USEFUL online journal (Volume 2, Issue 3, 2018)

VOLUME 2, ISSUE 3 | OCTOBER 2018

USEFUL Published by: SVP4U ®

ISSN 2574-4461 (online)

online journal

HTTPS://USEFUL.ACADEMY DOI HTTPS://DOI.ORG/10.32557/USEFUL

Table of contents 1. Target functionals of innovation-investment provider of development

of production-economic systems of infrastructure type.

By: Alla Bielova D.Sc. and Olexandra Orlovska and Alona Kochedikova doi:10.32557/useful-2-3-2018-0001 pages: 1-6 2. Characterization and substantiation of conflict situations in the

preparation of water for heat energy objects in the construction industry.

By: Nataliia Zhuravska doi:10.32557/useful-2-3-2018-0002 pages: 7-9

3. Experimental and theoretical studies of biaxially prestressed steel-

fiber-concrete slabs.

By: Oleksandr Zhuravskyia and Andriy Gorobetc doi:10.32557/useful-2-3-2018-0003 pages: 10-14 4. Eco-design: methodological approach in designing.

By: Anait Daniielian doi:10.32557/useful-2-3-2018-0004 pages: 15-28 5. Determination of support reactions of rod constructions obtained by

morphogenesis.

By: Volodymyr Skochko doi:10.32557/useful-2-3-2018-0005 pages: 29-42

Editor-in-chief Vitalii Ploskyi, D.Sc. October 03, 2018 doi of issue: https://doi.org/10.32557/useful-2-3-2018

Volume 2, Issue 3, 2018 USEFUL online journal

Volume 2, Issue 3, 2018 USEFUL online journal

Target functionals of innovation-investment provider of development of production-economic systems of infrastructure type.

Alla Bielova D.Sc.1* and

Olexandra Orlovska2 and

Alona Kochedikova3

Abstract The problems of innovation and investment development of the country’s economy, industrial-production and transport enterprises of the railway industry are considered. The main factors influencing the innovation and investment climate of industrial and economic systems are determined and substantiated. In particular, the present state of transport enterprises of the Western region of Ukraine is investigated, problems and perspectives of formation of innovation and investment development of enterprises of infrastructure type are outlined.

Keywords Innovation - investment climate - production-economic systems - innovation idea - strategic development transport sector - economy - innovations 1 Department

of Economics and Management of the GSP ”Institute of Innovation Education” of Kyiv National University of Construction and Architecture, 4, Education Department, Kyiv, 03037, Ukraine 2 Department of humanitarian and socio-economic training. Dniprovsky National University of Railway Transport. acad. V. Lazaryan, Lviv Branch, Lviv, Ukraine 3 Postgraduate student of the Kiev National University of Construction and Architecture, 4, Education, 4, Kyiv, 03037, Ukraine

*Corresponding author: E:alla64@ukr.net, Address: 03037, Povitroflotsky Avenue, 31, Kyiv, Ukraine Received: 09/20/2018, Accepted: 09/28/2018, Available online: 09/30/2018.

R https://doi.org/10.32557/useful-2-3-2018-0001 This is an open access article under the CC BY-NC-ND license. Maintained by SVP4U

Contents 1

Introduction

1

2

Experimental samples and the research program

2

3

Conclusions

5

Acknowledgments

6

References

6

1. Introduction One of the most important problems of the present and the development of the economy of innovation type is to ensure the effectiveness of the introduction of new forms of entrepreneurial activity where the formation and development of adequate innovative systems capable of effectively implementing innovative ideas of a higher level on the basis of high organization, effective use of resources of different nature that will ensure the achievement of balance their functioning and elimination of destructive consequences in the process of innovations from roses Regional Committee effective strategy development and investment attraction.

We consider it worthwhile to note that the inefficient use

of innovative ideas by enterprises is due to the need to improve the efficiency of realization of the problems of economic development of domestic industrial enterprises and enterprises of the infrastructure type of implementation of targeted innovative projects, which is why the increasing priority is the question of determining priority directions of strategic development in the conditions of global changes and integration of the country’s economy into the European dimension, taking into account the constructive and key factors behind conditioning their effects. Hence, the solving of the task that will allow the development and substantiation of theoretical and conceptual foundations for forming the basis of innovation development of construction and service enterprises to ensure their whole-oriented innovation development will be urgent.

At present, the transport industry is a key generator of the company, which ensures the efficient functioning of almost all sectors of the country’s economy, and in a free market, the question of finding ways to the economic growth of the country’s economy is urgent. With the introduction of a visa-free regime for our country with the countries of Western Europe, the necessary stage for socio-economic development is the process of increasing the economic indicators that are directly dependent on the effective operation of all components of

Target functionals of innovation-investment provider of development of production-economic systems of infrastructure type. — 2/6

the industrial complex of the country. This requires a corresponding increase in the capabilities and capacities of all its parts. In the face of acute competition in the global market for limited access to financial and other types of resources, the highest chances for minimizing losses and quick recovery of economic indicators will be, first of all, those countries that were able to ensure sustainable development and political stability within the state, and resulting in the effect of such functioning is the achievement of a high-performing economy, based on the continuous improvement of production processes, the creation of innovative products, the optimization of the systems Belief and high innovation culture of the population.

The development of infrastructural enterprises depends in no small extent on the ability to solve complex problem problems at the state level, where the creation of a basis for the innovation and investment climate, both for the country as a whole and for a separate branch, the production-economic system, and a different structural link, should be the first ones.

The issue of development of the transport sector of the economy, which is a balancing element of industrial-production systems, remains urgent, and it provides for creation of favorable conditions for the implementation of innovation and investment policy in the industrial-industrial sector of the economic complex, where it is expedient to highlight an essential factor - to promote the process of restoration of the national economy, which is indisputable is the role of transport, which serves the industrial complex of the country, providing it with the necessary resource potential for the continuous operation of the latter.

The gradual growth of volumes of production of industrial and construction purposes requires a process of expanding ties between enterprises, while the rules of the free market dictate increasing requirements for the quality of transport services and increasing the efficiency of the use of vehicles, it’s continuous updating, and improvement, improving technical and operational capabilities, as well as constant improvement of the level of service. The process mentioned above is connected with the search for effective current organizational and economic mechanisms of innovation and investment development of all the railways of the country. This requires revision of existing norms and standards for the development of methodological approaches, principles, measures, and forms of organization of innovation and investment processes in the context of the formation of the national transport services market.

2. Experimental samples and the research program The current stage of economic development of the country requires political will in matters of assistance to the state in developing and applying the mechanism of activation of the investment policy of the industrial-construction-industrial complex of the country. Infusion into the economy of any country of ”fresh money” allows us to implement and intensify production capacities, new technologies, introduce funda-

mentally new approaches using new knowledge, techniques, development of scientific centers and applied institutes.

An important stage in the development of enterprises of the industrial-industrial complex is the renovation of its material base, the replacement of equipment with a more modern innovative, which will allow the production-economic system to obtain the opportunity to produce competitive products of innovative type. In this way, the country faces some problems that need to be addressed comprehensively and systematically.

In order to develop further measures, the necessary stage is the analysis of the factors of the investment market, which influence the result on the efficiency of their attraction. Table: 1 Source: [1, 4] systematized and supplemented by the author.

At present, Ukraine has not developed a substantiated system of support for innovation and investment activity for industrial and construction enterprises of the industrial complex.

This has become one of the reasons for a catastrophic decline in innovation and investment activity in the country, deteriorating indicators of a favorable investment climate, which led to a drop in production and a decrease in competitiveness, and the possible emergence of the current situation should be the intensification of economic growth, the development of new technologies, technological lines, energy-saving technologies, new types of innovative products that can only be achieved under the conditions of an effectively selected development strategy and accelerating the innovation-investment process processes [6].

At present, the innovative development of enterprises of the industrial complex in the country is developing under extremely unfavorable conditions, where the main reason is the production of products with a low share of value added and significant depreciation of fixed assets and production infrastructure in almost all sectors and sectors of the country’s economy.

It should be noted that the current state policy does not contribute to the development and support of industrial and construction enterprises, railways enterprises and the maintenance of a favorable investment climate: there is a limited introduction of innovative projects, followed by a reduction in financing, and an increase in interest rates by banks, which creates a favorable basis for growth corruption in almost all sectors of the economy, the use of significant financial resources is not intended and low level of control over development provided funds. Under these conditions, the likelihood of a significant degree of threat of economic risk in long-term lending increases, which deters potential investors and lowers the investment climate as a positive phenomenon of economic growth of any production-economic system [5].

In addition to the above, we would like to emphasize that, with the help of specialists in international business, to overcome the scientific stagnation of the industrial complex and to develop new investment projects, we believe that the necessary action should be the activation of the EUREKA program, which ensures the implementation of innovation and investment projects, which various reasons can not be realized Volume 2, Issue 3, 2018 pages: 1-6 (02)

Target functionals of innovation-investment provider of development of production-economic systems of infrastructure type. — 3/6

Negative influence and hinder the development of innovation and investment potential.

Positively influence and promote the growth of innovation and investment potential

Technical

Limitation or lack of sources of project financing, insufficient development of scientific and logistical base, outdated technology and technology, reduction of scientific potential of the state due to the lack or lack of staffing of the scientific and practical base.

Public funding and availability of state orders for products and scientific developments, availability of necessary scientific potential, provision of scientific and technical base, development of competition and shortening the life cycle of science-intensive goods, state support for innovation, support for young scientists and specialists, and the rate for scientific development of the state.

Economic

Availability of high economic risk, lack of demand for products, lack of product orders, low financial support to the scientific and technical sector, high level of competition, high economic barriers and interest rates on loans, nontransparency of origin and distribution of financial flows.

Positive dynamics of the growth of socioeconomic indicators, increase in cash receipts and transparency of their distribution to the needs of the industry, low turnover of personnel at enterprises, a perfect legislative framework for the account of resources, the availability of professionals, favorable rates of payment for a loan for potential customers, etc.

Systems of factors

Systematic organizational structures, excessive centralization, lack of desire Organizational and for new innovative strategies, orientation managerial towards established markets, lack of international scientific and technical cooperation.

Adaptability and flexibility of organizational management structures, decentralization, self-control and selfdiscipline, international scientific and technical cooperation, creation of innovation infrastructure.

Sociopsychological

Active resistance to changes due to the fear of uncertainty, the change of persistent stereotypes, the low professional status of the innovator and his low level of professionalism, the outflow of scientific personnel due to unfavorable working conditions or insufficient level of remuneration.

Positive perception of changes, innovations, employee's personal qualities, fair moral and material rewards, the possibility of self-realization.

Informational and communicative

Insufficient information on innovations, sources of their development and distribution, lack or incomplete information about an investor or source of investment, inadequate information exchange for innovation management, closure and limited interbranch relations.

Ability to quickly obtain the necessary information, the right choice of information channels, the acquisition of licenses, patents, know-how, expansion of horizontal streams of information

Legal

The imperfection of the legislative framework for investment and Legislative measures (special benefits, innovation activities, the lack of laws) that encourage innovation. knowledge of the legal framework in the field of intellectual property protection.

Table 1. Factors inuencing innovation and investment potential

Volume 2, Issue 3, 2018 pages: 1-6 (03)

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by the efforts of one country. Accession of Ukraine to this Program will facilitate the implementation of cross-border projects, especially in the western region of the country, with the participation of Ukrainian companies, research organizations and universities specializing in innovative technologies.

Ukraine will have the opportunity to participate in this process and strengthen the position of Ukrainian technologies in the European space, which will contribute to the effective integration of Ukraine into the world’s European space [7].

It is worth mentioning the importance of the problems of innovative development of enterprises of the industrialconstruction complex, which requires the development of a comprehensive national innovation development program and would involve a coordinated solution of problems both at the state and regional levels. Therefore, it is advisable to thoroughly examine the features of investment attractiveness as separate enterprises, regional development, and special conditions inherent in each individual region. Table: 2 Source: [1, 6], systematized and supplemented by the author.

World experience shows that investment is a powerful driving force in economic development. Moreover, experience and examples of highly developed countries show that, with a well-balanced management of the process of attracting foreign and domestic investments, as well as the introduction of well-designed management, significant progress can be made in ensuring economic growth and providing competitive advantages. [5] It is worth noting that in the conditions of an unstable market environment and resource constraints, infrastructuretype enterprises can be generated as strong factors and the emergence of structural conditions for the deepening of weaknesses, which will impede the formation of capacities in production-economic systems for the development of a new type of industrial production - innovation-informational. [2].

It is worth pointing out that in order to increase the level of trust in the national economy of foreign and domestic partners, the abolition of state registration of foreign investments was a positive factor. It positively affected Ukraine’s position in the Doing Business rating and the International Business Compass index, and in the long run could lead to an increase in direct investment in Ukraine (in the World Bank’s ”Doing Business 2015” rating, Ukraine took an indicator of ease of doing business 96th place among 189 countries in the world (in 2014 - 112th place), according to the International Business Compass index, which is calculated by the international consulting network BDO, Ukraine in 2015 ranked 89th out of 174 countries (in 2014, - 109th place) [6].

In this regard, it is worthwhile to note that by the volume of attraction of capital investments, the leading spheres of economic activity in January-December 2016 remained: industry - 33.3%, construction - 12.6%, agriculture, forestry and fisheries - 13.8%, information and telecommunications 4.8%, transport, warehousing, postal and courier activities 7.6%, public administration and defense; compulsory social insurance - 5.9%. [4, 6]. Fig. 1

According to the chosen course of the Ukrainian economy, which is connected with the process of European integration, the issue of preparation of the legislative framework of the leading branches of the economy for the current norms of the European Union requires much attention. This means that the priority task should be to prepare for cooperation with the EU transport industry, as one of the key suppliers of almost all types of resources. The transport industry requires not only the updating of the fleet of locomotives and wagons, the provision of service, as well as significant innovative technologies and investments for the development of the industry.

In the European countries over the past 10 years, a program on interoperability in transport has been developed and implemented to ensure the effective cooperation of all railway systems in European countries. At the moment, the same program has been implemented in Ukraine. [8] For more than three years, the national rail network of the Western region has been operating on the principles of interoperability of transport systems, which enables the gradual adaptation of national railways to the principles, methods, and methods of functioning of European lines. The principles of interoperability are intended to promote the creation of mechanisms for cooperation between the EU and Ukraine, not only in the transport sector, but also in the field of higher education, which will enable the training of future specialists to work within this European Program, the strategy chosen should become a convincing factor for improving the investment climate. Both in the region, in particular, and in Ukraine as a whole. [9] With the activation of programs to increase the efficiency of cooperation of transport systems of our country with the countries of foreign countries, we can speak about creating positive conditions for injection into the transport industry of investments, which will allow to modernize the production base of transport enterprises, conduct scientific research, expert work, reduce the risk of transport by means of track improvement programs. The process of European integration requires bringing to a single European standard not only the rolling stock of the railway but also the railway industry, which will have a positive effect on the functioning of the leading sectors of the economy and enterprises of the industrial-construction complex. The foregoing requires significant funds that can be ensured by the logical consistent innovation and investment policy of the state taking into account the current market requirements.

However, the following fact should also be noted: the number of existing production-economic systems is determined taking into account their production and economic and technological characteristics and depends, first of all, on technological capabilities and the formation of the innovation and investment potential of the latter.

At present, Ukraine has not yet developed a single mechanism for effective investment for large-scale technological changes and improvements in technical bases for both industrial and transport enterprises. At the same time, the national Volume 2, Issue 3, 2018 pages: 1-6 (04)

Target functionals of innovation-investment provider of development of production-economic systems of infrastructure type. — 5/6

Table 2. Regional features of investment attractiveness of regions of Ukraine

Figure 1. Distribution of capital investment in the sphere of economic activity (2016) [6]

base has unique technologies, but the general state of production and the limited resources do not create positive conditions for their implementation, does not create opportunities for the production of products that would meet the European standards. The importance of today’s problems - the problem of activating innovation development requires the development and intensification of a comprehensive national program that would provide for their coordinated solution to regulate the innovation and investment market of almost all sectors of the

country’s economy.

3. Conclusions When analyzing the real situation of enterprises of the industrialconstruction complex and the transport component of Ukraine it is expedient to emphasize that for international investors there is a significant risk of loss of invested capital, with the above-mentioned negative factors, such as high interest rates, opacity of the legislative base, low purchasing power of the Volume 2, Issue 3, 2018 pages: 1-6 (05)

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population. In view of this, it is necessary to implement the program of promotion of international investments, state insurance of investment risks and other means to increase the investment attractiveness of innovative domestic enterprises, which will contribute to the unconditional development of the national economy.

Consequently, the competitiveness of the country in the world community is ensured by an active innovation policy, which must be formed at all levels of economic management, and also take into account the global nature of innovation. For the effective implementation of innovations it is necessary to create an innovation core of the national economy at the expense of foreign investments and, on favorable terms, to participate in international scientific and technical cooperation.

[7]

Ukraine joins EUREKA [Electronic resource]. - Access mode: http://eunews.unian.net/eng/detail/188601

[8]

MISCTIF project. [Electronic resource]. - Access mode: http://diit.edu.ua /sites/tempus/ukr/about.html

[9]

Mironenko V. Transport interoperability and multimodal solutions. [Electronic resource] - Access mode: https://mtu.gov.ua

Acknowledgments We thank our colleagues from Kyiv National University of Construction and Architecture who provided insight and expertise that greatly assisted the research, although they may not agree with all of the interpretations/conclusions of this paper.

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Analysis of investment attractiveness of cities and regions of Ukraine / Research of the state analytical center ”Institute of reforms”. [Electronic resource] - Access mode: http://www.bdo.com.ua/Default.aspx

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Bielova A. I. 2013. System-universal obstacles on the way of development in the regions of industrial production of innovation-informational type [Text] / A. I. Bielova // Sb. scientific works DUTUTE Series ”Economics and Management - 2013. - Issue. 26. - Kiev; View of the DETAIL of the Ministry of Education of Ukraine, - P. 196 - 200.

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Bielova A. I. 2014. System of Strategic Management of Enterprise Restructuring: Goals, Problems and Prospects: Monograph Kyiv; KNUBA Ministry of Education and Science of Ukraine, Vitya Vyshemirsky V.S., - 295 p

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National target program of industrial development of Ukraine for the period till 2017 [Electronic resource] Access mode: http://kmu.gov.ua

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Bielova A. I. 2013. Competitive advantages of the railways in the market of transport services in Ukraine [Text] / A. I. Bielova, O. Orlovskaya // Sb. sciences State Economy and Technology Transport University of the Ministry of Education and Science of Ukraine. Series ”Economics and Management”. - Whip 23-24 - Kiev; View of the DETAIL of the Ministry of Education of Ukraine, - P. 103 - 109.

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Statistical Yearbook of Ukraine for 2016: Statistical Material / State Statistics Committee of Ukraine. - K: Consultant, 2016 - 575 pp.

Volume 2, Issue 3, 2018 pages: 1-6 (06)

Volume 2, Issue 3, 2018 USEFUL online journal

Characterization and substantiation of conflict situations in the preparation of water for heat energy objects in the construction industry.

Nataliia Zhuravska 1* Abstract In this paper one of the issues of the thermodynamic concept of material flows in the system of thermal power objects (TEO) is considered. The emergence of potentially possible conflict situations in the preparation of technical water under critical operating conditions of the TEO systems is investigated. It is established that there is a decrease in the total concentration of integral material evidence. It is shown that biological changes in the TEO systems are formed at the temperature (mode) - specific electrical conductivity of biological overgrowing, which is confirmed by the presence of ”wildlife”, which necessitates the identification and establishment of optimal gradations in relation to the operation of the TEO systems. Confirmation of this is the biotic potential in the operation of the feasibility study: for optimal regimes -1,7; under critical conditions - 2,6 (the self-organization of material flows is violated).

Keywords potentially possible conflict situations - energy saving technologies - material flows - biotic potential - biological damage.

1 Department of Occupational Health and the Environment of Kyiv National University of Construction and Architecture *Corresponding author: E: nzhur@ua.fm, Address: 03037, Povitroflotsky Avenue, 31, Kyiv, Ukraine

Received: 09/20/2018, Accepted: 09/28/2018, Available online: 09/30/2018.

R https://doi.org/10.32557/useful-2-3-2018-0002 This is an open access article under the CC BY-NC-ND license. Maintained by SVP4U

Contents 1

Introduction

1

2

Experimental samples and the research program

2

3

Results and discussion

2

4

Conclusion

3

References

3

1. Introduction To the modern nanotechnologies of non-reagent treatment of technical water in heat energy facilities is the method of application of electromagnetic fields (EMF). At present, processes with the use of magnetized water at the stage of its receipt (stable installation of electro-physical parameters and their specific indicators) are not sufficiently studied and at the stage of determining the characteristics of material flows directly in the system of heat supply [1, 2, 3]. The need for these studies is also due to the lack of structural and functional evidence of the effectiveness of the use of EMF for the reagent water sampling in thermal energy objects (TEs). In this paper, the aspects of the role of biological impurities of material

flows will be considered, from the point of view of the emergence of conflict situations ”magnetized water in the TEO system - technological and energy resources” (bio-destruction - bio-formation - bio-corrosion).

The scientific interest in the development of these changes in material flows under the influence of EPM is due to the fact that biological changes occurring in the feasibility study systems are formed in temperature regimes (intervals) when the presence of biotic factors (microorganisms) in the feasibility study system promotes local changes physic-chemical structure of technical water [4] due to the appearance of a technologically-determined EMF of magnetized water. The proposed model of technical water structure [1] is also emphasized in the model of the existence of rotor interaction between particles and atoms in Fig.1 [4]. The strength of the magnetic coupling of the proton decreases, which allows the molecule to be prone to dissociation at the constant temperature, pressure.

In Fig. 2 the model of the associate, volumetric, with 5 molecules of water, developed taking into account the model of Fig.1 is given. Dots show hydrogen bonds between molecules.

It can be seen that the water molecule can have up to four hydrogen bonds. A very high increase in the efficiency of

Characterization and substantiation of conflict situations in the preparation of water for heat energy objects in the construction industry. — 2/3

Figure 1. Model of water molecule

the installation with the use of magnetic fields can be estimated with the help of known dependencies of the hydrogen indicator of the liquid temperature.

And the biotic component of material flows, as a result of the interaction of microparticles (organic, non-graphic and biological nature), ensures the interconnections between these particles. In our opinion, conducting research in this scenario expands knowledge based on synergistic interactions in the EMF in relation to the fundamental approaches of thermodynamic catalysis [5].

Figure 2. Model of the association of five molecules of water

2. Experimental samples and the research program The objects of research are heat energy objects of low power (local installations for obtaining heat for some residential regions of cities).

The driving force for obtaining the thermal effect in the feasibility study is electromagnetic fields.

To receive EMF was used electromagnetic device ”Ilios” [1].

Methods for mathematical modeling and interpretation of the obtained results were used to evaluate the EMF’s effectiveness. And the solution of the engineering problem regarding the probability distribution of EMF in the heating system showed their finding in material flows. The microbiological research was carried out in conjunction with us at the Institute of Microbiology and Virology D.K. Zabolotniy of the National Academy of Sciences of Ukraine. The establishment of the physical basis of electromagnetic fields was determined to take into account the thermodynamic foundations of the entropy process [1].

The establishment of the physical basis of electromagnetic fields was determined to take into account the thermodynamic foundations of the entropy process [1, 6].

3. Results and discussion Achievement of scientific and technological progress at the end of XX and the beginning of the XXI century created only the prerequisite for the economical use of energy resources in industrial production. Among such production are heat energy objects of the construction industry: low power thermal power, which is locally used in the municipal economy of Ukraine.

In addition, it should be noted that the technocratic paradigm of the modern direction of optimization of technological processes associated with cybernetization and automation, leads to the separation of rights from the object of energy resources (technological and resource issues), which means that from rational use energy resources. That is, there is a conflict situation on the way to rational use of energy at the socio-economic level. At the same time, the use of energy-saving technologies at the heat energy facilities of the municipal economy showed that modern heat supply technologies ceased to be a brake on their development, which increased the efficiency of heat supply pipelines, which is distributed by hot water, heat and power facilities. Of great importance in this direction (optimization of the TEO systems) are modern methods of monitoring the technological state of the heating systems.

Despite the indirect control, one can determine the dynamics of the state of material flows in them as the carriers of the running thermodynamic start in the feasibility study. However, a lot can be done in this direction by stabilizing the state of technical water in the feasibility study, if carried out by a staged analysis of the state of material flows in them when using non-reagent methods for processing technical water in them using EMF [1, 7]. That is, a certain role in the economy of energy saving on the feasibility study should be influenced by the redevelopment of urban heat supply in the feasibility study. The effectiveness of the TEO systems, as mentioned earlier, is influenced by biological processes, which are the satellites of the formation of bio damage, biocorrosion on the pipelines of the TEO systems due to their biogenesis. Bioconversion processes that are fixed when using non-reagent water treatment are associated with the ”activity” of certain groups of microorganisms (encoding of membrane microbiota or micromycetes). The damage to metal structures and other Volume 2, Issue 3, 2018 pages: 7-9 (08)

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materials occurs due to man-made impacts of components of material flows - CO2, 2S, NH3 as products metabolism of chemical compounds (technical water) of material flows and activity of microorganisms. We have, in artificial conditions (laboratory tests), created optimal and critical conditions of action of EMP on the industrial water of the TEO systems.

The obtained results are shown in Fig.3

4. Conclusion We can state that the violation of equivalence in the system of material flows TEO, in practice, did not affect the heat capacity of the system. Thus, the conflict of the situation between the optimal and critical conditions of the feasibility study system did not change its primary purpose - the net calorific value. And, in relation to biological influences, the total concentration of impurities (organic, inorganic, biological) can be noted that there is a rejection of biological overgrowth in material flows. The quantitative value of the biotic potential is, therefore, a confirmation.

References

Figure 3. Dynamics of the state of material flows under different operating conditions of the TEO systems: for optimum operating mode - ”1”; under critical operating conditions - ”2”. Graphical dynamics of the state of material flows, for: a) concentration of material impurities, b) for pH; c) for the OKV - potential, d) for the specific conductivity, e) for the specific heat.

Analysis Fig.1 shows that due to technologically determined magnetized flow, changes in such parameters and their specific parameters, such as pH, specific conductivity, the concentration of impurities in material flows occur.

Systematization and analysis of repeated studies showed the need to establish the optimal gradation of the action of electromagnetic fields on the feasibility study. Changes in biological impurities as a sign of biological overshoot in the system (and the component of material flows) were determined by the biotic potential, which was characterized by the optimal operation of the feasibility study according to the technical regulations, as - 1.7; according to the critical operating mode, respectively, 2.6 conventional units. Such figures indicate that the level of the feasibility of a feasibility study depends on the orderly organization of its material flows and, first of all, on the synergetic processes in them.

Under these conditions, the conflict is associated with increased energy costs, when the specific conductivity for the optimal mode of operation was equal to 0.92; and under the critical mode of operation of the TEO systems - 0,98.

We have the consistency of the state of material flows in the feasibility study on the capacity of the system (technicaldriven nature of the feasibility study systems due to physicochemical changes in technical water) and specific electrical conductivity (due to the partial loss of the level of selforganization of flows due to violation of the interaction between microparticles).

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155-162.

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- P. 187-191

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Draganov B.H. 1990. Engineering and application of heat in agriculture. Agropromizdat, - 463 p.

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Malkin E.S., Furtat I.E. Zhuravska N.Y. 2015. The system of preparation of magnetized water in the electromagnetic fields of water and solutions and compounds based on it.

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Volume 2, Issue 3, 2018 pages: 7-9 (09)

Volume 2, Issue 3, 2018 USEFUL online journal

Experimental and theoretical studies of biaxially prestressed steel-fiber-concrete slabs.

Oleksandr Zhuravskyi 1* and Andriy Gorobetc 2 Abstract The article presents the results of experimental and theoretical studies of strength and deformability of steel-fiber concrete double-sided pre-stress slabs under the action of transverse loading. The simulation of such plates in the software complex LIRA-SAPR was performed taking into account the physical nonlinearity of materials.

Keywords steel fiber reinforced concrete, double-sided pre-stress slabs, strength, losses of prestressing, finite elements.

1,2 Department

of Reinforced Concrete and Stone Structures of Kyiv National University of Construction and Architecture.

*Corresponding author: E-mail: azhur@ua.fm, Address: Povitroflotsky Avenue, 31, Kyiv, 03680 Ukraine Received: 09/20/2018, Accepted: 10/03/2018, Available online: 10/03/2018.

R https://doi.org/10.32557/useful-2-3-2018-0003 This is an open access article under the CC BY-NC-ND license. Maintained by SVP4U

2. Formulation of the problem

Contents 1

Introduction

1

2

Formulation of the problem

1

3

The purpose of research

1

4

Research methodology

1

5

Research results

2

6

Numerous studies

3

7

Conclusion

3

References

3

1. Introduction In connection with the search for new high-strength materials with high deformation characteristics, materials are now becoming increasingly popular with the use of composite impurities. These materials include steel-fiber reinforced concrete, which is characterized by increased bending strength and high deformability. These characteristics are very important for modern construction, which is characterized by an increase in spans and a decrease in the weight of building structures.

The use of large-sized elements in the form of plates, panels are economically proven. It is also proved that the use of biaxial structures is most appropriate. Therefore, the study of strength and deformation characteristics of double-sided pre-stressed steel-fiber reinforced concrete slabs is a rather actual and practically unresolved problem.

The study of the properties of steel fiber reinforced concrete was engaged in many domestic and foreign scientists [1, 3, 4, 5, 7, 10]. These studies have shown the prospect of using steel-fiber concrete in the construction industry. Most of the works were devoted to studying the characteristics of steel-fiber reinforced concrete, such as the influence of the form, number, orientation, strength and shape of steel fibers, as well as the structure and strength of the concrete matrix on strength, deformability, frost resistance, water resistance, crack resistance and other characteristics of steel fiber reinforced concrete. But works devoted to the study of pre-compressed steel-fiber concrete in one and two directions are not enough..

3. The purpose of research The urgency of the work carried out is that data for the use of steel-fiber concrete in pre-stressed structures is not enough, the more intense in two directions. To remedy this gap in the prospective study of material use, experimental studies were carried out on single- and double-acting pre-stressed steel-fiber reinforced concrete slabs with different levels of compression.

4. Research methodology Two series of slabs of 800x800x40 mm with different fibrous reinforcement were investigated. The I series was reinforced with a mixture of steel fibers of the STAFIB 50/1,0 and STAFIB 30/0,6 with anchors at the ends, the percentage

Experimental and theoretical studies of biaxially prestressed steel-fiber-concrete slabs. — 2/5

ratio of which was 0,5% by volume of each fiber. Samples of the II series contained 1,0% wavelength fibers of the brand NOVOKON URW 50/1,0. Portland cement brand M400 was used for all samples, and quartz sand as a fine aggregate.

The composition of steel-fiber reinforced concrete in volume was adopted in a proportion of 1:3 cement to the sand with water-cement ratio of 0,62.

Steel-fiber reinforced concrete slabs were made in inventory metal formwork. Concreting was carried out in a horizontal position. The seal of steel-fiber reinforced concrete took place within 2...3 minutes by a surface vibrator.

The study used plates after lengthy studies to determine the losses of previous stresses from shrinkage and creep steelfiber reinforced concrete. The volume and characteristics of the sample plates before the study are shown in Table 1. The levels of pre-bending of steel-fiber reinforced concrete at the time of applying a uniform load were determined to take into account the losses of the previous tension from shrinkage and creep of steel-fiber reinforced concrete.

The stamping of the plates was made hinged on four sides at a distance of 50 mm from the side faces of the slab. The loading of the plate was carried out by 16 lumped forces in accordance with the scheme shown in Figure 1.

During loading of slabs, deflections were measured in the center of the slabs. For this purpose, indicators of a clock type with a price of 0,01 mm were used. In addition, measurements were made of moving the plate over the supports.

The load was created by two hydraulic jacks with a strength of 200 kN each. After that, the load through the traverse system was transferred to the slab Figure 1.

The load was carried out in step P1 = 2 kN with a delay of 5...8 minutes at each step for measuring deflections. The value of the load was fixed according to the indicators of the sample manometer of the hydraulic pumping station. Before starting the experiments, the entire hydraulic system (pump station, jack, model gauge) was trolled using a master proving ring.

Table 1. The volume and characteristics of the sample plates

for research. 10.6084/m9.figshare.7158758

5. Research results During the study, efforts were made to crack cracks and destroy the efforts of pre-stressed steel-fiber reinforced concrete slabs and plates without prior tension Table 2.

Figure 1. Scheme of testing the plates: 1 – indicator-deflectometer. 10.6084/m9.figshare.7158860

From Table 2 it is evident that in the slabs I-PP-0,7/0,7 the visual appearance of cracks was recorded at loading of Pcrc,1 = 11, 8kN, in plates I-PP-0,7/0,3 - with Pcrc,1 = 15, 1kN in slabs I-PP-0,7/0,0 - at loading Pcrc,1 = 6, 85kN. The appearance of cracks in an I-PW-U slab is fixed at loading Pcrc,1 = 5, 4kN, which is earlier than in the pre-stressed plates.

Cracking in slabs of the second series II-PP-0,7/0,0 and II-PP-0,7/0,3 was fixed at loading Pcrc,1 = 9, 8kN, and in slabs II-PP-0,7/0,7 - at loading Pcrc,1 = 11, 3kN. The appearance of cracks in the non-pre-stressed slab II-PP-S was fixed at loading Pcrc,1 = 5, 4kN. Analyzing the obtained data, one can see the growth of the cracking effort when increasing the efforts of the previous tension, as well as the effect of lateral compression on these values.

It has been experimentally proved that the character of the cracks in the pre-stressed and non-pre-stressed slabs is different. In slabs I-PP-S, the first cracks arose in the zones of maximum bending moments in parallel with the rods of the valves of both directions. In slabs I-PP-0,7/0,0, similar cracks formed only in directions perpendicular to non-elastic rods.

Cracks in the corners of the lower surfaces formed in all slabs practically at the same load. In the direction of these cracks, there was a destruction of the slabs.

Slabs of the series II-PP-0,7/0,7 were destroyed at loading P1 = 25, 1kN, and slabs I-PP-0,7/0,3 - at P1 = 22, 1kN. The destructive load in the slabs I-PP-0,7/0,0 was P1 = 17, 6kN. The non- pre-stressed slab I-PP-S collapsed at P1 = 15, 7kN, but it should be noted that one of these slabs was destroyed at P1 = 19, 6kN, which is not much less than the pre-stressed slabs. Such a discrepancy among the sample slabs arose Volume 2, Issue 3, 2018 pages: 10-14 (11)

Experimental and theoretical studies of biaxially prestressed steel-fiber-concrete slabs. — 3/5

Table 2. Strength and crack resistance of the slabs of series I

and II. 10.6084/m9.figshare.7158884

as a result of the adopted step of reinforcement along the Y-axis. The destruction of the slabs of the second series II-PP-0,7/0,7 occurred at loading P1 = 26, 9kN, and slabs IIPP-0,7/0,3 - at P1 = 23, 1kN. The destructive load in slabs II-PP-0,7/0,0 was P1 = 18, 6kN. The uneven slab II-PP-S collapsed at P1 = 21, 1kN.

6. Numerous studies For verification of experimental studies, the calculation of steel-fiber reinforced concrete slabs using the software complex LIRA-SAPR was performed, which allows simulating the work of reinforced concrete structures taking into account physical and geometric nonlinearity [1, 4].

To determine the stress-strain state of steel-fiber reinforced concrete slabs a calculation scheme was created. The slabs were modeled with square and rectangular finite elements (KEs) according to the recommendations [1, 4]. Efforts of the previous tension were given in the form of concentrated external forces applied at the corresponding nodes (Fig. 2).

The step of reinforcement for different boards along the Xaxis was the same and amounted to 89 mm, and along the axis of Y ranged from 89 mm to 133 mm. Thus, two different calculation schemes were created to calculate the slabs, which took into account the uneven application of forces.

Scheme 1 is created for double-sided pre-stress slabs with the same level of compression and one-sided pre-stress slabs squeezed Figure 2 (a). Scheme 2 is designed for non-prestressed slabs and for pre-stressed slabs with different levels of compression Figure 2 (b). Scheme 1 consisted of 500 finite elements and 441 nodes, Scheme 2 - out of 624 finite elements and 675 nodes. The following directions of coordinate axes are taken: axis X is directed from node 1 to node 22 for circuit 1 and from node 1 to node 28 for circuit 2, axis Y - from node 1 to 2, and axis Z forms with axes X and Y right three.

The stiffness of the slabs was set from the library of finite elements of PC LIRA-SAPR, which allows taking into account the physical and geometric nonlinearity in the calculation. The following inputs are used for calculation: h - the

thickness of the slab; v - Poisson coefficient; Ec f - modulus of elasticity of steel fiber reinforced concrete; f f cd - design strength of steel-fiber reinforced concrete to compress (prism strength); f f ctd - design strength of steel-fiber reinforced concrete to tensile strength; Es - module of elasticity of reinforcement; fyd - design strength of steel reinforcement to tensile strength; Hi - height (thickness) of the plot; Asxi - the area of the steel reinforcement is located along the X-axis on a running meter of the section; Asyi - the same, along the axis of Y.[8, 9] To simulate the work of steel-fiber reinforced concrete, the piecewise-linear deformation law was used, and for the reinforcement, it was an exponential deformation law.

The calculation was made for loads that correspond to loading steps, and the magnitudes of the efforts of the previous tension were taken according to the already established before the study of stress in the valve, taking into account all losses [2].

Theoretical and experimental graphs of deflections are shown in Figure 3 and Figure 4. Analysis of the graphs shows a sufficient comparison of theoretical and experimental data.

7. Conclusion Thus, it can be stated that the cracking forces in double-sided pre-stress slabs are 2 times higher than one-sided pre-stress slabs and 2,5 times higher than in non-pre-stressed slabs. The forces of cracking one-sided pre-stress slabs are higher than non-pre-stressed slabs in 1,3...2,1 times. The previous tension does not significantly affect the size of the bearing capacity of the slabs. Moreover, there is a slight decrease in the strength with increasing intensity of concrete bending with pre-stressed reinforcement in the direction of the Y-axis. But these changes are not significant and are within the limits of permissible scattering of results, which allows us to make reliable conclusions.

PC LIRA-SAPR gives an opportunity with a sufficient degree of accuracy to simulate the work of double-sided prestress steel-fiber reinforced concrete slabs.

References [1]

Bocharnikov A.S., Korneev A.D., 2005. The zone of interaction of systems ”concrete - steel fiber” in steel-fiber concrete and the rational degree of disperse reinforcement of fine-grained concrete. Construction materials, equipment, technologies of the XXI century, 79 (8), pp.58-59.

Available at: http://bit.ly/2zO9F66

[2]

Gorobets A.M., Zhuravskyi O.D., 2007. Experimentaltheoretical studies of losses of the previous tension in steel-fiber-reinforced structures with one-piece and twoaxis bending. The theory and practice of construction: The Bulletin of the National University ”Lviv Polytechnic”, 600, pp. 68-74. Available at: http://bit.ly/2yalG3E

[3]

Gorobets A.M., Zhuravskyi O.D., 2017. Strength and fissure strength of two-tier pre-stressed steel-fiber reinforced Volume 2, Issue 3, 2018 pages: 10-14 (12)

Experimental and theoretical studies of biaxially prestressed steel-fiber-concrete slabs. — 4/5

concrete plates with transverse bending. Building structures. Theory and practice: Collection of scientific works, 1, pp. 181-186.

[4]

Korotyshevsky O.V., 2003. The calculation of steel-fiber reinforced concrete on the strength of axial stretching and tensile bending. Construction materials, 8, pp. 31-33.

[5]

Lysenko E.F., Hetun G.V., 1989. Design of steelreinforced concrete structures. Study allowance, 189.

[6]

Milovidov K.I., Mishukov N.E., 1980. Rational areas of application of fiber reinforced concrete in structures. Concrete and reinforced concrete, 5, pp. 29-30.

[7]

D.A.Gorodetsky, M.S.Barabash, R.Yu.Vodopianov, and others, 2013. Program complex LIRA-SAPR, 2013. Training manual edited by A.S. Gorodetsky, p.376.

[8]

Rabinovich F.N., 1989. Disperse-reinforced concrete, p.176.

[9]

Talantova K.V., 2003. Fundamentals for the production of steel-reinforced concrete structures with specified properties. Concrete and reinforced concrete, 5, pp. 4-8.

[10]

Cernant A.A., 2004. Estimation of the efficiency of steelfiber-reinforced concrete structures in the operational period. Transport construction, 10, pp. 31-32. a) doi:10.6084/m9.figshare.7158968

b) doi:10.6084/m9.figshare.7158986 Figure 2. Calculation scheme of slabs in PC LIRA-SAPR.

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Experimental and theoretical studies of biaxially prestressed steel-fiber-concrete slabs. — 5/5

a) I-PP-0,7/0,7; II-PP-0,7/0,7 doi:10.6084/m9.figshare.7159001

b) I-PP-0,7/0,3; II-PP-0,7/0,3 doi:10.6084/m9.figshare.7159007 Figure 3. Charts of deflection plates of series I and II:

a) I-PP-0,7/0,3; II-PP-0,7/0,3 doi:10.6084/m9.figshare.7159028

b) I-PP-U; II-PP-U doi:10.6084/m9.figshare.7159022 Figure 4. Charts of deflections of plates of series I and II:

Volume 2, Issue 3, 2018 pages: 10-14 (14)

Volume 2, Issue 3, 2018 USEFUL online journal

Eco-design: the methodological approach in designing.

Anait Daniielian 1* Abstract A system of concepts and definitions was generalized, the basic principles of creating eco- objects are defined and formulated in the article. A model of the system representation of the eco-object is proposed to determine the concepts of environmental sustainability and structure ecologization. The concept of ecologization of the life cycle of the object is formulated, the appropriative scheme is proposed. The concept of ecologization of the object’s functioning is presented. As a result of the structural analysis of the concept, three concepts of the functioning of the eco-object during the exploitation were formulated and highlighted. The factors that directly or indirectly influence decision-making in the process of shaping were systematized, formulated, presented in the schematic form. As a result of the dissertation research on the basis of analysis of examples from the design practice, systematization of data, development of the typology of objects of eco-architecture in terms of geometric schemes, constructive solutions and their interrelations, elaboration of a proposal for a new method of rational, multivariate use of ecological principles in the objects of different structure and purpose, the algorithm of modeling objects of eco-design as the basis of the decision-making system was developed.

Keywords eco-design, ecologization in design, structure ecologization, the life cycle of an object, the concept of functioning, the concept of ecologization of the object’s functioning, eco-concept, geometry, the interaction between the object and environment, form-forming concept, geometrization, hierarchical model, composition, algorithm.

1 Architectural

Structures Department of Kyiv National University of Construction and Architecture.

*Corresponding author: e-mail: anait.daniyelyan@gmail.com, address: Povitroflotsky Avenue, 31, Kyiv, 03680 Ukraine Received: 06/01/2018, Accepted: 10/03/2018, Available online: 10/05/2018.

R https://doi.org/10.32557/useful-2-3-2018-0004 This is an open access article under the CC BY-NC-ND license. Maintained by SVP4U

1. Introduction

Contents

The issue of preserving the natural environment for subsequent generations is caused by the rapid deterioration of the 2 A system approach to defining concepts for creating global environmental situation, the increase in the rate of eco-design objects 2 global warming, the rising energy crisis as a result of increased consumption and construction in metropolises. Migration of 3 Formation factors, models and algorithms for the forthe population and rapid development of the metropolises mation of eco-design objects.

6 create the necessity of building large fields of functional com4 Algorithmic scheme of modeling of eco-design objects as the basis of the corresponding decision-making plexes and high-rise buildings. Due to this, the issue of creating environmental objects became topical. system.

11 The design has become a sphere of human activity that ofReferences 13 fers solutions to global humanity problems, shaping a lifestyle and planning its future development. Since last few decades has seen a multifaceted and almost universal commitment to Purpose green design. Cities of the future, as represented by architects, System research of concepts and constructive-geometric means are the concentration of a new ecologically responsible social order, technologies for the use of renewable energy sources of eco-objects designing; creation of methodical bases for the and the rational use of natural resources. ”Sustainable develdevelopment of the corresponding decision-making system. opment” has become an extremely popular destination in all areas of human activity, in the architectural and construction 1

Introduction

1

Eco-design: the methodological approach in designing. — 2/14

industry and in particular in design.

In the field of design, there are such concepts as architectural and building bionics, Bio-architecture (bio-design), Bionics, Biotech, Eco-architecture, Eco-design, Eco-tech, Green building (green architecture, ecological building, etc.), Natural construction, Organic architecture (organic design), Sustainable development. However, a single glossary, conceptual apparatus as the basis for creating the theoretical basis for further system development of eco-direction in designing is not proposed. Further development of the industry requires the formulation of a single conceptual apparatus and the development of a methodology for designing of eco-objects.

Tools for designing eco-objects based on a systematic approach to design are proposed as a basis for the creation of a methodology for eco-design.

During the work on the dissertation ”Instrumental tools for the creation of eco-design objects on the basis of a methodological approach”, a system of concepts and definitions in the field of eco-design was generalized on the basis of a comprehensive analysis of research and design decisions in the field of eco-design [7], the basic principles of creating eco-design objects [1, 2, 3, 4] are defined and formulated.

After analysis of design practice and theoretical material of different authors [13, 14, 15, 16, 17, 18, 19, 21, 22, 23] basic principles of eco-objects creating where formulated.

- In the process of dismantling.

5. The objects which are isolated from the environment during the operation: - Development of environments not foreseen for human life (underground, underwater skyscrapers, utopian space projects); - Designing in an area with the unsatisfactory ecological situation (amount of harmful impurities in the air, radioactive background).

2. A system approach to defining concepts for creating eco-design objects A model of the system representation of the eco-object is proposed to determine the concepts of environmental sustainability and structure ecologization.

Structure ecologization - the implementation of the principle of environmental friendliness at the hierarchical levels of consideration of the object - urban development (as a set of objects associated with architectural and planning solutions and engineering infrastructure), and the levels of an individual object (as a set of subsystems of architectural, structural and engineering solutions). Figure 1.

Basic principles of eco-objects creating:

1. Buildings with a closed loop of resources - objects in which different systems of recirculation and cyclic use of resources are applied: - Systems and measures of water recirculation; - Systems and measures of energy recirculation; - Systems and measures of air circulation and purification; - Disposal and energy use systems of biological waste.

2. Energy-intensive and energy-zero buildings are objects that provide their energy needs through the use of alternative energy sources without connecting to city electricity networks.

3. Objects that ecologize the environment, that is, improve the ecological state of the environment as a result of maximizing the positive effect of the object on the environment.

- Air: absorption of carbon dioxide and oxygen production; application of technologies and means of cleaning from harmful impurities; application of technologies and means of absorption and neutralization of radiation.

- Water: purification of reservoirs; desalination of water.

- Soils: enrichment, fortification, the rebirth of soils.

- Planting greenery: restoring the local flora; flora enrichment; creation of artificial flora.

4. Minimization of emissions and impacts: - In the manufacture of materials; - During construction; - During operation;

Figure 1. Structure ecologization.

doi:10.6084/m9.figshare.7170122 The ecology of the structure, functioning and life cycle of the eco-object is investigated from the system positions.

The life cycle of an object is a multi-stage process of its design, creation, operation, completion of the operation (utilization or renovation). Ecologization of the life cycle of the object - the compliance of the object with the environmental conditions at all stages of its life cycle.

Ecology must be ensured at each stage of the life cycle of the object in such a way that the set of decisions taken has a positive environmental effect. Figure 2.

The life cycle of an object of construction consists of six stages: definition of the concept, designing (object, constructive, technological, engineering solutions), production of materials (natural or synthetic origin), construction, opVolume 2, Issue 3, 2018 pages: 15-28 (16)

Eco-design: the methodological approach in designing. — 3/14

Figure 2. Ecologization of the life cycle of the object. doi:10.6084/m9.figshare.7170140

eration (functioning, exploitation), expiration of operation (deconstruction or renovation and extension of service life).

There are environmental characteristics that were investigated, formulated and structured in a schematic form in Figure 3 for each stage of the life cycle.

One of the requirements for modern construction projects is extending the period of their operation, therefore it is obvious that the fifth stage of the life cycle of the eco-object functioning - occupies the largest part of the total life cycle.

Therefore, speaking about the environmental friendliness of one or another object, we mean, in fact, ecological compatibility of the operation of this object.

In order to determine the degree of environmental friendliness, we introduce the concept of ecologization of the object’s functioning. The concept of functioning (the concept of ecologization of the object’s functioning, eco-concept) - was chosen when designing the principles of the future functioning of the eco-object.

As a result of the structural analysis of the concept, three concepts of the functioning of the eco-object during the exploitation were formulated and highlighted Figure 4: 1. Isolation of the object from the environment (closed object according to the concept of functioning) - minimization of any influence of the object on the environment, both positive and negative. Such projects include skyscrapers-farms, skyscrapers-ecosystems with closed cycles of energy, water, biological waste, closed objects such as ”city in the city”, etc 2. Maximum interaction of the object with the environment (open object according to the concept of operation) - the maximum (positive) impact of the object on the environment, the creation of the artificial environment, change or replacement of the elements of the environment. Examples of such sites are projects that include the creation of artificial reservoirs, water livestock, the creation of arable land through irrigation of

Figure 3. Signs of the environmental friendliness of the life cycle of the eco-object.

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land, landing parks and gardens in uncharacteristic for these regions (such as desert, or developed industrial city).

3. Optimization of the object’s interaction with the environment (neutral object according to the concept of functioning) - the interaction of the object and the environment in which the positive effects are maximal, and the negative effects and consequences are minimal.

The mutual influence of the object and the surrounding environment was investigated and the object-environment interaction types were identified for each of the eco-concepts.

Fig. 5, 6, 7 Figure 6. Maximum interaction of the object’s with the environment.

Figure 4. Concepts of ecologization of the object’s functioning.

Figure 7. Optimization of the object’s interaction with the environment.

Figure 5. Isolation of the object from the environment.

Each of the eco-concepts considered has its own means of implementation.

The isolation concept is realized when designing in environments not intended for human life, environment with the unsatisfactory ecological situation when creating an object similar to a natural ecosystem. The means of realization of the concept of isolation of an object from the environment are the creation of objects of a closed zero cycle, the creation of closed complexes with internal processes of functioning, similar to natural ecosystems. Figure 8.

Maximum interaction of the object with the environment implies a change in the environment: its environment, the

improvement of natural conditions or the complete replacement of the environment. This concept is implemented by creating an artificial ecosystem of the environment or by integrating the object into the environment and further changing the conditions of this environment.

Optimization of the interaction of the object with the environment can be realized through the application of geometric principles, constructive structures and systems, technological solutions, engineering solutions; the principle of the choice of materials (materials to be disposed of and reused or materials that are decomposed in wildlife); principles and means of ensuring energy efficiency of the facility.

The concept of optimizing the interaction of the object with the environment can be realized by different geometric methods: classical methods of formulation, methods of bionic prototyping, methods of physical formulation. Figure 9.

Classical methods of shaping is a geometric modeling of an object, based on functional, typological needs, compositional and space-spatial laws of shaping and subjective author’s decisions.

Methods of bionic prototyping is a geometric shell modeling of an object by borrowing geometry, constructive scheme or principles of the structure of a living organism, a natuVolume 2, Issue 3, 2018 pages: 15-28 (18)

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Figure 9. Geometric methods for optimizing the interaction between the object and the environment.

Figure 8. Means of implementing the concepts of environmental friendliness.

ral ecosystem, a natural phenomenon, physical or chemical processes.

Methods of physical shaping is a simulation of the geometric shell of the object, taking into account the physical factors of the environment that affect it. The purpose of forming such a method is to create a geometry which is optimal for specific environmental conditions and functional requirements to the object.

The geometry of an eco-object can be represented by three geometric principles of interaction of an object with an environment: open, closed, neutral geometry. Figure 9. Open geometry - is the geometric form and/or structure of the object of eco-architecture (eco-design), in which the object interacts with the environment at the functional level, for example, actively ”works” in urban planning schemes and streams. These objects are open to the environment compositionally and according to the planning structure.

Closed geometry - is the geometric form and/or structure of an eco-architecture object (eco-design) that does not interact with the environment at the functional level which, in the process of using the object, remains closed to the environment. Neutral geometry - is the ”invisible” geometry of the object, which is aimed to merge with the environment or its

imitation. Neutrality of geometry can be achieved by various form-forming means (immersion under water, deep into the earth, imitation of natural forms of terrain, repetition of the landscape, etc.).

Having three main environmental parameters of an object: ecologization of the structure, ecologization of the life cycle and the concept of ecologization of the object’s operation, one can obtain a concrete state of the solution that will correspond to a certain definition of the environmental nature of the object of architecture in space and time, by setting the solution for each of the parameters. A graphic model of ecological compatibility has been constructed, which relates the structure, the concept of functioning and the life cycle of the eco-object.

Figure 10.

Figure 10. Ecology of the object in time.

Eco-designing is an integrated process that involves making decisions at all stages and in all components of the project (geometry, design, engineering, technological decisions, maVolume 2, Issue 3, 2018 pages: 15-28 (19)

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terials selection).

3. Formation factors, models and algorithms for the formation of eco-design objects.

The geometry of eco-objects depends on many factors that directly or indirectly influence decision-making in the process of shaping. During the course of work it became clear that in order to create an algorithm for decision making in the eco-design process, to create the possibility of a systematic approach to the geometric shaping of eco-objects, it is necessary to formulate and systematize the factors of influence. As a result, if we present it in the form of a schema, we have a ramified system of factors that influence the process of geometric modeling of an object and determine the shape of its shell Figure 11.

Figure 11. Factors that influence the geometry of the eco-object design.

All impact factors can be divided into three main groups: geometric, non-geometric, and factors that can be ”geometrically”, that is set in the form of mathematical numerical parameters.

Geometric factors of influence - factors that are characterized by numerical parameters and directly affect the shape of the shell in the process of geometric modeling. Geometric factors may include both the design conditions and requirements, as well as the environmental conditions.

Non-geometric factors affecting the geometry of an object - factors that influence the geometric model indirectly, have no numerical parameter and cannot be introduced into a mathematical computer model. This group includes the historical and cultural aspects of designing in the area, the factor of time and fashion, the economic feasibility of the object being created, the application of the idea of using environmental materials (of natural origin, or materials that are to be recycled or decomposed naturally), the application of the idea of greening large areas to create recreational zones, formal geometric approach to modeling.

Factors that can be geometric - factors that are not calculated, have no mathematical parameter, and therefore the direct impact on the geometry of the shell of the object, but can be represented as numerical values that can be set in a geometric model for obtaining a specific shell form.

Geometrization is the process of representing non - calculated design conditions and requirements in the form of numerical values that can be set in a geometric computer model.

The list of factors in this group includes the following requirements: architectural and planning conditions and requirements, requirements for the functioning of the structurefunctional requirements, the application of enclosing structures (complex multilayer shells, nano-leather, etc.), the object type, the use of systems of recirculation of resources inside the object, providing natural ventilation and thermal regulation of the building, and environmental conditions. [5] The creation of an optimal geometric model of an ecoobject reduces to the selection of priority factors influencing it, depending on the design conditions and the specific environment.

The components of the formation process of an object are the form-forming concept, the geometric model and constructive structure. [6] The choice of the author of the formulating concept determines the geometric model of the object, which in turn determines its constructive structure. However, the means of realization of the object are limited, depending on the technical, technological and constructive possibilities of the time.

Therefore, based on the possibilities of realization, clarification of the constructive structure leads to the correction of the geometric model. Figure 12.

Under such a scheme, the formation of an object occurs.

The process of shaping ecological objects is fundamentally different from the design of other objects because the priority in the model of the formation of an eco-object is the selection and tasking of the concept of the object’s functioning. It is the concept of functioning that determines the form-forming concept, from which the geometric modeling of the design Volume 2, Issue 3, 2018 pages: 15-28 (20)

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of functioning in the process of eco-design. Figure 14.

Figure 12. System hierarchical model of designing objects of

design. object, classified as ”eco” begins.

The choice of the concept of operation depends on the source data for design: the conditions and requirements for the design object, as well as the physical factors of the environment. Requirements for the object are formed on the basis of historical and cultural aspects, demographic requirements, functional requirements and, as a rule, included in the design task. When choosing the concept of functioning the factors and conditions of the environment, both natural (natural conditions and features, weather conditions, and phenomena) and urban (features of the urban development scheme, transport-pedestrian flows, condition, and conditions of urban engineering networks, etc.) are taken into account. Figure 13.

Figure 13. System hierarchical model of forming of eco-objects.

On this basis, the following system hierarchical model for the formation of eco-design objects is created taking into account the effect of the input data on the choice of the concept

Figure 14. System hierarchical model of geometric modeling

of eco-objects.

The process of geometric modeling of the eco-object can be represented in the form of four levels: the formulating concept, the means of implementing the concept, classes of means of implementation, options for implementing classes of means. [6] The concept of form-forming (the concept of the formation of an eco-object) is the chosen fundamental principle, which forms the basis of the formation of an ecoobject.

The choice of form-forming concept sets the approach to geometric modeling of an object, determines its space-spatial structure, architectural image, and geometric model. After determining the formulating concept, the author of the project goes to the second level of space-spatial modeling - the definition of the geometric means of the implementation of the form-forming concept. And depending on the chosen means of implementation, it is necessary to determine a class of means of realization of concepts, and give - the last level of geometric modeling - the choice of the option of the implementation of classes. Figure 15.

Figure 15. Levels of eco-object modeling process.

On the basis of system analysis of design materials, it was possible to differentiate between eight fundamentally different conceptions of forming objects of eco-objects. A sequence is proposed and analyzed from system positions Volume 2, Issue 3, 2018 pages: 15-28 (21)

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of form-forming concepts: geometric concept, constructive concept, computer-mathematical concept, formal-geometric concept, functional-geometric concept, the concept of the prototype for operation, concept of non-interference, concept of historical and cultural imitation. Figure 16.

Figure 16. Form-forming concepts of eco-object design.

The chosen concept of shaping determines the volumespatial model and the architectural image of the object.

The geometric form-forming concept - is the search for geometry as a balanced form for taking into account and providing optimal solutions for all elements of the eco-object.

In essence, it is a search of the ideal form of construction, established in a particular city under specific environmental conditions: both physical factors, architectural and planning, ecological, urban, demographic, ethnic, historical, etc.

The constructive form-forming concept - is the concept of shapes formation when the basic principle in designing becomes the constructive calculation of the form or the task of a specific constructive structure, as well as the possibility and maximum simplicity of the project implementation.

The computer-mathematical concept - is the idea of creating objects on the basis of mathematical calculations, the introduction of numerical parameters of environmental conditions (both physical and urban) into a computer program and obtaining the corresponding geometry of the object, a kind of digital architecture.

Formal-geometric concept - the definition of formal geometry (not related to the function or optimal structure) as the basis of design. This is a formal approach to shaping, which is subjective and reflects the wishes of the author.

Functional-geometric concept - the idea of providing a functional purpose or process of the object’s operation through geometry in the first place. In such projects, the function or process of the building is directly reflected in its geometry or enclosing structures.

The concept of a prototype for functioning – meaningful, analytical approach to design, when the author examines not only the external component but the complex interrelationships of chemical, physical and biological processes characteristic of living organisms, natural phenomena.

The concept of noninterference - the creation of ecoobjects that are hidden, invisible to the eye, which does not

change the natural landscape or urban silhouette, do not attract attention to geometry and design - objects of neutral geometry.

This approach deserves to be isolated in an independent form since architects often try to create anti-architectural objects in their essence, objects that do not change the visual perception of the environment. In the design practice, there are many different solutions for geometric modeling of objects with the concept of non-interference. Underground, underwater objects, aerial ground projects, proposals for the selection of such a form and composition in which the object visually dissolves in the environment, etc.

The concept of noninterference is interesting in terms of the variety of geometric implementations, architecturalcompositional and space-spatial solutions.

The concept of historical and cultural imitation - the concept of creating an object of an organic and comfortable for a person at the level of its historical traditions, religious views and mental and ethnic preferences.

Taking into account the tradition of the nationality, the historical features of the area in which the object of design is located, the imitation of traditional forms and constructive techniques, the use of local materials and traditional systems - all these features determine the principle of historical and cultural imitation, which can be differentiated depending on the emphasis placed on four levels of imitation: geometric, constructive, use of building materials, functional.

The geometric level of historical and cultural imitation is the imitation of traditional forms and silhouettes. The constructive level is the imitation of traditional constructive techniques and structures. The level of use of building materials involves the use of building materials of local origin or the use of traditional materials, which are typical for residents of a particular area over a long period of time. The functional level is a traditional mode of operation and functional appointment in the traditions of the people, for example, religious buildings, and others like that.

Any of the formative concepts can be realized by various geometric means. The typology and specific varieties of geometric means for implementing form-forming concepts are established. There are eight different means of realization of the concepts of form-forming (prototype, frame, kinematics, module, combinatorics, fractals, self-regulation, line-plural sets), as well as their combination.

Geometric means for the implementation of form-forming concepts: • • • • • • • • •

Form as s prototype Frame Module Kinematics Combinatorics Fractals Self-regulation Linear plurals Combination of methods Volume 2, Issue 3, 2018 pages: 15-28 (22)

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The choice of means is the subjective process, based on many factors and design conditions: architectural and planning requirements, urban planning requirements and constraints, physical factors of the environment, the concept of the object’s operation is chosen, and so on. These means of implementing form-forming concepts, each of which are divided into classes and implementation options classes, give developers many opportunities for geometric modeling in the design of eco-design objects.

Form-prototype - one of the means of implementing formforming concepts, this is a method of prototyping as a basis in the formation process. Depending on the type of selected prototype, this tool can be divided into the following classes: bioprototype, abstract prototype, prototype-lattice, irregular prototype form.

The bioprototype - is the use of a prototype of natural origin. These can be natural phenomena, biological organisms, living creatures, other inanimate forms of natural origin. The class of the medium has several classes of implementation options, namely the use of a bio prototype can be implemented in several ways: form, function, and design. Prototype in form is a variant of prototyping at the level of imitation of the general form (geometry) of the object of natural origin.

In this approach, the form and geometry of the object are copied or imaged, and its physical properties, mode of action and functioning, as well as the principles of the structure of a natural object may not be taken into account. The function is prototyping on the principle of operation or functioning of an object of natural origin or phenomenon. When using such a prototype option, the visual perception of an architectural object may not cause the viewer associations and figurative connection with a natural object. The realization option (the design) - is the use of a prototype based on the principle of the structure of a natural object or phenomenon and the transfer of these principles of the structure to the constructive structure of the building.

An abstract prototype - is a means of implementation that involves the use of a geometric form or body as a prototype.

An abstract prototype can be any geometric body, figure, line - any geometric object.

Prototype - lattice is a class of geometric means of prototyping when a lattice acts as a form-forming prototype: a flat, spatial, or complex-structured lattice.

Another way of implementing form-forming concepts is to create a framework. This can be constructed as a regular lattice or an irregular lattice. A regular lattice can have a triangular, quadrangular or hexagonal shape. The irregular lattice has a large variation of the solutions. Figure 17.

Using the module as a means of implementing a formforming concept has a large variety of architectural compositions. The module can be either a primitive project unit of minimum dimension or a whole block or even a building as a whole (with the application of the module at the urban planning level). However, the dimension, scale, and scale of the module are irrelevant when classifying geometric means

Figure 17. Geometric means for implementation of form-forming concepts.

of implementing architectural form-forming concepts. In this case, it is important to identify and simulate the types of modules and further combine them with the frame or free.

Creating a frame and further filling it with modules will be called the frame filling class. Frame filling can be performed in several ways: the same types of modules, the combination of several different types of modules, filling modules by variables according to a certain law (algorithm), filling the modules with variables randomly.

Types of modules can differ in all or several parameters: different geometrically (open, closed neutral), different in typological purposes (architectural and planning decisions, scale), different in geometric model, different in concept of functioning (different constituent structures can function for different principle and in different modes), different in function of purpose.

The free combination of modules can occur chaotically, according to a certain algorithm or according to the needs in the process of functioning, that is, in accordance with the Volume 2, Issue 3, 2018 pages: 15-28 (23)

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functional requirements in the operation of the object. For example, if during the operation of a modular residential building the creation of additional areas becomes necessary, the required type of module is added to the general composition.

As already noted above, the dimension of a module does not matter to determine the geometric modeling tools when designing eco-design objects. This can be both the dimension of one room and the whole apartment, or a residential block of flats.

Kinematics - a tool for geometric modeling (implementation of form-forming concepts), which includes classes of means: motion along guides, discrete kinematics, a combination of different kinematic surfaces.

Combinatorics - a geometric means of applying the patterns of location, arrangement, selection, and distribution of elements of an architectural object. Fractals - a tool for geometric modeling, which is based on the idea of using fractals.

Architectural expressiveness and composite solutions can be very diverse.

Self -regulation is one of the geometric means for the implementation of form-forming concepts of modeling of eco-objects. Can be considered as geometric self-regulation forms and self-regulation of development. These are the objects in which the idea of an independent, non-architectural development of the structure is laid.

Linear plurals are a geometric simulation tool that allows you to create objects by multiplying line sets and the inverse problem - the congruences of normals.

The combination of several geometric means is the most common type of implementation of form-forming concepts in the process of modeling eco-objects because the combination of several methods gives the author a large variety of composite and geometric solutions. Figure 18.

Figure 18. Geometric means for implementation of form-forming concepts. Geometric classes and options for their realization.

The object of eco-design differs from the usual design object not only by the use of technologies, means and methods of preservation of the environment but above all by the logic of design: the formulating concept is an integral part of eco-design. Determination of geometrical methods of mod-

eling as a means of implementing a form-forming concept is the next stage in the process of creating a three-dimensional model of eco-object. The application of a particular compositional technique completes the logical sequence of creating an architectural image of an object. Figure 19.

Figure 19. Logic of spatial modeling of eco-object.

During the course of the work a number of objects of the eco class were analyzed and a database of eco-objects (126 representatives) was developed that contains the features: ecoconcept, form-forming concept, geometric means, constructive system, compositional method, eco-effect, technological solution, typological variety and allows to systematize, group and find project decisions on a certain basis, identify and catalog new project solutions, supplementing the database, obtaining analytical information, etc.

On the basis of the systematic analysis of a plurality of design solutions, the combinatorial models of the interconnection geometric means - the formulating concept and the geometric means - compositional technique, which allow to generate new solutions of eco-objects and fill the corresponding database, were constructed. Compositional techniques typology is founded on Yakovlev N., Golubeva O., Ikonnikov A. Textbooks [10, 11, 12].

Using combinatorial models, it is possible to create a new type of space-spatial solution of eco-object, using a pair of features in the process of geometric simulation, not used up to now. [9] With such combinatorial models, in addition to generating new solutions for geometric shells, you can receive a variety of analytical information. In this work, the tables of comparison of pairs of attributes not used in the design practice, but possible for use - combinations for creating new solutions are given. The expediency of using one or another concept is analyzed, it is determined that the most effective and variability is the geometric form-forming concept that can be realized by any of the means of geometric modeling.

On the basis of the combinatorial models mentioned above, one can also obtain information on the universality of each of the geometric means of concept implementation.

Each of the models (form-forming concept - a geometric means, a geometric means - a compositional technique) is presented in two versions. The first is the indication of concrete examples from the design practice and the distinction of some of the empty cells - combinations that have not yet been addressed in the project materials but are possible to apply. Combinations, which today do not make sense, according to modern technologies, technological capabilities, and knowledge, remain empty. The second is combinatorial models, which show the combinations of features that are most commonly used in the design practice. Based on these data, we can draw conclusions about the universality and variability Volume 2, Issue 3, 2018 pages: 15-28 (24)

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of the use of a particular tool or concept.

Conformity of the formulating concept and the possibility of its realization by means of geometric modeling, which have not yet been used in the design of eco-design objects:

Just as in the previous combinatorial model, the analysis is conducted and combinations are indicated graphically that are most commonly used in the design practice. Below is a table of matching combination of geometric means - compositional method, using which it is possible to get a new design decision, not presented before.

Combination of geometric means and compositional reception possible to use:

Figure 20. Combinator model of analysis and generation of project decisions of eco-objects. GEOMETRIC MEANS FORMATIC CONCEPT. Examples from project practice.

4. Algorithmic scheme of modeling of eco-design objects as the basis of the corresponding decision-making system.

As a result of the dissertation research on the basis of analysis of examples from the design practice, systematization of data, development of the typology of objects of eco-architecture in terms of geometric schemes, constructive solutions and their interrelations, elaboration of a proposal for a new method of rational, multivariate use of ecological principles in the objects of different structure and purpose, the algorithm of modeling objects of eco-design as the basis of the decisionmaking system was developed.

Figure 21. Combinator model of analysis and generation of project decisions of eco-objects. GEOMETRIC MEANS FORMATIC CONCEPT. The most popular combinations.

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The design process is divided into four stages: the analytical stage of the object, design and engineering design, the stage of obtaining the finished project and the stage of analysis of the finished project in terms of its environmental friendliness. Figure 24.

Figure 24. Logic of spatial modeling of eco-object.

Figure 22. Combinator model of analysis and generation of project decisions of eco-objects. GEOMETRIC MEANS COMPOSITE RECEIPT. Examples from project practice.

Figure 23. Combinator model of analysis and generation of project decisions of eco-objects. GEOMETRIC MEANS COMPOSITE RECEIPT. The most popular combinations.

The analytical stage is a theoretical - conceptual level of design, in which, with the task of designing, the author begins analytic work. Based on the analysis of the natural environment and taking into account the typological and functional requirements of the customer, the designer chooses the priority factors of the environmental impact on the object, determines the terms of operation and construction of the object, chooses the concept of the operation of the object. The last step in the analytical stage of designing is the choice of the geometric principle of interaction between the object and the environment.

The stage of object, design and engineering designing is a direct working process of designing, in which, in addition to designers and architects, related professions are already involved: designers, engineers, technologists, as well as environmentalists and botanists. At this stage, first of all, the author sets geometry and constructive scheme that invariably affect one another, and therefore work on geometry and construction is carried out simultaneously and in parallel. After clarifying the geometry and design scheme, the author develops a typology according to the design task. When the architectural parts of the project are already mutually agreed and solved, the equipment will be equipped with engineering systems, technological equipment, technical solutions (here we will also take the decision on greening and other tasks).

The final stage in the process of object design is the selection of materials: bearing and enclosing structures, external and internal decoration.

Stage of the finished object is the third level of the algorithmic scheme. Ideally, a ready-made project should be a combination of making optimal decisions at each stage and for each design component.

To determine the level of environmental friendliness of an object and the effectiveness of its interaction with the surrounding environment, the fourth stage of the design-analysis of the finished object is required. In the analysis of the project, all its components are evaluated simultaneously, taking into account the environmental effects of the measures taken both separately and in the complex.

The first step in the analysis process is the definition of the principle: the concept of the functioning of the object and the geometric principle of the interaction of the object with the environment. The following is an analysis of constructive, engineering, typological, technological solutions. A Volume 2, Issue 3, 2018 pages: 15-28 (26)

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very important part of the analysis of the finished project is the construction and utilization, in which the utilization of both waste construction and utilization or renovation of the building after the expiration of its exploitation. And the last level of object analysis is the evaluation of the materials used.

Materials should be evaluated according to the parameters: • natural and environmentally friendly (non-toxic material for humans and nature), • missions and depletion of natural resources during production and utilization, • the durability of materials and expediency of their use.

If in the process of analysis of the finished project, it is not possible to obtain a balanced optimized object that meets all the environmental requirements at each stage of the design and requirements for the functioning of the building, then the work starts from the beginning, from the first stage of designing.

During the work on the dissertation, testing and introduction into practice of system methodology, decision-making system and appropriate methods of designing eco-architecture objects were conducted: a project of environmental reconstruction of the Lviv square and the complex of buildings of the Kiev Theatrical University. As a result of the application of combinatorial models, a new geometric model of the eco-object was obtained. [8] In the framework of the dissertation research, the system model with the definition of the concept of functioning and the formulating concept for the project ”Biotecton” was developed by Lazarev O.I. The object analysis was completed, eco-concept was supplemented, the form-forming concept was formulated, geometric means, compositional method, the type of constructive system was determined, identified and entered into the database of eco-objects.

Figure 26. Logic of spatial modeling of eco-object.

References

Figure 25. Logic of spatial modeling of eco-object.

[1]

Daniielian A. (2010). The essence of the implementation of the principles of environmental friendliness in projects of eco-architecture. The Interdepartmental Collection of Proceedings. The industrial art and design. 8, K .: KNUBA - P. 98-101. (in Ukrainian).

[2]

Daniielian A., Bilous S. Eco-Architecture: Definition, Concepts, Objectives and Schemes of Realization. The Interdepartmental Collection of Proceedings. The industrial art and design. 9, - K .: KNUBA - P. 78-88. ISBN 5-8238-0731-7 (in Ukrainian).

[3]

Ploskiy V. PhD, Daniielian A. (2011). Geometric principles in the formatting of the ecology architecture’s objectsSciences. Special Issue of the Proceedings of the Taurian State Agro-Technical University, Issue 4: Example on Geometry and Engineering Graphics, Vol. 50, Melitopol, 2011. - P. 18-22. ISSN 2078-0877. (in Ukrainian).

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[4]

[5]

[6]

[7]

Daniielian A. Guidelines for implementing the methods of formation of concepts of eco-architecture. The Interdepartmental Collection of Proceedings. The industrial art and design. 10, - K .: KNUBA. P. 42-50. ISSN 2221-9293 (in Ukrainian).

[17]

Al-Kodmany Kheir. (2016). New suburban: Sustainable tall building development. New York: Routledge. 283 p.

[18]

Al-Kodmany Kheir (2015). Eco towers: Sustainable cities in the sky. Chicago: WIT Press. 483 p. ISBN-10: 1784660175

Daniielian A. (2013). Geometric model as a means of selecting the priority environmental factors taken into account during the design process of eco-architecture objects. Materials of the 3rd International Scientific and Practical Conference April 9-10, 2013, Bryansk, ”Problems of Innovative Biosphere-Compatible Socially -economic development in the construction, housing and communal and road complexes ”. BGITA. Volume 2. P. 26-32. ISBN 978-5-98573-138-5 (in Russian).

[19]

Riba Hicham. Al-Kodmany Kheir. (2014). Innovative approaches to sustainable design: The riba green city model. Amazon Digital Services LLC. 241 p. ASIN: B00HUV525K

[20]

Gissen David. (2003). Big and Green: Toward Sustainable Architecture in the 21st Century. New York: Princeton Architectural Press. 196 p. ISBN-10: 1568983611

[21]

Lynn S. Beedle., Lynn S. Beedle, Mir M. Ali, Paul J. Armstrong. (2007). Skyscraper and the City Design, Technology and Innovation, Book 1. NY: Edwin Mellen Press.

750 p. ISBN-10: 0773453334

[22]

Wells Matthew. (2005). Skyscrapers: Structure and Design. London: Laurence King Publishing. 791 p. ISBN10: 0300106793

[23]

Howard . Mumford L., Leopold A., Jacobs J., . McHarg Ian L , Andre Gunder Frank (Author), Herman E. Daly (Author), and 3 more. (2004). Routledge: Taylor and Francis Group. 1st edition. 392 p.

Daniielian A. (2016). Decision-making system for ecoobjects designing. Scientific and technical collection ”Modern problems of architecture and urban planning”, issue 45. Kyiv: KNUBA. P. 36-44. (in Ukrainian).

Ploskiy V. PhD, Daniielian A. (2017). Some features and principles of formation conceptual eco-design apparatus.

The Interdepartmental Collection of Proceedings. The industrial art and design. 13, - K .: KNUBA. P. 130-135.

ISSN 2221-9293 (in Ukrainian).

[8]

Daniielian A. Implementation of the Systems Engineering Methodology of Eco-Design Object: Ecological Reconstruction of Lviv Square. (2018). Scientific and technical collection “Urban and territorial planning” issue 66. Kyiv: KNUBA. P. 123-133. ISSN 2076-815X.

[9]

Daniielian A. (2018). Combinatorial models for analysis and generation of eco-objects geometry. Scientific and technical journal “Energy-efficiency in civil engineering and architecture”, issue 10 Kyiv: KNUBA. P. 85-95 (in Ukrainian).

[10]

Myhailenko V., Yakovlev N. (2017). Basics of composition. Textbook. - K.: Caravela. P. 304 (in Ukrainian).

[11]

Golubeva O. L. (2004). Composition basics. textbook. 2nd edition - M .: Art. P.120. (in Russian).

[12]

Ikonnikov A., G. Stepanov. (1971). Fundamentals of architectural composition. Tutorial. M: Art. P. 225. (in Russian).

[13]

Bergman David. (2012). Sustainable Design: A Critical Guide (Architecture Briefs). NY: Princeton architectural press. 144 p.

[14]

Williams Daniel E., Orr David W., Watson Donald.

(2007). Sustainable Design: Ecology, Architecture, and Planning. Hoboken, New Jersey: John Wiley and Sons.

320 p.

[15]

Yeang Ken. (2007). Eco skyscrapers. Ken Yeang,Ivor Richards. Australia: The Images Publishing Group Pty Ltd. third edition. P. 165.

[16]

Engel Hiino. (2007). Bearing Systems - Moscow: AST: Astrel. P. 344. (in Russian).

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Volume 2, Issue 3, 2018 USEFUL online journal

Determination of support reactions of rod constructions obtained by morphogenesis.

Volodymyr Skochko 1* Abstract This article shows a method for determining the support reactions of rod architectural structures, the position of free nodes of which was previously obtained by discrete geometrical morphogenesis (shaping). At the same time, the order and principle of defining and forming all the necessary discrete data on the topological features of the model, its boundary conditions, external loads and the distribution of internal forces are considered. The algorithm is based on logical operations and matrix transformations.

Keywords rod structures, geometrical morphogenesis, discrete geometry.

1Kyiv

National University of Construction and Architecture.

*Corresponding author: E-mail: vladimir .and.friends@gmail.com, Address:Povitroflotsky Avenue, 31,Kyiv, 03680 Ukr aine Received:09/20/2018,Accepted:10/03/2018, Availableonline:10/07/2018.

R https://doi.org/10.32557/useful-2-3-2018-0005 This is an open access article under the CC BY-NC-ND license. Maintained by SVP4U

Contents 1

Introduction

1

2

Mathematical apparatus of morphogenesis

2

3

Generating data about model parameters.

2

3.1 Construction of matrix of correspondence of sequence numbers of rods and. . . . . . . . . . . . . . . .

3 3.2 Construction of matrix [ℵ] and vectors {g x }, {g y }, {g z }, {ℑ x }, {ℑ y } and {ℑ z }. . . . . . . . . . . . . . . 3 4

Determination of the coordinates of free nodes of the model.

8

5

Calculation of support reactions.

6

Examples of the calculations.

10

7

Conclusion

13

References

13

8

1. Introduction Most of the rod structures of coatings have a difficult configuration and are statically indeterminate due to the large number of supports and the specifics of their fixing. Therefore, the calculation of support reactions in order to determine the potential loads from these structures can become a challenge and require the use of modern numerical methods of complex modelling. If all internal forces in the rods (connections) that are attached to the supporting nods are known, then the determination of the values of the support reactions themselves is

reduced to the operation of adding the vector components of the internal forces in these rods. Thus, the search of support reactions again indicates the need to calculate the components of the stress-strain state of the entire structure. For the use of numerical methods and the corresponding software, it is necessary to specify all the geometric, physical and mechanical structure parameters, which, as a rule, requires considerable effort and time. However, if it is a question of pre-design works or sketch designing, the time spent by designers on the evaluation of non-final solutions may not be appropriate.

In this case, methods of geometrical morphogenesis can be useful. One of these methods is the static geometric method of discrete geometry [1], which allows realizing morphogenesis of the rod structure with pre-defined topological features and expected pattern of distribution of the density of internal forces in all rods. This method is applicable for momentless constructions, the rods of which work only on compression or tension. However, there is no unified methodology for the shaping of all necessary discrete data, relating to topological features, boundary conditions, external loads and the distribution of internal forces of the computational model. At the same time, the development of a systematic approach to the preparation of all these data is a very urgent task, since on the basis of the results of solving it, the tasks of shaping and determining support reactions can also be solved.

Determination of support reactions of rod constructions obtained by morphogenesis. — 2/14

2. Mathematical apparatus of morphogenesis

The matrix [â„‘] looks like this:

Morphogenesis of mechanical rod construction with hinged nodes is based on the system calculation of such coordinates of free nodes of the model, under which the construction will maintain the state of static equilibrium under the inuence of given loads [2, 3, 4]. In this case, there must be predeďŹ ned topological structural features, coordinates of the supporting (ďŹ xed) nodes, nodal loads−ℑ and parameters of conditional rigidity â„ľa,b that should be set as ratios of internal efforts in rods Ra,b to their lengths δ a,b :

ℾa,b = ℾb,a = Ra,b δa,b .

[â„‘] =

â„‘x

â„‘y

â„‘z

,

(6)

where: {ℑ x }, {ℑ y } and {ℑ z } – vectors of the external inuences components.

Taking into account the structure of the matrices [s], [g] and [ℑ], equality (3) can be written in the form of a system of 3 separate matrix equations: [ ℾ] ¡ { x} + { gx } + { ℑx } = 0

(7)

[ ℾ] ¡ { y} + { gy } + { ℑy } = 0

(8)

[ ℾ] ¡ { z} + { gz } + { ℑz } = 0

(9)

(1)

The parameters â„ľa,b according to [5] can also be called as density coefďŹ cients of internal forces. This name corresponds to the physical meaning of this quantity [6, 7, 8], and therefore in this paper we will use just such a deďŹ nition.

The system of equilibrium equations for each free node will have the following form:

3. Generating data about model parameters.

m

∑ (si − sa ) ¡ â„ľa,i + â„‘s a = 0,

(s = x, y, z),

(2)

i=1

where: s – generalized coordinate designation (x, y and z); â„‘s a – projections of vector of the inuence−ℑ in ath node (â„‘x a , â„‘y a and â„‘z a ); m – amount of nodes, adjacent to ath node.

In matrix form, the complete system of equations of static equilibrium of all free nodes of the construction will be as follow: [ ℾ] ¡ [ s] + [ g] + [ ℑ] = 0,

(3)

where: [s] – matrix of coordinates (with dimension k×3 for 3-dimensional case, where k – amount of free nodes of the system); [g] – matrix of boundary conditions (with dimension k×3 for 3-dimensional case); [ℑ] – matrix of external inuences (with dimension k×3 for 3-dimensional case); [ℾ] – force density matrix of rod structure (with dimension k×k).

The matrix [s] has the form:

[ s] =

x

y

z

,

(4)

where: {x}, {y} and {z} – coordinate vectors of the nodes.

The matrix [g] has the following form:

[g] =

gx

gy

gz

,

(5)

where: {g x }, {g y } and {g z } – vectors of the boundary conditions.

As a rule, for the formation of matrix identities (3) or (7) – (9), researchers or engineers resort to a series of logical reasoning, as a result of which equations of the type (2) can be constructed. And only after this constructing can be built matrix identities (3) or (7) – (9). In doing so, it is necessary to rely on a priori information about which nodes are free, and which are supporting (ďŹ xed), as well as the order and number of rod connections. Let‘s analyze all these reasoning and write them down in the form of mathematical expressions and operations.

First of all, in order to form a system of equations of the type (2), it is necessary to construct a discrete image, that will be equivalent from the topological point of view of the rod structure, the shape of which is planned to be found. We need this image for the template analysis of the boundary conditions of the model and for determining the number of unknown variables (coordinates of free nodes) in each equation of the type (2). In addition, the discrete image of the model must contain the numbers of all nodes.

At ďŹ rst glance, the formation of the vectors {â„‘ x }, {â„‘ y } and {â„‘ z }, does not cause any complications. However, the number of elements in these vectors, just as in the vectors {x}, {y} and {z}, is k and corresponds to the number of free nodes, while the total number of nodes (including support nodes) is q.

At the same time, in the process of software implementation of computational algorithms, it is much more convenient to generate coordinate and nodal load vectors at once for all model nodes, since in this case we can assign sequence numbers to the nodes and corresponding data simultaneously. It remains an open question, how to extract from the common vectors of all nodal data the vectors of only free nodes for equations (7) – (9).

Volume 2, Issue 3, 2018 pages: 29-42 (30)

Determination of support reactions of rod constructions obtained by morphogenesis. — 3/14

The next important question is how to number the rods of the model, and in what form it is necessary to store information about the order of connecting the nodes with rods? This information must be used to form the matrix [ℵ], the vectors {g x }, {g y } and {g z }, and, as a consequence, the matrices [g].

So, at the first stage we will single out two main tasks: 1. construction of a matrix of correspondence of the sequence numbers of rods and nodes; 2. the formation of the matrix [ℵ], vectors {g x }, {g y }, {g z }, {ℑ x }, {ℑ y } and {ℑ z }.

3.1 Construction of matrix of correspondence of sequence numbers of rods and.

Analyzing the topology of a discrete image, we construct its adjacency matrix [A]. This matrix will have the following form: ⎡ ⎢ ⎢ [A] = ⎢ ⎣

Ai, j =

0 A2,1 . ..

A1,2 0 . ..

··· ··· .. .

A1,q A2,q . ..

Aq,1

Aq,2

···

0

0, → i = j, 1 ∨ 0, → i 6= j,

0 Ahal f 2,1

⎢ ⎢ Ahal f = ⎢ .. ⎣ .

Ahal f q,1

0 0 ...

⎥ ⎥ ⎥, ⎦

(10)

(i = 1, q; j = 1, q). (11)

Ahal f q,2

0, → i ≤ j, Ai, j , → i > j,

Ahal f i, j =

··· ··· .. . ···

q

0 0 .. .

0

⎥ ⎥ ⎥, ⎦

(12)

(i = 1, q; j = 1, q). (13)

q

i=1 j=1

0 0 .. .

··· ··· .. .

0 0 .. .

Ahal f .numq,2

···

0

0

⎢ ⎢ Ahal f .num2,1 Ahal f .num = ⎢ .. ⎣ .

Ahal f .numq,1

⎤ ⎥ ⎥ ⎥ , (15) ⎦

Ahal f .numi, j = ( =

0, → Ahal fi, j = 0, j ∑im=1 ∑n=1 Ahal fm,n, , → Ahal fi, j = 1, (i = 1, q; j = 1, q). (16)

max(Ahal f .num1, j ) = h, (i = 1, q; j = 1, q).

(14)

(17)

Using matrix [Ahal f .num ], we construct the matrix of the correspondence of sequence numbers of rods and nodes [N], with dimension h×3. The first column of the matrix [N] will include the list of rod numbers in ascending order. The second column of the matrix [N] will include the numbers of beginning nodes of the rods. The third column of the matrix [N] will include the numbers of end nodes of the rods. This matrix will have the following form: ⎡ ⎢ ⎢ [N] = ⎢ ⎣

⎤

Since the positive elements of matrix [A] indicate nodes, connected by rods, the order of joining nodes can be determined equally by the elements below and above the diagonal, since the incidence matrix is symmetric. Consequently, the matrix [Ahal f ] contains half of positive elements of the matrix [A], and the sum of the elements of the matrix [Ahal f ] corresponds to the number of rods of the model h:

h = ∑ ∑ Ahal f i, j .

⎡

Obviously, the maximal member of the matrix [Ahal f .num ] is equal to h:

⎤

The symbol ”→” means the logical consequence ”if” (implication). The symbol ”∨” means the operation ”or” (disjunction).

We replace the ”0” elements of the matrix [A], which are above the main diagonal. And we get the matrix [Ahal f ]: ⎡

Let’s replace the positive elements of the [Ahal f ] matrix by sequence numbers, so that the number of the first node (beginning) of each rod will be lower than the number of its end (the second node). We get the matrix [Ahal f .num ], which will have the following form:

N1,1 N2,1 ...

N1,2 N2,2 . ..

N1,3 N2,3 . ..

Nh,1

Nh,2

Nh,3

⎤

⎡

⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎦ ⎣

1 N1,2 2 N2,2 .. ... . h Nh,2

N1,3 N2,3 ...

⎤ ⎥ ⎥ ⎥ , (18) ⎦

Nh,3

Nm,n = ⎧ ⎨ Ahal f .numi, j , → (m = Ahal f .numi, j > 0) ∧ (n = 1), j, → (m = Ahal f .numi, j > 0) ∧ (n = 2), = ⎩ i, → (m = Ahal f .numi, j > 0 ) ∧ (n = 3), (m = 1, h; n = 1, 3; i = 1, q; j = 1, q). (19) The symbol ”∧” means the operation ”and” (conjunction).

In fact, the matrix [N] allows us to go from two-character to one-character indexation of the density coefficients ℵ of internal forces, using as indexes a numbers from its first column.

3.2 Construction of matrix [ℵ] and vectors {g x }, {g y }, {g z }, {ℑ x }, {ℑ y } and {ℑ z }.

Having matrix [ℵ], we can form the vector {ℵ}, containing h density coefficients of internal forces with single-symbol Volume 2, Issue 3, 2018 pages: 29-42 (31)

Determination of support reactions of rod constructions obtained by morphogenesis. — 4/14

indexation. The vector {ℵ} will contain the values of the internal force density coefficients, and determine the initial position of the model nodes as a result of the shaping. Taking into account (20) and (21), the vector {ℵ} will have the following form:

We replace the elements of the numbered adjacency matrix [Anum ], on the corresponding by numbers of the cells values of the vector {ℵ}. We get the matrix [A0 ], which will look like this:

⎡ { ℵ} T = { ℵN1,1

ℵN2,1

... ℵNh,1 } =

= { ℵ1

ℵ2

{ F0.x } T = { F0.x 1

F0.x 2

... F0.x q } ,

(21)

{ F0.y } T = { F0.y 1

F0.y 2

... F0.y q } ,

(22)

A0 i, j =

{ F0.z } = { F0.z 1

F0.z 2

... F0.z q } .

(23)

F0.s i = 0 ∨ ℑs i , (i = 1, q; s = x, y, z).

⎢ Anum2,1 ⎢ [Anum ] = ⎢ .. ⎣ .

Anumq,1

Anum1,2 0 .. .

Anumq,2

··· ··· .. . ···

Anum1,q Anum2,q ...

0

⎥ ⎥ ⎥ , (27) ⎦

Using matrix [A0 ], we construct a diagonal matrix [Adiag ], whose elements contain negative sums of elements of the corresponding rows of the matrix [A0 ]. In fact, in each element of the diagonal matrix [Adiag ] will be placed the sum of the coefficients of the force density of the rods, incident to each node. This matrix will have the following form: Adiag1,1 0 .. .

0

⎢ ⎢ Adiag = ⎢ ⎣

0 Adiag2,2 .. .

0

··· ··· .. . ···

0 0 ...

⎤ ⎥ ⎥ ⎥ , (29) ⎦

Adiagq,q

0, → i= 6 j, q ∑m=1 A0i,m , → i = j, (i = 1, q; j = 1, q). (30)

We construct the full matrix of force density coefficients of the rod structure [A], summing the matrices [A0 ] and [Adiag ]:

[A ℵ ] = [A 0 ] + Adiag .

⎤ ⎥ ⎥ ⎥ ⎦

Adiagi, j =

(25)

→ i = j, → (i > j) ∧ (Ahal f .numi, j → (i > j) ∧ (Ahal f .numi, j → (i < j) ∧ (Ahal f .num j,i → (i < j) ∧ (Ahal f .num j,i

> 0), = 0), > 0), = 0),

(i = 1, q; j = 1, q). (26)

(31)

Besides this method, it is possible to construct matrix [A], using following formula [5]: [A ℵ ] = − [IO ] · [D] · [IO ] T ,

Anumi, j = ⎧ 0, ⎪ ⎪ ⎪ ⎪ A ⎨ hal f .numi, j , 0, = ⎪ ⎪ A ⎪ hal f .num j,i , ⎪ ⎩ 0,

⎤

(i = 1, q; j = 1, q). (28)

0

A0 1,q A0 2,q .. .

0

(24)

Let’s construct a symmetric matrix [Anum ]. This matrix should include all elements of the matrix [Ahal f .num ] and symmetrically equal to them relative to the main diagonal. The matrix [Anum ] should be an adjacency matrix, whose nonzero cells will contain the numbers of the rods, connecting the nodes, instead of ”1” (ones). This matrix will have the following form: ⎡

··· ··· .. . ···

0, → Anumi, j = 0, ℵm , → (m = Anumi, j ) ∧ (Anumi, j > 0),

⎡ T

A0 1,2 0 .. .

A0 q,2

⎢ A0 2,1 ⎢ [A0 ] = ⎢ . ⎣ ..

A0 q,1

... ℵh } . (20)

Also, the external influence vectors {F 0.x }, {F 0.y } and {F 0.z } must be previously formed for all q (free and reference) nodes. If the value of the cell of the vector corresponds to a constant number or is written in the form of some function, then the node is free. At this stage, assume that if the node is a support node, then the value of the cell of the vector is ”0”, but in general the cell value should be equal to the vector components of the suport reactions. Vectors {F 0.x }, {F 0.y } and {F 0.z } will have the following form:

0

(32)

where: [IO ] – is the incidence matrix of the discrete image of the model as an oriented graph, and [D] – the diagonal matrix of the force density coefficients.

To construct the matrix [IO ], first of all, using the matrix [Anum ], we construct the incidence matrix [I] of the discrete image, assuming it to be an undirected graph. For each cell of matrix [I], the number of row should correspond to the number of the cell of matrix [Anum ], and the column number Volume 2, Issue 3, 2018 pages: 29-42 (32)

Determination of support reactions of rod constructions obtained by morphogenesis. — 5/14

then the node is free (unfixed). The vector {t} will have the following form:

should correspond to the value of the current cell of the matrix [Anum ]. The matrix [I] will have the following form: ⎡ ⎢ ⎢ [I] = ⎢ ⎣

Ii, j =

0, 1,

I1,1 I2,1 .. .

I1,2 I2,2 . ..

··· ··· .. .

I1,h I2,h .. .

Iq,1

Iq,2

···

Iq,h

→ →

{t}T = { t1

⎤ ⎥ ⎥ ⎥, ⎦

ti = 1 ∨ 0, (i = 1, q)

(i = m) ∧ ( j = 6 Anum m,n = 0), (m = 1, q; n = 1, q), (i = m) ∧ ( j = Anum m,n > 0), (m = 1, q; n = 1, q),

⎢ ⎢ [IO ] = ⎢ ⎣

IO 1,1 IO 2,1 .. .

IO q,1

··· ··· .. . ···

IO 1,2 IO 2,2 .. .

IO q,2

(i = 1, q; j = 1, h).

(34)

IO 1,h IO 2,h .. .

IO q,h

⎤ ⎥ ⎥ ⎥, ⎦

(35)

{ Nnods }T = { Nnods 1

Nnods i = i, (i = 1, q).

The matrix [D] can be constructed using vector {ℵ}, and will have the following form:

⎢ ⎢ [D] = ⎢ ⎣

0 D2,2 ...

0

0 ⎡ ⎢ ⎢ =⎢ ⎣

Di, j =

··· ··· .. .

0 0 . ..

···

Dh,h

⎤ ⎥ ⎥ ⎥= ⎦

ℵ1 0 . ..

0 ℵ2 . ..

··· ··· .. .

0 0 . ..

0

0

···

ℵh

0, → i 6= j, ℵi , → i = j,

(42)

We‘ll create one more vector {N f ree.nods }, in which we leave the numbers of only free nodes, replacing the sequence numbers of the support nodes of the model by ”0”:

{ N f ree.nods }T =

(i = 1, q; j = 1, h). (36)

D1,1 0 ...

... Nnods q } =

Nnods 2

= { 1 2 ... q }, (41)

⎧ ⎨ 0, → (Ii, j = 0) ∧ (∑im−=11 Im, j ≥ 0), = 1, → (Ii, j = 1) ∧ (∑i−1 m=1 Im, j = 0), ⎩ −1, → (Ii, j = 1) ∧ (∑im−=11 Im, j > 0),

⎡

(40)

Now we create a vector {Nnods } of the sequence numbers of all nodes of the model:

= { N f ree.nods 1 IO i, j

(39)

(33)

Using matrix [I], we construct the matrix [IO ] for the discrete image of the model, as an oriented graph, in which the beginning and the end of each arc of the graph are known (in our case – of each rod). The matrix [IO ] will have the following form: ⎡

... tq },

t2

N f ree.nods 2

= { 1∨0

N f ree.nods i =

... N f ree.nods q } =

2 ∨ 0 ... q ∨ 0 } (43)

0, → ti = 0, Nnods i , → ti = 1,

(i = 1, q) (44)

At this stage, it is necessary to specify the coordinate vectors of all q nodes {X 0 }, {Y 0 } and {Z 0 }, establishing the exact coordinates of the supporting nodes and the approximate values of the free nodes: { X0 }T = { x1

x2

... xq },

(45)

{Y0 }T = { y1

y2

... yq },

(46)

{ Z0 }T = { z1

z2

... zq }.

(47)

⎤ ⎥ ⎥ ⎥ , (37) ⎦

(i = 1, h; j = 1, h) (38)

Let‘s introduce the data about which nodes are fixed and which are free. To do this, we form the vector {t}. If the value of the cell of the vector is ”0”, then the node is supporting (fixed). If the value of the cell of the vector is equal to ”1”,

We perform the ranking of the cells of the sequence number vector {N f ree.nods } in ascending order, and along with this ranking of the corresponding rows (in full composition) of the matrix [A], and of the elements of vectors {Nnods }, {F 0.x }, {F 0.y }, {F 0.z }, {X 0 }, {Y 0 } and {Z 0 }. We will designate the sorted (ranked) matrix and vectors as follows: [A/ ], {N / f ree.nods }, {N / nods }, {F / x }, {F / y }, {F / z }, {X / 0 }, Volume 2, Issue 3, 2018 pages: 29-42 (33)

Determination of support reactions of rod constructions obtained by morphogenesis. — 6/14

{Y / 0 } and {Z / 0 }, counting them as equal to the matrix and vectors [A], {N f ree.nods }, {Nnods }, {F 0.x }, {F 0.y }, {F 0.z }, {X 0 }, {Y 0 } and {Z 0 } at the beginning moment of the ranking. Mathematical algorithms for ranking these data can be written as follows:

/

⎫ ⎪ ⎪ ⎬

C2 = N f ree.nods j , /

/

N f ree.nods j = N f ree.nods j+1 , → ⎪ ⎪ ⎭ N = C2 . / f ree.nods j+1

→ N f ree.nods j > N f ree.nods j+1 , (i = 1, q − 1; j = 1, q − i), (48)

⎫ / ⎪ C6 = Fz j , ⎬ / / → Fz j = Fz j+1 , ⎪ ⎭ / Fz j+1 = C6 . → N f ree.nods j > N f ree.nods j+1 , (i = 1, q − 1; j = 1, q − i), (53) /

⎫ ⎪ ⎪ ⎬

C7 = X0 j , /

/

X0 j = X0 j+1 , / X0 j+1

→

⎪ ⎪ = C7 . ⎭

→ N f ree.nods j > N f ree.nods j+1 , (i = 1, q − 1; j = 1, q − i), (54) /

C1 = Aℵ j,m , /

/

Aℵ j,m = Aℵ j+1,m , /

Aℵ j+1,m = C1 .

⎫ ⎪ ⎪ ⎬

→

⎪ ⎪ ⎭

/

/

→ N f ree.nods j > N f ree.nods j+1 , (m = 1, q; i = 1, q − 1; j = 1, q − i), (49)

⎫ ⎪ ⎪ ⎬

C8 = Y0 j , /

Y0 j = Y0 j+1 , → ⎪ ⎭ Y / = C8 . ⎪ 0 j+1

→ N f ree.nods j > N f ree.nods j+1 , (i = 1, q − 1; j = 1, q − i), (55)

/

C3 = Nnods j , /

/

Nnods j = Nnods j+1 , /

Nnods j+1 = C3 .

⎫ ⎪ ⎪ ⎬

/

→

⎪ ⎪ ⎭

→ N f ree.nods j > N f ree.nods j+1 , (i = 1, q − 1; j = 1, q − i), (50)

C9 = Z0 j , /

/

Z0 = Z0 j+1 , / Z0 j+1

⎫ ⎪ ⎪ ⎬

→

⎪ ⎪ = C9 . ⎭

→ N f ree.nods j > N f ree.nods j+1 , (i = 1, q − 1; j = 1, q − i). (56)

/

Here the coefficients Ci – are the temporary constants, required in the intermediate stages of the calculus.

We select from the matrix [A/ ] the submatrix [A/ .K .g ] that doesn‘t include data on fixed model nodes:

⎫ ⎪ ⎪ ⎬

C4 = Fx j , / / Fx j = Fx j+1 , → ⎪ F / = C4 . ⎪ ⎭ x j+1

→ N f ree.nods j > N f ree.nods j+1 , (i = 1, q − 1; j = 1, q − i), (51)

⎫ / ⎪ C5 = Fy j , ⎪ ⎬ / Fy j = F / , y j+1 ⎪ → ⎪ ⎭ / Fy j+1 = C5 .

h i / Aℵ.K.g = ⎡ / Aℵ.K.g1,1 ⎢ / ⎢ A ⎢ ℵ.K.g2,1 =⎢ ⎢ ... ⎣ / Aℵ.K.gk,1 /

→ N f ree.nods j > N f ree.nods j+1 , (i = 1, q − 1; j = 1, q − i), (52)

/

/

···

Aℵ.K.g1,q

/

··· .. .

Aℵ.K.g2,q .. . / Aℵ.K.gk,q

Aℵ.K.g1,2 Aℵ.K.g2,2 . .. / Aℵ.K.gk,2

···

/

⎤

/

⎥ ⎥ ⎥ ⎥ , (57) ⎥ ⎦

Aℵ.K.gi, j = Aℵ i+q−k, j , (i = 1, k; j = 1, q).

(58)

Now we perform the ranking of the columns of the obtained matrix [A/ .K .g ] in ascending order of the values of the Volume 2, Issue 3, 2018 pages: 29-42 (34)

Determination of support reactions of rod constructions obtained by morphogenesis. — 7/14

cells of vector {N f ree.nods }. The resulting matrix is denoted as: [A// .K .g ]. The ranking must be performed by analogy with algorithms (48) – (56). This algorithm will have the following form: // C = Aℵ.K.gm, j , // // Aℵ.K.gm, j = Aℵ.K.gm, j+1 , // Aℵ.K.gm, j+1 = C.

⎫ ⎪ ⎪ ⎬

→

⎪ ⎪ ⎭

→ N f ree.nods j > N f ree.nods j+1 ,

{ ℑx } T = { ℑ x1

ℑx2

... ℑxk } ,

(68)

{ ℑy } T = { ℑy1

ℑy2

... ℑyk } ,

(69)

{ ℑ z } T = { ℑ z1

ℑz2

... ℑzk } ,

(70)

ℑsi = Fsi+q−k , (i = 1, k; s = x, y, z).

(71)

/

(m = 1, k; i = 1, q − 1; j = 1, q − i). (59)

Using the matrix [g] and the vectors {X 0 }, {Y 0 } and // {Z 0 }, we construct the matrices [Gx ], [Gy ] and [Gz ] as folFrom the obtained matrix [Aℵ.K.g ] we select the following submatrixes: [ℵ] – the matrix of the coefficients of the system lows: (3) equilibrium, and [gℵ] the matrix of coefficients of the boundary ⎡ ⎤ conditions: Gx1,1 Gx1,2 · · · Gx1,q−k ⎢ Gx2,1 Gx2,2 · · · Gx2,q−k ⎥ ⎢ ⎥ ⎡ ⎤ (72) [Gx ] = ⎢ . ⎥, .. .. ℵ1,1 ℵ1,2 · · · ℵ1,k ... ⎣ .. ⎦ . . ⎢ ℵ2,1 ℵ2,2 · · · ℵ2,k ⎥ ⎢ ⎥ Gxk,1 Gxk,2 · · · Gxk,q−k [ℵ] = ⎢ .

(60) .. .. ⎥ , .. ⎣ .. . . . ⎦ ⎡ ⎤ ℵk,1 ℵk,2 · · · ℵk,k G G ··· G //

ℵi, j = Aℵ.K.gi, j+q−k , (i = 1, k; j = 1, k);

⎡

gℵ1,1 gℵ2,1 . .. gℵ k,1

⎢ ⎢ [gℵ ] = ⎢ ⎣

gℵ1,2 gℵ2,2 . .. gℵk,2

··· ··· .. . ···

gℵ1,q−k gℵ2,q−k . .. gℵk,q−k

(61)

y1,1

y1,2

⎢ Gy2,1 ⎢ [Gy ] = ⎢ . ⎣ ..

Gyk,1

Gy2,2 ...

Gyk,2

··· .. . ···

Gy2,q−k . ..

Gyk,q−k

⎥ ⎥ ⎥, ⎦

⎡

Gz1,2 Gz2,2 ...

Gzk,2

··· ··· .. . ···

Gz1,q−k Gz2,q−k . ..

Gzk,q−k

⎤

⎤

Gz1,1 Gz2,1 ...

Gzk,1

⎥ ⎥ ⎥, ⎦

(62)

⎢ ⎢ [Gz ] = ⎢ ⎣

gℵi, j = Aℵ.K.gi, j , (i = 1, k; j = 1, q − k).

(63)

Gs i, j = gℵ i, j · sm ,

//

Also, from the vector {N / nods } we select the vectors of the numbers of free nodes {N K } and fixed nodes {N g}, and from the vectors {F / x }, {F / y }, {F / z } – vectors of the external influences {ℑ x }, {ℑ y } and {ℑ z }: {NK }T = { NK1

NK2

. . . NKk },

/

NKi = Nnodsi+q−k , (i = 1, k);

T

{Ng } = { Ng1

Ng2

. . . Ngq−k },

y1,q−k

⎥ ⎥ ⎥, ⎦

(73)

(74)

(m = Ng j ; i = 1, k; j = 1, q − k; s = x, y, z). (75) Summarize the elements of the matrices rows [Gx ], [Gy ] and [Gz ] and obtain the vectors of the boundary conditions {gx }, {gy } and {gz }:

(64)

{gx }T = { gx1

gx2

. . . gxk },

(76)

(65)

{gy }T = { gy1

gy2

. . . gyk },

(77)

{gz }T = { gz1

g z2

. . . gzk },

(78)

(66)

q−k / NKi = Nnodsi ,

(i = 1, q − k);

(67)

gsi =

∑ Gsi, j ,

(i = 1, k; s = x, y, z).

(79)

j=1

Volume 2, Issue 3, 2018 pages: 29-42 (35)

Determination of support reactions of rod constructions obtained by morphogenesis. — 8/14

4. Determination of the coordinates of free nodes of the model.

/

Having the matrix [ℵ], as well as the vectors {gx }, {gy }, {gz }, {ℑ x }, {ℑ y } and {ℑ z }, we determine the vectors {x}, {y} and {z} from systems (7) – (9), or the matrix [s] from the system (3). The corresponding solutions will look like this: { x} = [ ℵ]−1 · (−{ ℑx } − { gx }),

{Y / }T = {

{ x} = { x1

(80)

... xk };

x2

/

y f ix.2

· · · y f ix.q−k

y

···

/

f ree.2

= { Y1/

/

( / Yi

T

/

y f ix.1

=

/

/ yi−q+k = y f ree.i

(82)

/

/

... yk };

{ z} = [ ℵ]−1 · (−{ ℑz } − { gz }),

(84) /

Zi = { z} = { z1

... zk };

z2

/

[ s] = [ ℵ]−1 · (− [ ℑ] − [ g]).

(86)

Expressions (80), (82), and (84) or (86) are the solutions of the problem of morphogenesis for rod structures under the influence of given nodal loads.

5. Calculation of support reactions.

Having determined coordinates of the free nodes of the model, we can proceed to the calculation of the support reactions.

First, we need to specify the vectors {X / }, {Y / } and {Z / }, whose cells will contain the unordered coordinates of all q (free and support) nodes of the model. These vectors will have the following form: /

/

x f ix.1

x

/

f ree.1

( =

/

/

X0i = x f ix.i

/ xi−q+k = x f ree.i

x f ix.q−k

/ x f ree.2

= { X1/

/ Xi

/

···

x f ix.2

Z / = z f ix.i

⎩

/ zi−q+k = z f ree.i

/

0i

→

/

X2

···

}=

}=

/

z f ree.k

/ · · · Zq }, (91)

1 ≤ i ≤ (q − k),

(i = 1, k). (92) Here: s / f ix. i and s / f ree. i – the conditionally introduced notations for the coordinates of the supporting and free nodes in the vectors {X / }, {Y / } and {Z / }.

We perform the reverse sorting of the obtained vectors {X / }, {Y / } and {Z / }, using as a vector for ranking the vector {N / nods }, which contains the numbers at first fixed and then free nodes in ascending order. We will designate the sorted vectors as {Nnods }, {X }, {Y } and {Z }, counting them equal to vectors {N / nods }, {X / }, {Y / } and {Z / } at the beginning of the ranking. Mathematical algorithms for ranking these vectors can be (by analogy with algorithms (48) – (56)) written as follows: ⎫ C1 = Nnods j , ⎪ ⎬ Nnods j = Nnods j+1 , → ⎪ ⎭ Nnods j+1 = C1 . /

/ x f ree.k

/

Z2

···

→ (q − k + 1) ≤ i ≤ k,

(85)

or:

{ X / }T = {

⎧ ⎨

z f ix.q−k

z f ree.2

= { Z1/

(83)

(90)

/

···

z f ix.2 z f ree.1

T

1 ≤ i ≤ (q − k),

→ (q − k + 1) ≤ i ≤ k,

/

z f ix.1

}=

/ · · · Yq }, (89)

(i = 1, k);

{ Z / }T = { y2

/

Y2

(81)

{ y} = [ ℵ]−1 · (−{ ℑy } − { gy }),

{ y}T = { y1

/

y f ree.k

→

Y0i = y f ix.i

/

y f ree.1

/

→ Nnods j > Nnods j+1 , (i = 1, q − 1; j = 1, q − i), (93)

/ · · · Xq }, (87)

→

1 ≤ i ≤ (q − k),

→

(q − k + 1) ≤ i ≤ k, (i = 1, k); (88)

⎫ C2 = X j , ⎬ X j = X j+1 , → ⎭ X j+1 = C2 . /

/

→ Nnods j > Nnods j+1 , (i = 1, q − 1; j = 1, q − i), (94) Volume 2, Issue 3, 2018 pages: 29-42 (36)

Determination of support reactions of rod constructions obtained by morphogenesis. — 9/14

The matrix [F] has the following form: ⎫ ⎬

C3 = Y j , Y j = Y j+1 , → ⎭ Y j+1 = C3 . /

[F] =

/

→ Nnods j > Nnods j+1 , (i = 1, q − 1; j = 1, q − i), (95)

⎫ C4 = Z j , ⎬ Z j = Z j+1 , → ⎭ Z j+1 = C4 . /

/

→ Nnods j > Nnods j+1 , (i = 1, q − 1; j = 1, q − i).

Fx

Fy

Fz

,

(102)

where: {F x }, {F y } and {F z } – vectors of the influences components: { Fx } T = { Fx1

Fx2

... Fxq } ,

(103)

{ Fy } T = { Fy1

Fy2

... Fyq } ,

(104)

{ Fz } T = { Fz1

Fz2

... Fzq } ,

(105)

(96) As a result of ranking, the vector {Nnods } will get its original form (36) – (37), and the vectors {X }, {Y } and {Z } will have the following form: { X}T = { x1

x2

· · · xq },

(97)

{Y }T = { y1

y2

· · · yq },

(98)

{ Z}T = { z1

· · · zq }.

z2

Fs i = Psi ∨ ℑsi , (i = 1, q; s = x, y, z).

(106)

where: Ps i – designation of the projection of the supporting force of the ith node on the coordinate axes.

Taking into account the structure of the matrices [S], [F] and [Aℵ ], equality (3) can be written in the form of a system of 3 separate matrix equations: [A ℵ ] · {X} + {Fx } = 0,

(107)

[A ℵ ] · {Y } + {Fy } = 0,

(108)

[A ℵ ] · {Z} + {Fz } = 0.

(109)

(99)

Now we return to the analysis of the static equilibrium of the model. Before that, we considered the state of static equilibrium of only free nodes. However, if we replace the fixed nodes by free ones with the supporting reactions−Pi , applied to them, then the entire rod system will remain in the equilibrium state in the future. In this case, for fixed nodes, we can also build the equations of equilibrium of the type (2).

Then, for supporting and free nodes, we can build a system of q such equations. In the matrix form, the corresponding system can be written as follows:

It is obvious, that in order to determine the values of the of the support reactions components in the fixed nodes the model and of external load vectors (which were specified as the formative ones), the equations (100) or (107) – (109) must be solved with respect to the matrix [F] or the vectors {F x }, {F y } and {F z }, respectively. Such solving will have the following form: [F] = − [A ℵ ] · [S] ,

(110)

or: [A ℵ ] · [S] + [F] = 0,

(100)

where: [S] – matrix of coordinates of all q nodes of the model (with dimension q×3); [F] – matrix of influences on all q nodes of the model (with dimension q×3); [Aℵ ] – complete force density matrix of rod structure (with dimension q×q), which will be determined in accordance with (33).

The matrix [S] has the form: [ S] =

X

Y

Z

,

(101)

where: {X }, {Y } and {Z } – coordinate vectors of the nodes (92) – (94).

[Fx ] = − [A ℵ ] · {X},

(111)

[Fy ] = − [A ℵ ] · {Y },

(112)

[Fz ] = − [A ℵ ] · {Z}.

(113)

The absolute values of the support reactions, as well as the vectors of external forces, should be determined by the formula: Fi = (Fx2i + Fy2i + Fz2i ) 1/2 , (i = 1, q − k).

(114)

Volume 2, Issue 3, 2018 pages: 29-42 (37)

Determination of support reactions of rod constructions obtained by morphogenesis. — 10/14

6. Examples of the calculations.

Let’s consider examples of calculation of support reactions of the momentless rod structure after its shaping. As an object of research, we will consider a triple-bearing spatial frame, whose topological equivalent in the projection to the coordinate plane XOY is shown in Figure (1). The model has 11 nodes, of which 8 are free, and 3 are supporting and represented by fixed hinges. Therefore, the system is statically indeterminate.

[N] T =

h

1 1 2

2 1 7

3 1 8

4 1 9

5 2 8

6 2 9

7 3 4

8 3 7

9 3 8

10 3 10

11 4 8

12 4 10

13 5 6

14 5 7

15 5 8

16 5 11

17 6 8

18 6 11

19 7 8

(116) The numbering of the rods in Figure 1 was made on the basis of the matrix (115).

The exact coordinates of the model supporting nodes and the initial coordinates of the free nodes in their arbitrary first approximation (these data actually form the vectors {X 0 }, {Y 0 } and {Z 0 }) are presented in Table 1. Also Table (1) contains data about the which nodes are fixed, and which are free, as a vector {t} in the form (39) and (40).

Table 1. The initial coordinates of the model nodes and data

about their fixing. doi:10.6084/m9.figshare.7177145

Figure 1. The topological equivalent of the model in the projection on the coordinate plane XOY. doi:10.6084/m9.figshare.7177136

Using the scheme in Figure 1, we construct the adjacency matrix of the model [A] in the form (10) and (11): ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ [A] = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0 1 0 0 0 0 1 1 1 0 0

1 0 0 0 0 0 0 1 1 0 0

0 0 0 1 0 0 1 1 0 1 0

0 0 1 0 0 0 0 1 0 1 0

0 0 0 0 0 1 1 1 0 0 1

0 0 0 0 1 0 0 1 0 0 1

1 0 1 0 1 0 0 1 0 0 0

1 1 1 1 1 1 1 0 0 0 0

1 1 0 0 0 0 0 0 0 0 0

0 0 1 1 0 0 0 0 0 0 0

0 0 0 0 1 1 0 0 0 0 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ . (115) ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Using the matrix (115), we construct a matrix of correspondence of the sequence numbers of rods and nodes [N], with dimension 11×3 in the form (18) and (19):

Using matrix (116), it is possible to construct a vector {ℵ} of the internal force density coefficients of all 19 rods, on the basis of which the further shaping of the model will be performed. It should be remembered that if the sign of the coefficient of force density is ”+”, then after shaping the corresponding rod will work for tension, and if the sign of the force density coefficient is ”–”, then the rod will work for compression. Table (2) gives examples of different distributions of the force density coefficients, as well as the initial vectors of external nodal loads {F 0.x }, {F 0.y } and {F 0.z }, which determine different resulting forms and support reactions of the structure, respectively.

Example No. 1. Let’s consider the first of the examples presented in Table 2 in more detail. For this example, the vector {ℵ} will have the form:

{ ℵ} T = −{

5

3

1

4

2

2

5

3

1

4

2

2

5

3

1

4

(117) Here the sign ”–” is set deliberately, so that all the rods of the model after shaping work only on compression. Having made all the necessary transformations, we construct the matrix of force density coefficients of the rod structure [Aℵ ], using formulas (31) or (32). Regardless of the chosen way, the result is:

Volume 2, Issue 3, 2018 pages: 29-42 (38)

2

2

7

}

i

Determination of support reactions of rod constructions obtained by morphogenesis. — 11/14

Vectors {ℑ x }, {ℑ y } and {ℑ z }: ⎡ ⎢ ⎢ ⎢ [Aℵ ] = ⎢ ⎢ ⎣

13 −5 0 0 0 0 −3 −1 −4 0 0

−5 9 0 0 0 0 0 −2 −2 0 0

0 0 13 −5 0 0 −3 −1 0 −4 0

0 0 −5 9 0 0 0 −2 0 −2 0

0 0 0 0 13 −5 −3 −1 0 0 −4

−3 0 −3 0 −3 0 16 −7 0 0 0

0 0 0 0 −5 9 0 −2 0 0 −2

−1 −2 −1 −2 −1 −2 −7 16 0 0 0

−4 −2 0 0 0 0 0 0 6 0 0

0 0 −4 −2 0 0 0 0 0 6 0

0 0 0 0 −4 −2 0 0 0 0 6

⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎦

{ ℑx } T = { 0

0

0

0

0

0

0

0 },

(126)

{ ℑy } T = { 0

0

0

0

0

0

0

0 },

(127)

−3

0

−3

(118) The initial vectors of external nodal loads {F 0.x }, {F and {F 0.z } in the form (21)– (24) will look like this: { F0.x } T = { 0 0 0 0 0

0

0

0

0

0.y }

0 },

0

{ ℑz } T = { 0 −3

0

−4 }. (128)

0

Using the matrix and vectors (122) – (128), we define the vectors {x}, {y} and {z} by formulas (80), (82) and (84). We will get:

(119) { x}T = { 2 { F0.y } T = { 0 0 0 0 0

0

0

0

0

{ F0.z } T = { 0 −3 0 −3 0 −3 0 −4 0 0 0 }.

(121) The components of the identities (80), (82) and (84) will have the following form: Matrix [ℵ]:

⎢ ⎢ ⎢ ⎢ ⎢ [ℵ] = ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

13 −5 0 0 0 0 −3 −5 9 0 0 0 0 0 0 0 13 −5 0 0 −3 0 0 −5 9 0 0 0 0 0 0 0 13 −5 −3 0 0 0 0 −5 9 0 −3 0 −3 0 −3 0 16 −1 −2 −1 −2 −1 −2 −7

⎤ −1 −2 ⎥ ⎥ −1 ⎥ ⎥ −2 ⎥ ⎥. −1 ⎥ ⎥ −2 ⎥ ⎥ −7 ⎦ 16 (122)

{gy }T = { −4

{gz }T = { 0

−2

−20

−2

0

−6

0

0

4

3

3

3 },

3

(129)

{ y}T = { 1.75 1.75 2 2 3.75 3.75 2.5

{ z}T = {

0.6244

0.9179

0.6244

0.9179

0.6244

0.9179

2.5 }, (130)

}.

(131)

0.8192

1.0697

With vectors {x}, {y} and {z}, we construct the vectors of all coordinates of the model {X}, {Y} and {Z} after the formation. They will have the following appearance: { X}T = { 2

2

4

4

3

3

3

3

1

3 }, (132)

5

{Y }T = { 1.75 1.75 2 2 3.75 3.75 2.5

2.5

1

0.6244

0.9179

0.6244

0.9179

0.6244

0.9179

0.8192

1.0697

0

(134) −10 −12 −6

0

−3

0 0 }, (124)

0

−20

0

0

−10

0 }.

0 }, (123)

(125)

5 },

1.5

(133)

{ Z}T = {

Vectors {gx }, {gy } and {gz }: {gx }T = { −4

4

0 },

0

(120)

⎡

2

Using the vectors {X }, {Y } and {Z }, as well as the matrix [ℵ], determine the necessary vectors of the support reactions and the nodal loads {F x }, {F y } and {F z } by the formulas (111) – (113): { Fx }T = { 0

{ Fy }T = { 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

4.5

−6

6

3

0 }, (135)

−7.5 },

Volume 2, Issue 3, 2018 pages: 29-42 (39)

0

0

}.

Determination of support reactions of rod constructions obtained by morphogenesis. — 12/14

(136)

the results of the calculations are contained in Tables (2) and (3) (Variant No. 2).

Example No. 3. This example demonstrates the shaping and results of calculations of support reactions of similar in { Fz }T = { 0 −3 0 −3 0 −3 0 −4 4.3333 4.3333 4.3333 }. topological features construction, in which the rods of the upper belts do not work for compression but for tension. In (137) this case, the signs of the density coefficients of the rods of Taking into account the data in Table 1, the components the upper belts and the central riser are changed to positive, of the vectors of the support reactions are located in the last instead of negative (see Table 2). The resulting shape of the three cells of the vectors (135) – (137). Vectors of the support rod structure is shown in Figure 4, and the corresponding reactions will be as follows: results of the calculations are contained in Tables (2) and (3) (Variant No. 3).

Obviously, it is possible to use other approaches for determining the support reactions of complex rod structures, which { F9 }T = { Fx 9 Fy 9 Fz 9 }T = { 6 4.5 4.3333 }, from the point of view of mechanics are statically indeter(138) minate systems. However, in most cases these approaches require the re-construction of models of already formed structures in the environment of computer programs that work T T on the basis of universal numerical modelling methods (in { F10 } = { Fx 10 Fy 10 Fz 10 } = { −6 3 4.3333 }, particular, variational methods of construction mechanics), (139) as well as preliminary calculations of internal forces in the rods [9, 10]. This, in turn, requires considerable time and labour resources, which is not always advisable, especially at T T { F11 } = { Fx 11 Fy 11 Fz 11 } = { 0 −7.5 4.3333 }. the stages of preliminary design and feasibility studies of the proposed design solutions.

(140) The absolute values of these reactions, according to formula (114), will be as follows:

F9 = (Fx2 9 + Fy2 9 + Fz2 9 ) 1/2 = = (62 + 4.52 + 4.33332 ) 1/2 = 8.6619, (141)

F10 = (Fx2 10 + Fy2 10 + Fz2 10 ) 1/2 = = ((−6)2 + 32 + 4.33332 ) 1/2 = 7.9861, (142)

F9 = (Fx2 9 + Fy2 9 + Fz2 9 ) 1/2 = = (0 + (−7.5)2 + 4.33332 ) 1/2 = 8.6619. (143) The model of the rod structure, constructed basing on the results of the calculations of Example No. 1, is shown in Figure (1), and the results of calculations are contained in Tables (2) and (3) (Variant No. 1).

Example No. 2. In this example, nodal loads are set not strictly vertical, but at an oblique angle to the structure, to simulate the morphogenesis of the model, which should resist not only the force of attraction, but also lateral (for example from wind) loads. At the same time, the density coefficients of internal forces have also been changed (see Table 2). The resulting shape of the rod structure is shown in Figure (3), and

Table 2. iants of initial conditions and results of coordinates

calculations. doi:10.6084/m9.figshare.7177148 Volume 2, Issue 3, 2018 pages: 29-42 (40)

Determination of support reactions of rod constructions obtained by morphogenesis. — 13/14

Table 3. The calculation results: vectors of support reactions

and external influences. doi:10.6084/m9.figshare.7177151

7. Conclusion Demonstrated examples show the correctness of all calculations and, as a consequence, the efficiency of proposed algorithm, regardless of the nature of external loads and the distribution of the internal forces density coefficients.

It should be noted, that this algorithm is simple enough from the point of view of software implementation and allows solving two problems at once: geometrical morphogenesis of the momentless rod structures and the mechanical calculation of the reactions of support attachments on the basis of the free nodes of the model, obtained in the defining of coordinates.

This makes the proposed algorithm valuable to both architects and construction designers.

On the one hand, this algorithm demonstrates the simplicity of using the static-geometric method of discrete geometry and can allow researchers and architects to experiment with various shaping normative loads. On the other hand, this algorithm makes it possible to accurately calculate the loads, that will be transferred by the generated statically indeterminate structures to underlying structural elements, regardless of the number of supports and the complexity of the topological configuration of the model. With that, the time, required for carrying out the calculations, is significantly reduced.

Figure 3. Rod structure, formed and calculated in accordance with the conditions No. 2 of Table (2). doi:10.6084/m9.figshare.7177157

Figure 4. Rod structure, formed and calculated in accordance with the conditions No. 3 of Table (2). doi:10.6084/m9.figshare.7177163 Figure 2. Rod structure, formed and calculated in accordance with the conditions No. 1 of Table (2). doi:10.6084/m9.figshare.7177154

Volume 2, Issue 3, 2018 pages: 29-42 (41)

Determination of support reactions of rod constructions obtained by morphogenesis. — 14/14

References [1]

S. Kovalev, ”The formation of discrete surface models of spatial architectural structures”. Dissertation of the Dr.Sc. Tech. Moscow, MAI, 348 p., 1986.

[2]

V. Skochko, L. Skochko, ”The Equation of State and Condition Parameters of the Mesh Structure Relationships”.

Grounds and Foundations. Vol. 34. Kyiv: KNUCA, pp.

47-57., 2013.

[3]

V. Skochko, ”Morphogenesis and Correction of Planar Rod Constructions with a Small Amount of Free Nodes”.

Polish Academy of Sciences. Lublin-Rzeszow. Motrol.

Vol. 17 (8), pp. 35-42., 2015.

[4]

P. Kulikov, V. Ploskiy, V. Skochko, ”The Principles of Discrete Modeling of Rod Constructions of Architectural Objects”. Polish Academy of Sciences. Lublin-Rzeszow.

Motrol. Vol. 16 (8), pp. 3-10., 2014.

[5]

R. Motro, ”Tensegrity: structural systems for the future”.

Kogan Page Science, UK and USA, 238 p., 2013.

[6]

K. Snelson, ”Continuous tension, discontinuous compression structures”. U.S. Patent No 3,169,611,16, Feb.1965.

[7]

D. G. Emmerich, ”Reseaux”, in Space Structures: A study of methods and developments in three-dimensional construction”. Proceedings of the International Conference on Space Structures, Guildford 1966, edited by R.M. Davies, Blackwell Scientific Publications, pp.1059-1072., 1967.

[8]

A. Franklin Graybill, ”Introduction to matrices with applications in statistics”. Wrels Worth Publishing Company, Inc., Belmont, California, 1969.

[9]

P. Marti, ”Theory of Structures: Fundamentals, Framed Structures, Plates and Shells”. John Wiley & Sons, Inc., Ernst & Sohn GmbH & Co. KG., XVI. 679 p., 2013.

[10]

I. Rabinovich, ”Fundamentals of Structural Mechanics of Rod Systems”. – M.: Gostroizdat, 1960.

Volume 2, Issue 3, 2018 pages: 29-42 (42)

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