Optimizing structural roof form for life-cycle energy efficiency

Energy and Buildings 104 (2015) 336–349

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Optimizing structural roof form for life-cycle energy efficiency N. Huberman a , D. Pearlmutter a,∗ , E. Gal b , I.A. Meir a a b

Jacob Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Israel Department of Structural Engineering, Ben-Gurion University of the Negev, Israel

a r t i c l e

i n f o

Article history: Received 16 February 2014 Received in revised form 18 June 2015 Accepted 4 July 2015 Available online 8 July 2015 Keywords: Structural form Life-cycle energy efficiency Embodied energy Optimization

a b s t r a c t This study evaluates the potential for life-cycle energy savings in buildings through the use of efficient structural roof form. A simulation-based optimization methodology was developed for comparing the energy requirements of reinforced concrete structures based on conventional flat slabs with those employing alternative vaulted spanning elements, in which the required quantities of high embodiedenergy materials like steel and cement may be significantly reduced. The modeling framework combines structural and thermal analyses for the respective quantification of embodied and operational energy. It accounts for local conditions influencing material production and transport, code requirements for structural reliability and serviceability, and heating and cooling demands over a 50-year life span. Results clearly show the potential of non-flat structural roof forms to reduce life-cycle energy consumption – with maximum savings of over 40% in embodied energy and of nearly 25% in cumulative life-cycle energy. This model may be implemented from the early design phases for achieving environmentally responsible buildings. © 2015 Elsevier B.V. All rights reserved.

1. Introduction 1.1. Energy efficiency in the whole life of buildings About 40% of the total energy consumed in industrialized countries is required for the ongoing operation of residential and commercial buildings [1–5]. This proportion increases significantly, however, when considering the energy which is embedded in the manufacturing and transport of the building’s materials, and in its initial construction. This embodied energy can be quantified using life-cycle assessment (LCA) [6,7] methods which account for the inputs required throughout the whole life of a product as an indicator of its environmental impacts and resource efficiency [8–11]. The complete life-cycle energy consumption of buildings includes three phases: the pre-use phase, including initial embodied energy (EE), the use phase, including operational energy (OE) as well as “recurring” EE (for ongoing construction and maintenance processes), and the post-use phase, including demolition or possible recycling and reuse. In the past, initial EE has often been considered negligible when compared with the operational energy used in buildings throughout their useful lifetime. As the thermal efficiency of building

∗ Corresponding author. E-mail address: [email protected] (D. Pearlmutter). http://dx.doi.org/10.1016/j.enbuild.2015.07.008 0378-7788/© 2015 Elsevier B.V. All rights reserved.

envelopes and systems increases, however, pre-use energy is proportionally amplified – particularly since this improved efficiency is often achieved through the use of high embodied-energy materials such as thermal insulation and advanced windows. A detailed LCA study focusing on a thermally efficient residential building with a reinforced concrete (RC) envelope found that initial EE may represent as much as 60% of the overall energy consumption over a 50-year life cycle [12]. In addition, it showed that for conventional concrete frame and block construction, about two thirds of the initial EE could be attributed to the building’s reinforced concrete structure. Horizontal spanning elements alone were found to represent over 50% of the total EE – and the largest portion of this (about 30% of the total) was accounted for by reinforcing steel (Fig. 1). These findings emphasize the importance of energy-efficient structural systems for achieving sustainable built environments.

1.2. Identifying energy-efficient structural approaches Structural efficiency is generally linked with savings in the use of materials, achieved by taking advantage of their strength and stiffness properties. Efficient structural forms are designed with attention to the geometry of the overall structure and its parts, providing the greatest structural performance with the least material. When the shape of a structural element corresponds to the flow of forces, loads are transmitted only axially, with constant stresses over its cross section; hence bending, which is inherently

N. Huberman et al. / Energy and Buildings 104 (2015) 336–349

80

300 60

columns + beams

200 Concrete Structure 100

Structural Horizontal Slabs

slab cement + aggregates

40

Slab Reinforcing Steel

20

0

Embodied Energy (% of total)

non-structural (wall infill + finish) Embodied Energy (GJ)

1.3. Roof form and thermal energy efficiency

100

400

0 Building Components

Structural Elements

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Raw Materials

Fig. 1. Breakdown of initial embodied energy in typical low-rise, reinforced concrete frame construction. Based on [12].

inefficient, may be avoided. Two basic structural types meet these conditions [13]: pure tension structures (e.g. tensile membranes and cables), and pure compression structures (e.g. arches and vaults). In a practical situation, in which lateral wind and seismic stresses must also be considered, these constant loading conditions are not strictly possible. However, this concept can still be exploited to maximize structural efficiency. In the past, when structures were built from available masonry materials such as stone or brick – which may have considerable compressive strength, but negligible tensile strength – the importance of basic geometry as a touchstone for efficient structures was crucial. This led to the discovery that the “ideal” geometry for achieving uniform axial compressive forces is the catenary arch, as the inverse shape of a freely “hanging chain” [14]. The advent of new building materials such as high-strength steel and reinforced concrete, along with subsequent technologies based on them, has largely masked the importance of efficient structural form in modern design practice. Structural members which resist bending are the norm, while compression-only and tension-only structures are relatively uncommon. Reinforced concrete structures with rectilinear forms, which are commonly used today, are a typical example of this inefficient design. Large bending moments in flat slabs generate uneven stresses, and since material which is neither in compression or tension is not utilized, the quantity of materials required is inflated. Most importantly, the particular materials that must be employed in larger quantities – such as steel and cement – are especially voracious in their consumption of pre-use energy, and thus the life-cycle energy requirements of contemporary buildings has increased dramatically. The manufacture of steel products, including reinforcing bars for poured concrete, is responsible for over 5% of all global anthropogenic greenhouse gas emissions [15]. Its EE is approximately 35 MJ/kg, or the equivalent of 280,000 MJ/m3 [16], and reinforcing in structural concrete commonly consumes up to 1500 MJ per square meter of built area [12]. The EE of concrete (without reinforcing) is typically in the range of 2070–4180 MJ/m3 [12], where cement is the largest contributor and is known as a notorious source of CO2 emissions. At the same time, and despite its reliance on highly energyintensive materials, reinforced concrete can allow for dramatically thinner sections than are possible with masonry compressiononly curved spanning elements. It may also be indispensable for earthquake resistance and for providing a level of durability and serviceability demanded in modern buildings.

The form of the building envelope can also significantly influence its thermal behavior – and the roof in particular, since its surface is exposed to intense energy exchange by incoming solar radiation during the day and out-flow of long-wave radiation at night [17]. It has been found that in arid areas, the roof may be responsible for 50% of the building’s total heat load [18]. Both physical modeling [17] and computational analysis [19] have shown that free-running buildings with semi-cylindrical vaulted roofs maintain lower summer daytime temperatures than those with flat roofs, due to a combination of radiative and convective energy exchanges affecting the overall heat balance. While these advantages are especially pronounced in hot-arid regions, they become more subtle as the thermal insulation of roofs increases. Moreover, for air-conditioned buildings, the comparison between flat and vaulted roofs is less clear-cut and depends on the particular geometry of the whole envelope. Therefore, and considering the arguments laid out previously regarding structurally efficient form, it has been proposed that the extensive use of vaults and domes in traditional Middle Eastern architecture may be attributed first and foremost to the availability of high compressive strength masonry materials, and their thermal advantage represents a secondary benefit [17].

2. Analysis of structural energy efficiency A number of studies have assessed embodied energy in buildings with standard structural systems, with the search for improvements directed in three directions: (1) selecting the least harmful of standard practices [20,21]; (2) adapting vernacular technologies by reviving traditional materials [22], combining them with new techniques and efficient forms [23], evaluating their EE [24], or analyzing them with actual tools [25,26]; and (3) developing new materials or construction practices [27]. New movements have appeared in a number of disciplines that search for efficient design practices by observing and taking examples from nature, including “emergent architecture” [28,29], “new organic architecture” [30], and “biomimetics” [31] or “biomimmicry” [32] in engineering, and “constructal theory” [33]. The use of non-planar forms for achieving efficient structures has been proposed in the form of lightweight concrete vaults [34], earthquake-resistant optimized vaults and domes for earth houses [35], or curved roof forms made of newly developed materials such as FRP [36]. However, the focus of these efforts is not necessarily to reduce EE or total environmental impacts. While this additional objective has indeed been recognized and applied in practice, the focus has been on long-span buildings (especially lightweight structures), and quantitative data on the actual magnitude of energy savings is lacking [37,38]. Mathematical structural optimization aids in identifying a structural form that achieves a specific objective in the best possible way through an automated iterative process. Recent optimization techniques combined with finite element modeling (FEM) have been successful in minimizing bending, though since they start from an inefficient form, improvements are localized. Structural optimization, however, is often pursued using criteria such as minimum total weight or initial monetary costs, which are not necessarily correlated in a simple way with actual structural efficiency or with the ‘costs’ of embodied energy and related environmental impacts. This is especially significant in reinforced concrete structures, where the embodied energy of reinforcing steel may not be fully reflected in terms of weight or cost. Only recently has the use of environmental objectives in structural optimization been addressed [39–41].

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The use of optimization models, generally using genetic algorithms, has recently gained momentum for improving energy efficiency in buildings [42–45]. Pushkar et al. [46] proposed a method to be used in the design stage for a building’s environmental optimization, including production, operation and maintenance. One ceiling form optimization was performed for improving daylighting performance [47], and another study [48] applied an evolutionary algorithm for maximizing solar irradiation on both roof and vertical facades. Still another recent study [49] presented a performance-driven geometric design using genetic algorithms for long span roofs, including finite element analysis (FEA) for structural performance. However, none of these studies quantified actual EE (or life-cycle energy) values. A number of studies present examples of multi-objective simulation-based optimization frameworks using different types of algorithms aimed at minimizing life cycle costs and different life cycle environmental impacts including energy. Some of them consider as variables the building shape [50], envelope features [51], or both [52,53], though none of these studies incorporate structural analysis. Another study [39] presents a software model for multidisciplinary design optimization (MDO) of steel structures, evaluating daylighting and life-cycle energy consumption and costs. The example application combines thermal and structural analysis but does not have capabilities for evaluating reinforced concrete frames. A number of studies have evaluated recycled or waste materials and particular internal shapes for environmental optimization of standard RC slabs [54–56], though without analyzing the building’s overall form. Previously published studies only deal with part of the problem of life-cycle energy efficiency in standard reinforced concrete horizontal roof forms. The potential energy savings of using efficient overall RC structural forms (as opposed to component geometry) has not been quantified. Accordingly, no automated optimization routine has been developed that includes structural and thermal analysis and life-cycle energy assessment for common RC building types. The present study examines the structural efficiency of buildings as a means for improving their life-cycle energy efficiency, and examines to what extent curved roof forms, i.e. vaults, may represent an energy-efficient approach to reducing whole life cycle energy consumption in common buildings that are produced on a large scale. The hypothesis underlying these inquiries is that a significant potential for energy savings and reductions in the exploitation of natural resources lie in reducing the embodied energy of buildings by means of alternative non-planar and efficient structural forms. The ultimate research goal is to propose a set of alternative structural vaulted roof geometries that meet the

region-specific needs/demands for buildings, while reducing the use of highly energy-intensive manufactured materials. To achieve these objectives, an optimization methodology was developed which integrates essential simulation methods (structural and thermal) in a framework for analyzing and quantifying potential savings generated by alternative efficient structural roof forms for low-rise buildings in arid and seismic regions. The life-cycle energy use considered here includes both the energy embodied in the building’s materials, with an emphasis on its load-bearing structure, and that required for maintaining the building’s thermal environment at a level which is conducive to occupant activities over its useful life. In this way the analysis can be systematized, and geometric changes may be automated for achieving rapid evaluation of optional roof configurations and obtaining results. In addition, it provides information on the extent to which different roof variables affect the total energy consumption.

3. Methods 3.1. Model of study Potential energy savings of hypothetical configurations are estimated relative to a base case scenario which has a flat slab roof. Horizontal reinforced concrete (RC) slabs commonly used for exterior roof elements and/or intermediate floor-ceilings with external and partition block walls, mainly of hollow concrete blocks and/or aerated concrete blocks [57] serve as a reference (control) for investigating a range of non-planar RC alternatives which fulfill an equivalent function. A diverse set of vaulted, or upwardly curved forms is considered by modifying the geometric attributes of the basic spanning element (Fig. 2), while a standard reinforced-concrete structural column-beam skeleton serves as the common supporting structure for all the alternatives, based on common local practice in Israel. Regarding roof typologies, surface-active systems (with in-plane stresses) – or shells – were analyzed, principally vaulted forms including both parabolic and segmental (circular) sections. As compared with flat slabs, both structural shapes are expected to reduce bending stresses and in turn the required amount of steel. The parabola is considered to have an inherent advantage in terms of structural efficiency due to its close similarity to an inverted catenary, which is free of bending and therefore subject to minimal tensile stresses. In a semicircular vault the line of support does not necessarily run in the middle third of its cross-section, which may increase tensile stresses.

H2

th

H1

R

a

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Fig. 2. Schematic building sections with conventional flat slab and vaulted roofs, illustrating critical parameters.

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The structural efficiency of a shell is based on its curvature (which increases structural stiffness), giving the opportunity to reduce the thickness, and thus diminishing concrete quantities. However, when the thickness (th) is not significantly lower than a flat slab, either shape may have a greater total volume of concrete due to its enlarged surface. For this reason, the analysis must take into account concrete and steel quantities required in each specific case, balancing additions or reductions, and evaluating overall energy expenditures. For that purpose a model of study was adopted for establishing dimensions and orientations within which the different structural systems are applied and evaluated. The building represents a lowrise residential configuration that is typical in the area. It has a rectangular or square layout depending on the span dimensions. A study region was selected for the purpose of obtaining appropriate climatic and seismic data, as well as for calculating or selecting site dependent values such as EE coefficients of materials. The context for the study is the arid Negev region of southern Israel. This location establishes for the analysis the source locations and transport distances for building materials (embodied energy), and the climatic conditions for the thermal simulations (operational energy), based on data from the city of Beer Sheva. Construction in the Negev typically requires longer transportation distances from Israel’s commercial and industrial centers, increasing energy requirements for physical development. The area’s climate is distinguished by sharply changing daily and seasonal temperatures, intense solar radiation, very low humidity, mostly clear skies, and characteristic strong N–W winds at evenings. The harshness of the desert climate also affects energy consumption, due to the heavy heating and cooling loads. By and large, planning and design follow practices that are standard in the country’s more temperate regions, and particular adaptation to local conditions is the exception rather than the rule [58]. Israel is situated at the confluence of tectonic plates, in an active earthquake zone. Implementing the requirements of the code for earthquake-resistant construction entails enormous expenditures, but is considered necessary for the protection of thousands of human lives [59,60]. 3.2. Integral optimization methodology In order to achieve the above goals, a multi-faceted optimization framework was developed, which uses an automated process to assess the performance of the alternatives and allows for the evaluation of numerous configurations more efficiently than by manual trial-and-error testing. Performance simulations are run in successive iterations to assess the effects of different variables and select the best performing option. In order to avoid conflicting and often subjective weighting criteria, the design problem was simplified as a single-objective optimization focusing on life-cycle energy consumption, by converting structural requisites into functional constraints. The objective, variables and constraints of the optimization model were defined as follows: 3.2.1. Objective The principal objective of the optimization framework is to minimize the lifetime energy requirement of the building – to reduce consumption in the two life-cycle building phases analyzed (i.e. pre-use and use phases) while maintaining structural stability and serviceability. The objective function includes a calculation and summation of EE and OE requirements for each roof configuration, in primary energy terms. Recurring EE for building maintenance and post-use energy consumption for demolition and disposal are not included in the analysis, due to the considerable degree of uncertainty involved.

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3.2.2. Design variables Two of the geometric parameters defining the structural roof shape are thickness and rise (measured from the spring-point, at the center of the span), both of which are taken as continuous variables for all types of roofs. Additional discrete variables may be selected as input by the assessor/user from a predefined list of alternatives, and not automatically by the algorithm. This is in order to isolate the variables and evaluate the consequences of decisions on the energy performance of the building. Once they are selected, before the optimization is run, they are transformed into ‘non-design’ constant parameters. Examples of required input that are not optimization variables include plan dimensions (“a” = width and “b” = length), type of roof form (flat, parabolic or segmental), beam and column dimensions, and number of finite elements to be included in the structural analysis. Default parameters include location, number and type of supports, wall type, layers and properties of envelope materials, building orientation, plant system, etc.

3.2.3. Constraints Each variable may be assigned a value within a given range, defined by the maximum and minimum acceptable roof thickness and vault rise (i.e. the height difference from spring point to apex). Moreover, there are constraints that must be considered in order to ensure that the structure fulfills existing design code requirements. For instance, assuming linear response and ignoring the dynamic characteristics of seismic action during the design phase may lead to structural configurations that are highly vulnerable to earthquake damage [61]. In addition, building regulations are taken as indirect (or functional) constraints, since they are not limiting the variable values, but are included as part of the calculations of the objective function. Indirect functional constraints include, for example, code requirements such as minimum steel reinforcement for slabs, beams and columns, and minimum thermal insulation (R-value). The effects of implementing a minimum thickness as required by the Israeli standard for the flat slab is presented here in the results and analysis, though taking this as a variable in the optimization framework allows for a greater number of comparisons. This is possible since a structural analysis is performed for each configuration, satisfying maximum permissible values to ensure serviceability. Other constraints are imposed by the architectural design, relating esthetic or functional concerns. Geometrical functional constraints, not related to code requirements, are minimum springpoint (above the height of window and door openings), minimum ceiling height at the center of the span (H1 and H2 in Fig. 2), and constant volume of the space for all the cases within each span (calculated from the maximum rise alternative), in order to make “objective” comparisons in the thermal analysis. Structural functional constraints are the basic stability requirements (equilibrium equations) and resistance to dynamic lateral seismic loads, which are included as part of the structural analysis and design of components. Based on the defined optimization problem, the multidisciplinary simulation-based form optimization framework includes a performance evaluation process and an optimizer (Fig. 3). The performance evaluation process assesses the total energy consumption (value of the objective function) of the alternatives by running several external and internal simulations. It is divided into three interconnected modules: (1) embodied energy (EE), including structural analysis; (2) operational energy (OE), and (3) life-cycle energy assessment (LCEA). Hence, the output of these modules is the value of the objective function which calculates the cumulative life-cycle energy (CLCE) consumption and is expressed in mathematical terms as:

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algorithm) for the alternative that presents the highest degree of savings within the objective function.

Definition of the Optimization Model Assessor selection of inputs

3.3. Computational optimization tool

Geometric Model Generation Automatic selection of variables

Performance evaluation

Optimizer

1- Embodied energy analysis Structural analysis Static and Dynamic Loads Design of structural components Dimensions of elements and quantification of material required Embodied energy calculation Primary energy in pre-use phase

2 - Operational energy analysis Thermal simulation Calculates summer and winter loads Operational energy calculation Heating and cooling primary energy

3 - Life-cycle energy assessment Cumulative energy consumption calculation

no

Minimum CLCE? yes End

Fig. 3. Flowchart of the optimization framework.

CLCE = EE(X) + OE(X)



= [˙m (Qm ∗ EECm )] + le ∗ ˙f

 ˛f ∗

OECf COPf

 (1)

where X is a vector of possible variables, and EE and OE are respectively the embodied energy and operational energy of the entire building for variable X. Qm is the quantity of material m in m3 , calculated from the output of the structural analysis. EECm is the embodied energy coefficient for material m, in primary energy (MJ/m3 ) [11], le is the life expectancy of the building in years, and ˛f is the efficiency of production and delivery of fuel f (MJconsumer /MJprimary ) – which is also known as the fugitive energy coefficient, and converts energy consumed from the specific source (e.g. electricity) into primary energy [62]. OECf is the annual energy consumption of fuel f in MJ for the operation of the building, due to heating and cooling, obtained from thermal simulation. COPf is the coefficient of performance, or the energy-efficiency measurement of heating, cooling, and refrigeration appliances, which represents the ratio of useful energy output (heating or cooling) to the amount of energy that is input. A higher COP indicates a more efficient device, and it depends on the fuel utilized. The optimizer is responsible for finding optimized solutions by searching within the design space (guided by a numerical

The automated tool is based on a search algorithm for achieving optimized results with a minimum of computational resources. The model builds automatic chains that control two external application programs and internal sub-programs using scripting. It is a graphically driven tool which functions as a single-objective optimization environment, designed to couple stand-alone simulation tools for finite element structural analysis and thermal analysis. It performs the optimization by systematically modifying the values assigned to input variables, running the analysis programs for performance evaluation, calculating and aggregating outputs, and then analyzing and reporting the results as objectives of the design problem. Therefore, this LCEA optimization toolset integrates software applications for parametric design, thermal energy simulation, and finite element analysis (FEA) into one cohesive software environment, which includes computing techniques for search space selection, sampling and output data aggregation, analysis and visualization. The structure of the software implementation, as described schematically in Fig. 4, is configured as a hybrid interface which compiles both external and internal simulations for the purpose of computing objective function values (after Wang et al. [63]). The optimizer communicates with external simulations through text files [64], and transfers the selected variable values to the internal simulation – which calculates the objective function by writing input files, running an external simulation, and reading and interpreting the simulation output files, returning the calculated final result to the optimizer. The external simulation program is called via the operating system. The head program, which integrates all the sub-programs and automates the process, is a custom application written in MATLAB [65], which was selected for its capabilities in scripting the required interfaces, working with numerical operations and array organization and calling stand-alone analysis tools, and for its practical built-in functions and toolboxes. A graphical user interface (GUI) was designed to facilitate the input of data which are required to define a given design alternative. Input parameter values are stored and organized by the program as vectors or matrices to be used later by the optimizer or the translator. The GUI also serves as a framework for visualizing the optimization process while it is running, and for presentation of the final results. A numerical algorithm guides the search toward the optimized solution [66]. Input values are sent to the simulation modules, which process the input and return objective function values. Depending on these results, the optimizer determines whether a minimum solution has been reached, and if not continues the process. The solver “fmincon” from the Optimization Toolbox of MATLAB, which finds the minimum of non-linear multivariate constrained problems, is used in this study for solving the optimization problem based on a sequential quadratic programming method [67]. The external stand-alone simulations used in this study are commercial software applications (STRAP [68] for structural analysis and EnergyPlus [69] for thermal analysis) which are configured with a translator for pre-processing and an internal simulation for post-processing. The translator writes the input file(s) for each optimization run, including the detailed description of the building’s geometry (coordinates of element nodes are established based on numerical equations for the parabola or segment of a semicircle), materials, structural system, loads, and of the local climatic conditions (in TMY2 weather data format), occupancy and HVAC setpoints. External simulation programs are executed using a batch mode, with input files in ASCII text format.

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MATLAB MATLAB head program head program

MATLAB sub-programs

Communication files

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External programs

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Algorithm variables

Performance evaluation Module 1 – EE Interface 1

write

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Module 2 – OE Interface 2

write

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read EnergyPlus simulation

Internal OE simulation results

Plot Final results

read

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Module 3 – LCE

Internal LCE simulation

Fig. 4. Flowchart of the program architecture, showing hybrid interface between software applications.

Output data from the simulations are post-processed to obtain values of the objective function, including EE and OE calculations. To obtain total EE values, output data from the FEA (which accounts for applied loads) are used to compute quantities of reinforcing steel and other materials, which are then multiplied by corresponding EE coefficients. In this way, the structural analysis is translated into structural design, whereby the components of the system are dimensioned. Based on the thermal simulation output, yearly OE is computed by summing monthly requirements for heating and/or cooling in primary energy terms, accounting for the efficiency of the mechanical system utilized (COP) and the fugitive energy coefficient (the energy cost of producing and transporting the energy). Finally, life-cycle operational energy requirements are assessed for each option by considering the expected lifetime of the building. Total life-cycle energy (CLCE) is also assessed in primary energy terms, as the sum of all partial energy inputs, and finally life-cycle values are divided by the building’s floor area to obtain results in GJ/m2 . 3.4. Implementation of the optimization tool Since the optimization process presents only the “best” performances of the iterations, it does not provide additional information on effects and consequences of roof form changes. For this

reason, a detailed examination known as form exploration was initially applied to an experimental model, executing the performance evaluation (see Fig. 3) but not the optimizer. Although this assessment is done without the search algorithm, it is still carried out automatically, including geometry definition and the running of external as well as internal simulations. This analysis explores a system of discrete rather than continuous values, with preset intervals (0.25 m increments for rise, and 2 cm increments for thickness), in order to reduce the amount of feasible options, and thus the number of evaluations required. Additionally, it serves as a validation tool to verify that the optimization algorithm reliably identifies the true optimum among the design options (see Section 4.3). In the full optimization, input variables are continuous – they may take any value within the defined maximum and minimum bounds, which are entered from the GUI. The optimization is allowed to act only the two variables which affect the energy performance of the building, i.e. the roof thickness and rise. As presented in Section 4.3, results of the optimization are given for parabolic roof alternatives only. In the analysis (for both discrete and continuous variables), the vault rise (h) was allowed to range from “0” (flat roof) to 2 m. The roof thickness (th) varied from a minimum value of 8 cm to a maximum of 20–36 cm, depending on the roof span. The span (a) was not included as a variable; however all the alternatives were

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spring point (hsp)

thickness (th) rise (h)

span (a) Fig. 5. Schematic section of the vaulted roof used in discrete analyses, showing principal dimensions.

evaluated each time for a series of possible spans from 3 to 10 m (with 1 m increments) – see Fig. 5. In all cases the length of the building is 10 m, beam dimensions are 0.50 m × 0.25 m, and the column section is 0.20 m × 0.20 m. The finite element size was fixed at 0.5 m in both the spanwise and lengthwise horizontal directions. For the structural analysis, two applied loads were considered in the vertical direction: a dead load, including the self weight and a distributed load of 2 kN/m2 , as well as a live load of 1.5 kN/m2 as specified for residential buildings [70]. Partial safety factors applied are as follows: F = 1.4D + 1.6L (F = Design forces, D = dead load, L = live load)

Construction details and thermal properties of materials as defined in the input file can be found in [72]. A COP-value of 3.0 was taken for HVAC equipment, and the primary energy coefficient (˛f ) for electricity was 2.23 MJ/MJ. Windows were defined with a fixed area of 2 m2 on each side of the building. The building is assumed to be cooled between June and August, including natural ventilation and open shutters at night (from 6 p.m. to 6 a.m.), with shutters closed for shading in the daytime (6 a.m.–6 p.m.). Heating is assumed to operate in winter, between December and February, with shutters open during the day and closed during night hours. The “Thermal Analysis Research Program” (TARP) was selected as the default outside surface convective heat transfer algorithm, since it takes into account the tilt angle of the roof segment by correlating the convective heat transfer coefficient to the surface orientation and the difference between the surface and zone air temperatures [73]. The internal air in the room is entirely mixed. All of these selections can be easily modified by the user, for instance by using a constant window-to-wall-ratio, or stratification of internal air. 4. Results and discussion 4.1. “Form exploration” – segmental roofs

(2)

The “Equivalent Static Analysis” was used here to evaluate the effects of seismic actions [71]. The seismic load was included as a joint load in the middle of the roof, displaced by the required eccentricity e to simulate the torsional effect T, in both directions (one at each time), for flat roofs. In non-flat roofs, it was included as a proportion of the load in every node and in both sides. The maximum EE result (between both sides) was taken for the subsequent calculations. For the thermal simulation, the city of Beer-Sheva, Israel, was established as the climate reference site. Orientation of the main building facades is N–S (i.e. the long axis of the vault is aligned E–W, as shown in Fig. 6). Insulation thickness was determined by the code requirements of Israel Standard 1045, stipulating the equivalent of 3 cm and 2 cm of expanded polystyrene for roofs and walls respectively. Roof solar absorptance is set at a fixed value of 0.65. To determine the distribution of thermal loads within the building, a simple single-zone model without interior partitions was defined, and for each span the wall height was adjusted to maintain a constant internal volume regardless of vault rise. Thermal runs simulate the energy needed to maintain a temperature set point of 18.5 ◦ C and 25 ◦ C in winter and summer respectively, without setback. All walls and roof are defined as external (sun and wind exposed) heat transfer surfaces, hence affected by solar radiation, wind speed and direction and outdoor temperature.

As described above, the entire discrete spectrum of the design space was automatically simulated and assessed for segmental vaulted roofs with varying rise and compared with horizontal slabs, both over a range of thicknesses. All of the results shown here (Figs. 7–13) represent configurations using the optimal roof thickness for each span and rise in terms of CLCE: in all the segmental vaults analyzed, overall maximum CLCE savings are achieved with a thickness of 8 cm, while the thickness of the flat roof is determined as the minimum value which meets the requirements of the Israeli standard [74]. Results of cumulative life-cycle energy (CLCE) consumption over an assumed 50-year life span are illustrated in Fig. 6 for building configurations with spans ranging from 3 to 10 m. For each span, CLCE values are shown on the vertical axis for a range of roof vault rises, from a minimum of 0 (representing a flat slab) to a maximum of 2 m. Points representing the rises which yield the lowest CLCE values for each of the roof spans are marked with a circle, and for each span the maximal savings yielded by a vaulted roof relative to a flat roof of the same span is labeled as a percent reduction. It can be seen that in all cases, a segmental vault does provide some measure of CLCE savings relative to a flat roof of the same span, with thicknesses as described above. For all spans larger than 3 m, minimal CLCE is reached with the highest vault rise – that is, as the rise increases, the cumulative energy consumed over the building’s life cycle decreases accordingly. The relative magnitude of CLCE savings for each span, illustrated as a percentage reduction

Fig. 6. Flat and segmental models. From DesignBuilder.

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Fig. 7. Cumulative life-cycle energy (CLCE) comparison for segmental 3–10 m span roofs with vault rise ranging from zero (flat) to 2 m. Results represent vault thicknesses yielding the lowest CLCE (8 cm in all cases), and for flat roofs (rise = 0) the minimum thickness required by code.

Fig. 8. Embodied energy (EE) comparison for segmental 3–10 m span roofs with vault rise ranging from zero (flat) to 2 m.

in Fig. 13a, increases from approximately 4% at 3 m spans to 24% at 10 m spans. Put another way, these savings represent the equivalent of between 5 and 35 years’ worth of the base-case building’s operational energy for heating and cooling (see Fig. 13b). Fig. 8 shows a comparison of rises and spans similar to that in Fig. 7, but for initial embodied energy (EE) of the building’s reinforced concrete structure alone. It can be seen that the pattern described earlier for cumulative energy consumption emerges for EE as well: at all spans larger than 3 m, energy requirements decrease as the height of the vault increase. In this case, however, the scale of savings is more dramatic – with vaults of wider spans

achieving embodied energy reductions of over 40%. It may also be seen that the largest savings are not necessarily attributable to the highest vault rises, as the “optimal” rise increases from less than 1 m at short spans (of 3–4 m) to 1.5 m at intermediate spans (6–8 m), and only reaches the full 2.0 m increment at the widest span (of 10 m). The fact that the percent reductions in EE are considerably larger than those in overall CLCE makes it clear that initial embodied energy is the primary contributor to overall potential energy savings. Fig. 9, which gives a breakdown of EE and OE using a span of 6 m as an example, shows that while EE declines with vault rise

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3 2.5 2 1.5 1

Heating

Cooling

1.6

Operational Energy [GJ/m²]

Cumulative Life-cycle Energy [GJ/m²]

1.8

Operational Embodied

3.5

0.5

1.4 1.2 1 0.8 0.6 0.4 0.2 0

0 Flat

0.25

0.5

0.75

1

1.25

1.5

1.75

Flat

2

0.25

0.5

0.75

Rise [m] Fig. 9. CLCE by vault rise, broken down into EE and OE (50 years) for a 6-m span.

1.5

1.75

2.0

Fig. 12. Operational energy for summer cooling and winter heating over a 50-year period, for a 6-m span.

a

1.2

1.0 1.25 Rise [m]

1.0

1.0

22.6%

Segmental roof Parabolic roof

0.8 CLCE savings [GJ/m²]

Embodied Energy [GJ/m²]

24.0%

Concrete Steel

0.8 0.6 0.4 0.2

21.0% 19.4% 18.5% 17.4%

0.6 14.8% 12.8%

0.4 9.0%

13.1%

10.0%

6.7%

0.2

6.2%

4.2%

5.0%

2.3%

0

Flat

0.25

0.5

0.75

1.0 1.25 Rise [m]

1.5

1.75

2.0

0 3

4

5

6 7 Span [m]

8

9

10

6 7 Span [m]

8

9

10

Fig. 10. EE by vault rise, broken down into concrete (without reinforcing) and steel, for a 6-m span.

b

35

0.4 Steel − columns Steel − beams Steel − roof

0.3

EE Savings [years of OE]

Embodied Energy [GJ/m²]

40

0.2

Segmental roof Parabolic roof

30 25 20 15 10

0.1 5

0

0

Flat

0.25

0.5

0.75

1.0 1.25 Rise [m]

1.5

1.75

2.0

Fig. 11. EE for reinforcing steel by vault rise, broken down into different structural elements, for a 6-m span.

relative to a flat slab, OE remains fairly constant with span. In other words, operational energy savings are negligible in this analysis and the life-cycle energy savings yielded by vaulted roof forms are predominantly due to EE efficiency. Given the dominant role of EE in potential energy savings, it is informative to observe in detail the different materials

3

4

5

Fig. 13. Comparative energy savings for buildings with segmental and parabolicsection roof forms, expressed in terms of (a) CLCE reduction relative to a flat slab (with the percent reduction for each span labeled), and (b) EE savings expressed as an energy equivalent, in terms of years of OE.

responsible for embodied energy savings. Fig. 9 gives a breakdown of EE for 6 m span, for the two materials which comprise the building’s reinforced concrete structural frame: the concrete itself (without reinforcing), and the steel bars used for reinforcement. It can be seen that taken together, these materials account for nearly

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Fig. 14. GUI snapshots of running optimization for 6 m span roof, showing initial stage (top), intermediate stage (middle) and final stage (bottom).

1.2 GJ/m2 in a flat-roofed building, and thus constitute over 60% of that building’s total EE (shown in Fig. 9). The bulk of this initial EE is for the concrete, whose high value is a reflection of the relatively large minimum thickness (20 cm for a 6-m span) required for flat roofs by the Israeli code. With vaulted forms this concrete EE declines sharply due to the reduced minimum thickness (8 cm) allowed in the model.

It can be seen that as the rise of the vault increases, the embodied energy of steel declines – from nearly 0.4 GJ/m2 at a minimal rise to just over half that amount at the full rise of 2 m. This reduction in energy consumed is a direct expression of the reduced volume of steel required, due to the greater structural efficiency of the roof form. At the same time, the EE of concrete increases with rise, due to the larger volume of material. When combined, these two opposing

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Fig. 15. Comparison of discrete exploration (left) and optimization (right), with CLCE results as a function of thickness and rise for a 6 m span roof.

Table 1 Discrete form exploration vs. optimization results for three roof spans. For each span and method of analysis, results are given for the optimal rise, the maximum cumulative life-cycle energy (CLCE), the percent reduction in CLCE relative to a flat roof, number of iterations performed in the optimization run, and the number of functions evaluations. CLCE (GJ/m2 )

Span

Method

Rise (m)

3m

Discrete Optimization

0.75 0.875

3.7312 3.7000

Difference

0.125

Discrete Optimization

1.00 1.200

Difference

0.200

−0.0058

0.17

Discrete Optimization

1.25 1.434

2.9367 2.9371

22.64 22.62

– 8

138 33

Difference

0.184

0.0004

−0.02

–

−105

6m

10 m

Reduction (%)

Iterations

Function evaluations

2.35 3.16

– 5

63 19

−0.0312

0.82

–

−44

3.0470 3.0412

10.00 10.17

– 6

63 28

trends yield local minima for total EE at intermediate rises in the range of 1–1.5 m. An additional breakdown of steel requirements for the building’s various structural elements (roof, columns and beams) is given in Fig. 11. It is clear that with increased vault rise there is a pronounced reduction in the steel required for the roof, though this is slightly offset by additional steel required for columns and beams, due to the extra weight and lateral thrust. On the operational energy side, it was observed that an increase in the rise of the vault leads to a noticeable reduction in the energy requirement for heating in winter, but not for cooling in summer – with the latter being largely insensitive to roof form (Fig. 12). This result is counter to earlier findings (e.g. [17]) which showed a significant thermal advantage for vaulted roofs in similar summer conditions, but such an advantage was found for passive, or free-running (i.e. non-air conditioned) buildings, rather than for an actively cooled building such as that simulated here.

4.2. Segmental vs. parabolic roofs Fig. 13 depicts the comparative energy performance of buildings with segmental and parabolic roof forms, in terms of both CLCE savings (relative to a flat slab) and EE savings as measured in years’ worth of OE equivalent. Overall it can be observed that segmental and parabolic roof forms have similar performance, although savings for parabolic sections are slightly lower. Unlike segmental roofs, for which the optimum CLCE is reached at a full rise of 2 m for all but the smallest spans (see Fig. 7), parabolic roofs actually achieve the optimum at lower rises in the range of 1–1.25 m.

−35

It must be taken into consideration that for the same span and rise, the segmental roof has a larger area than the parabolic roof. Hence, its internal volume, calculated from the highest rise analyzed (2 m in this case), and taken as a constant for smaller rises, is also larger for segmental roofs (by about 12% for a 3 m span, decreasing to 1.25% for a 10 m span). Therefore, comparisons must be performed relative to the corresponding base case (horizontal roof) with the same volume as each vaulted alternative. In an additional comparison of the two alternative roofs, in which segmental roofs have the same volume and rise as the best performing parabolic roof for each span, it was observed that the two types of roof are highly similar in their energy performance. Steel required for segmental roof elements are found to be slightly higher than for parabolic ones, though this is outweighed by smaller EE for other elements in segmental cases, since the height of the walls is reduced to achieve the same volume. Of the two alternatives, segmental roofs present not only a slight advantage in CLCE terms, but also offer practical benefits in terms of ease of construction. All and all, these form exploration examples clearly show the potential savings made possible by alternative roof forms, as well as the importance of weighing the effects of different materials. 4.3. Optimization: results and validation Results of the full optimization process over a continuous domain were performed for three simulation runs, covering spans of 3, 6 and 10 m respectively and evaluating parabolic roof geometry only. In Fig. 14, a series of three “snapshots” is shown for the 6 m run to illustrate the process: (1) the starting point, (2) an intermediate evaluation, and (3) the final GUI with optimal alternatives

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For a 10 m span, the number of evaluations was reduced by almost 80%, with an error of only 0.13% in optimal CLCE. The optimization algorithm was thus demonstrated to effectively find accurate “best” alternatives with high efficiency (relatively rapidly, with no need to explore every option). In this sense, the optimization framework represents an advantage as compared with the practical limitations of discrete form exploration and obviously to standard trial-and-error. The results show that optimization can be used to find high-quality optimal roof form alternatives, and also confirms the significant benefits of nonplanar parabolic roofs. 5. Conclusions Results of the evaluation presented here lead to the following main conclusions: Fig. 16. CLCE performance for flat roofs and for optimal parabolic vaulted solutions found using discrete variables vs. optimization, for 3, 6 and 10 m span roofs.

selected in all the iterations, including the “best” solution. In the diagram on the left side of each snapshot frame, colors represent the quantity of reinforcing steel required, in units of cross-sectional area (cm2 /m, interpolated between elements). The bar graph on the right side of each frame indicates calculated CLCE values for the configurations analyzed by the algorithm, combining different options of rise and thickness (x and y axes, respectively). Stacked bars are divided into separate EE and OE values, except for those in the last view, which represent the whole cumulative value. Similarly to the discrete analysis, the optimization analyses illustrate clearly that vaulted roof solutions result in CLCE values that are significantly lower than for horizontal flat roofs. The framework method managed to find an optimal cut-off solution relatively rapidly, by evaluating performance trends with no need to explore every alternative in the whole design space. Since simulation usually dominates the computation time in simulation-based optimization problems, the number of function evaluations (which in this study includes two external simulations – for structural and thermal analysis) can be utilized as a measure for computational efficiency [75]. Fig. 15 compares results of optimization (left) with discrete outcomes (right), for the 6 m span. It can be seen that the algorithm for multivariate single objective optimization finds an optimal solution which is very close to that identified through discretized form exploration, but with a finer level of resolution. The fact that the optimization is executed automatically, in a small fraction of the time required, demonstrates the effectiveness and efficiency of the approach. As shown in Table 1 and Fig. 16, the correlation between the two assessments is extremely high, in terms of both design characteristics – with differences of 10–20 cm for the vault rise, and under 1% for CLCE. At the same time, the number of function evaluations is reduced by over half. For two of the three spans evaluated, the optimization allowed for identification of better-performing options which reside in between the discrete values assessed in the form exploration (and hence not computed there). As an example, for a 3 m span the “best” alternative was identified by the search engine at a 87.5 cm rise (instead of at 75 cm, the closest 25-cm increment) and at thickness of 8 cm (as before). This configuration offered a percentage of reduction from the base case flat roof (with thickness as required by code) of about 3% in the objective function metrics – slightly more than the option found in the discrete search. The optimization run for this case performed five iterations, and the number of function evaluations (or simulations) required to achieve the optimum design was reduced from 63 (total discrete options evaluated in the form exploration) to 19 (around 70% less).

• In virtually all cases, a vaulted roof does in fact provide some measure of life-cycle energy savings relative to a flat roof of the same span. The use of alternative roof forms can reduce the initial production energy of a building (EE) by anywhere between 5% and 48% (over a range of spans from 3 to 10 m), or the equivalent of between 2.5 and 36 years’ worth of operational energy. The energy saved cumulatively over a 50-year life cycle by this substitution is on the order of 2–24%, depending on the span analyzed. The greatest relative reductions in EE, and in turn CLCE, are obtained in wider spans. • Operational energy savings due to roof vault geometry are negligible (and marginal relative to EE), unless the roof is very and/or uninsulated. The observed thermal advantage of a vault was mainly limited to passive, rather than active (energy-consuming) operation. • Overall achieved savings by optimal cases are explained by the application of efficient structural forms and the reduced thickness they enable, which on the whole were found to reduce both steel and concrete requirements and hence initial EE, translated into CLCE reductions – when compared with flat roofs with thickness as mandated by the local code [74]. • In short spans, on the order of 3 m, curved roofs offer only modest energy reductions, which may be overshadowed by practical considerations not taken into account in this study, such as possible construction limitations of non-standard forms. • Results are very sensitive to parameters of several types, including dimensional (e.g. beam size), environmental (climate, orientation) and constructional (materials selected). • At all spans and thicknesses, parabolic vaults with a modest rise of about 1 m were found to provide superior lifetime energy performance relative to flat roofs with no rise at all. This suggests that efficient structural forms may be implemented not only as external roofs, but as intermediate structural ceiling elements supporting overlying floors in multi-storey buildings. Overall it can be observed that both segmental and parabolic typologies have similar performance, although segmental roofs with a large rise (2 m) present a slight advantage in CLCE terms (from 1.1 to 2.8% more), as well as practical benefits in terms of ease of construction. • In general, it may be concluded that design decisions involving the use of efficient roof forms could significantly reduce embodied energy and in fact cumulative life-cycle energy in buildings. • The validation study clearly illustrates the ability of the automated tool to find optimal solutions for roof structures with different vaulted geometries, in terms of CLCE. These results demonstrate the value of the methodology as a powerful tool for exploring and visualizing the role of non-planar structural roof forms in energy efficient buildings, with a minimal requirement

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