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Approximated Calculation of the Kirchhoff Plate Resting on the Vlasov Foundation with Selected Boundary Conditions

Approximated Calculation of the Kirchhoff Plate Resting on the Vlasov Foundation with Selected... Acta Sci. Pol. Architectura 20 (3) 2021, 11–18 content.sciendo.com/aspa ISSN 1644-0633 eISSN 2544-1760 DOI: 10.22630/ASPA.2021.20.3.21 ORIGINAL P APER Received: 05.07.2021 Accepted: 24.08.2021 APPROXIMATED CALCULATION OF THE KIRCHHOFF PLATE RESTING ON THE VLASOV FOUNDATION WITH SELECTED BOUNDARY CONDITIONS Konstantin Rusakov, Mykola Nagirniak Institute of Civil Engineering, Warsaw University of Life Sciences – SGGW, Warsaw, Poland ABSTRACT The paper presents the problem of bending of the Kirchhoff plate resting freely on the elastic Vlasov subsoil with additional external load g to the subsoil applied near the transverse edge of the plate. The presented example is a special case of a plate resting freely on an elastic subsoil, it is common in construction industry. It was considered approximately, how an additional soil load g applied along the y-axis affects the deflection of a plate resting freely on the Vlasov foundation. Deflection diagrams of the plate and the surface of the elastic foundation outside the plate boundaries have been obtained. The diagrams of deflection of the plate middle surface and the displacements of the soil surface beyond the plate boundaries (in the transverse and longitudinal directions, taking into account the additional load g beyond the plate boundary) depending on the distance in the x-axis direction of this load were calculated. Key words: Kirchhoff plate, Vlasov foundation, work of forces, plate deflection INTRODUCTION ways of solving such complex problems are approxi- mate methods. In addition, the hypotheses assumed In the structural mechanics, the theory of plates is one with regard to the behavior of a ground soil (Vlasov of the most important issues keeping researchers inter- & Leontiev, 1960) cannot be recognized as perfect ested in this topic. either. The analysis of the literature on the theory of The issues of interaction of the structure and the plates can be concluded that the theory of homogene- ground most often concern the foundations and lin- ous thin plates can be used when the quotient of the ing of excavations. It is known that the foundations plate thickness h and its width a is lower than 1/10 of high buildings are calculated by the finite element (Jemielita, 2001). In the case of homogeneous plates method with use of an elastic-plastic soil model. For with h/a > 1/10 and investigations of the boundary ef- smaller objects, programs with the Winkler foundation fect (Bolle, 1947) or stress concentration, the theories model are used but this model does not take into ac- of middle thickness plates must be used (for example count the variability of the soil elasticity and the dis- the Hencky–Bolle model). In the case of plates resting placement of the soil outside the place of load. Due on a subsoil, existing calculation methods are not yet to this fact, it is reasonable for simple calculations to perfect and in general do not allow for the calculation use the Vlasov elastic foundation for thin Kirchhoff of complex spatial systems. One of the most effective plates (Ozgan, 2013; Höller et al., 2019; Yue, Wang, Konstantin Rusakov https://orcid.org/0000-0002-6683-0547; Mykola Nagirniak https://orcid.org/0000-0003-4996-7397 mykola_nagirniak@sggw.edu.pl © Copyright by Wydawnictwo SGGW Rusakov, K., Nagirniak, M. (2021). Approximated calculation of the Kirchhoff plate resting on the Vlasov foundation with selected boundary conditions. Acta Sci. Pol. Architectura, 20 (3) 2021, 11–18, doi: 10.22630/ASPA.2021.20.3.21 Jia, Wu & Wang, 2020), what allows to calculate the The function ϑ(z) is the function of displace- cross-sectional forces and to change the parameters of ment distribution in the ground, it is assumed ϑ(0) = the plate and the subsoil in a wide range. = 1, ϑ(H) = 0. In the monograph of Vlasov and Leon- tiev (1960), the following functions ϑ(z) of the disap- pearance of displacements along depth were proposed: TWO-PARAMETER MODEL OF THE ELASTIC VLASOV SOIL [ ( )] (2) ( ) =1 − , ( ) = , ( ) = [ ] Two-dimensional models of elastic foundation are di- vided into two groups: where γ is the rate of displacement distribution along 1) models resulting from the equations of the theory –1 depth [N·m ]. of elasticity after introducing certain simplifica- tions – they are called structural models, DIFFERENTIAL EQUATION OF KIRCHHOFF 2) models created by means of combination of layers PLATE RESTING ON VLASOV FOUNDATION with different material characteristics − these are the so-called multiparameter phenomenological Assume that the contact between the plate and the soil models (Jemielita, 1992, 1994). always exists. It means that it is satisfied an equality The Vlasov elastic foundation model is a structural w(x, y) = w (x, y, 0). Under this assumption, there is model. Denoting by q(x, y) and w (x, y, z) the load act- always an interaction between the plate and the foun- ing on the ground and the vertical displacement, respec- dation. The load acting on the plate (related to the tively, one can write the equation of the two-parameter midplane) p (x, y) is equal to Vlasov foundation as (Vlasov & Leontiev, 1960): ( , ) = ( , ) − ( , ) ( ) ( ) ( ) (1) , = , −2 ∇ , where: where: w(x, y) – plate midplane deflection [m], –2 k – stiffness coefficient of the foundation, which char- p(x, y) – external load [N·m ]. acterizes the compressive work of the foundation –3 [N·m ], The differential equations of deflection of a plate t – stiffness coefficient of the foundation, which resting on the Vlasov foundation (1) in the Cartesian –1 characterizes the foundation shear work [N·m ]. coordinate system may be written as (Vlasov & Leon- The elastic constants k and t can be determined tiev, 1960): from the formulas (Vlasov & Leontiev, 1960): ( ) (3) ∇ ( , ) −2 ∇ ( , ) + ( , ) = ( ) 1− where: = ( ) 4( 1 + ) where: 1− – elastic modulus by bending of the plate 1− [N·m], –2 E – Young modulus of the plate [N·m ], H – foundation thickness [m], p ν – Poisson ratio of the plate [–], ν – Poisson ratio of the soil [–], –2 h – thickness of the plate [m]. E – Young modulus of the soil [N·m ]. 12 architectura.actapol.net Rusakov, K., Nagirniak, M. (2021). Approximated calculation of the Kirchhoff plate resting on the Vlasov foundation with selected boundary conditions. Acta Sci. Pol. Architectura, 20 (3) 2021, 11–18, doi: 10.22630/ASPA.2021.20.3.21 APPROXIMATED SOLUTION OF THE PLATE FREELY RESTING ON VLASOV SUBSOIL WITH ADDITIONAL LOAD NEAR ONE EDGE Consider a rectangular plate, symmetrically loaded, resting freely on a single-layer elastic subsoil as shown in Figure 1. Additionally, the load g is applied to the subsoil at a distance a from the transverse edge of the plate. It has been investigated the plate deflection under an influence of the vertical load (rigid beam) applied at the distance a along the y-axis. Fig. 1. Plate freely resting on an elastic foundation with additional load g near one edge The problem under consideration is solved on the basis of the principle of superposition and can be divided into two stages: the first one – deflection of the plate resting freely on the elastic foundation, and the second one – deflection of an elastic foundation loaded uniformly along the y-axis by a rigid beam. First stage. Deflection of the plate resting freely on the elastic foundation (Fig. 2) Fig. 2. Plate freely resting on an elastic foundation architectura.actapol.net 13 Rusakov, K., Nagirniak, M. (2021). Approximated calculation of the Kirchhoff plate resting on the Vlasov foundation with selected boundary conditions. Acta Sci. Pol. Architectura, 20 (3) 2021, 11–18, doi: 10.22630/ASPA.2021.20.3.21 The plate displacements w (x, y) are presented as 1p (6) (Vlasov & Leontiev, 1960): π π ( , ) = + cos +cos + (7) (4) π π + cos cos where C , C , C and C are constant coefficients 1 2 3 4 measured in units of length. where w is a value of the deflection function at a plate corner points. The assumption of displacements w (x, y) given The index b in Eq. (6) means that the values of the 1p by Eq. (4) meets the geometrical conditions of the deflection function w (x, y) and its derivatives should 1p problem under consideration. The first component is be calculated at the points on the longitudinal side of defined as settling of the plate as a rigid element, the the plate (x = ±a). The index a in Eq. (7) means that second and third components are the cylindrical bend- the values of the deflection function w (x, y) and its 1p ing of the plate in the x and the y directions, respec- derivatives should be calculated at the points on the tively, and the last component characterizes the plate transverse side of the plate (y = ±b). bending in both directions. To determine the coefficients C we use the condi- tion that the total work of internal and external plate forces on possible unit displacements is equal to zero: (5) where (i = 1, 2, 3, 4) is a virtual displacement of the plate. f f Fig. 3. Additional reactions Q and corner forces R of Apart from these forces, it is necessary to take into a plate freely resting on the elastic foundation account additional reactions Q along the edges of the plate, what requires to consider the work of the elastic Writing down the expression for the work of all foundation outside the plate boundaries (Fig. 3). In the internal and external forces on possible displacements case of rectangular Kirchhoff plates, corner forces R (5) for the Kirchhoff plate resting on elastic foundation, are generated at the corners of the plates. The reactions one obtains a system of four equations from which the f f Q and R can be written as (Vlasov & Leontiev, 1960): coefficients C , C , C and C can be determined: 1 2 3 4 14 architectura.actapol.net Rusakov, K., Nagirniak, M. (2021). Approximated calculation of the Kirchhoff plate resting on the Vlasov foundation with selected boundary conditions. Acta Sci. Pol. Architectura, 20 (3) 2021, 11–18, doi: 10.22630/ASPA.2021.20.3.21 The displacements of the soil surface outside the where: plate boundaries will be assumed in the form: b – half length of the plate [m], –1 g – load magnitude applied to the rigid beam [N·m ], (8) Note that in Eq. (8) denotes the value of the deflection function w (x, y) at the points on the lon- 1p gitudinal side of the plate (x = ±a), while denotes A final solution to the problem presented in the deflection function w (x, y) at the points on the 1p Figure 1 is the sum of the first and the second solu- transverse side of the plate (y = ±b). tion. Ultimately, the displacement of the soil surface Second stage. Deflection of an elastic over the entire area under consideration is written as: foundation loaded uniformly along the y-axis by a rigid beam (Fig. 4) (10) CALCULATION EXAMPLE Fig. 4. The elastic foundation loaded uniformly along the y-axis by a rigid beam As an example, consider the plate shown in Figure 1. Assume the following geometrical dimensions and Assuming that the settlement under this load is stiffness characteristics of the plate and the soil: constant over the entire length on which it is applied, a = 7 m, b = 1.5a, H = 2a, h = 0.5 m the displacement ν(x, y) may be written as: 3 –2 6 –2 ν = 0.25, ν = 0.3, E = 50·10 kN·m , E = 27·10 kN·m s s p (9) The function of displacement distribution along depth is assumed in a form: The coefficient C from Eq. (9) is determined from the condition that the projections of all forces on the (11) z-axis acting on the rigid beam are equal to zero: For the linear displacement distribution function along depth (11), the displacements (10) are as fol- lows: architectura.actapol.net 15 Rusakov, K., Nagirniak, M. (2021). Approximated calculation of the Kirchhoff plate resting on the Vlasov foundation with selected boundary conditions. Acta Sci. Pol. Architectura, 20 (3) 2021, 11–18, doi: 10.22630/ASPA.2021.20.3.21 The diagrams of the deflection of the middle surface of the plate and the displacement of the soil surface outside the plate boundaries are presented in Figures 5–10. Fig. 6. Deflection w(x, 0) of Kirchhoff plate on the Vlas- Fig. 5. Deflection w(x, 0) of Kirchhoff plate on the Vlas- ov elastic foundation for a = 2a ov elastic foundation for a = a + 1 Fig. 7. Deflection w(x, 0) of Kirchhoff plate on the Vlas- Fig. 8. Deflection w(0, y) of Kirchhoff plate on the Vla- ov elastic foundation for a = 3a sov elastic foundation for a = a + 1 16 architectura.actapol.net Rusakov, K., Nagirniak, M. (2021). Approximated calculation of the Kirchhoff plate resting on the Vlasov foundation with selected boundary conditions. Acta Sci. Pol. Architectura, 20 (3) 2021, 11–18, doi: 10.22630/ASPA.2021.20.3.21 Fig. 9. Deflection w(0, y) of Kirchhoff plate on the Vlas- Fig. 10. Deflection w(0, y) of Kirchhoff plate on the Vla- ov elastic foundation for a = 2a sov elastic foundation for a = 3a 1 1 CONCLUSIONS The paper presents results of an approximated calcu- Figures 5−7 show the diagrams of the plate deflection lation for Kirchhoff plate resting freely on the Vlasov and surface displacements of the elastic foundation lo- elastic foundation with additional load g near plate edge. cated outside the plate boundaries for y = 0 at various Plate deflection function w(x, y) was expressed in such distances a of the additional load (rigid beam) from a way that it satisfies the geometrical boundary condi- the longitudinal edge of the plate. In the case where tions, namely, the deflection function is different from the load is applied in the immediate vicinity of the lon- zero on the plate edge. The static boundary conditions gitudinal edge of the plate a = (a + 1 m), strong influ- have been met as approximated. The calculations of ence of the load g on the plate deflection is observed. simple approximated displacement function of the soil Increasing the distance a for the same load values) the surface beyond the plate boundaries were performed by deflection of the plate decreases and the character of CAS Mathematica. The selection of the plate deflection the deflection curve changes. Figures 5−7 show that function in the form of Eq. (4) is not only one possible. if the distance increases, there is an increasingly pro- The displacement of the foundation surface beyond the nounced cylindrical bending of the plate in the trans- plate boundaries was assumed as approximation – the verse direction. For sufficiently large values of a , the settlement of the elastic foundation in spatial conditions effect of the g on the deformed state of the plate tends outside the plate boundaries will be more complex. to zero and the considered case of symmetric loading The system of algebraic equations allowing to de- of the plate causes its cylindrical bending. termine the constants C was obtained by equating to Figures 8−10 show the deflections of the plate and zero the work of all the forces in the plate movements the elastic foundation beyond its boundaries in the di- and the possible continuity conditions. The displace- rection of the y-axis at x = 0. ment distribution function ϑ(z) along depth was linear. It should be noted that the considered method is The constants C were determined for the assumed pa- applicable only for determining displacements. Static rameters of the plate and soil. Diagrams of the plate equilibrium conditions of the plate are met approxi- deflection and the displacements of soil surface for g mately. In sections x = ±a, the bending moment M –1 equals to 20 and 50 kN·m were presented. In case should be zero, but this condition is not satisfied in the when the additional load g → 0 or a → ∞, the deflec- considered superposition method. tion character in the transverse direction is the same as in the longitudinal direction. architectura.actapol.net 17 Rusakov, K., Nagirniak, M. (2021). Approximated calculation of the Kirchhoff plate resting on the Vlasov foundation with selected boundary conditions. Acta Sci. Pol. Architectura, 20 (3) 2021, 11–18, doi: 10.22630/ASPA.2021.20.3.21 Jemielita, G. (2001). Teorie płyt sprężystych. In C. Woźniak REFERENCES (Ed.), Mechanika techniczna. Vol. 8. Mechanika Bolle, L. (1947). Contribution au problème linèare de sprężystych płyt i powłok. Warszawa: Wydawnictwo flexion d’une plaque èlastique. Bulletin Technique de la Naukowe PWN. Suisse Romande, 73, 293–298. Ozgan, K. (2013). Dynamic analysis of thick plates includ- Höller, R., Aminbaghai, M., Eberhardsteiner, L., Eberhards- ing deep beams on elastic foundations using modified teiner, J., Blab, R., Pichler, B. & Hellmich, C. (2019). Vlasov model. Shock and Vibration, 20 (1), 29–41. Rigorous amendment of Vlasov’s theory for thin elastic Vlasov, V. & Leontiev, N. (1960). Balki, plity i oboločki na plates on elastic Winkler foundations, based on the Prin- uprugom osnovanii. Moskva: Gosudarstvennoe izdatel- ciple of Virtual Power. European Journal of Mechanics stvo fiziko-matemetičeskoj literatury. / A Solids, 73, 449–482. Yue, F., Wang, F., Jia, S., Wu, Z. & Wang, Z. (2020). Bending Jemielita, G. (1992). Generalization of the Kerr foundation analysis of circular thin platesresting on elastic founda- model. Journal of Theoretical and Applied Mechanics, tions using two modified Vlasov models. Mathematical 4 (30), 843–853. Problems in Engineering, 2020, 2345347. https://doi. Jemielita, G. (1994). Governing equations and boundary org/10.1155/2020/2345347 conditions of a generalized model of elastic foundation. Journal of Theoretical and Applied Mechanics, 4 (32), 887–901. PRZYBLIŻONE OBLICZENIE PŁYTY KIRCHHOFFA SPOCZYWAJĄCEJ NA PODŁOŻU SPRĘŻYSTYM WŁASOWA O WYBRANYCH WARUNKACH BRZEGOWYCH STRESZCZENIE W pracy przedstawiono zagadnienie zginania płyty Kirchhoffa swobodnie spoczywającej na sprężystym podłożu Własowa z dodatkowym obciążeniem zewnętrznym podłoża g, przyłożonym w pobliżu krawędzi poprzecznej płyty. Podany przykład jest szczególnym przypadkiem płyty swobodnie spoczywającej na sprę- żystym podłożu, występującym w praktyce budowlanej. Rozpatrzono w przybliżeniu, jak przyłożone wzdłuż osi y dodatkowe obciążenie gruntu g wpływa na ugięcie płyty swobodnie spoczywającej na podłożu Właso- wa. Przedstawiono wykresy ugięcia powierzchni środkowej płyty i warstwy powierzchniowej gruntu poza granicami płyty (w kierunku poprzecznym i wzdłużnym z uwzględnieniem dodatkowego obciążenia g poza granicą płyty) w zależności od odległości w kierunku osi x tego obciążenia. Słowa kluczowe: płyta Kirchhoffa, podłoże Własowa, praca sił, ugięcie płyty 18 architectura.actapol.net http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Scientiarum Polonorum Architectura de Gruyter

Approximated Calculation of the Kirchhoff Plate Resting on the Vlasov Foundation with Selected Boundary Conditions

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de Gruyter
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© 2021 Konstantin Rusakov et al., published by Sciendo
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2544-1760
DOI
10.22630/aspa.2021.20.3.21
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Abstract

Acta Sci. Pol. Architectura 20 (3) 2021, 11–18 content.sciendo.com/aspa ISSN 1644-0633 eISSN 2544-1760 DOI: 10.22630/ASPA.2021.20.3.21 ORIGINAL P APER Received: 05.07.2021 Accepted: 24.08.2021 APPROXIMATED CALCULATION OF THE KIRCHHOFF PLATE RESTING ON THE VLASOV FOUNDATION WITH SELECTED BOUNDARY CONDITIONS Konstantin Rusakov, Mykola Nagirniak Institute of Civil Engineering, Warsaw University of Life Sciences – SGGW, Warsaw, Poland ABSTRACT The paper presents the problem of bending of the Kirchhoff plate resting freely on the elastic Vlasov subsoil with additional external load g to the subsoil applied near the transverse edge of the plate. The presented example is a special case of a plate resting freely on an elastic subsoil, it is common in construction industry. It was considered approximately, how an additional soil load g applied along the y-axis affects the deflection of a plate resting freely on the Vlasov foundation. Deflection diagrams of the plate and the surface of the elastic foundation outside the plate boundaries have been obtained. The diagrams of deflection of the plate middle surface and the displacements of the soil surface beyond the plate boundaries (in the transverse and longitudinal directions, taking into account the additional load g beyond the plate boundary) depending on the distance in the x-axis direction of this load were calculated. Key words: Kirchhoff plate, Vlasov foundation, work of forces, plate deflection INTRODUCTION ways of solving such complex problems are approxi- mate methods. In addition, the hypotheses assumed In the structural mechanics, the theory of plates is one with regard to the behavior of a ground soil (Vlasov of the most important issues keeping researchers inter- & Leontiev, 1960) cannot be recognized as perfect ested in this topic. either. The analysis of the literature on the theory of The issues of interaction of the structure and the plates can be concluded that the theory of homogene- ground most often concern the foundations and lin- ous thin plates can be used when the quotient of the ing of excavations. It is known that the foundations plate thickness h and its width a is lower than 1/10 of high buildings are calculated by the finite element (Jemielita, 2001). In the case of homogeneous plates method with use of an elastic-plastic soil model. For with h/a > 1/10 and investigations of the boundary ef- smaller objects, programs with the Winkler foundation fect (Bolle, 1947) or stress concentration, the theories model are used but this model does not take into ac- of middle thickness plates must be used (for example count the variability of the soil elasticity and the dis- the Hencky–Bolle model). In the case of plates resting placement of the soil outside the place of load. Due on a subsoil, existing calculation methods are not yet to this fact, it is reasonable for simple calculations to perfect and in general do not allow for the calculation use the Vlasov elastic foundation for thin Kirchhoff of complex spatial systems. One of the most effective plates (Ozgan, 2013; Höller et al., 2019; Yue, Wang, Konstantin Rusakov https://orcid.org/0000-0002-6683-0547; Mykola Nagirniak https://orcid.org/0000-0003-4996-7397 mykola_nagirniak@sggw.edu.pl © Copyright by Wydawnictwo SGGW Rusakov, K., Nagirniak, M. (2021). Approximated calculation of the Kirchhoff plate resting on the Vlasov foundation with selected boundary conditions. Acta Sci. Pol. Architectura, 20 (3) 2021, 11–18, doi: 10.22630/ASPA.2021.20.3.21 Jia, Wu & Wang, 2020), what allows to calculate the The function ϑ(z) is the function of displace- cross-sectional forces and to change the parameters of ment distribution in the ground, it is assumed ϑ(0) = the plate and the subsoil in a wide range. = 1, ϑ(H) = 0. In the monograph of Vlasov and Leon- tiev (1960), the following functions ϑ(z) of the disap- pearance of displacements along depth were proposed: TWO-PARAMETER MODEL OF THE ELASTIC VLASOV SOIL [ ( )] (2) ( ) =1 − , ( ) = , ( ) = [ ] Two-dimensional models of elastic foundation are di- vided into two groups: where γ is the rate of displacement distribution along 1) models resulting from the equations of the theory –1 depth [N·m ]. of elasticity after introducing certain simplifica- tions – they are called structural models, DIFFERENTIAL EQUATION OF KIRCHHOFF 2) models created by means of combination of layers PLATE RESTING ON VLASOV FOUNDATION with different material characteristics − these are the so-called multiparameter phenomenological Assume that the contact between the plate and the soil models (Jemielita, 1992, 1994). always exists. It means that it is satisfied an equality The Vlasov elastic foundation model is a structural w(x, y) = w (x, y, 0). Under this assumption, there is model. Denoting by q(x, y) and w (x, y, z) the load act- always an interaction between the plate and the foun- ing on the ground and the vertical displacement, respec- dation. The load acting on the plate (related to the tively, one can write the equation of the two-parameter midplane) p (x, y) is equal to Vlasov foundation as (Vlasov & Leontiev, 1960): ( , ) = ( , ) − ( , ) ( ) ( ) ( ) (1) , = , −2 ∇ , where: where: w(x, y) – plate midplane deflection [m], –2 k – stiffness coefficient of the foundation, which char- p(x, y) – external load [N·m ]. acterizes the compressive work of the foundation –3 [N·m ], The differential equations of deflection of a plate t – stiffness coefficient of the foundation, which resting on the Vlasov foundation (1) in the Cartesian –1 characterizes the foundation shear work [N·m ]. coordinate system may be written as (Vlasov & Leon- The elastic constants k and t can be determined tiev, 1960): from the formulas (Vlasov & Leontiev, 1960): ( ) (3) ∇ ( , ) −2 ∇ ( , ) + ( , ) = ( ) 1− where: = ( ) 4( 1 + ) where: 1− – elastic modulus by bending of the plate 1− [N·m], –2 E – Young modulus of the plate [N·m ], H – foundation thickness [m], p ν – Poisson ratio of the plate [–], ν – Poisson ratio of the soil [–], –2 h – thickness of the plate [m]. E – Young modulus of the soil [N·m ]. 12 architectura.actapol.net Rusakov, K., Nagirniak, M. (2021). Approximated calculation of the Kirchhoff plate resting on the Vlasov foundation with selected boundary conditions. Acta Sci. Pol. Architectura, 20 (3) 2021, 11–18, doi: 10.22630/ASPA.2021.20.3.21 APPROXIMATED SOLUTION OF THE PLATE FREELY RESTING ON VLASOV SUBSOIL WITH ADDITIONAL LOAD NEAR ONE EDGE Consider a rectangular plate, symmetrically loaded, resting freely on a single-layer elastic subsoil as shown in Figure 1. Additionally, the load g is applied to the subsoil at a distance a from the transverse edge of the plate. It has been investigated the plate deflection under an influence of the vertical load (rigid beam) applied at the distance a along the y-axis. Fig. 1. Plate freely resting on an elastic foundation with additional load g near one edge The problem under consideration is solved on the basis of the principle of superposition and can be divided into two stages: the first one – deflection of the plate resting freely on the elastic foundation, and the second one – deflection of an elastic foundation loaded uniformly along the y-axis by a rigid beam. First stage. Deflection of the plate resting freely on the elastic foundation (Fig. 2) Fig. 2. Plate freely resting on an elastic foundation architectura.actapol.net 13 Rusakov, K., Nagirniak, M. (2021). Approximated calculation of the Kirchhoff plate resting on the Vlasov foundation with selected boundary conditions. Acta Sci. Pol. Architectura, 20 (3) 2021, 11–18, doi: 10.22630/ASPA.2021.20.3.21 The plate displacements w (x, y) are presented as 1p (6) (Vlasov & Leontiev, 1960): π π ( , ) = + cos +cos + (7) (4) π π + cos cos where C , C , C and C are constant coefficients 1 2 3 4 measured in units of length. where w is a value of the deflection function at a plate corner points. The assumption of displacements w (x, y) given The index b in Eq. (6) means that the values of the 1p by Eq. (4) meets the geometrical conditions of the deflection function w (x, y) and its derivatives should 1p problem under consideration. The first component is be calculated at the points on the longitudinal side of defined as settling of the plate as a rigid element, the the plate (x = ±a). The index a in Eq. (7) means that second and third components are the cylindrical bend- the values of the deflection function w (x, y) and its 1p ing of the plate in the x and the y directions, respec- derivatives should be calculated at the points on the tively, and the last component characterizes the plate transverse side of the plate (y = ±b). bending in both directions. To determine the coefficients C we use the condi- tion that the total work of internal and external plate forces on possible unit displacements is equal to zero: (5) where (i = 1, 2, 3, 4) is a virtual displacement of the plate. f f Fig. 3. Additional reactions Q and corner forces R of Apart from these forces, it is necessary to take into a plate freely resting on the elastic foundation account additional reactions Q along the edges of the plate, what requires to consider the work of the elastic Writing down the expression for the work of all foundation outside the plate boundaries (Fig. 3). In the internal and external forces on possible displacements case of rectangular Kirchhoff plates, corner forces R (5) for the Kirchhoff plate resting on elastic foundation, are generated at the corners of the plates. The reactions one obtains a system of four equations from which the f f Q and R can be written as (Vlasov & Leontiev, 1960): coefficients C , C , C and C can be determined: 1 2 3 4 14 architectura.actapol.net Rusakov, K., Nagirniak, M. (2021). Approximated calculation of the Kirchhoff plate resting on the Vlasov foundation with selected boundary conditions. Acta Sci. Pol. Architectura, 20 (3) 2021, 11–18, doi: 10.22630/ASPA.2021.20.3.21 The displacements of the soil surface outside the where: plate boundaries will be assumed in the form: b – half length of the plate [m], –1 g – load magnitude applied to the rigid beam [N·m ], (8) Note that in Eq. (8) denotes the value of the deflection function w (x, y) at the points on the lon- 1p gitudinal side of the plate (x = ±a), while denotes A final solution to the problem presented in the deflection function w (x, y) at the points on the 1p Figure 1 is the sum of the first and the second solu- transverse side of the plate (y = ±b). tion. Ultimately, the displacement of the soil surface Second stage. Deflection of an elastic over the entire area under consideration is written as: foundation loaded uniformly along the y-axis by a rigid beam (Fig. 4) (10) CALCULATION EXAMPLE Fig. 4. The elastic foundation loaded uniformly along the y-axis by a rigid beam As an example, consider the plate shown in Figure 1. Assume the following geometrical dimensions and Assuming that the settlement under this load is stiffness characteristics of the plate and the soil: constant over the entire length on which it is applied, a = 7 m, b = 1.5a, H = 2a, h = 0.5 m the displacement ν(x, y) may be written as: 3 –2 6 –2 ν = 0.25, ν = 0.3, E = 50·10 kN·m , E = 27·10 kN·m s s p (9) The function of displacement distribution along depth is assumed in a form: The coefficient C from Eq. (9) is determined from the condition that the projections of all forces on the (11) z-axis acting on the rigid beam are equal to zero: For the linear displacement distribution function along depth (11), the displacements (10) are as fol- lows: architectura.actapol.net 15 Rusakov, K., Nagirniak, M. (2021). Approximated calculation of the Kirchhoff plate resting on the Vlasov foundation with selected boundary conditions. Acta Sci. Pol. Architectura, 20 (3) 2021, 11–18, doi: 10.22630/ASPA.2021.20.3.21 The diagrams of the deflection of the middle surface of the plate and the displacement of the soil surface outside the plate boundaries are presented in Figures 5–10. Fig. 6. Deflection w(x, 0) of Kirchhoff plate on the Vlas- Fig. 5. Deflection w(x, 0) of Kirchhoff plate on the Vlas- ov elastic foundation for a = 2a ov elastic foundation for a = a + 1 Fig. 7. Deflection w(x, 0) of Kirchhoff plate on the Vlas- Fig. 8. Deflection w(0, y) of Kirchhoff plate on the Vla- ov elastic foundation for a = 3a sov elastic foundation for a = a + 1 16 architectura.actapol.net Rusakov, K., Nagirniak, M. (2021). Approximated calculation of the Kirchhoff plate resting on the Vlasov foundation with selected boundary conditions. Acta Sci. Pol. Architectura, 20 (3) 2021, 11–18, doi: 10.22630/ASPA.2021.20.3.21 Fig. 9. Deflection w(0, y) of Kirchhoff plate on the Vlas- Fig. 10. Deflection w(0, y) of Kirchhoff plate on the Vla- ov elastic foundation for a = 2a sov elastic foundation for a = 3a 1 1 CONCLUSIONS The paper presents results of an approximated calcu- Figures 5−7 show the diagrams of the plate deflection lation for Kirchhoff plate resting freely on the Vlasov and surface displacements of the elastic foundation lo- elastic foundation with additional load g near plate edge. cated outside the plate boundaries for y = 0 at various Plate deflection function w(x, y) was expressed in such distances a of the additional load (rigid beam) from a way that it satisfies the geometrical boundary condi- the longitudinal edge of the plate. In the case where tions, namely, the deflection function is different from the load is applied in the immediate vicinity of the lon- zero on the plate edge. The static boundary conditions gitudinal edge of the plate a = (a + 1 m), strong influ- have been met as approximated. The calculations of ence of the load g on the plate deflection is observed. simple approximated displacement function of the soil Increasing the distance a for the same load values) the surface beyond the plate boundaries were performed by deflection of the plate decreases and the character of CAS Mathematica. The selection of the plate deflection the deflection curve changes. Figures 5−7 show that function in the form of Eq. (4) is not only one possible. if the distance increases, there is an increasingly pro- The displacement of the foundation surface beyond the nounced cylindrical bending of the plate in the trans- plate boundaries was assumed as approximation – the verse direction. For sufficiently large values of a , the settlement of the elastic foundation in spatial conditions effect of the g on the deformed state of the plate tends outside the plate boundaries will be more complex. to zero and the considered case of symmetric loading The system of algebraic equations allowing to de- of the plate causes its cylindrical bending. termine the constants C was obtained by equating to Figures 8−10 show the deflections of the plate and zero the work of all the forces in the plate movements the elastic foundation beyond its boundaries in the di- and the possible continuity conditions. The displace- rection of the y-axis at x = 0. ment distribution function ϑ(z) along depth was linear. It should be noted that the considered method is The constants C were determined for the assumed pa- applicable only for determining displacements. Static rameters of the plate and soil. Diagrams of the plate equilibrium conditions of the plate are met approxi- deflection and the displacements of soil surface for g mately. In sections x = ±a, the bending moment M –1 equals to 20 and 50 kN·m were presented. In case should be zero, but this condition is not satisfied in the when the additional load g → 0 or a → ∞, the deflec- considered superposition method. tion character in the transverse direction is the same as in the longitudinal direction. architectura.actapol.net 17 Rusakov, K., Nagirniak, M. (2021). Approximated calculation of the Kirchhoff plate resting on the Vlasov foundation with selected boundary conditions. Acta Sci. Pol. Architectura, 20 (3) 2021, 11–18, doi: 10.22630/ASPA.2021.20.3.21 Jemielita, G. (2001). Teorie płyt sprężystych. In C. Woźniak REFERENCES (Ed.), Mechanika techniczna. Vol. 8. Mechanika Bolle, L. (1947). Contribution au problème linèare de sprężystych płyt i powłok. Warszawa: Wydawnictwo flexion d’une plaque èlastique. Bulletin Technique de la Naukowe PWN. Suisse Romande, 73, 293–298. Ozgan, K. (2013). Dynamic analysis of thick plates includ- Höller, R., Aminbaghai, M., Eberhardsteiner, L., Eberhards- ing deep beams on elastic foundations using modified teiner, J., Blab, R., Pichler, B. & Hellmich, C. (2019). Vlasov model. Shock and Vibration, 20 (1), 29–41. Rigorous amendment of Vlasov’s theory for thin elastic Vlasov, V. & Leontiev, N. (1960). Balki, plity i oboločki na plates on elastic Winkler foundations, based on the Prin- uprugom osnovanii. Moskva: Gosudarstvennoe izdatel- ciple of Virtual Power. European Journal of Mechanics stvo fiziko-matemetičeskoj literatury. / A Solids, 73, 449–482. Yue, F., Wang, F., Jia, S., Wu, Z. & Wang, Z. (2020). Bending Jemielita, G. (1992). Generalization of the Kerr foundation analysis of circular thin platesresting on elastic founda- model. Journal of Theoretical and Applied Mechanics, tions using two modified Vlasov models. Mathematical 4 (30), 843–853. Problems in Engineering, 2020, 2345347. https://doi. Jemielita, G. (1994). Governing equations and boundary org/10.1155/2020/2345347 conditions of a generalized model of elastic foundation. Journal of Theoretical and Applied Mechanics, 4 (32), 887–901. PRZYBLIŻONE OBLICZENIE PŁYTY KIRCHHOFFA SPOCZYWAJĄCEJ NA PODŁOŻU SPRĘŻYSTYM WŁASOWA O WYBRANYCH WARUNKACH BRZEGOWYCH STRESZCZENIE W pracy przedstawiono zagadnienie zginania płyty Kirchhoffa swobodnie spoczywającej na sprężystym podłożu Własowa z dodatkowym obciążeniem zewnętrznym podłoża g, przyłożonym w pobliżu krawędzi poprzecznej płyty. Podany przykład jest szczególnym przypadkiem płyty swobodnie spoczywającej na sprę- żystym podłożu, występującym w praktyce budowlanej. Rozpatrzono w przybliżeniu, jak przyłożone wzdłuż osi y dodatkowe obciążenie gruntu g wpływa na ugięcie płyty swobodnie spoczywającej na podłożu Właso- wa. Przedstawiono wykresy ugięcia powierzchni środkowej płyty i warstwy powierzchniowej gruntu poza granicami płyty (w kierunku poprzecznym i wzdłużnym z uwzględnieniem dodatkowego obciążenia g poza granicą płyty) w zależności od odległości w kierunku osi x tego obciążenia. Słowa kluczowe: płyta Kirchhoffa, podłoże Własowa, praca sił, ugięcie płyty 18 architectura.actapol.net

Journal

Acta Scientiarum Polonorum Architecturade Gruyter

Published: Sep 1, 2021

Keywords: Kirchhoff plate; Vlasov foundation; work of forces; plate deflection

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