Geometric Aspects of Assessing the Amount of Material Consumption in the Construction of a Designed Single-Family House

: In this paper, we present a new approach for the analysis of the dependence of construction costs on the geometric shape of a building. Instead of di ﬃ cult or even impossible-to-establish uniform prices and costs, we propose a cost analysis concerning the amount of materials needed for construction. We show that the basic parameters are the base area of the building (plan), assumed in the study as the building area, and the area of the external walls of the building. The amount of consumption of most materials is proportional to the base area and the area of the external walls. The materials required for construction consume large amounts of energy during their manufacture. Therefore, shape optimization is not only economically signiﬁcant for the investor but is also important in terms of the energy consumption, i.e., embodied energy. We propose a set of indicators to help a designer optimize the shape of the building at the initial design stage.


Introduction
Not so long ago, the common belief was that the amount of embodied energy was small compared to the operational energy needed to operate a building throughout its lifetime. Hence, measures were mainly taken to reduce the operating energy by increasing the energy efficiency of the building envelope. This hypothesis has been challenged after extensive research.
For 1996-2019, Hu and Milner [1] found 320 publications in the Web of Science on embodied energy and its environmental impact in the building and construction industry. In particular, the analysis of these works showed that buildings consume huge amounts [2], around 40% [3], over 50% [4], over 33% [5], and almost 50% [6] of the annual global energy and, thus, increase the concentration of carbon dioxide in the atmosphere. The total energy consumption for the entire life cycle of a building consists of embodied and operational energy. Embodied energy comprises all the energy needed to extract raw materials; produce and transport building materials; and to construct, maintain, and demolish a building. Embodied energy can be the equivalent of many years of operational energy. Research by the Australian National Research Agency Commonwealth Scientific and Industrial Research Organization (CSIRO) showed that the average house contains about 1000 GJ of energy hidden in the materials used to build it. This corresponds to approximately 15 years of normal energy consumed during operation. In the case of a 100-year-old house, this is more than 10% of the energy

The Objective of the Work
The dependence of construction costs on the shape of a building is within our area of expertise and research [15,17,18,28,29]; textbooks have devoted considerable space to this field, enough to fill nine monograph editions [16,19]. The designer of a building should be aware from the beginning of the design process of the extent to which the adopted first compositions concerning the geometrical shape (plan) of the building will result in the structure, construction costs, operating costs, aesthetics, functionality, etc. The designer should not compromise any of the above-mentioned criteria according to a formally or intuitively made multicriteria analysis. The literature [2][3][4][5][6][7][8][9][10][11] advises that an important criterion should be the limitation of the consumption of building materials, resulting, in particular, in the reduction of the embodied energy, regardless of the need to constantly search for low-energy technologies for the production of building materials [30][31][32][33].
Energies 2020, 13, 5382 3 of 19 In our opinion, the best solution is to provide the designer with as simple as possible mechanisms to check the dependence of construction costs on the shape of the building after adopting the projection of the designed house. Previous studies known from the cited literature concerned the analyses of several formulated theoretical models of buildings. In this study, we examined 30 ready-made designs of single-family houses and formulated a short set of indicators, helpful for the designer, which are easy to determine at the beginning of the design process based only on the base (floor plan) area, perimeter, and height of the story.

Relationships between the Building Compactness Indicators Known in the Literature
For A-the area of the plan (building projection) of the building (floor area), P-the perimeter of the plan of the building (P s -the perimeter of a square of the same area A), and h-the height of the external walls, we can define the indicators: which is the ratio of external wall area to floor area [17], which is the Cooke wall/floor ratio index [17,19], which is the relative compactness indicator with respect to a square [14], which is the relative compactness indicator with respect to the cuboid [14], and which is the Banks length/breadth index [17,19].
The LBI index is defined as the quotient of the elements of the pair (a and b) of the solution of the system of equations: a·b = A, 2a + 2b = P. This is the ratio of the sides of the rectangle with dimensions a and b.
Energies 2020, 13, 5382 4 of 19 The indicators (1), (2), (3), and (4) are linearly dependent. However, there is no simple relationship between EWA/FA and the other indicators. Namely, However, if, in the design process, we assume that area A is fixed (building size) and the height h is determined by nature, then the dependencies (RC sq and EWA/FA) and (RC cd and EWA/FA) are linear, because then 4h 4 √ A is a constant value. Thus, the RC sq and RC cd ratios can serve as a measure of the EWA/FA ratio. These indicators express a clear measure: values of the indicator greater than 1 (the building plan is a rectangular polygon, other than a square) and equal to 1 are ideal (the building plan is square), and multiplied by 100% indicates the percentage deviation from the ideal dimensions in terms of the measure of the compactness of the object. The relative defects of an area (RDA) (RDA') and RDP indicators show the percent loss of deviation from the rectangle.

Indicators Characterizing Building Blocks on a Rectangular Polygon Plan
Most buildings, especially single-family houses, are built on a polygonal plan with right convex or concave angles. A polygon with this property is called a rectangular polygon; it has an even number of sides, and the difference between the number of convex and concave angles is 4 (see Figures 1 and 2). A-the area of the plan of a building, P-the perimeter of the plan of building, A R -the area of the rectangle circumscribed on the plan of a building, and P R -the perimeter of the rectangle circumscribed on the plan of a building can be defined by the indicators: which are the relative defects of an area (Figures 1 and 2), and which is the relative defect of the perimeter (Figures 1 and 2). A rectangular polygon with more than four sides must have concave angles and RDA > 0.  A rectangular polygon with more than four sides must have concave angles and RDA > 0.

Selected Designs of Single-Family Houses-the General Characteristics
We researched 30 single-family house designs [34], which were subjected to a comparative analysis, according to the following criteria: usable area of 100-150 m 2 (data according to the Central Statistical Office-the most statistically popular range of selected houses in Poland and Europe); -free-standing buildings with a rectangular plan; -single-story (one-story) buildings with no usable attic, no basement, and no garage; -basic building materials for the housing: H + H blocks, concrete, and steel;  A rectangular polygon with more than four sides must have concave angles and RDA > 0.

Selected Designs of Single-Family Houses-the General Characteristics
We researched 30 single-family house designs [34], which were subjected to a comparative analysis, according to the following criteria:

Selected Designs of Single-Family Houses-The General Characteristics
We researched 30 single-family house designs [34], which were subjected to a comparative analysis, according to the following criteria: usable area of 100-150 m 2 (data according to the Central Statistical Office-the most statistically popular range of selected houses in Poland and Europe); -free-standing buildings with a rectangular plan; -single-story (one-story) buildings with no usable attic, no basement, and no garage; -basic building materials for the housing: H + H blocks, concrete, and steel; -parameters of windows and external doors the same for all projects; -and roof shape-gable or hipped with a wooden roof truss structure (allowing for possible installation of a photovoltaic installation-roof slope angle 30-45 • ).
The above-mentioned projects, H1-H30, are listed in Appendix A (Tables A1-A3 and  Figures A1 and  is accompanied by a drawing of a rectangle described on the polygon of the base of the building. After some adaptations, Tables A1-A3 contains data on the dimensions of the buildings, costs, quantities of selected materials, calculated values of parameters, and indicators discussed in Section 3.

Analysis of Selected Designs of Single-Family Houses
These considerations do not include the area of the plot on which the project is implemented, the location (climatic zones), the location in relation to the directions of the world (insolation), the annual distribution of external temperatures, etc. The cost was calculated based on the available data (www.archon.pl) and the minimum prices for the second quarter of 2020 [35,36]. However, due to major changes in the Polish, European, and world economy, the presented calculations should be adopted as estimates.
Before further analyses, it is worth paying attention to how the indicators work together on the data from the H1-H30 projects. Here are their mutual correlation coefficients: cor(RC sq , JC) = 1.0000 (obvious, full correlation between RC sq , JC (see (6) The last dependence does not have to hold. The RDP index, where RDP > 0, shows the existence of a "niche" in the plan of the building. In the H1-H30 project set, only nine buildings (H1, H2, H6, H12, H16, H23, H27, H28, and H29) have a small niche in their plan. For others, RDP = 0.
Comparing the shell construction costs of many buildings in terms of the shape of the body makes sense with many limitations: the same building area; the same materials and technologies; the same percentage of individual materials (e.g., reinforced steel and concrete in the same ratio); and the same prices of materials, equipment, and labor. However, considering prices means that the analysis of the shape of the building is no longer universal. The prices of materials, equipment, and labor depend on so many factors that this type of comparative analysis no longer makes sense.
These results confirm that including costs in the considerations distorts any dependence of these costs on shape indicators (see Figure 3).
For similar reasons as the prices and costs, we basically omitted the problem of the dependence of the amount of labor in the considerations. This is because it is quite well-characterized by the A parameter and the shape of the base polygon (plan) of the building (including the RDA and RDP parameters and the number of vertices of the base polygon). Due to the inability to separate the steel and concrete used in the construction of horizontal and vertical elements, these materials were also omitted from the considerations. With these data, such an analysis is possible, and similar results should be expected.
Therefore, it remained to analyze the consumption of individual materials depending on the shape of the building, which was performed separately for each type of material. It is a completely sufficient analysis to show the close relationship between the consumption of materials and the values of indicators discussed in Section 3. The consumption of many materials essential in the adopted technology depends on the area of the building. These include building blocks, ready-mixed concrete, reinforcing steel, insulation materials, etc., and, therefore, all the most important building Energies 2020, 13, 5382 7 of 19 materials. Some materials (concrete, reinforcing steel, insulating foils, etc.) concern horizontal elements, such as the floor and ceiling; others (e.g., masonry blocks) concern external walls; others (concrete and reinforced steel) concern vertical elements (columns, lintels, and window sills). For similar reasons as the prices and costs, we basically omitted the problem of the dependence of the amount of labor in the considerations. This is because it is quite well-characterized by the A parameter and the shape of the base polygon (plan) of the building (including the RDA and RDP parameters and the number of vertices of the base polygon). Due to the inability to separate the steel and concrete used in the construction of horizontal and vertical elements, these materials were also omitted from the considerations. With these data, such an analysis is possible, and similar results should be expected.
Therefore, it remained to analyze the consumption of individual materials depending on the shape of the building, which was performed separately for each type of material. It is a completely sufficient analysis to show the close relationship between the consumption of materials and the values of indicators discussed in Section 3. The consumption of many materials essential in the adopted technology depends on the area of the building. These include building blocks, ready-mixed concrete, reinforcing steel, insulation materials, etc., and, therefore, all the most important building materials. Some materials (concrete, reinforcing steel, insulating foils, etc.) concern horizontal elements, such as the floor and ceiling; others (e.g., masonry blocks) concern external walls; others (concrete and reinforced steel) concern vertical elements (columns, lintels, and window sills).
The analysis discussed above is also possible for various building areas, but then, the number of materials used per 1 m 2 of building area should be analyzed. This approach also shows the possibility of calculating the number of materials per 1 m 2 of the flat area and, thus, for the collective estimation of the cost of 1 m 2 of the flat area. In the presented set of buildings H1-H30, we had the opportunity to analyze the size of, among others, the consumption of concrete blocks for the construction of external walls of buildings. The analysis of the consumption of concrete or steel was not possible due to the lack of information: how much concrete (steel) was used for the construction of horizontal elements (broken down into floor and ceiling) and for the construction of vertical elements (columns and lintels).
Therefore, the number of materials required to construct individual elements of the building should be considered as broken down into horizontal elements, depending on the size of the building area (floor, ceiling, and roof) and vertical elements (building walls), depending on the ratio of the wall area to the building area (plan). The built-up area A (Section 3) adopted to describe the shape of The analysis discussed above is also possible for various building areas, but then, the number of materials used per 1 m 2 of building area should be analyzed. This approach also shows the possibility of calculating the number of materials per 1 m 2 of the flat area and, thus, for the collective estimation of the cost of 1 m 2 of the flat area. In the presented set of buildings H1-H30, we had the opportunity to analyze the size of, among others, the consumption of concrete blocks for the construction of external walls of buildings. The analysis of the consumption of concrete or steel was not possible due to the lack of information: how much concrete (steel) was used for the construction of horizontal elements (broken down into floor and ceiling) and for the construction of vertical elements (columns and lintels).
Therefore, the number of materials required to construct individual elements of the building should be considered as broken down into horizontal elements, depending on the size of the building area (floor, ceiling, and roof) and vertical elements (building walls), depending on the ratio of the wall area to the building area (plan). The built-up area A (Section 3) adopted to describe the shape of the building allows for the formulation of a direct correlation with the number of materials needed to make horizontal elements, such as the floor and ceiling, and considers the slope angle and the roof.
Derived dependencies (10) and (11), assuming h = const and A = const, define the linear relationship between the EWA/FA index and the RC cd , RC sq , and JC indices, respectively. In the range of H1-H30, the area varies from 134.41 to 198.51 m 2 ; thus, a high correlation coefficient cannot be expected (Table 1, Figure 4); in the H1-H15 range, the area varies from 134.41 to 163.73 m 2 , and the correlation coefficient is much higher ( Figure 5); after selecting the range H10-H17, the area changes from 159.74 to 167.03 m 2 , and then, the relationship is almost linear (correlation coefficient close to 1.0000). If A = const, the dependence of RC cd , RC sq , and JC on EWA/FA is exactly linear (see (10), (11), and Figure 6). The RC cd index is most closely correlated with the EWA/FA parameter (and with the "H + H blocks per 1 m 2 " parameter), due to its three-dimensional nature. Thus, it can serve as the best measure of the EWA/FA parameter with a fixed (or slightly changing) A. The smaller (closer to one) the RC cd value, the closer the building shape to the ideal shape of a square cuboid with the greatest compactness.
Considering the sensitivity of the RC cd and EWA/FA parameters in relation to the shape of the building (building plan), the sensitivity of the EWA/FA and RC cd parameters to the change in the shape of the building base is visible in Figures 5 and 6 in comparison with the shapes of buildings H6, H8, H13, and H14 (Tables A1 and A3 and Figure A1). The complexity of the shape (discrepancy with the square shape) translates into the values of the parameters in Figures 5 and 6. The EWA/FA parameter indicates the material consumption per m 2 , and RC cd shows the measure of this material consumption as a deviation from the 1 (reference) value. Table 1. The dependence of the correlation coefficient of the relative compactness indicator with respect to the cuboid (RC cd ), the relative compactness indicator with respect to a square (RC sq ), Banks length/breadth index (LBI), and relative defects of an area (RDA) indices on the range of changes in the built-up area of buildings belonging to H1-H30. EWA/FA: the ratio of the external walls area to the floor area. the building allows for the formulation of a direct correlation with the number of materials needed to make horizontal elements, such as the floor and ceiling, and considers the slope angle and the roof. Derived dependencies (10) and (11), assuming h = const and A = const, define the linear relationship between the EWA/FA index and the RCcd, RCsq, and JC indices, respectively. In the range of H1-H30, the area varies from 134.41 to 198.51 m 2 ; thus, a high correlation coefficient cannot be expected (Table 1, Figure 4); in the H1-H15 range, the area varies from 134.41 to 163.73 m 2 , and the correlation coefficient is much higher ( Figure 5); after selecting the range H10-H17, the area changes from 159.74 to 167.03 m 2 , and then, the relationship is almost linear (correlation coefficient close to 1.0000). If A = const, the dependence of RCcd, RCsq, and JC on EWA/FA is exactly linear (see (10), (11), and Figure 6). The RCcd index is most closely correlated with the EWA/FA parameter (and with the "H + H blocks per 1 m 2 " parameter), due to its three-dimensional nature. Thus, it can serve as the best measure of the EWA/FA parameter with a fixed (or slightly changing) A. The smaller (closer to one) the RCcd value, the closer the building shape to the ideal shape of a square cuboid with the greatest compactness. Table 1. The dependence of the correlation coefficient of the relative compactness indicator with respect to the cuboid (RCcd), the relative compactness indicator with respect to a square (RCsq), Banks length/breadth index (LBI), and relative defects of an area (RDA) indices on the range of changes in the built-up area of buildings belonging to H1-H30. EWA/FA: the ratio of the external walls area to the floor area.   (Table 1). RCsq: relative compactness indicator with respect to a square.  (Table 1). RC sq : relative compactness indicator with respect to a square. building (building plan), the sensitivity of the EWA/FA and RCcd parameters to the change in the shape of the building base is visible in Figures 5 and 6 in comparison with the shapes of buildings H6, H8, H13, and H14 (Tables A1 and A3 and Figure A1). The complexity of the shape (discrepancy with the square shape) translates into the values of the parameters in Figures 5 and 6. The EWA/FA parameter indicates the material consumption per m 2 , and RCcd shows the measure of this material consumption as a deviation from the 1 (reference) value.  (Table 1).

Figure 6.
An almost perfect correlation between the RCcd and EWA/FA indicators at the level cor(RCcd, EWA/FA) = 0.9832 in the limited range of the H10-H17 area variation (Table 1).

Indicators Helpful in the Initial Geometric Analysis of the Project
The results of the research demonstrated that the parameter that best characterizes the amount of material consumed in the shell home is the area of the external envelope of the building, i.e., the sum of the areas of the horizontal elements (floor and ceiling) and vertical elements (external walls of the building) divided into horizontal and vertical elements. In accordance with the adopted notations, these are, therefore, the area A and the EWA/FA ratio. We know from the considerations  (Table 1).
building (building plan), the sensitivity of the EWA/FA and RCcd parameters to the change in the shape of the building base is visible in Figures 5 and 6 in comparison with the shapes of buildings H6, H8, H13, and H14 (Tables A1 and A3 and Figure A1). The complexity of the shape (discrepancy with the square shape) translates into the values of the parameters in Figures 5 and 6. The EWA/FA parameter indicates the material consumption per m 2 , and RCcd shows the measure of this material consumption as a deviation from the 1 (reference) value.   (Table 1).

Indicators Helpful in the Initial Geometric Analysis of the Project
The results of the research demonstrated that the parameter that best characterizes the amount of material consumed in the shell home is the area of the external envelope of the building, i.e., the sum of the areas of the horizontal elements (floor and ceiling) and vertical elements (external walls of the building) divided into horizontal and vertical elements. In accordance with the adopted notations, these are, therefore, the area A and the EWA/FA ratio. We know from the considerations Figure 6. An almost perfect correlation between the RC cd and EWA/FA indicators at the level cor(RC cd , EWA/FA) = 0.9832 in the limited range of the H10-H17 area variation (Table 1).

Indicators Helpful in the Initial Geometric Analysis of the Project
The results of the research demonstrated that the parameter that best characterizes the amount of material consumed in the shell home is the area of the external envelope of the building, i.e., the sum of the areas of the horizontal elements (floor and ceiling) and vertical elements (external walls of the building) divided into horizontal and vertical elements. In accordance with the adopted notations, these are, therefore, the area A and the EWA/FA ratio. We know from the considerations in Section 3.2, the value of EWA/FA parameter is best characterized by the RC cd index (Table 1 and Figures 4 and 5 and formula (11)). According to the definition of the RC cd index, in terms of the building compactness, the area of the entire building envelope is best characterized by the RC cd index [14].
The RC cd indicator has clear scaling. The reference value is equal to 1, and the product (1 − RC cd ) × 100% shows the percentage deviation of the shape from the reference value. Due to the relationship between the EWA/FA and RC cd parameters, the RC cd indicator describes the amount of material consumption and, thus, the expected construction cost. The size of the RC cd indicator allows the designer to optimize the geometric efficiency of the building. In addition, a helpful indicator is the RDA indicator, showing the loss on the surface of the house with the given materials used for the execution of external walls. The LBI indicator informs the shape of a rectangle with the area and perimeter equal to the area and perimeter of the planned building plan.
In summary, the RC cd , LBI, and RDA indicators easily obtained based on A, P, and h data will be helpful to the designer in the first stage the building geometry. Additionally, the RDP indicator can be helpful, especially when RDP > 0. Determining the RDA and RDP indices requires simple calculations of the A R area value and the P R perimeter of the rectangle described on the polygon of the building plan (Section 3). The number of sides (vertices) of the building base polygon is important. The larger the number of vertices, the more complicated the shape generated, among others, increases the labor.
Finally, many new results were obtained in terms of the influence of the shape of the building on the consumption of materials and, thus, on the energy consumption and the size of costs. Namely, the previously known EWA/FA index and the RC cd geometric compactness indicator introduced in [14] were used to analyze real designs, not model diagrams as before. Thus, a reliable method of assessing material consumption in the initial phase of the building design was constructed. A linear relationship between the EWA/FA and RC cd parameters was demonstrated. The RC cd index has a readable and natural scale (value 1 for a building with a square base and natural height) and, thus, measures the EWA/FA index values well (material consumption per 1 m 2 of building area). Thus, both indicators become more useful for designing buildings than those proposed so far. Research showed that the cost can be objectively analyzed only after characterizing the relationship between the amount of material consumption and the value of the compactness index. This ensures the independence of the analysis from material prices, as well as the direct measurement of embodied energy after using the appropriate tables of embodied energy materials.
The extensive literature on the subject so far lacked simple procedures for the preliminary analysis of the influence of the shape of the building on the consumption of materials and energy in the first phase of building design. The considerations on this subject did not provide unambiguous constructive and simple conclusions for the designer. We hope that this work will fill this gap and that the results of further research on the effect of building shape on operational energy demand will provide a complete solution to the problem.

Conclusions
Research on the relationship between the shape of a building and the costs of its erection can be reduced to the analysis of the consumption of materials. Then, the amount of material consumption is independent of the prices, costs, and, thus, of the country and its currency.
The geometric shape of the building, which can be determined by the system of building compactness indicators proposed in the article, has a significant impact on the amount of material consumption when erecting a building. The amount of building materials directly translates to the amount of embodied energy.
The proposed system of indicators (RC cd , LBI, RDA, and RDP) may be helpful for a designer in terms of the shape optimization, in particular, at the initial stage of building design. This should be understood so that the architect chooses the option with the lowest RC cd (LBI, RDA, and RDP) value from among the acceptable shapes of buildings in terms of the functionality, aesthetics, maintenance, etc.
Reducing the consumption of materials results in a reduction in the embodied energy consumption.
Author Contributions: E.K., first conceptualization and methodology and prepared and wrote Section 3; K.B., prepared the data and calculations; formulated Tables 1, A1 and A2; and prepared Figures 3-6; E.K. and K.B. together discussed and perfected the methodology, conducted the analyses, and wrote the manuscript. All authors have read and agreed to the published version of the manuscript.
Funding: This work was supported by the Bialystok University of Technology grant WZ/WBiIS/6/2019, financed from the funds for the science of MNiSW, and an agreement on the implementation of the scientific grant in 2020 for the employees of the Warsaw University of Technology supporting scientific activities in the discipline of Civil Engineering and Transport-504/04505/1110/43.070004.

Acknowledgments:
The authors would like to thank the anonymous reviewers for their remarks, suggestions, and comments that have improved the manuscript.

Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.   Figure A1. Building plans of the H1-H15 set and rectangles described on rectangular polygons of the plans of these buildings.