FINITE ELEMENTS OF THE PLANE PROBLEM OF THE THEORY OF ELASTICITY WITH DRILLING DEGREES OF FREEDOM
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Abstract
Twelve new finite elements with drilling degrees of freedom have been developed: triangular and quadrangular elements based on a modified hypothesis about the value of approximating functions on the sides of the element, which made it possible to avoid dimensional instability when all rotation angles are zero; incompatible and compatible triangular and quadrangular elements which can have additional nodes on the sides. Approximating functions satisfy the following condition: the value of the rotational degree of freedom of a node is nonzero and equal to one only for one of them. Numerical examples illustrate estimated minimum orders of convergence for displacements and stresses. All created elements retain the existing symmetry of the design models.
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Karpilovskyi, V. (2020). FINITE ELEMENTS OF THE PLANE PROBLEM OF THE THEORY OF ELASTICITY WITH DRILLING DEGREES OF FREEDOM. International Journal for Computational Civil and Structural Engineering, 16(1), 48–72. https://doi.org/10.22337/2587-9618-2020-16-1-48-72
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REFERENCES
1. Novozhilov V.V. Osnovy nelineynoy teorii uprugosti [Foundations of the Nonlinear Theory of Elasticity]. Moskow, Gostekhteorizdat, 1948, 333 pages (in Russian).
2. Allman D.J. A compatible triangular element including vertex rotations forplane elasticity analysis. //Computers and structures, 1984, Vol. 19(1-2), pp. l-8. DOI: 10.1016/0045-7949(84)90197-4
3. Allman D.J. A quadrilateral finite element including vertex rotations for plane elasticity analysis. //International journal for numerical methods in engineering, 1988, Vol. 26(3), pp. 717-730. DOI: 10.1002/nme.1620260314
4. Cook R.D. On the Allman triangle and a related quadrilateral element. //Computers and structures, 1986, Vol. 22(6), pp. 1065-1067. DOI: 10.1016/0045-7949(86)90167-7
5. MacNeal R.H., Harder, R.L. A refined four-noded membrane element with rotational degrees of freedom. //Computers and structures, 1988, Vol. 28(1), pp. 75-84. DOI: 10.1016/0045-7949(88)90094-6
6. Karpilovskyi V.S. Novyy sovmestnyy chetyrekhugolnyy konechnyy element balki-stenki s vrashchatelnymi stepenyami svobody [New compatible quadrangular deep beam finite element with rotational degrees of freedom]. // Materials of the II International Scientific and Practical Conference “Suchasni` metodi i problemno-oriyintovani kompleksi rozrakhunku konstrukczij i yikh zastosuvannya u proyektuvanni i navchalnomu proczesi” [“Modern Methods and Problem-Oriented Complexes for Structural Analysis and Their Application in Design and Educational Process”]. Kyiv, september 2018, pp. 57-59 ( in Russian)
7. Wilson E.L., Ibrahimbegovic A. Thick shell and solid elements with independent rotation fields. //International journal for numerical methods in engineering, 1991, Vol. 31(7), pp. 1393-1414. DOI: 10.1002/nme.1620310711
8. Zhang H., Kuong J.S. Eight-node membrane element with drilling degrees of freedom for analysis of in-plane stiffness of thick floor plates. //International journal for numerical methods in engineering, 2008, Vol. 76(13). pp. 2117-2136. DOI: 10.1002/nme.2395
9. Yunus S., Saigal S., Cook R. On improved hybrid finite elements with rotational degrees of freedom. //Interntional journal for numerical methods in ingineering, 1989, Vol. 28(4), pp. 785-800. DOI: 10.1002/nme.1620280405
10. Choo Y.S., Choi N., Lee B.C. Quadrilateral and triangular plate elements with rota-tional degrees of freedom based on the hybrid Trefftz method. //Finite Elements in Analysis and Design, 2006, Vol. 42(11), pp. 1002-1008. DOI: 10.1016/j.finel.2006.03.006
11. Cook R. A plane hybrid element with rotational d.o.f. and adjustable stiffness. //International Journal for Numerical Methods in Engineering, 1987, Vol. 24(8), pp. 1499-1508. DOI: 10.1002/nme.1620240807
12. Fellipa C.A. A study of optimal membrane triangles with drilling freedoms. //Computer Methods in Applied Mechanics and Engineering, 2003, Vol. 192(16-18), pp. 2125-2168. DOI: 10.1016/S0045-7825(03)00253-6
13. Chen X.M., Cen S., Sun J.Y., Li, Y.G. Four-Node Generalized Conforming Membrane Elements with Drilling DOFs Using Quadrilateral Area Coordinate Methods. //Mathematical Problems in Engineering, 2015, Vol. 3-4, pp. 1-13. DOI: 10.1155/2015/328612
14. Britvin E.I., Peysin, A., Eisenberger, M. Deformiruyemyy v svoyey ploskosti chetyrekhugolnyy konechnyy element s vrashchatelnymi stepenyami svobody v uzlakh. Chast 2. Proizvolnyy chetyrekhugolnik [Quadrangular finite element deformable in its plane with rotational degrees of freedom in the nodes. Part 2. Arbitrary quadrangle]. // Structural Mechanics and Analysis of Constructions, 2018, 4, pp. 50-54. (in Russian)
15. Karpilovskyi V.S. Issledovanie i konstruirovanie nekotorykh tipov konechnykh elementov dlya zadach stroitel`noj mekhaniki. [Study and design of some types of finite elements for the structural mechanics problems]. // PhD diss., National transport university(KADI), Kyiv, 1982, 179 pages (in Russian)
16. Evzerov I.D. Oczenki pogreshnosti po peremeshheniyam pri ispol`zovanii nesovmestny`kh konechny`kh e`lementov [Estimates of the error in displacements when using incompatible finite elements] // Chislennye metody mekhaniki sploshnoj sredy [Numerical methods of continuum mechanics], Novosibirsk, 1983, Vol 14(5), pp. 24-31 (in Russian)
17. Gorodetsky A.S., Karpilovskyi V.S. Metodicheskiye pekomendatsii po issledovaniyu i konstpuipovaniyu konechnyx elementov [Methodological recommendations for the study and construction of finite elements]. Kyiv, NIIASS, 1981. 48 pages (in Russian).
18. Irons, B.M., Razzaque, A. Experience with the path test. // In: The Matematical Foundations of the Finite Element Method with Application to Partial Differential Equations, Academic Press, 1972, pp. 557-587.
19. Strang G., Fix G. An Analysis of the Finite Element Method. Prentice Hall, Englewood Cliffs, N.J., 1973.
20. Aubin J.P. Approximation of Elliptic Boundary-Value Problems. John Wiley & Sons, N.Y., 1972.
21. Macneal R.H., Harder R.L. A proposed standard set of problems to test finite element accuracy. // Finite elements in analysis and design, 1985, Vol. 1, pp. 3-20. DOI: 10.1016/0168-874X(85)90003-4
22. Rekach V.G. Rukovodstvo k resheniyu zadach po teorii uprugosti [Guide to Solving Problems of Elasticity]. Moscov, Vysha shkola, 1977, 216 pages (in Russian).
23. Kalmanok A.S. Raschet balok-stenok [Analysis of Deep Beams]. Moscov, Gosstroyizdat, 1956. 151 pages (in Russian).
24. Gorodetsky A.S., Karpilovskyi V.S. O svyazi metoda konechnykh elementov s variatsionno-raznostnymi metodami [On the relation between the finite element method and the variation difference methods]. Strength of Materials and Theory of Structures, 1974, Vol. 24, pp. 32 –42. (in Russian)
СПИСОК ЛИТЕРАТУРЫ
1. Новожилов В.В. Основы нелинейной теории упругости. – М.: Гостехтеориздат, 1948. – 333с.
2. Allman D J. A compatible triangular element including vertex rotations forplane elasticity analysis // Computers and structures, 1984, Vol. 19, pp. l-8. DOI: 10.1016/0045-7949(84)90197-4
3. Allman D.J. A quadrilateral finite element including vertex rotations for plane elasticity analysis // International journal for numerical methods in engineering, 1988, Vol. 26, pp. 717-730. DOI: 10.1002/nme.1620260314
4. Cook R.D. On the Allman triangle and a related quadrilateral element // Computeis andstructures. 1986, Vol. 22(6), pp. 1065-1067. DOI: 10.1016/0045-7949(86)90167-7
5. MacNeal R.H., Harder R.L. A refined four-noded membrane element with rotational degrees of freedom // Computers and structures, 1988, Vol. 28(1), pp. 75-84. DOI: 10.1016/0045-7949(88)90094-6
6. Карпиловский В.С. Новый совместный четырехугольный конечный элемент балки-стенки с вращательными степенями свободы. // Сборник трудов ІІ Международной научно-практической конференции «Сучасні методи і проблемно-орієнтовані комплекси розрахунку конструкцій і їх застосування у проектуванні і навчальному процесі» [Современные методы и проблемно-ориентированные комплексы расчета конструкций и их применение в проектировании и учебном процессе]. (Киев, 26-27 сентября 2018 года), Киев, 2018, с.57-59.
7. Wilson E.L, Ibrahimbegovic A. Thick shell and solid elements with independent rotation fields //International journal for numerical methods in engineering, 1991, Vol. 31(7), pp. 1393-1414. DOI: 10.1002/nme.1620310711
8. Zhang H., Kuong J.S. Eight-node membrane element with drilling degrees of freedom for analysis of in-plane stiffness of thick floor plates// International journal for numerical methods in engineering, 2008, Vol. 76(13), pp. 2117-2136. DOI: 10.1002/nme.2395
9. Yunus .M., Saigal S., Cook R.D. On improved hybrid finite elements with rotational degrees of freedom// Interntional journal for numerical methods in ingineering, 1989, Vol. 28, pp.785-800. DOI: 10.1002/nme.1620280405
10. Choo Y.S., Choi N., Lee B.C. Choo Y.S., Choi N., Lee B.C. Quadrilateral and triangular plate elements with rota-tional degrees of freedom based on the hybrid Trefftz method. // Finite Elements in Analysis and Design, 2006, Vol. 42(11), pp. 1002-1008. DOI: 10.1016/j.finel.2006.03.006
11. Cook R. A plane hybrid element with rotational d.o.f. and adjustable stiffness. // International Journal for Numerical Methods in Engineering, 1987, Vol. 24(8), pp. 1499-1508. DOI: 10.1002/nme.1620240807
12. Fellipa C.A. A study of optimal membrane triangles with drilling freedoms. //Computer Methods in Applied Mechanics and Engineering, 2003, Vol. 192(16-18), pp. 2125-2168. DOI: 10.1016/S0045-7825(03)00253-6
13. Chen X.M., Cen S., Sun J.Y., Li Y.G. Four-Node Generalized Conforming Membrane Elements with Drilling DOFs Using Quadrilateral Area Coordinate Methods. //Mathematical Problems in Engineering, 2015, Vol. 3-4, pp. 1-13. DOI: 10.1155/2015/328612
14. Бритвин Е.И., Пейсин А., Эйсенбергер М. Деформируемый в своей плоскости четырехугольный конечный элемент с вращательными степенями свободы в узлах. Часть 2. Произвольный четырехугольник. // Строительная механика и расчет сооружений, 2018, №4, c.50-54.
15. Карпиловский В.С. Исследование и конструирование некоторых типов конечных элементов для задач строительной механики. Диссертация на соискание ученой степени кандидата технических наук по специальности 01.02.03 – «Строительная механика». – Киев: Национальный транспортный университет (КАДИ), 1982, 179 с.
16. Евзеров И.Д. Оценки погрешности по перемещениям при использовании несовместных конечных элементов // Численные методы механики сплошной среды, Новосибирск, 1983, том 14, №5, c. 24-31.
17. Городецкий А.С., Карпиловский В.С. Методические рекомендации по исследованию и констpуиpованию конечных элементов. Киев: NIIASS, 1981. – 48с.
18. Irons B.M., Razzaque A. Experience with the path test. // In: The Matematical Foundations of the Finite Element Method with Application to Partial Differential Equations, Academic Press, 1972, pp. 557-587.
19. Стренг Г., Фикс Дж. Теория метода конечных элементов – М.: Мир, 1977. – 349 с.
20. Обен Ж.-П. Приближенное решение эллиптических краевых задач. – М.: Мир, 1977. – 384с.
21. Macneal R.H., Harder, R.L. A proposed standard set of problems to test finite element accuracy. // Finite elements in analysis and design, 1985, Vol. 1, pp. 3-20. DOI: 10.1016/0168-874X(85)90003-4
22. Рекач В.Г. Руководство к решению задач по теории упругости. М.: Высшая школа, 1977. – 216с.
23. Калманок А.С. Расчет балок-стенок. М.: Госстройиздат, 1956. – 151с.
24. Городецкий А.С., Карпиловский В.С. О связи метода конечных элементов с вариационно-разностными методами. // Сопротивление материалов и теория сооружений, вып. 24, Киев, Будiвельник, 1974, с.32 –42.
1. Novozhilov V.V. Osnovy nelineynoy teorii uprugosti [Foundations of the Nonlinear Theory of Elasticity]. Moskow, Gostekhteorizdat, 1948, 333 pages (in Russian).
2. Allman D.J. A compatible triangular element including vertex rotations forplane elasticity analysis. //Computers and structures, 1984, Vol. 19(1-2), pp. l-8. DOI: 10.1016/0045-7949(84)90197-4
3. Allman D.J. A quadrilateral finite element including vertex rotations for plane elasticity analysis. //International journal for numerical methods in engineering, 1988, Vol. 26(3), pp. 717-730. DOI: 10.1002/nme.1620260314
4. Cook R.D. On the Allman triangle and a related quadrilateral element. //Computers and structures, 1986, Vol. 22(6), pp. 1065-1067. DOI: 10.1016/0045-7949(86)90167-7
5. MacNeal R.H., Harder, R.L. A refined four-noded membrane element with rotational degrees of freedom. //Computers and structures, 1988, Vol. 28(1), pp. 75-84. DOI: 10.1016/0045-7949(88)90094-6
6. Karpilovskyi V.S. Novyy sovmestnyy chetyrekhugolnyy konechnyy element balki-stenki s vrashchatelnymi stepenyami svobody [New compatible quadrangular deep beam finite element with rotational degrees of freedom]. // Materials of the II International Scientific and Practical Conference “Suchasni` metodi i problemno-oriyintovani kompleksi rozrakhunku konstrukczij i yikh zastosuvannya u proyektuvanni i navchalnomu proczesi” [“Modern Methods and Problem-Oriented Complexes for Structural Analysis and Their Application in Design and Educational Process”]. Kyiv, september 2018, pp. 57-59 ( in Russian)
7. Wilson E.L., Ibrahimbegovic A. Thick shell and solid elements with independent rotation fields. //International journal for numerical methods in engineering, 1991, Vol. 31(7), pp. 1393-1414. DOI: 10.1002/nme.1620310711
8. Zhang H., Kuong J.S. Eight-node membrane element with drilling degrees of freedom for analysis of in-plane stiffness of thick floor plates. //International journal for numerical methods in engineering, 2008, Vol. 76(13). pp. 2117-2136. DOI: 10.1002/nme.2395
9. Yunus S., Saigal S., Cook R. On improved hybrid finite elements with rotational degrees of freedom. //Interntional journal for numerical methods in ingineering, 1989, Vol. 28(4), pp. 785-800. DOI: 10.1002/nme.1620280405
10. Choo Y.S., Choi N., Lee B.C. Quadrilateral and triangular plate elements with rota-tional degrees of freedom based on the hybrid Trefftz method. //Finite Elements in Analysis and Design, 2006, Vol. 42(11), pp. 1002-1008. DOI: 10.1016/j.finel.2006.03.006
11. Cook R. A plane hybrid element with rotational d.o.f. and adjustable stiffness. //International Journal for Numerical Methods in Engineering, 1987, Vol. 24(8), pp. 1499-1508. DOI: 10.1002/nme.1620240807
12. Fellipa C.A. A study of optimal membrane triangles with drilling freedoms. //Computer Methods in Applied Mechanics and Engineering, 2003, Vol. 192(16-18), pp. 2125-2168. DOI: 10.1016/S0045-7825(03)00253-6
13. Chen X.M., Cen S., Sun J.Y., Li, Y.G. Four-Node Generalized Conforming Membrane Elements with Drilling DOFs Using Quadrilateral Area Coordinate Methods. //Mathematical Problems in Engineering, 2015, Vol. 3-4, pp. 1-13. DOI: 10.1155/2015/328612
14. Britvin E.I., Peysin, A., Eisenberger, M. Deformiruyemyy v svoyey ploskosti chetyrekhugolnyy konechnyy element s vrashchatelnymi stepenyami svobody v uzlakh. Chast 2. Proizvolnyy chetyrekhugolnik [Quadrangular finite element deformable in its plane with rotational degrees of freedom in the nodes. Part 2. Arbitrary quadrangle]. // Structural Mechanics and Analysis of Constructions, 2018, 4, pp. 50-54. (in Russian)
15. Karpilovskyi V.S. Issledovanie i konstruirovanie nekotorykh tipov konechnykh elementov dlya zadach stroitel`noj mekhaniki. [Study and design of some types of finite elements for the structural mechanics problems]. // PhD diss., National transport university(KADI), Kyiv, 1982, 179 pages (in Russian)
16. Evzerov I.D. Oczenki pogreshnosti po peremeshheniyam pri ispol`zovanii nesovmestny`kh konechny`kh e`lementov [Estimates of the error in displacements when using incompatible finite elements] // Chislennye metody mekhaniki sploshnoj sredy [Numerical methods of continuum mechanics], Novosibirsk, 1983, Vol 14(5), pp. 24-31 (in Russian)
17. Gorodetsky A.S., Karpilovskyi V.S. Metodicheskiye pekomendatsii po issledovaniyu i konstpuipovaniyu konechnyx elementov [Methodological recommendations for the study and construction of finite elements]. Kyiv, NIIASS, 1981. 48 pages (in Russian).
18. Irons, B.M., Razzaque, A. Experience with the path test. // In: The Matematical Foundations of the Finite Element Method with Application to Partial Differential Equations, Academic Press, 1972, pp. 557-587.
19. Strang G., Fix G. An Analysis of the Finite Element Method. Prentice Hall, Englewood Cliffs, N.J., 1973.
20. Aubin J.P. Approximation of Elliptic Boundary-Value Problems. John Wiley & Sons, N.Y., 1972.
21. Macneal R.H., Harder R.L. A proposed standard set of problems to test finite element accuracy. // Finite elements in analysis and design, 1985, Vol. 1, pp. 3-20. DOI: 10.1016/0168-874X(85)90003-4
22. Rekach V.G. Rukovodstvo k resheniyu zadach po teorii uprugosti [Guide to Solving Problems of Elasticity]. Moscov, Vysha shkola, 1977, 216 pages (in Russian).
23. Kalmanok A.S. Raschet balok-stenok [Analysis of Deep Beams]. Moscov, Gosstroyizdat, 1956. 151 pages (in Russian).
24. Gorodetsky A.S., Karpilovskyi V.S. O svyazi metoda konechnykh elementov s variatsionno-raznostnymi metodami [On the relation between the finite element method and the variation difference methods]. Strength of Materials and Theory of Structures, 1974, Vol. 24, pp. 32 –42. (in Russian)
СПИСОК ЛИТЕРАТУРЫ
1. Новожилов В.В. Основы нелинейной теории упругости. – М.: Гостехтеориздат, 1948. – 333с.
2. Allman D J. A compatible triangular element including vertex rotations forplane elasticity analysis // Computers and structures, 1984, Vol. 19, pp. l-8. DOI: 10.1016/0045-7949(84)90197-4
3. Allman D.J. A quadrilateral finite element including vertex rotations for plane elasticity analysis // International journal for numerical methods in engineering, 1988, Vol. 26, pp. 717-730. DOI: 10.1002/nme.1620260314
4. Cook R.D. On the Allman triangle and a related quadrilateral element // Computeis andstructures. 1986, Vol. 22(6), pp. 1065-1067. DOI: 10.1016/0045-7949(86)90167-7
5. MacNeal R.H., Harder R.L. A refined four-noded membrane element with rotational degrees of freedom // Computers and structures, 1988, Vol. 28(1), pp. 75-84. DOI: 10.1016/0045-7949(88)90094-6
6. Карпиловский В.С. Новый совместный четырехугольный конечный элемент балки-стенки с вращательными степенями свободы. // Сборник трудов ІІ Международной научно-практической конференции «Сучасні методи і проблемно-орієнтовані комплекси розрахунку конструкцій і їх застосування у проектуванні і навчальному процесі» [Современные методы и проблемно-ориентированные комплексы расчета конструкций и их применение в проектировании и учебном процессе]. (Киев, 26-27 сентября 2018 года), Киев, 2018, с.57-59.
7. Wilson E.L, Ibrahimbegovic A. Thick shell and solid elements with independent rotation fields //International journal for numerical methods in engineering, 1991, Vol. 31(7), pp. 1393-1414. DOI: 10.1002/nme.1620310711
8. Zhang H., Kuong J.S. Eight-node membrane element with drilling degrees of freedom for analysis of in-plane stiffness of thick floor plates// International journal for numerical methods in engineering, 2008, Vol. 76(13), pp. 2117-2136. DOI: 10.1002/nme.2395
9. Yunus .M., Saigal S., Cook R.D. On improved hybrid finite elements with rotational degrees of freedom// Interntional journal for numerical methods in ingineering, 1989, Vol. 28, pp.785-800. DOI: 10.1002/nme.1620280405
10. Choo Y.S., Choi N., Lee B.C. Choo Y.S., Choi N., Lee B.C. Quadrilateral and triangular plate elements with rota-tional degrees of freedom based on the hybrid Trefftz method. // Finite Elements in Analysis and Design, 2006, Vol. 42(11), pp. 1002-1008. DOI: 10.1016/j.finel.2006.03.006
11. Cook R. A plane hybrid element with rotational d.o.f. and adjustable stiffness. // International Journal for Numerical Methods in Engineering, 1987, Vol. 24(8), pp. 1499-1508. DOI: 10.1002/nme.1620240807
12. Fellipa C.A. A study of optimal membrane triangles with drilling freedoms. //Computer Methods in Applied Mechanics and Engineering, 2003, Vol. 192(16-18), pp. 2125-2168. DOI: 10.1016/S0045-7825(03)00253-6
13. Chen X.M., Cen S., Sun J.Y., Li Y.G. Four-Node Generalized Conforming Membrane Elements with Drilling DOFs Using Quadrilateral Area Coordinate Methods. //Mathematical Problems in Engineering, 2015, Vol. 3-4, pp. 1-13. DOI: 10.1155/2015/328612
14. Бритвин Е.И., Пейсин А., Эйсенбергер М. Деформируемый в своей плоскости четырехугольный конечный элемент с вращательными степенями свободы в узлах. Часть 2. Произвольный четырехугольник. // Строительная механика и расчет сооружений, 2018, №4, c.50-54.
15. Карпиловский В.С. Исследование и конструирование некоторых типов конечных элементов для задач строительной механики. Диссертация на соискание ученой степени кандидата технических наук по специальности 01.02.03 – «Строительная механика». – Киев: Национальный транспортный университет (КАДИ), 1982, 179 с.
16. Евзеров И.Д. Оценки погрешности по перемещениям при использовании несовместных конечных элементов // Численные методы механики сплошной среды, Новосибирск, 1983, том 14, №5, c. 24-31.
17. Городецкий А.С., Карпиловский В.С. Методические рекомендации по исследованию и констpуиpованию конечных элементов. Киев: NIIASS, 1981. – 48с.
18. Irons B.M., Razzaque A. Experience with the path test. // In: The Matematical Foundations of the Finite Element Method with Application to Partial Differential Equations, Academic Press, 1972, pp. 557-587.
19. Стренг Г., Фикс Дж. Теория метода конечных элементов – М.: Мир, 1977. – 349 с.
20. Обен Ж.-П. Приближенное решение эллиптических краевых задач. – М.: Мир, 1977. – 384с.
21. Macneal R.H., Harder, R.L. A proposed standard set of problems to test finite element accuracy. // Finite elements in analysis and design, 1985, Vol. 1, pp. 3-20. DOI: 10.1016/0168-874X(85)90003-4
22. Рекач В.Г. Руководство к решению задач по теории упругости. М.: Высшая школа, 1977. – 216с.
23. Калманок А.С. Расчет балок-стенок. М.: Госстройиздат, 1956. – 151с.
24. Городецкий А.С., Карпиловский В.С. О связи метода конечных элементов с вариационно-разностными методами. // Сопротивление материалов и теория сооружений, вып. 24, Киев, Будiвельник, 1974, с.32 –42.