there exists several formulations for multi-material continuum topology optimization (e.g. [29], [30], [31]), most truss topology optimization focuses on single-material design. The main research emphasis has been on designing for minimum compliance or weight subject to stress and local buckling constraints [32], [33], [34], [35], [36]. Extensions have also addressed global buckling considerations [37], [38], [39]. The few works that consider multiple material properties include Achtziger [40] that used a single base material with different allowable stress limits in tension and compression. Stolpe and Svanberg [41] extended single material truss design to enable the selection between a finite number of predefined materials of each bar in minimum weight problems. This was done by introducing additional design variables. Stolpe and Svanberg [41] proved that at most two materials are sufficient in an optimal truss. Rakshit and Ananthasuresh [42] later allowed material selection of truss members from a database, with the goal of having the final design consist of a single material system.
This work will develop a new truss topology optimization framework for two-material design and demonstrate its applicability for several objectives and constraints. The two materials used for demonstration in the current work will be glue-laminated timber (GLT) and steel. The paper is organized as follows: for completeness a brief introduction of single-material truss topology optimization with the ground structure approach is given in Section 2. This is followed by an extension to two-material design for stiffness objectives with either weight or constraints in Section 3. Section 4 extends to designing with two materials for a minimum objective subject to stress constraints. Finally, Section 5 recasts the stress-constrained problem as a Mixed Integer Program and discusses possible future extensions.
Section snippets
Single-material truss topology optimization
This section gives a brief introduction to truss topology optimization using the ground structure approach [43] and a single material for all truss members. The ground structure approach is based on defining a dense ground structure with many potential elements and subsequently performing a generalized sizing optimization, where the minimum bar area is allowed to approach zero. The design engineer must define a design domain with applied loads and boundary conditions. This domain is then
Two-material truss topology optimization for minimum compliance
In this work we extend the single material truss topology optimization problem from Section 2 by introducing a new set of continuous design variables . For each element, will determine the material composition. The design problem formulation will thus aim at determining both the material composition and the cross-sectional area of each element. The number of design variables is therefore doubled in comparison to the single-material design problem in Eq. (2). For the timber–steel
Two-material trusses with stress constraints
Since the problem formulation in Eq. (9) only considers the material stiffness and does not consider the strength of each material, the optimizer is not encouraged to identify solutions that consists of multiple materials. In this work we have therefore formulated a design problem with the objective of minimizing the subject to stress constraints. The following problem statement is used:
The
Two-material trusses with stress constraints as a mixed integer quadratic program
Despite this preliminary success, it is speculated that the increase of nonlinearity introduced through the presented framework may make it more difficult for the optimizer to identify quality solutions. This could especially pose problems if additional constraints must be considered e.g. local or global buckling or displacement constraints. Solving potential extensions of this work that include additional relevant design requirements such as multiple load cases and code constraints will likely
Conclusion
A technique is proposed for topology optimization of truss structures with two materials for a range of objectives and constraints, including the embodied carbon. The embodied carbon is herein accounted for through the of the first stage of the structures life cycle. The proposed framework allows the designer to automatically generate topology-optimized truss designs that incorporate elements of two distinct material systems, herein demonstrated for timber and steel. The design framework
CRediT authorship contribution statement
Ernest Ching: Conceptualization, Methodology, Software, Writing – original draft. Josephine V. Carstensen: Conceptualization, Methodology, Supervision, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
References (68)
- et al.
Energy Build
(2013) - et al.
Energy Build
(2007) - et al.
Energy Procedia
(2015) - et al.
Build Environ
(1996) - et al.
Energy Build
(2017) - et al.
Build Environ
(2018) - et al.
J Environ Manag
(2016) - et al.
J Build Eng
(2015) - et al.
Energy Build
(2018) - et al.
Build Environ
(2016)
Energy Build
Eng Struct
Eng Struct
Struct Saf
Comput Struct
Comput Struct
Int J Solids Struct
Struct Saf
Int J Mech Sci
Eng Struct
Constr Build Mater
Waste Manage
U N Environ Program
Sustainability of construction works – assessment of environmental performance of buildings – calculation method
Proceedings of the institution of civil engineers-engineering sustainability
National design specification (NDS) supplement: Design values for wood construction
Embodied carbon: The inventory of carbon and energy (ICE)
Topology optimization: Theory, methods and applications
Struct Multidiscip Optim
Struct J
Struct J
J Struct Eng
Nature Commun
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