Complex assessment of structural systems used for high-rise buildings.

Tamosaitiene, Jolanta ; Gaudutis, Ernestas

1. Introduction

Design of structural frames for high-rise buildings is one of the most complex design tasks in structural engineering. The design of a structural system must result in an appropriate solution. Besides, it must correspond to the construction and demolition processes. Hence, it is not only important to calculate loads that impact on the structural system of a building, but also estimate other factors, such as architectural solutions, engineering systems, construction process features and price. However, the design stage of a structural system often fails to apply various sustainable principles. There is a need to develop a method that would simultaneously reflect the impacts of decision-making on the cost and environment. Comparing the research results with sustainable design principles for selection of structural systems, this article investigates the effect of the existing sustainability requirements on structural systems of high-rise buildings and selection of materials. The decision-making process involves selection of the best alternative from several possible options. The selection is based on evaluation of relevant qualitative and quantitative criteria.

2. Complex assessment of structural systems used for high-rise buildings

Different normative documents and literature sources offer a wide choice of criteria that can be used to describe a high-rise building (Hang et al. 2012). Usually, a contemporary high-rise building is described in terms of metres above the ground or the number of storeys. Today, normative documents of different countries are one of the major sources for such information. Definitions of high-rise buildings differ from country to country (Parasonis, Gaudutis 2009) (Fig. 1). In Lithuania as well as in other countries, the definition of a high-rise building is determined by the ultimate height of the fire fighting equipment such as ladders and hoists. The Technical Construction Regulation STR 1.01.06:2002 "Structures of Exceptional Significance" of the Republic of Lithuania (2002) defines a high-rise building as a structure, the height of which from the ground to the highest point amounts to 30 metres. Another Technical Construction Regulation STR 2.02.01:2004 "Residential Buildings" (2004) contains a provision that describes a high-rise building as a structure with the upper storey including a mansard at the surface altitude of 26.5 metres and more. Consequently, the normative documents contain a contradiction, which impedes on the decision-making of local authorities. Rules on Preparation of Detailed Plans for Layout of High-Rise Buildings issued by the Ministry of Environment of the Republic of Lithuania define a high-rise building as a structure, the height of which from the mean altitude of the surface of the land plot to the highest point of the roof structure must be no less than 30 metres, unless a local municipal council regulates differently. In the city of Vilnius, a high-rise building must be at least 35 metres above the ground. This illustrates that different Lithuanian regulations contain small contradictions. As no uniform definition of a high-rise building exists, we defined our research object on the basis of the definition adopted by the Council of Vilnius.

The design of a building that would be efficient throughout the entire lifecycle requires rationality from the beginning to the end. All stages of a building lifecycle are closely interrelated; therefore, each and every of them need to be evaluated in order to achieve the maximum result (Fig. 2). The entire design of a building structure starts from selection of the key system. In the initial stage, it is sufficient to produce schematics of the structural system, regarding the type of structural elements, joints and materials. Usually, this task can be resolved on the basis of previous experience. To reduce the number of possible options, the approximate data on the use of different structural elements can be used as provided in tables below. The next step is the selection of the final structural system by modelling the load schemes for structural elements which are determined on the basis of normative documents and structural system calculation to select cross-sections of rational structural elements (Fig. 2).

The following basic requirements must be taken into account during the design of a structural system for a building (Razaitis 2004):

Strength

The strength of the structural system must be ensured during the design stage. It should be achieved by selecting the appropriate geometry of the structural system as well as the types of supports and joints of structural elements.

Stability

A building structure can horizontally move or collapse under wind load. Thus, it is especially important to ensure the structural stability of a building. This stability depends on the weight of the structure as well as soil on which the foundation rests.

Stiffness

It is the ability of a structure to resist deformation under loads; it can be ensured by increasing the cross-sectional dimensions of structural elements as well as stiffness of joints.

Efficiency

Efficient selection of a structural system covers the overall rationality and functionality of the architectural and structural solution.

Workability

Building process duration and economy in large part depend on workability of the selected structural system.

Price

Cost has traditionally been considered the most important factor in the decision-making process (Shen et al. 2010). The total price depends on numerous factors such as the price for construction works that consist of materials and labour costs.

Analysis of a construction scheme allows estimating and simulating external loads and effects as well as identifying the limits. In structural system design, the next step is to find the best cross-sections of structural elements that would fully satisfy the requirements.

Many sustainable design principles have not been properly observed during the design stage of structural systems of different high-rise buildings. Project stakeholders fail to understand the importance of sustainable development principles for project feasibility studies (Shen et al. 2010). Mostly, this can be explained by an increase in the amount of required investments, which do not result in a financial benefit. The process of design not only requires estimating the construction and use stages, but also the demolition of the building. The main tasks of sustainable design are as follow:

--Reduction of environmental pollution;

--Reduction of energy consumption.

[FIGURE 2 OMITTED]

The building sector is one of the biggest energy consumers and carbon emitters (Zuo et al. 2012). The carbon footprint may be reduced by reusing the structural system, separate structural elements or materials of a building (Hong et al. 2012; Lee et al. 2012; Kua, Wong 2012). At the end of their useful life, construction materials could be reused (Fujita 2012; Berge 2012; Pecas et al. 2013). Reuse refers to the ability to take parts of the structure and employ them elsewhere. However, such opportunity without the knowledge of the future demands is difficult to predict. The design stage of a structural system provides a possibility to take structural elements that remain at the end of a building lifecycle and turn them into other products (Ali, Moon 2007).

Buildings consume approximately 40 percent of total global energy: during the construction phase in the form of embodied energy and during the operation phase as operating energy (Dixit et al. 2010; Fiaschi et al. 2012). Without a doubt, energy efficiency is one of the most important aspects to be considered in a sustainable model of a building lifecycle. Embodied energy is expended in the processes of building material production (mining and manufacture), on-site delivery, construction and assembly on-site, renovation and final demolition (Dixit et al. 2010). Separate sustainable design concepts are based on reduction of embodied energy during different building lifecycle stages (Yuan et al. 2012; Dixit et al. 2012). Embodied energy accounts for a large proportion of lifecycle energy utilization in the building sector, and the estimation of this embodied energy is often difficult (Jiao et al. 2012; Hearn et al. 2012; Qian et al. 2012). Methodology aimed at minimising the embodied energy typically neglects the maximisation of the efficiency of the structural system. Although they do not play an active role in the energy design plan, the structural strategy and materials should be designed to respond to the overarching sustainability idea (Akadiri et al. 2012; Bojkovic et al. 2010).

A result is a triple bottom line, which refers to the three prongs of social, environmental and financial performance, which are directly tied to concept of complex assessment model and goal of sustainable development (Fig. 2).

3. Methodology

Complex assessment model of a structural system used in a high-rise building using MCDM (multi-criteria decision-making) remains somewhat different from the standard structural system assessment process (Fig. 3). In order to select the best alternative, it is necessary to have formed the decision matrix and to perform the multi-criteria analysis of the project. MCDM refers to making preference decision on the alternatives in terms of multi-criteria. Typically, each alternative is evaluated on the established set/system of criteria.

Multi-criteria analysis is a popular tool used to resolve various economical, managerial, constructional and other types of problems. This method has been successfully used in research by various authors since 1987 to determine the quality criteria of significance in construction. The theoretical aspects and practical application of the expert judgment method have been investigated by many different areas shown in Table 1.

The main problem involving multi-criteria is often too complex for a decision-maker (Choi et al. 2012). The assessment of selection of an efficient structural system is made with the help of the COPRAS-G method with the values expressed in intervals. The idea of the COPRAS-G method comes from real conditions of decision-making and from applications of the grey systems theory.

The objective of this research is to demonstrate how a simulation can be used to reflect grey inputs, which allows more complete interpretation of model results. COPRAS method was developed by Zavadskas and Kaklauskas (1996). The COPRAS method determines a solution with the ratio to the rational solution.

4. Case study: selection of a sustainable structural system for a high rise building

The main problem is that different structural systems can be used for the same high-rise building. The research aims to select the most efficient structural system from several possible alternatives defined with the help of intervals. In Vilnius, a 24-storey administrative building was selected as a research object, which has a framed structural system, vertical concrete

plate elements and a glass curtain wall.

The main steps of multi-criteria decision-making start with establishing evaluation criteria that relate the capabilities of the system to the goals. First, possible options of the structural system of a high-rise building have to be selected on the basis of the shape and height of the building. Possible structural system alternatives are provided in Table 2 and Figs 4, 5. On this building design stage we do not have precise building structural elements sizes therefore for different structural systems comparison we use approximate data taken from manuals for structural engineers which data expressed in intervals (Taranath 1998; Razaitis 2004; Parasonis 2008). According to this data were calculated amounts of wastes and energy, building design and construction price. Next, set of alternatives have to be developed to reach the goals. In this case, it is possible to use a methodology that allows making a decision on the basis of process-related qualitative and quantitative criteria. In order to select the best alternative, it is necessary to create the decision-making matrix and to perform the multi-criteria analysis of the project, accepting one alternative as the optimal one (Kendall 1970).

[FIGURE 3 OMITTED]

The expert judgment method was used to determine the significance of quantitative criteria and form the order of priority. The task had to be completed using various criteria of effectiveness with different dimensions, significances and direction of optimization. The criteria define the positive and the negative characteristics of an object under investigation. A survey was made to ask experts to prioritize 11 criteria (the rating scale ranged from 1 to 11, where 11 meant "very important" and 1 meant "not important at all"):

[x.sub.1]--effective height of the structural system (storeys);

[x.sub.2]--typical floor-to-floor height (m);

[x.sub.3]--lengthwise step of a column (m);

[x.sub.4]--transverse step of a column (m);

[x.sub.5]--length of a slab span (m);

[x.sub.6]--price for the design of the structural system ([euro]/[m.sup.3]);

[x.sub.7]--terms of performance ([m.sup.3]/w.d.);

[x.sub.8]--price for the construction of the building ([euro]/[m.sup.3]);

[x.sub.9]--embodied energy (kJ/kg);

[x.sub.10]--embodied carbon (kgC[O.sub.2]/kg);

[x.sub.11]--price for the demolition of the building ([euro]/[m.sup.3]).

[FIGURE 4 OMITTED]

The team of 12 experts was comprised of civil engineers with a long-term experience in design of structural systems for high-rise buildings. The experts had to use their knowledge, experience and intuition and rate criteria of effectiveness starting with the most important ones. The optimization directions of selected criteria and expert priorities were given to structural systems of the high-rise building on the basis of the data, important parameters of which are given in Table 3.

Out of all possible options, the final alternative was selected with the help of the COPRAS-G method. On the basis of the efficiency priority of alternatives, a rank R for each alternative was established (Table 4). According to calculation results, alternative A1 was identified as the best one. The first alternative was also the best in terms of its utility degree that equals 100%. The second alternative with the utility degree of 77.2% was ranked second. The forth alternative with the utility degree of 76.9% was ranked third. The fifth alternative with the utility degree of 69.0% was ranked fourth. The third alternative with the utility degree of 51.9% was the worst choice and ranked fifth. The vector of optimality criterion values was [N.sub.j] = [100; 77.2; 51.9; 76.9; 69.0]. The ranking of alternatives according to the results of the research are presented in the Figure 5.

According to the vector N the alternatives ranked as follows: A1 > A2 > A4 > A5 > A3.

According to the analysis results, structural engineers can choose the most effective alternative. The next step in the design of the structural system of a building should be the development of a calculation scheme for a selected structural system that would help determining loads and impacts as well as assessing the precise of geometrical characteristics of structural elements.

5. Discussion

In the future, this academic task could be transformed into an expert system, which--based on knowledge and applied analysis rules--would make it possible to identify certain field problems. It could transform into a practically used structural analysis and design programs. Besides, it could be used by structural engineers as yet another step toward automated design of a structural system and the whole building based on the life-cycle model as well as, possibly for the development of artificial intelligence.

6. Conclusions

The research showed that the integration of expert judgment and COPRAS-G methods can be used by structural engineers during the design stage of a building to select the most efficient structural system, when initial data expressed in grey numbers.

The selection of the structural system is approximate and the final decision can be taken after the final selection of the best structural system, taking into account structure affecting load values and selection of geometrical characteristics of structural elements. This methodology could help to reduce the number of options on the basis of a large number of criteria.

A case study demonstrated that contemporary environmental aspects have little importance for the design of structural systems.

The analysis of the problem on the basis of the selected criteria demonstrated that the semi-rigid frame [A.sub.1], which consists of prefabricated reinforced concrete products, is more preferable than the remaining four alternatives under investigation.

doi: 10.3846/13923730.2013.772071

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Jolanta Tamosaitiene (1), Ernestas Gaudutis (2)

(1) Department of Construction Technology and Management, Vilnius Gediminas Technical University, Sauletekio al. 11, LT-10223 Vilnius, Lithuania

(2) Department of Architectural Engineering, Vilnius Gediminas Technical University, Sauletekio al. 11, LT-10223 Vilnius, Lithuania

E-mails: (1)jolanta.tamosaitiene@vgtu.lt (corresponding author); (2) ernestas.gaudutis@vgtu.lt

Received 4 Aug. 2012; accepted 21 Dec. 2012

Jolanta TAMOSAITIENE. Assoc. Prof., Vice-Dean of Civil Engineering Faculty, Department of Construction Technology and Management at Vilnius Gediminas Technical University, Lithuania. Since 2013 she is a member of Editorial Board of the Journal of Engineering, Project, and Production Management", since 2011 she is a member of Editorial Board of the Journal, Technological and Economic Development of Economy", since 2009 she is a member of EURO Working Group OR in Sustainable Development and Civil Engineering, EWG-ORSDCE. She published more than 50 scientific papers. Research interests: civil engineering, many miscellaneous management areas (enterprise, construction project and etc.), risk assessment, construction project administration, building life-cycle, construction technology and organisation, decision-making and grey system theory, decision making (DM), statistics, optimization, strategies, game theory, intelligent support system, sustainable development: developing of alternative construction processes, economic and other aspects, sustainable development challenges for business and management in construction enterprises, environmental impact processes and etc.

Ernestas GAUDUTIS. PhD student, Department of Architectural Engineering, Vilnius Gediminas Technical University (VGTU), Sauletekio al. 11, LT-10223 Vilnius, Lithuania. MSc (2007) from VGTU. Author and co-author of 5 scientific articles and 1 study book.

Table 1. Use of MCDM in the analysis of a building life cycle

Stage            Methods                     Article title and authors

Building         AHP (Analytic Hierarchy     Multi-criteria
  design         Process)                    Optimization System for
                                             Decision-making in
                                             Construction design and
                                             management (Turskis et al.
                                             2009)

                 Expert judgment method      Assessment of the indoor
                                             environment of dwelling
                                             houses by applying the
                                             COPRAS-G method: Lithuania
                                             case study (Zavadskas et
                                             al. 2011)

                 COPRAS (COmplex             Passive house model for
                 PRoportional ASsessment     quantitative and
                 of alternatives)            qualitative analyses and
                                             its intelligent system
                                             (Kaklauskas et al. 2012)

                                             An assessment of
                                             sustainable housing
                                             affordability using a
                                             multiple criteria
                                             decision-making method
                                             (Mulliner et al. 2013)

                 COPRAS-G (COmplex           Assessment of the indoor
                 PRoportional ASsessment     environment of dwelling
                 of alternatives)            houses by applying the
                                             COPRAS-G method: Lithuania
                                             case study (Zavadskas et
                                             al. 2011)

Building         SAW (Simple Additive        Safety of technological
  construction   Weighting) method           projects using multi-
                                             criteria decision-making
                                             methods (Dejus 2011)

                 TOPSIS (Technique for       Complex estimation and
                 Order Preference by         choice of resource saving
                 Similarity to Ideal         decisions in construction
                 Solution)                   (Zavadskas 1987)

                                             Groundwater quality
                                             assessment based on rough
                                             sets of criteria reduction
                                             and TOPSIS method in a
                                             semi-arid area China (Li
                                             et al. 2012)

                 PROMETHEE (The Preference   Selection of logistic
                 Ranking Organization        service provider using
                 MeTHod for Enrichment       fuzzy PROMETHEE for a
                 Evaluations)                cement industry (Gupta et
                                             al. 2012)

                                             PROMETHEE with Precedence
                                             Order in the Criteria
                                             (PPOC) as a New Group
                                             Decision-making Aid: An
                                             Application in Urban Water
                                             Supply Management
                                             (Roozbahani et al. 2012)

                 Expert judgment method      Complex estimation and
                                             choice of resource saving
                                             decisions in construction
                                             (Zavadskas 1987)

                                             Multiple criteria
                                             evaluation of buildings
                                             (Zavadskas, Kaklauskas
                                             1996)

                                             Risk assessment of
                                             construction projects
                                             (Zavadskas et al. 2010)

                                             Application of Expert
                                             Evaluation Method to
                                             Determine the Importance
                                             of Operating Asphalt
                                             Mixing Plant Quality
                                             Criteria and Rank
                                             Correlation (Sivilevicius
                                             2011)

                 COPRAS (COmplex             Multiple criteria
                 PRoportional ASsessment     evaluation of buildings
                 of alternatives)            (Zavadskas, Kaklauskas
                                             1996)

                                             COPRAS based comparative
                                             analysis of the European
                                             country management
                                             capabilities within the
                                             construction sector in the
                                             time of crisis (Kildiene
                                             et al. 2011)

                                             Materials selection using
                                             complex proportional
                                             assessment and evaluation
                                             of mixed data methods
                                             (Chatterjee et al. 2011)

                                             Material selection using
                                             preferential ranking
                                             methods (Chatterjee,
                                             Chakraborty 2012)

                                             Evaluating the
                                             construction methods of
                                             cold-formed steel
                                             structures in
                                             reconstructing the areas
                                             damaged in natural crises,
                                             using the methods AHP and
                                             COPRAS-G (Bitarafan et al.
                                             2012)

                                             Owner preferences
                                             regarding renovation
                                             measures--the
                                             demonstration of using
                                             multi-criteria decision-
                                             making (Medineckiene,
                                             Bjork 2011)

                                             Multiple criteria decision
                                             support system for
                                             assessment of projects
                                             managers in construction
                                             (Zavadskas et al. 2012)

                 COPRAS-G                    Risk assessment of
                                             construction projects
                                             (Zavadskas et al. 2010)

Building         SAW (Simple Additive        Multi-criteria assessment
  renovation     Weighting) method           of alternatives for built
                                             and human environment
                 TOPSIS (Technique for       renovation (Tupenaite et
                 Order Preference by         al. 2010)
                 Similarity to Ideal
                 Solution)

Building         AHP (Analytic Hierarchy     Life-Cycle Analysis of A
  life-cycle     Process)                    Sustainable Building,
                                             Applying Multi-

                 COPRAS (COmplex             Criteria Decision-making
                 PRoportional ASsessment     Method (Medineckiene et
                 of alternatives)            al. 2011)

Stage            Methods                     Results of the calculation

Building         AHP (Analytic Hierarchy     Alternatives importance
  design         Process)                    relative to one other

                 Expert judgment method      In determining the
                                             significance of
                                             quantitative indicators,
                                             the order of priority was
                                             arranged.

                 COPRAS (COmplex             The optimal alternative is
                 PRoportional ASsessment     at the minimum distance
                 of alternatives)            from the ideal solution
                                             while the maximum distance
                                             from the ideal solution
                                             means the worst option.

                 COPRAS-G (COmplex
                 PRoportional ASsessment
                 of alternatives)

Building         SAW (Simple Additive        The order of priority of
  construction   Weighting) method           alternatives

                 TOPSIS (Technique for
                 Order Preference by
                 Similarity to Ideal
                 Solution)

                 PROMETHEE (The Preference   Prove the significance of
                 Ranking Organization        each criterion and define
                 MeTHod for Enrichment       it on the scale of an
                 Evaluations)                interval.

                 Expert judgment method      In determining the
                                             significance of
                                             quantitative indicators,
                                             the order of priority was
                                             arranged

                 COPRAS (COmplex             Optimal alternative is the
                 PRoportional ASsessment     minimum distance from
                 of alternatives)            ideal solution and maximum
                                             distance from ideal
                                             solution is the worst

                 COPRAS-G                    The optimal alternative is
                                             at the minimum distance
                                             from the ideal solution
                                             while the maximum distance
                                             from the ideal solution
                                             means the worst option.

Building         SAW (Simple Additive        The order of priority of
  renovation     Weighting) method           alternatives

                 TOPSIS (Technique for       The order of priority of
                 Order Preference by         alternatives
                 Similarity to Ideal
                 Solution)

Building         AHP (Analytic Hierarchy     Alternatives importance
  life-cycle     Process)                    relative to one other

                 COPRAS (COmplex             Optimal alternative is the
                 PRoportional ASsessment     minimum distance from
                 of alternatives)            ideal solution and maximum
                                             distance from ideal
                                             solution is the worst

Table 2. Establishment of weight of structural system criteria

                 Effective                  Building frame
                structural     Floor-to-      lengthwise
               system height     floor       columns step
Experts             (m)        height (m)        (m)

Expert 1            11             10             5
Expert 2            11             10             7
Expert 3            10             9              6
Expert 4            11             10             5
Expert 5             8             7              9
Expert 6             8             7              10
Expert 7            11             10             8
Expert 8            11             10             6
Expert 9            11             10             7
Expert 10           11             10             5
Expert 11           11             10             8
Expert 12           10             9              6
Average            10.33          9.33           6.83
rank
Sum of              124           112             82
ranks
Order of             1             2              6
priority
Significance       0.098         0.097          0.090

               Building frame    Slab       Building
                 transverse      span        design
                columns step    length   price ([euro]/
Experts             (m)          (m)       [m.sup.3])

Expert 1             6            9            4
Expert 2             8            6            4
Expert 3             7            5            4
Expert 4             6            8            4
Expert 5             10           11           4
Expert 6             11           9            4
Expert 7             7            9            4
Expert 8             7            5            4
Expert 9             8            6            4
Expert 10            6            8            4
Expert 11            7            9            4
Expert 12            7            5            4
Average             7.50         7.50         4.00
rank
Sum of               90           90           48
ranks
Order of             5            4            8
priority
Significance       0.088        0.090        0.094

                                     Building
                   Terms of        construction
                 performance      price ([euro]/      Embodied
Experts        ([m.sup.3]/w.d.)     [m.sup.3])     energy (kJ/kg)

Expert 1              7                 8                3
Expert 2              5                 9                3
Expert 3              8                 11               3
Expert 4              7                 9                3
Expert 5              6                 5                3
Expert 6              5                 6                3
Expert 7              6                 5                3
Expert 8              8                 9                3
Expert 9              5                 9                3
Expert 10             7                 9                3
Expert 11             6                 5                3
Expert 12             8                 11               3
Average              6.50              8.00             3.00
rank
Sum of                78                96               36
ranks
Order of              7                 3                9
priority
Significance        0.089             0.098            0.095

                                      Building
                   Embodied          demolition
                    carbon         price ([euro]/
Experts        (kgC[O.sub.2]/kg)     [m.sup.3])

Expert 1               2                 1
Expert 2               1                 2
Expert 3               2                 1
Expert 4               2                 1
Expert 5               2                 1
Expert 6               2                 1
Expert 7               2                 1
Expert 8               1
Expert 9               2                 1
Expert 10              2                 1
Expert 11              2                 1
Expert 12              1                 2
Average              1.83               1.17
rank
Sum of                22                 14
ranks
Order of              10                 11
priority
Significance         0.086             0.084

Concordation ratio W = 0.846.

Sum of the deviations square S = 13400. Significance of the
concordation ratio [chi] = 101.52.

Significance of the concordation ratio [[chi].sub.[alpha],v] = 23.210.

If [chi square] > [[chi square].sub.[alpha],v] expert opinion
consistent and criteria weights are recommended to apply calculation.

Table 3. Initial decision-making matrix with criteria values expressed
in intervals

Alternative

Structural      Structural     Material    Optimisation
system           elements       of the      direction
alternatives                  structural     Criteria
                              system of       weight
                              a building     Criteria

                                             Criteria
                                              values
                                            expressed
                                           in intervals

Semi-rigid     Beams          Sectional    [A.sub.1]
  frame        Columns        monolithic
               Span           concrete
               Beams          Monolithic   [A.sub.2]
               Columns        concrete
               Span
               Beams          Steel        [A.sub.3]
               Columns
               Span           Concrete
Rigid          Beams          Concrete     [A.sub.4]
  frame        Columns
               Span
               Beams          --

                      Effective
                     structural                   Floor-to
                       system                      -floor
                       height                      height
Alternative              (m)                         (m)

Structural
system                   max                         min
alternatives
                        0.098                       0.097
               [cross product][x.sub.1]    [cross product][x.sub.2]

               [[w.sub.1];   [b.sub.1]]    [[w.sub.2];   [b.sub.2]]

Semi-rigid         20            30            3.7           4.1
  frame

                   20            30            3.7           4.1

                   20            30            3.3           3.9

Rigid              20            40            3.6           3.9
  frame

                   20            35            3.4           3.6

                     Building                    Building
                       frame                       frame
                    lengthwise                  transverse
                      columns                     columns
                       step                        step
Alternative             (m)                         (m)

Structural
system                  max                         max
alternatives
                       0.090                       0.086
               [cross product][x.sub.3]    [cross product][x.sub.4]

               [[w.sub.3];   [b.sub.3]]    [[w.sub.4];   [b.sub.4]]

Semi-rigid          6            12             6             9
  frame

                    6            12             6             9

                    6            12             6            12

Rigid              4.5            9            4.5            9
  frame

                   4.5            9            4.5            9

                                                 Structural
                        Slab                       system
                        span                       design
                       length                       price
Alternative              (m)                 ([euro]/[m.sup.3])

Structural
system                   max                         min
alternatives
                        0.090                       0.094
               [cross product][x.sub.5]    [cross product][x.sub.6]

               [[w.sub.5];   [b.sub.5]]    [[w.sub.6];   [b.sub.6]]

Semi-rigid          4            12           27.5           35
  frame

                    6            18            45            55

                    6            18            50            65

Rigid              4.5            9            35            40
  frame

                   4.5            9            45            60

                      Terms of                    Building
                       perfor-                   construction
                        mance                       price
Alternative       ([euro]/[m.sup.3])          ([euro]/[m.sup.3])

Structural
system                   min                         min
alternatives
                        0.089                       0.098
               [cross product][x.sub.7]    [cross product][x.sub.8]

               [[w.sub.7];   [b.sub.7]]    [[w.sub.8];   [b.sub.8]]

Semi-rigid         0.5            1            275           350
  frame

                    3             4            450           550

                    2             3            500           650

Rigid               4             5            350           400
  frame

                    4             5            450           600

                      Embodied                     Embodied
                       energy                       carbon
Alternative            (kJ/kg)                 (kgC[O.sub.2]/kg)

Structural
system                   min                         min
alternatives
                        0.095                       0.082
               [cross product][x.sub.9]    [cross product][x.sub.10]

               [[w.sub.9];   [b.sub.9]]    [[w.sub.10];   [b.sub.10]]

Semi-rigid        1.11            2           0.139          0.176
  frame

                  1.11            2           0.139          0.176

                   32           56.7          1.317          1.936

Rigid             1.11            2           0.139          0.176
  frame

                  1.11            2           0.139          0.176

                       Building
                      demolition
                        price
Alternative       ([euro]/[m.sup.3])

Structural
system                   min
alternatives
                        0.081
               [cross product][x.sub.11]

               [[w.sub.11];   [b.sub.11]]

Semi-rigid         165            210
  frame

                   270            330

                   300            390

Rigid              210            240
  frame

                   270            360
               Columns        Concrete
               Span

Table 4. Calculation results

Alternative   Total sum of maximizing   Total sum of minimizing
No              normalized criteria       normalized criteria
                     [P.sub.i]                 [R.sub.i]

1                      0.143                     0.141
2                      0.159                     0.217
3                      0.164                     0.487
4                      0.132                     0.198
5                      0.129                     0.230

Alternative   Alternative's      Alternative's         Rank
No            significance         degree of         [R.sub.i]
                [Q.sub.i]     efficiency [N.sub.i]

1                 0.533              100.00              1
2                 0.412              77.18               2
3                 0.277              51.92               5
4                 0.410              76.90               3
5                 0.368              69.02               4

Fig. 1. Different definition of a high-rise building

Russia                  75
Ukraine                 73.5
Lithuania               30
United States           23
Germany                 22
France                  22
United Kingdom          22

Note: Table made from bar graph.

Fig. 5. Ranking of alternatives

1      100.0
2       73.3
3       51.9
4       76.9
5       69.0

Note: Table made from bar graph.