- Email: [email protected]
RSM Method in Probabilistic Analysis of the Foundation Plate
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 190 (2017) 516 – 521
Structural and Physical Aspects of Construction Engineering
RSM Method in Probabilistic Analysis of the Foundation Plate Katarina Tvrdaa,* a
Slovak University of Technology in Bratislava, Faculty of Civil Engeenering, Radlinského 11, 81101 Bratislava, Slovakia
Abstract Many new approaches are currently used in the design and assessment of building structures. One of these approaches is the probabilistic analysis allowed by the European standards for the assessment of structural design. Input parameters are given as stochastic ones and they are characterized by different distribution types. During a probabilistic analysis, software executes multiple analysis loops to compute the random output parameters as functions of the set of random input variables. Uncertainties in input variables (load, material properties and dimensions structures) are reflected in the RSM method based on approximation of Monte Carlo simulations. Finally, some results of these probabilistic analyses are presented. © Published by Elsevier Ltd. This © 2017 2017The TheAuthors. Authors. Published by Elsevier Ltd. is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the issue editors. Peer-review under responsibility of the organizing committee of SPACE 2016 Keywords: foundation plate; Responce Surface Method; probability design; Monte Carlo Simulation; Elastic Half Space; bilateral bounds
1. Introduction The design practice, more and more statically indeterminate building structures are increasingly suggested. In the design of such structures it is necessary to take into account the complex material, geometry and boundary conditions. In solving and modeling tasks of soil-structure interaction various idealized and simplified procedures are reported in the literature that only approximately describe the actual boundary conditions, what affects compliance of achieved results with reality. Every building structure is imposed on the certain type of soil. At the beginning simple approaches, as the Winkler model (one-parametric), Pasternak model (two-parametric background model) and the Boussinesq model (elastic half-space), were used in modeling of subsoil. Complex processes began with appearing and developing of numerical methods, the use of which was dependent on the level of computer equipment and software. For example, not all commercial software application can specify two-parametric soil
* Corresponding author. Tel.: +421-2-59274291 E-mail address: [email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of SPACE 2016
doi:10.1016/j.proeng.2017.05.372
517
Katarina Tvrda / Procedia Engineering 190 (2017) 516 – 521
model. In the field of interaction structures with the subsoil, there are numerous papers [1, 2] published in journals and in the books [3]. In modeling of an elastic half-space, it is necessary to model a relatively large portion of ground mass, and outside the active depth of deformation. In assessing the structure, international standards and regulations, together with the National Annexes, are used. Eurocodes also permit assessment of building structures based on probabilistic analysis. Many papers, dealing with problems, mentioned above, show, that reliability (safety, serviceability, durability) of the design of structure throughout the service life is also influenced by its correct model of foundation [4, 5, 6, 7, 8, 9, 10]. 2. Interaction of the plate with subsoil An inseparable part of the building structure is ground mass in which the structure is built. This massive area is infinitive, but in modeling of structures in the interaction with the subsoil it must have its limits. It is therefore important to choose the right mathematical and material model, which describes the background of the problem. The elastic foundation can be modeled with these types of models: x One-parametric model (Winkler model) x Two-parametric model (Pasternak) x Elastic half-space model (Boussinesq). 2.1. One-parametric model - Winkler model Winkler model is the first model of the subsoil, background on the hypothesis of setting coefficient (coefficient of subgrade stiffness), which assumes that at any point of the subsoil there is the load p directly proportional to deflection w at this point, but it is not dependent on the deflection at the other points. Another drawback is the instability of the setting coefficient (subgrade stiffness) k which depends not only on the material properties of the subsoil, but the dimensions and the shape of the foundation structure as well. Plate on an elastic foundation is defined by partial differential equation (1). D 22 w( x, y) k w( x, y)
(1)
q( x, y)
where D is a plate constant (2), E - modulus of elasticity of the material of plate, h - thickness, Q - Poisson´s ratio. D
E h3
(2)
12 1 Q 2
2.2. Two-parametric model – Pasternak model Two-parametric model of the subsoil is defined by Pasternak so that the contact stress p is defined according to the deflection surface of w (x, y) and coefficients C1 - coefficient of friction in the vertical direction (N / m3), and C2 - coefficient characterizing the shear spreading of the effects of the load (N / m) obtained using energy principles. This model allows to take into account deformations beyond the foundation. ª w 2 w( x, y ) w 2 w( x, y ) º D 22 w( x, y) C1 w( x, y) C2 « » 2 wy 2 ¼ ¬ wx
q( x, y)
(3)
2.3. Elastic half-space model - Boussinesq Elastic half-space is another type of mathematical - material model of the elastic subsoil. The theory of elastic continuous half-space implies that the background forms a continuous, homogeneous and perfectly elastic body of
518
Katarina Tvrda / Procedia Engineering 190 (2017) 516 – 521
an infinitely large size, limited from above by the plane on which lies the underlying building structure. Sometimes it is called the Boussinesq soil model. The model is characterized by two material constants derived from experimental measurements (E - modulus of elasticity and Q - Poisson´s constant). Numerical methods are primarily used to solve contact problems (subgrade model - elastic half-space) to appropriately extend the closed form solutions while offering the possibility of more realistic capturing the interaction characteristics with the subsoil building structure for different boundary conditions. The most famous is FEM. Theoretical assumptions of Boundary Element Method - BEM and Finite Element method - FEM in interaction were dealt by Zienkiewicz and Taylor [11], Langen [12], Kollař et al. [13], Bittnar, Šejnoha [14], Jendželovský, Baláž [15], Sumec et al. [16], Kotrasova, Kormanikova [17], and others. Elastic foundation is modeled by spatial (3D) finite elements. Finite elements of different shapes can be used for the modeling. The most commonly used form are solid block elements with 8 nodes, and three displacements at each node. In modeling of elastic half-space it is necessary to create a sufficiently large volume of grown earth, which depends on supports, boundary conditions. The support may be rigid, flexible or usage of infinite finite elements. For the analysis of interaction plate ground with the subsoil (static binding on the contact surface) two basic models can be defined: computational model of bilateral bonds - continuous model, calculation model with a unilateral bonds - discrete model. 3. Probability and Reliability Under the term of the reliability of the design we understand the ability of the structure or structural member to fill special requirements, including design service life for which the structure was designed. Reliability implies safety, durability and serviceability of the structure and it is expressed through probabilistic terms. The structure shall be designed and carried out over the estimated useful life with the appropriate degree of reliability and economy resist all loads and influences that may occur in the operation and use of the structure and serves that purpose. When designing the building structure or its section, the reliability must be taken in particular to the design, i.e. it must fill the required properties throughout the service life of the structure. Design methods and assessment structures will evolve and grow gradually as the theoretical and practical possibilities proposal (introduction of computers into practice). In the development of different design methods, the practices included uncertainty and ensuring reliability of structures in close dependence on theoretical knowledge of elasticity, mathematical statistics and experimental measurements. Probabilistic methods may be divided into: x The first order reliability method – FORM x The second order reliability method – SORM x Simulation methods. 3.1. Probabilistic methods of reliability assessment To determine the reliability of probabilistic methods first performance criteria on the basis of functional relationship between the n-input quantities are defined referred to as the basic random variables Xi. This relationship is called the reliability function Fs (RF - reliability function), safety, serviceability, reliability and function of the reserves can be expressed generally as a function of stochastic parameters X1 , X 2 , , Xn .
Fs
RF
g ( X1 , X 2 ,
, Xn )
(4)
where the functional dependence g ( X1 , X 2 , , X n ) is a computational model (i.e. idealization of reality). Failure functions g ( X i ) express the conditions of preliability (reserve) and can be represented as a function of stochastic parameters. They may be defined as a simple (e.g. for one section) or as complex structures, for more cross-sections (e.g. all finite-element model). The reliability function can be defined as:
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Katarina Tvrda / Procedia Engineering 190 (2017) 516 – 521
Fs
RF
RS
(5)
where R is the resistance of the structure function (yield strength, allowable deflection, stress), S is a function of load (max. stress in the structure, max. deflection). The function is reliable if Fs ≥ 0, failure occurs if Fs <0 and Fs = 0 is the limit function. The probability of failure can be simply defined as: Pf
P>R S @
P > R S 0@
(6)
where Pf is generally given by the expression
Pf
³³
³³
f X ( X1 , X 2 ,
, X n ) dX1 dX 2
(7)
dX n .
g(X ) 0
The function f x ( X1 , X 2 ,..., X n ) represents the joint density of probability function for continuous randomly variables or multiply probability function for discrete random variables, where integration is transferred over the entire area of failure g (X) <0. According to the method of integral solutions of the equation (7) and reliability assessment the probabilistic methods may be divided to: x Analytical - a direct solution of the integral in relation (7) is impossible, x Simulation methods - direct and approximation. As simulation methods we may mention Monte Carlo – MC method, LHS - Latin Hypercube Sampling and SC Stratified Sampling. There are also advanced simulation methods as: Adaptive Sampling - AS, Direction Sampling DS, IS - Importance Sampling, LS - Line Sampling, RSM - Response Surface Method. MC - Monte Carlo method can be classified within the simulation methods with direct simulation. More comprehensive overview of the methods may be found e.g. in: Krejsa [18]. 4. Assessment of the foundation plate A foundation plate under the building is a subject of an assessment. The foundation plate was built using new technologies. Cobiax Technology is based on generating specific hollows inside a reinforced concrete slab. Massive concrete is replaced by synthetic void formers and remains only in statically relevant areas. a
b
Fig. 1. (a) model of foundation plate; (b) the load of plate.
Thus, it is possible to construct buildings with flat slabs while allowing for remarkable span width. The slab is made of concrete of class C25-30, thickness of 80 cm. Concrete modulus of elasticity is 31 GPa. When modelling
520
Katarina Tvrda / Procedia Engineering 190 (2017) 516 – 521
the slab, according to the manufacturer's recommendations, the elastic modulus in the area of incorporating plastic balls is reduced according to slab thickness and Cobiax balls type (Fig.1a). The elastic modulus of 27.59 GPa was used in the calculation, and Poisson's ratio 0.2. The subsoil consists of gravel G2. They were taken target standardized gravelly soil characteristics E = 180 MPa, Poisson's ratio 0.2. The foundation plate was of the size of 29.7 x 16.2 m. Ground mass was modeled in sufficient width, length and depth of 89.7 x 48.2 x 15 m. Then follows the rock massive. The foundation plate is loaded in the area of columns, which is calculated according to the relevant area of the load and transfers the load of 8 floors. Uniformly distributed load is 12 kN / m 2 (Fig.1b). The easiest way to model plate on elastic foundation was to use elements SHELL 63 in ANSYS [19], with the possibility of introducing the coefficient of stiffness of the subsoil via EFS command. Unfortunately in newer versions of ANSYS this element is not supported. In our case, the soil was modeled as an elastic half-space by Solid 185 elements, plate by Shell 181elements. For the analysis of interaction foundation plate with the subsoil, depending on the static binding of a contact area between the bedrock and the foundation the model with two-sided couplings, enabling the transmission of pressure and tension forces at the contact area was used – a continuous model Fig. 2.
Fig. 2. A complex model - plate and ground mass. Table 1. Random Input Variable Specifications. Material
Name
Characteristic value
Value
Variable parameter
Histogram
Mean value or Minimum value
Stand.deviation or
Concrete plate
Young´s modulus
EX1 [kPa]
3.100E+07
E1var_
GAUS
1
0.05
Poissons ratio
mi1 [-]
0.2
mi1var_
GAUS
1
0.05
Geometric
thickness
H1 [m]
0.8
H1var_
UNIF
-0.01
+0.01
Cobiax plate
Young´s modulus
EX2 [kPa]
2.759E+07
E2var_
GAUS
1
0.05
Poissons ratio
mi2 [-]
0.2
mi2var_
GAUS
1
0.05
Geometric
thickness
H2 [m]
0.8
H2var_
UNIF
-0.01
+0.01
Young´s modulus
EX3 [kPa]
1.800E+5
E3var_
GAUS
1
0.05
Poissons ratio
mi3 [-]
0.2
mi3var_
GAUS
1
0.05
Uniformly distributed
q [ kN/m2 ]
12
qvar_
LOG1
1.0
0.1
Forces
F1 – F4 [kN]
962 - 3888
Fvar_
LOG1
1.0
0.1
Soil
Load
Max. value
The individual input parameters varied according to the Tab. 1. Change of geometric characteristics of the plate was defined via H1, H2 (plate thickness) H1var_, H2var_ (change in thickness of the plate). The stiffness of the plate was determined through Young´s modulus EX1, EX2 and variable factor E1var_, E2var_, stiffness of the elastic foundation via EX3 and through a variable factor E3var_.
Katarina Tvrda / Procedia Engineering 190 (2017) 516 – 521
The load was determined through the q and F and through variables qvar_ and Fvar_. Resulting from variability of input quantity 45 simulations on the base of RSM method were realized. The probability of exceeding the limit deflection of plate structures was calculated using one million Monte Carlo simulations for 45 simulations of approximation method RSM on the structural FEM model. The probability of failure is equal to 6.79e-2 for the limit deflection - 0.0105 m exceed. 5. Summary The main aim of this article was to analyze, lightweight foundation plate rested on an elastic half-space. Elastic half-space was modeled via element Solid185 and foundation plate via Shell181 among which was modeled solid contact. The base plate was made using Cobiax technology, ground mass of the earth G2 lying on a rock. After the deterministic analysis the probabilistic analysis was carried out. The aim was to determine the probability of exceeding marginal deflection in the foundation plate, according to ten input parameters defined in Table 1. The calculation was carried out in Ansys 14.5 software application, based on stochastic FEM using RSM approach. Acknowledgements This contribution is the result of the research supported by the grant from VEGA Slovak Grant Agency Project No. 1/0544/15. References [1] R. Klucka, R. Frydrysek, M. Mahdal, Measuring the deflection of a circular plate on an elastic foundation and comparison with analytical and FE approaches, Applied Mechanics and Materials (2014). pp.407-412. [2] K. Kotrasova, E. Kormanikova, The ground plate on the Winkler foundation. in: Modeling in mechanics, 2009, VSB TU, Ostrava, pp. 1-6. [3] M. J. Tomlinson, Foundation Design and Construction, Pearson Education Ltd, England, 2001. [4] P. Marek, J. Brozzetti, M. Gustav, Probabilistic Assessment of Structures Using Monte Carlo Simulation Background, Exercises and Software, ITAM CAS, Prague, Czech Republic, 2003. [5] J. Duan, G. Duan, W.L.J. Jin, Probabilistic approach for durability design of concrete structures in marine environments, Journal of Materials in Civil Engineering. Vol. 27, Iss. 2, 2015. [6] J. Kralik, Reliability analysis of structures using stochastic finite element method, Edition of scientific papers, STU Bratislava, Iss. 77 (2009). [7] J. Kralik, J., A RSM approximation in probabilistic nonlinear analysis of fire resistance of technology support structures, Advanced Materials Research, Volume 969, 2014, Pages 1-8, 2nd International Conference on Structural and Physical Aspects of Civil Engineering, SPACE 2013; High Tatras; Slovakia; 2013. [8] D. Novak, M. Vorechovsky, B. Teply, FReET: Software for the statistical and reliability analysis of engineering problems and FReET-D: Degradation module, Advances in Engineering Software, 2014. [9] M. Krejsa, P. Janas, V. Krejsa, Structural reliability analysis using DOProC method, Procedia Engineering, Volume 142, 2016, Pages 34-41, Sustainable Development of Civil, Urban and Transportation Engineering Conference, 2016; Ton Duc Thang Univeristy19 Nguyen Huu Tho Street, Tan Phong WardHo Chi Minh City; Viet Nam; 2016; [10] A. Haldar, S. Mahadevan, Probability, reliability and Statistical Methods in Engineering Design. John Viley&Sons. New York, 2000. [11] O. C. Zienkiewicz, R L. Taylor, The Finite Element Method. Volume 2: Solid Mehanics. 5rd ed. Oxford: BUTTERWORTH-HEINEMANN, 2000. 479 pp. [12] H. Langen, Numerical analysis of Soil Structure interaction. 1.ed. Delf, university of Technology; 1991, 141pp. [13] V. Kolář, I. Němec, V. Kanocký, Principles and Practice of Finite Element Method. Praha: Computer Press; 1997, 402 pp. (in Czech) [14] Z. Bittnar, J. Šejnoha, Numerical Methods in Mechanics - 2. Praha: ČVUT, 1992. 261 pp. (in Czech) [15] N. Jendželovský, L. Baláž, Modeling of a gravel base under the cylindrical tank, Advanced Materials Research, Volume 969, 2014, Pages 249-252, 2nd International Conference on Structural and Physical Aspects of Civil Engineering, SPACE 2013; High Tatras; Slovakia; 2013 [16] J. Sumec, J. Kuzma, L Hruštinec, Experimental tests of the deformation and consolidation properties of the cohesive soils and their use in the geotechnical calculations, EAN 2010: 48th International Scientific Conference on Experimental Stress Analysis, 2010, Pages 415-420, Velke Losiny; Czech Republic; 31 May 2010 through 3 June 2010; [17] K. Kotrasova, E. Kormanikova, A case study on seismic behavior of rectangular tanks considering fluid - Structure interaction, International Journal of Mechanics, Volume 10, 2016, Pages 242-252 [18] M. Krejsa, J.Kralik, Probabilistic Computational Methods in Structural Failure Analysis Multiscale Modelling 6 (2015): 1550006 DOI: http://dx.doi.org/10.1142/S1756973715500067 [19] ANSYS ® User’s Manual for Revision 14.5, Swanson Analysis Systems, Inc.
521
ScienceDirect Procedia Engineering 190 (2017) 516 – 521
Structural and Physical Aspects of Construction Engineering
RSM Method in Probabilistic Analysis of the Foundation Plate Katarina Tvrdaa,* a
Slovak University of Technology in Bratislava, Faculty of Civil Engeenering, Radlinského 11, 81101 Bratislava, Slovakia
Abstract Many new approaches are currently used in the design and assessment of building structures. One of these approaches is the probabilistic analysis allowed by the European standards for the assessment of structural design. Input parameters are given as stochastic ones and they are characterized by different distribution types. During a probabilistic analysis, software executes multiple analysis loops to compute the random output parameters as functions of the set of random input variables. Uncertainties in input variables (load, material properties and dimensions structures) are reflected in the RSM method based on approximation of Monte Carlo simulations. Finally, some results of these probabilistic analyses are presented. © Published by Elsevier Ltd. This © 2017 2017The TheAuthors. Authors. Published by Elsevier Ltd. is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the issue editors. Peer-review under responsibility of the organizing committee of SPACE 2016 Keywords: foundation plate; Responce Surface Method; probability design; Monte Carlo Simulation; Elastic Half Space; bilateral bounds
1. Introduction The design practice, more and more statically indeterminate building structures are increasingly suggested. In the design of such structures it is necessary to take into account the complex material, geometry and boundary conditions. In solving and modeling tasks of soil-structure interaction various idealized and simplified procedures are reported in the literature that only approximately describe the actual boundary conditions, what affects compliance of achieved results with reality. Every building structure is imposed on the certain type of soil. At the beginning simple approaches, as the Winkler model (one-parametric), Pasternak model (two-parametric background model) and the Boussinesq model (elastic half-space), were used in modeling of subsoil. Complex processes began with appearing and developing of numerical methods, the use of which was dependent on the level of computer equipment and software. For example, not all commercial software application can specify two-parametric soil
* Corresponding author. Tel.: +421-2-59274291 E-mail address: [email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of SPACE 2016
doi:10.1016/j.proeng.2017.05.372
517
Katarina Tvrda / Procedia Engineering 190 (2017) 516 – 521
model. In the field of interaction structures with the subsoil, there are numerous papers [1, 2] published in journals and in the books [3]. In modeling of an elastic half-space, it is necessary to model a relatively large portion of ground mass, and outside the active depth of deformation. In assessing the structure, international standards and regulations, together with the National Annexes, are used. Eurocodes also permit assessment of building structures based on probabilistic analysis. Many papers, dealing with problems, mentioned above, show, that reliability (safety, serviceability, durability) of the design of structure throughout the service life is also influenced by its correct model of foundation [4, 5, 6, 7, 8, 9, 10]. 2. Interaction of the plate with subsoil An inseparable part of the building structure is ground mass in which the structure is built. This massive area is infinitive, but in modeling of structures in the interaction with the subsoil it must have its limits. It is therefore important to choose the right mathematical and material model, which describes the background of the problem. The elastic foundation can be modeled with these types of models: x One-parametric model (Winkler model) x Two-parametric model (Pasternak) x Elastic half-space model (Boussinesq). 2.1. One-parametric model - Winkler model Winkler model is the first model of the subsoil, background on the hypothesis of setting coefficient (coefficient of subgrade stiffness), which assumes that at any point of the subsoil there is the load p directly proportional to deflection w at this point, but it is not dependent on the deflection at the other points. Another drawback is the instability of the setting coefficient (subgrade stiffness) k which depends not only on the material properties of the subsoil, but the dimensions and the shape of the foundation structure as well. Plate on an elastic foundation is defined by partial differential equation (1). D 22 w( x, y) k w( x, y)
(1)
q( x, y)
where D is a plate constant (2), E - modulus of elasticity of the material of plate, h - thickness, Q - Poisson´s ratio. D
E h3
(2)
12 1 Q 2
2.2. Two-parametric model – Pasternak model Two-parametric model of the subsoil is defined by Pasternak so that the contact stress p is defined according to the deflection surface of w (x, y) and coefficients C1 - coefficient of friction in the vertical direction (N / m3), and C2 - coefficient characterizing the shear spreading of the effects of the load (N / m) obtained using energy principles. This model allows to take into account deformations beyond the foundation. ª w 2 w( x, y ) w 2 w( x, y ) º D 22 w( x, y) C1 w( x, y) C2 « » 2 wy 2 ¼ ¬ wx
q( x, y)
(3)
2.3. Elastic half-space model - Boussinesq Elastic half-space is another type of mathematical - material model of the elastic subsoil. The theory of elastic continuous half-space implies that the background forms a continuous, homogeneous and perfectly elastic body of
518
Katarina Tvrda / Procedia Engineering 190 (2017) 516 – 521
an infinitely large size, limited from above by the plane on which lies the underlying building structure. Sometimes it is called the Boussinesq soil model. The model is characterized by two material constants derived from experimental measurements (E - modulus of elasticity and Q - Poisson´s constant). Numerical methods are primarily used to solve contact problems (subgrade model - elastic half-space) to appropriately extend the closed form solutions while offering the possibility of more realistic capturing the interaction characteristics with the subsoil building structure for different boundary conditions. The most famous is FEM. Theoretical assumptions of Boundary Element Method - BEM and Finite Element method - FEM in interaction were dealt by Zienkiewicz and Taylor [11], Langen [12], Kollař et al. [13], Bittnar, Šejnoha [14], Jendželovský, Baláž [15], Sumec et al. [16], Kotrasova, Kormanikova [17], and others. Elastic foundation is modeled by spatial (3D) finite elements. Finite elements of different shapes can be used for the modeling. The most commonly used form are solid block elements with 8 nodes, and three displacements at each node. In modeling of elastic half-space it is necessary to create a sufficiently large volume of grown earth, which depends on supports, boundary conditions. The support may be rigid, flexible or usage of infinite finite elements. For the analysis of interaction plate ground with the subsoil (static binding on the contact surface) two basic models can be defined: computational model of bilateral bonds - continuous model, calculation model with a unilateral bonds - discrete model. 3. Probability and Reliability Under the term of the reliability of the design we understand the ability of the structure or structural member to fill special requirements, including design service life for which the structure was designed. Reliability implies safety, durability and serviceability of the structure and it is expressed through probabilistic terms. The structure shall be designed and carried out over the estimated useful life with the appropriate degree of reliability and economy resist all loads and influences that may occur in the operation and use of the structure and serves that purpose. When designing the building structure or its section, the reliability must be taken in particular to the design, i.e. it must fill the required properties throughout the service life of the structure. Design methods and assessment structures will evolve and grow gradually as the theoretical and practical possibilities proposal (introduction of computers into practice). In the development of different design methods, the practices included uncertainty and ensuring reliability of structures in close dependence on theoretical knowledge of elasticity, mathematical statistics and experimental measurements. Probabilistic methods may be divided into: x The first order reliability method – FORM x The second order reliability method – SORM x Simulation methods. 3.1. Probabilistic methods of reliability assessment To determine the reliability of probabilistic methods first performance criteria on the basis of functional relationship between the n-input quantities are defined referred to as the basic random variables Xi. This relationship is called the reliability function Fs (RF - reliability function), safety, serviceability, reliability and function of the reserves can be expressed generally as a function of stochastic parameters X1 , X 2 , , Xn .
Fs
RF
g ( X1 , X 2 ,
, Xn )
(4)
where the functional dependence g ( X1 , X 2 , , X n ) is a computational model (i.e. idealization of reality). Failure functions g ( X i ) express the conditions of preliability (reserve) and can be represented as a function of stochastic parameters. They may be defined as a simple (e.g. for one section) or as complex structures, for more cross-sections (e.g. all finite-element model). The reliability function can be defined as:
519
Katarina Tvrda / Procedia Engineering 190 (2017) 516 – 521
Fs
RF
RS
(5)
where R is the resistance of the structure function (yield strength, allowable deflection, stress), S is a function of load (max. stress in the structure, max. deflection). The function is reliable if Fs ≥ 0, failure occurs if Fs <0 and Fs = 0 is the limit function. The probability of failure can be simply defined as: Pf
P>R S @
P > R S 0@
(6)
where Pf is generally given by the expression
Pf
³³
³³
f X ( X1 , X 2 ,
, X n ) dX1 dX 2
(7)
dX n .
g(X ) 0
The function f x ( X1 , X 2 ,..., X n ) represents the joint density of probability function for continuous randomly variables or multiply probability function for discrete random variables, where integration is transferred over the entire area of failure g (X) <0. According to the method of integral solutions of the equation (7) and reliability assessment the probabilistic methods may be divided to: x Analytical - a direct solution of the integral in relation (7) is impossible, x Simulation methods - direct and approximation. As simulation methods we may mention Monte Carlo – MC method, LHS - Latin Hypercube Sampling and SC Stratified Sampling. There are also advanced simulation methods as: Adaptive Sampling - AS, Direction Sampling DS, IS - Importance Sampling, LS - Line Sampling, RSM - Response Surface Method. MC - Monte Carlo method can be classified within the simulation methods with direct simulation. More comprehensive overview of the methods may be found e.g. in: Krejsa [18]. 4. Assessment of the foundation plate A foundation plate under the building is a subject of an assessment. The foundation plate was built using new technologies. Cobiax Technology is based on generating specific hollows inside a reinforced concrete slab. Massive concrete is replaced by synthetic void formers and remains only in statically relevant areas. a
b
Fig. 1. (a) model of foundation plate; (b) the load of plate.
Thus, it is possible to construct buildings with flat slabs while allowing for remarkable span width. The slab is made of concrete of class C25-30, thickness of 80 cm. Concrete modulus of elasticity is 31 GPa. When modelling
520
Katarina Tvrda / Procedia Engineering 190 (2017) 516 – 521
the slab, according to the manufacturer's recommendations, the elastic modulus in the area of incorporating plastic balls is reduced according to slab thickness and Cobiax balls type (Fig.1a). The elastic modulus of 27.59 GPa was used in the calculation, and Poisson's ratio 0.2. The subsoil consists of gravel G2. They were taken target standardized gravelly soil characteristics E = 180 MPa, Poisson's ratio 0.2. The foundation plate was of the size of 29.7 x 16.2 m. Ground mass was modeled in sufficient width, length and depth of 89.7 x 48.2 x 15 m. Then follows the rock massive. The foundation plate is loaded in the area of columns, which is calculated according to the relevant area of the load and transfers the load of 8 floors. Uniformly distributed load is 12 kN / m 2 (Fig.1b). The easiest way to model plate on elastic foundation was to use elements SHELL 63 in ANSYS [19], with the possibility of introducing the coefficient of stiffness of the subsoil via EFS command. Unfortunately in newer versions of ANSYS this element is not supported. In our case, the soil was modeled as an elastic half-space by Solid 185 elements, plate by Shell 181elements. For the analysis of interaction foundation plate with the subsoil, depending on the static binding of a contact area between the bedrock and the foundation the model with two-sided couplings, enabling the transmission of pressure and tension forces at the contact area was used – a continuous model Fig. 2.
Fig. 2. A complex model - plate and ground mass. Table 1. Random Input Variable Specifications. Material
Name
Characteristic value
Value
Variable parameter
Histogram
Mean value or Minimum value
Stand.deviation or
Concrete plate
Young´s modulus
EX1 [kPa]
3.100E+07
E1var_
GAUS
1
0.05
Poissons ratio
mi1 [-]
0.2
mi1var_
GAUS
1
0.05
Geometric
thickness
H1 [m]
0.8
H1var_
UNIF
-0.01
+0.01
Cobiax plate
Young´s modulus
EX2 [kPa]
2.759E+07
E2var_
GAUS
1
0.05
Poissons ratio
mi2 [-]
0.2
mi2var_
GAUS
1
0.05
Geometric
thickness
H2 [m]
0.8
H2var_
UNIF
-0.01
+0.01
Young´s modulus
EX3 [kPa]
1.800E+5
E3var_
GAUS
1
0.05
Poissons ratio
mi3 [-]
0.2
mi3var_
GAUS
1
0.05
Uniformly distributed
q [ kN/m2 ]
12
qvar_
LOG1
1.0
0.1
Forces
F1 – F4 [kN]
962 - 3888
Fvar_
LOG1
1.0
0.1
Soil
Load
Max. value
The individual input parameters varied according to the Tab. 1. Change of geometric characteristics of the plate was defined via H1, H2 (plate thickness) H1var_, H2var_ (change in thickness of the plate). The stiffness of the plate was determined through Young´s modulus EX1, EX2 and variable factor E1var_, E2var_, stiffness of the elastic foundation via EX3 and through a variable factor E3var_.
Katarina Tvrda / Procedia Engineering 190 (2017) 516 – 521
The load was determined through the q and F and through variables qvar_ and Fvar_. Resulting from variability of input quantity 45 simulations on the base of RSM method were realized. The probability of exceeding the limit deflection of plate structures was calculated using one million Monte Carlo simulations for 45 simulations of approximation method RSM on the structural FEM model. The probability of failure is equal to 6.79e-2 for the limit deflection - 0.0105 m exceed. 5. Summary The main aim of this article was to analyze, lightweight foundation plate rested on an elastic half-space. Elastic half-space was modeled via element Solid185 and foundation plate via Shell181 among which was modeled solid contact. The base plate was made using Cobiax technology, ground mass of the earth G2 lying on a rock. After the deterministic analysis the probabilistic analysis was carried out. The aim was to determine the probability of exceeding marginal deflection in the foundation plate, according to ten input parameters defined in Table 1. The calculation was carried out in Ansys 14.5 software application, based on stochastic FEM using RSM approach. Acknowledgements This contribution is the result of the research supported by the grant from VEGA Slovak Grant Agency Project No. 1/0544/15. References [1] R. Klucka, R. Frydrysek, M. Mahdal, Measuring the deflection of a circular plate on an elastic foundation and comparison with analytical and FE approaches, Applied Mechanics and Materials (2014). pp.407-412. [2] K. Kotrasova, E. Kormanikova, The ground plate on the Winkler foundation. in: Modeling in mechanics, 2009, VSB TU, Ostrava, pp. 1-6. [3] M. J. Tomlinson, Foundation Design and Construction, Pearson Education Ltd, England, 2001. [4] P. Marek, J. Brozzetti, M. Gustav, Probabilistic Assessment of Structures Using Monte Carlo Simulation Background, Exercises and Software, ITAM CAS, Prague, Czech Republic, 2003. [5] J. Duan, G. Duan, W.L.J. Jin, Probabilistic approach for durability design of concrete structures in marine environments, Journal of Materials in Civil Engineering. Vol. 27, Iss. 2, 2015. [6] J. Kralik, Reliability analysis of structures using stochastic finite element method, Edition of scientific papers, STU Bratislava, Iss. 77 (2009). [7] J. Kralik, J., A RSM approximation in probabilistic nonlinear analysis of fire resistance of technology support structures, Advanced Materials Research, Volume 969, 2014, Pages 1-8, 2nd International Conference on Structural and Physical Aspects of Civil Engineering, SPACE 2013; High Tatras; Slovakia; 2013. [8] D. Novak, M. Vorechovsky, B. Teply, FReET: Software for the statistical and reliability analysis of engineering problems and FReET-D: Degradation module, Advances in Engineering Software, 2014. [9] M. Krejsa, P. Janas, V. Krejsa, Structural reliability analysis using DOProC method, Procedia Engineering, Volume 142, 2016, Pages 34-41, Sustainable Development of Civil, Urban and Transportation Engineering Conference, 2016; Ton Duc Thang Univeristy19 Nguyen Huu Tho Street, Tan Phong WardHo Chi Minh City; Viet Nam; 2016; [10] A. Haldar, S. Mahadevan, Probability, reliability and Statistical Methods in Engineering Design. John Viley&Sons. New York, 2000. [11] O. C. Zienkiewicz, R L. Taylor, The Finite Element Method. Volume 2: Solid Mehanics. 5rd ed. Oxford: BUTTERWORTH-HEINEMANN, 2000. 479 pp. [12] H. Langen, Numerical analysis of Soil Structure interaction. 1.ed. Delf, university of Technology; 1991, 141pp. [13] V. Kolář, I. Němec, V. Kanocký, Principles and Practice of Finite Element Method. Praha: Computer Press; 1997, 402 pp. (in Czech) [14] Z. Bittnar, J. Šejnoha, Numerical Methods in Mechanics - 2. Praha: ČVUT, 1992. 261 pp. (in Czech) [15] N. Jendželovský, L. Baláž, Modeling of a gravel base under the cylindrical tank, Advanced Materials Research, Volume 969, 2014, Pages 249-252, 2nd International Conference on Structural and Physical Aspects of Civil Engineering, SPACE 2013; High Tatras; Slovakia; 2013 [16] J. Sumec, J. Kuzma, L Hruštinec, Experimental tests of the deformation and consolidation properties of the cohesive soils and their use in the geotechnical calculations, EAN 2010: 48th International Scientific Conference on Experimental Stress Analysis, 2010, Pages 415-420, Velke Losiny; Czech Republic; 31 May 2010 through 3 June 2010; [17] K. Kotrasova, E. Kormanikova, A case study on seismic behavior of rectangular tanks considering fluid - Structure interaction, International Journal of Mechanics, Volume 10, 2016, Pages 242-252 [18] M. Krejsa, J.Kralik, Probabilistic Computational Methods in Structural Failure Analysis Multiscale Modelling 6 (2015): 1550006 DOI: http://dx.doi.org/10.1142/S1756973715500067 [19] ANSYS ® User’s Manual for Revision 14.5, Swanson Analysis Systems, Inc.
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