A fast fuzzy finite element approach for laterally loaded pile in layered soils

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A fuzzy finite element approach for static analysis of laterally loaded pile in multi-layer soil with uncertain properties is presented. The finite element (FE) formulation is established using a beam-on-two-parameter foundation model. Based on the developed FE model, uncertainty propagation of the soil parameters to the pile response is evaluated by mean of the α-cut strategy combined with a response surface based optimization technique. Journal of Science and Technology in Civil Engineering NUCE 2018 12 (3): 1–9 A FAST FUZZY FINITE ELEMENT APPROACH FOR LATERALLY LOADED PILE IN LAYERED SOILS Pham Hoang Anha,∗ a Faculty of Building and Industrial Construction, National University of Civil Engineering, 55 Giai Phong road, Hai Ba Trung district, Hanoi, Vietnam Article history: Received 05 October 2017, Revised 05 March 2018, Accepted 27 April 2018 Abstract A fuzzy finite element approach for static analysis of laterally loaded pile in multi-layer soil with uncertain properties is presented The finite element (FE) formulation is established using a beam-on-two-parameter foundation model Based on the developed FE model, uncertainty propagation of the soil parameters to the pile response is evaluated by mean of the α-cut strategy combined with a response surface based optimization technique First order Taylor’s expansion representing the pile responses is used to find the binary combinations of the fuzzy variables that result in extreme responses at an α-level The exact values of the extreme responses are then determined by direct FE analysis at the found binary combinations of the fuzzy variables The proposed approach is shown to be accurate and computationally efficient Keywords: laterally-loaded pile; uncertainty; fuzzy finite element analysis; α-cut strategy; response surface method; optimization c 2018 National University of Civil Engineering Introduction Piles subjected to lateral loadings can be found in many civil engineering structures such as offshore platforms, bridge piers, and high-rise buildings For the design of pile foundations of such structures, special attention needs to be concentrated not only on the bearing capacity but also on the behavior (horizontal displacement, stress) of the piles under lateral loading conditions The deterministic analysis of lateral loading behavior of piles is complicated and in general requires a numerical solution procedure (e.g., the finite difference method, finite element method) On the other hand, uncertainty is often present in the input data, especially in geotechnical engineering data These uncertainties can be accounted for by using probabilistic methods, e.g., methods proposed in [1–6] However, very often the input data fall into the category of non-statistical uncertainty The reason for this uncertainty is that the made observations could be best categorized with linguistic variables (e.g., the soil may be described with linguistic variables such as “very soft,” “soft,” or “stiff”; “loose”, “dense”, or “very dense”), or that only a limited number of samples are available ∗ Corresponding author E-mail address: anhph2@nuce.edu.vn (Anh, P H.) Anh, P H / Journal of Science and Technology in Civil Engineering and a particular soil properties are unknown or vary from one location to another location These types of uncertainties can be appropriately represented in the mathematical model as fuzziness [7] In recent years, non-probabilistic FE methods based on fuzzy set theory have been introduced to the analysis of uncertain structural systems The fuzzy FE methods have been applied for both static and dynamic analysis of various structures [8–11] In this paper, an efficient fuzzy FE approach is developed to analyze the response of laterally loaded pile in multi-layer soils It is assumed that only rough estimates of the soil parameters are available and these are modeled as fuzzy values The analysis of the pile-soil interaction is based on a “Beam-on-two-parameter-linear-elastic-foundation” FE model The fuzzy pile response is estimated by a response surface based optimization technique using first order Taylor’s expansion of the pile response The accuracy and computational efficiency of the proposed approach are illustrated in a numerical example xxx 2.1 General fuzzy structural analysis Fuzzy model of uncertainty practical engineering problems, 2.1.In Fuzzy model of uncertainty there are randomness and fuzziness associated with the model parameters (e g material properties, geometrical dimensions, loads) These uncertainties can be modeled randomness andasfuzziness theand model X (Xare , Xassociated ) with X with inAmong form of practical fuzzy setsengineering [7] Accordingproblems, to [7] a fuzzy set is defined is a set parameters (e.g material properties, geometrical dimensions, loads) These uncertainties can be [0,1] X ,µthe function Corresponding to each element ˜x value X modeled in form isofcalled fuzzythe setsmembership [7] According to [7] a fuzzy set is defined as X = (X, X ) with X is a set Xand → [0,membership 1] is calledlevel the of membership totoeach element X, the (x )µisX called defines theCorresponding level of x belong the fuzzy set Xx ∈The x ; X (x ) function ˜ value µX (x) is called membership level of x; µX (x) defines the level of x belonging to the fuzzy set X value state that x is not belong to X ; the value means that x is definitely belong to X ; the value ˜in ˜ The value states that x does not belong to X; the value means that x definitely belongs to X; the X is uncertain interval to shows the level x belonging value in interval to 1that shows thatofthe level of xtobelonging to X˜ is uncertain The α-cut, X of the fuzzy set X˜ is a set of elements xX∈ X with the membership level α: (x )µX (x) ≥ The α-cut, X αof the fuzzy set X is a set of elements x with the membership level : X X{ ∈X 1] α x= {xX : : Xµ(Xx(x) ) ≥ α}, }, α ∈ [0,[0,1] X Fig.1 1illustrates illustrates membership function of a triangular Fig the the membership function and an and α-cutan of α-cut a triangular fuzzy set (1) (1) fuzzy set μX (x) Xα α xα,min x, X xα,max Figure andthe theα-cut α-cutofofa afuzzy fuzzy Figure1.1.Membership Membership function function and setset 2.2 The α-level optimization 2.2.Consider The α-level optimization a model output y given by y f (x1, x 2, , xn ) with x i being n fuzzy input variables, output by y = f (x1 , xbe , xfunction being n model, fuzzy input variables, , any n ) withorxinumerical xConsider Xi : a model [0,1]y given The function f ( ) can e.g the finite i Xi (x ) xi ∈ Xi : µXi (x) → [0, 1] The function f (·) can be any function or numerical model, e.g the finite element model Through the mapping function f ( ) , the output y is also a fuzzy quantity represented by its output fuzzy setY {y Y : Y (y ) [0,1]} A practical mean to determine the membership function of y , Y (y ) , is the α-cut strategy [8] Here, the fuzzy input variables are discretized into levels, k ,(k = 1,2, ,m ) Corresponding to each level k m , we have crisp sets of values of inputs, Anh, P H / Journal of Science and Technology in Civil Engineering element model Through the mapping function f (·), the output y is also a fuzzy quantity represented by its output fuzzy set Y˜ = {y ∈ Y : µY (y) → [0, 1]} A practical mean to determine the membership function of y, µY (y), is the α-cut strategy [8] Here, the fuzzy input variables are discretized into m levels, αk , (k = 1, 2, , m) Corresponding to each level αk , we have crisp sets of values of ˜ is then inputs, Xi,αk ⊂ Xi The output interval of y corresponding to level αk (the αk -cut Yαk of Y) determined by interval analysis of the input sets Xi,αk through the mapping model f (·) Thus, a discrete approximation of the membership function of the output can be obtained by repeating the interval analysis on a finite number of αk -levels Fig illustrates the fuzzy analysis using the α-cut xxx strategy for a function of two input variables αk Interval analysis of level α x1 αk y αk x2 Figure Illustration of fuzzy analysis by α-cut strategy Figure Illustration of fuzzy analysis by α-cut strategy The smallest and largest values (the extreme values) of the α-cut Y define two points of the membership k The smallest and largest values (the extreme values) of the α-cut Yαk define two points of the function of the fuzzy output, Y The exact extreme values of the α-cut Y are often determined by ˜ The exact extreme valuesk of the α-cut Yαk are often membership function of the fuzzy output, Y solving two optimization problems, which referred as the α-level optimization [12]: determined by solving two optimization problems, which are referred as the α-level optimization [12]: y y ,min f (x1, x , , x n ) Xi , ykαk ,min =xi k ( f (x1 , x2 , , xn )) xi ∈Xi,αk m ax k ,m ax f (x1, x , , x n ) (2) (2) Xi , k ( f (x , x , , x )) yαk ,max = ximax n xi ∈Xi,αk The solution for the optimization problems of Eq 20 can be numerical demanding In order to reduce the computational burden, researchers have focused on efficient procedures to reduce the number of function The solution for the optimization problems Eq (2) can be numerical demanding In order evaluations in performing these optimization problemsof[8,10,11] to reduce the computational burden, researchers have focused on efficient procedures to reduce the This paper introduces a fast solution for the above optimization problems based on a response surface number of function evaluations in performing optimization problems [8, 10, method, which is applicable for the fuzzy analysisthese of laterally loaded piles with uncertain soil 11] parameters The methodology is presentedainfast the followings This paper introduces solution for the above optimization problems based on a response Fuzzy finite element analysis of laterally loadedanalysis pile surface method, which is applicable for the fuzzy of laterally loaded piles with uncertain soil parameters The methodology is presented in the followings 3.1 Model of analysis Consider a vertical pile embed in aof soillaterally deposit containing n layers, with the thickness of layer i Fuzzy finite element analysis loaded pile 3.1 given by Hi (Fig 1(a)) The top of the pile is at the ground surface and the bottom end of the pile is considered embedded in the nof -thanalysis layer Each soil layer is assumed to behave as a linear, elastic material with the compressive Model resistance parameter ki and shear resistance parameter ti The pile is subjected to a lateral force F0 and a Consider a vertical pile embed in a soil deposit containing n layers, with the thickness of layer i moment M at the pile top The pile behaves as an Euler–Bernoulli (EB) beam with length Lp and a constant given by Hi (Fig 3(a)) The top of the pile is on the ground surface and the bottom end of the pile flexural rigidity EI The governing differential equation for pile deflection wi within any layer i is given in [13]: is considered embedded in the n-th layer Each soil layer is assumed to behave as a linear, elastic material with the compressive resistance parameter ki 2and shear resistance parameter ti The pile EI d wi dz kiwi 2ti d wi dz (3) The equation (3) is exactly the same as the equation for the “Beam-on-two-parameter-linear-elasticfoundation” model introduced by Vlasov and Leont’ev [14] The use of linear elastic analysis in the laterally loaded pile problem, especially in the prediction of deformations at working stress levels, has become a widely accepted model in geotechnical engineering Also in the real problem where nonlinear stress-strain relationships for the soil must be used, linear elastic solution provides the framework for the analysis, in which Anh, P H / Journal of Science and Technology in Civil Engineering is subjected to a lateral force F0 and a moment M0 at the pile top The pile behaves as an Euler– Bernoulli (EB) beam with length L p and a constant flexural rigidity EI The governing differential equation for pile deflection wi within any layer i is given in [13]: EI d4 wi d wi + k w − 2t =0 i i i dz4 dz2 (3) Eq (3) is exactly the same as the equation for the “Beam-on-two-parameter-linear-elasticfoundation” model introduced by Vlasov and Leont’ev [14] The use of linear elastic analysis in the laterally loaded pile problem, especially in the prediction of deformations at working stress levels, has become a widely accepted model in geotechnical engineering Also in the real problem where nonlinear stress-strain relationships for the soil must be used, linear elastic solution provides the framework for the analysis, in which the elastic properties of the soil will be changed xxxwith the changing deformation of the soil mass (e.g., the “p–y” method [15]) this paper, this Beam-on-linear-elastic-foundation model is the basis for the finite element formulation of xxx xxx InIn this paper, this Beam-on-linear-elastic-foundation model is the basis for the finite element the laterally loaded pile problem which will be presented in the next section formulation of the laterally loaded pile problem which will be presented in the next section InInthis paper, this Beam-on-linear-elastic-foundation model is the basis for the finite element formulation of this paper, this Beam-on-linear-elastic-foundation model is the basis for the finite element formulation of the problem which in in thethe next section thelaterally laterallyloaded loadedpile pile which presented next section M 0problem F0 willwillbebepresented MM 0 Layer F0F0 H1 Layer 11 Layer H1H1 Layer 22 Layer H2H2 Layer … w ww ww Beam-type element H2 qjθ … Lp LpLp …… w node j …… Layer i qjθqjθ nodej j node w2, Q2 Hi Layer Layer i i HiHi … …… w2w , Q2,2Q2 ,M2,M θ θθ22,M 22 … …… Layer Layer Layern nn zz z z zz zz z (a) (a) (a) (a) Figure (a) qjw qjwqjw Beam-type Beam-type θelement M1 1, element θ1,M θ1,M 1 we w1, Q1 wewe w1w ,Q11,Q1 le le le (b) (b) (b) (b) (c) (c)(c) (c) Figure3.3.(a)(a)A A laterally-loaded pile a layered soil; discretization; laterally-loaded pile in in asoil; layered (b)(b) FEFE discretization; AFigure laterally-loaded pile in a layered (b) soil; FE discretization; (c) Beam-type Figure (a) A laterally-loaded pile in a layered soil; (b) FE discretization; (c) Beam-type element (c) Beam-type element 3.2 Finite element modeling 3.2 Finite element modeling element (c) Beam-type element 3.2 The Finite element 3.2 Finite elementmodeling modeling pile is divided into m finite elements and to each j -th node of their interconnection, two degrees of The pile is divided into m finite elements and to each j -th node of their interconnection, two degrees of q j each The pileare divided m deflection finite elements and each node of with theirpositive interconnection, freedom allowed: –finite the andq to – thej to rotation ofj-th cross section directionofas two q into The pile isisallowed: divided into elements and -th node theirsection interconnection, twodirection degrees m freedom are – the deflection and – the rotation of of cross with positive as q jwjw j degrees of freedom are allowed: q the deflection and q the rotation of cross section with jw is chosen jθ q in Figure 1(b) Element of EB-beam type for each pile element with length and two nodes, l freedom are allowed: – the deflection and theeach rotation ofelement cross section with positive as qofjw EB-beam j –for eand direction in Figure 1(b) Element type is chosen pile with length two nodes, l e positive direction in element Fig 3(b) Elementtoofother EB-beam forToeach element one at each end.asThe is connected elementstype only is at chosen the nodes eachpile element, two with one at each end.Element The element is connected to other elements only at the with nodes To each element, two at the in Figure 1(b) of EB-beam type is chosen for each pile element length and two nodes, l length l and two nodes, one at each end The element is connected to other elements only e degrees of freedom are allowed at both ends: deflection, w and rotation, , and we , respectively, degrees of freedom are allowed at both ends: deflection, w 1and rotation, , 1and w , 2 2respectively, oneTo at each each element, end The element is connected to otherare elements only atboth the ends: nodes deflection, To each element, tworotation, nodes two degrees of freedom allowed at w1 and positive in the system of local axes as shown in Figure 1(c) The element nodal displacement vector q e positive in the system of local axes as shown in Figure 1(c) The element nodal displacement vector q degrees are allowed at both deflection, rotation, , 3(c) respectively, θ1 , and w2 , θof2 freedom respectively, positive in ends: the system of wlocal as shown Thee element w 2Fig andaxes , and in and the element nodal force vector r with respect to the system of local axes are defined: nodal displacement vector element nodal force vector with respect to the system of {q} {r}edisplacement ewith and the element nodal force vector respect to1(c) the system of local axes are defined: e and r the positive in the system of local axes as shown in Figure The element nodal vector q e e local axes are defined: T T (4) w r w withT respect , rto the system Q1 M Q2 M and the element nodalqforce vector of1local axes 2T are defined: (4) q e e w1 1 1w2e 2 , T r e e Q1 M Q2 M T (4) {q}e = {w1 θ1 w2 θ2 } , {r}e = {Q1 M1 Q2 M2 } It is noted that Q and Q from (4) include shear T force in the pile section and also T shear force in the soil It is noted that Q and Q shear ,forcerin also shear force in the soil.(4) q efrom (4) w1include M1 Qand ethe pileQ1section w2 2 M2 The equilibrium equation of an element has the form: The equilibrium equation of an element has the form: It is noted that Q1 and Q from (4) include kshearqforce in the r pile section and also shear force in the soil (5) k eq e r e (5) e e e The equilibrium has form: the stiffness matrix of one-dimension finite element k kof an element k k the In equation (5) equation represents Anh, P H / Journal of Science and Technology in Civil Engineering It is noted that Q1 and Q2 from (4) include shear force in the pile section and also shear force in the soil The equilibrium equation of an element has the form: [k]e {q}e = {r}e (5) In Eq (5) [k]e = [k]b + [k]w + [k]t represents the stiffness matrix of one-dimension finite element of pile on two-parameter elastic foundations The terms of [k]b , [k]w , [k]t matrices have been established in [16] as:    12 −6le −12 −6le  EI  −6le 4le2 6le 2le2  [k]b =  (6)  12 6le  le  −12 6le  −6le 2le2 6le 4le2   54 13le   156 −22le  kle  −22le 4le2 −13le −3le2  [k]w = (7)   −13le 156 22le  420  54  13le −3le2 −3le2 4le2    36 −3le −36 −3le   2t  −3le 4le2 3le −le2  [k]t = (8)   36 3le  30le  −36 3le  −3le −le2 3le 4le2 The system equation is obtained by assembly of all elements, implementation of boundary conditions, and introduction of loads 3.3 Proposed fuzzy analysis Assume that a pile response y is monotonic with respects to the fuzzy soil parameters , i = 1, 2, , n, (here can be compressive parameters or shear parameters) A first order Taylor’s expansion of y at the soil parameter value (a01 , a02 , , a0n ) given by n y(a1 , a2 , , an ) y(a01 , a02 , , a0n ) + y˙ 0i (ai − a0i ) (9) i=1 where y˙ 0i is the partial derivative of y with respect to the parameter , taken at (a01 , a02 , , a0n ) The extreme values of y at an α-level can be determined then as n ymin = y(a01 , a02 , , a0n ) + y˙ 0i (ai − a0i ) i=1 n ymax = y(a01 , a02 , , a0n ) + (10) max y˙ 0i (ai − a0i ) i=1 or for monotonic function, n ymin = y(a01 , a02 , , a0n ) + y˙ 0i (ai,min − a0i ), y˙ 0i (ai,max − a0i ) i=1 n ymax = y(a01 , a02 , , a0n ) + (11) max i=1 y˙ 0i (ai,min − a0i ), y˙ 0i (ai,max − a0i ) Anh, P H / Journal of Science and Technology in Civil Engineering where ai,min and ai,max are the lower and upper bound of , respectively, corresponding to that α-level Since Eq (9) is only an approximation of the actual response, the extreme values obtained by (11) xxx not represent the real bounds of the response To calculate the exact bounds of y, we directly evaluate not represent the real bounds of the response To calculate the exact bounds of y , we directly evaluate y using FE analysis at the binary combinations of the fuzzy parameter values that result in the extreme y using FE analysis at the binary combinations of the fuzzy parameter values that result in the extreme responsesresponses of (11).of (11) Furthermore, the partial derivative y˙ 0i is approximated as: Furthermore, the partial derivative yi is approximated as: y(a0 , a0 , , a0 + δai , , a0n ) − y(a01 , a020, , a0i − 0δai , , a0n ) ai , , an ) y˙ 0i y 1y(a102, a20, , aii0 , , an0 ) y(a10, a20, , 2δai i (12) (12) where δai is a small variation of , taken as 0.001a0 0i in this study The determination of y˙ is carried where 𝛿𝑎𝑖 is a small variation of 𝑎𝑖 , taken as 0.001ai in this study The determination of yi0 is carriedi 0 out once for each , with (a1 , a2 , , an ) to be the value of the fuzzy variable having the memberonce for each 𝑎𝑖 , with (a10 , a20 , , an0 ) to be the value of the fuzzy variable 𝑎𝑖 having the membership ship of 1.outThus, the proposed approach requires 2(n + m) + model analysis to approximate the fuzzy of Thus, the proposed approach requires 2(𝑛 + 𝑚) + model analysis to approximate the fuzzy membership function of aofpile where mthe is number the number of discretized membership levels membership function a pileresponse, response, where 𝑚 is of discretized membership levels The flowchart of the proposed fuzzy analysis is presented in Fig The flowchart of the proposed fuzzy analysis is presented in Fig Begin Pile, soil data, n, m FE Modeling Calculate y(a10, a20, , an0 ); y i0 ; k=1 FALSE k≤n TRUE Determine Determine ai,min , ,max at level αk y k ,min ;y k ,m ax ; k = k+1 Membership function End Figure Flowchart proposedfuzzy fuzzy analysis for pile Figure Flowchartof ofthe the proposed analysis for pile Application JOURNAL OF SCIENCE AND TECHNOLOGY IN CIVIL ENGINEERING xxx To verify the above approach, a laterally-loaded pile taken from [17] is analyzed The pile of length L p = 20 m, cross-section radius r p = 0.3 m and modulus E p = 25 × 106 kN/m2 is subjected Application Application To verify the above approach, a laterally-loaded pile taken from [17] is analyzed The pile of length Lp =20 To verify the above approach, a laterally-loaded pile taken from [17] is analyzed The pile of length Lp =20 m, cross-section radius rp =0.3 m and modulus E p =25×106 kN/m is subjected to a lateral force F is subjected E pand m, cross-section radius rpAnh, =0.3P m modulus =25×106 kN/m H and / Journal of Science Technology in Civil Engineeringto a lateral force F0 =300kN and a moment M =100 kNm at the pile head The soil deposit has four layers with =300kN and a force moment the pile deposit has four with to a lateral F0 =M300 kN kNm and a at moment M0 head = 100The kNmsoil at the pile head Thelayers soil deposit =100 has four with H and = H m, H4 =are∞ soil properties H 2layers H = H =5 m, =The soiland properties areThe uncertain and givenare by uncertain triangular and fuzzy 1H H1 H H3 H =53 m, and1 H 24 3The soil properties uncertain and given by triangular fuzzy given2 by triangular fuzzy numbers: k1 = (33.6, 56.0, 78.4) MPa, k2 = (84.0, 140.0, 196.0) MPa, numbers: k1 =(33.6, 56.0, 78.4) MPa, k =(84.0, 140.0, 196.0) MPa, k3 =(93.0, 155.0, 217.0) MPa and k1 =(33.6, k3 =(93.0, numbers: 56.0, 78.4) MPa, 140.0, 196.0) MPa,MPa, MPa and k3 = (93.0, 155.0, 217.0) MPa andkk2 4=(84.0, = (120.0, 200.0, 280.0) and t1 155.0, = (6.6,217.0) 11.0, 15.4) MN, t k t t =(120.0, 200.0, 280.0) MPa, and =(6.6, 11.0, 15.4) MN, =(16.8, 28.0, 39.2) MN, =(24.0, 40.0, t = (16.8, 28.0, 39.2) MN, t = (24.0, 40.0, 56.0) MN and t = (36.0, 60.0, 84.0) MN Each fuzzy 4 k4 =(120.0, 200.0, 280.0) MPa, and t1 =(6.6, 11.0, 15.4) MN, t =(16.8, 28.0, 39.2) MN, t =(24.0, 40.0, parameter has the relative variation at different levels of membership with respect to the main value 56.0) MN and t =(36.0, 60.0, 84.0) MN Each fuzzy parameter has the relative variation at different levels t =(36.0,of60.0, 56.0) MN.40% Each fuzzy parameter has the relative variation at different levels atMN the and membership not84.0) exceed of membership with respect to the main value at the membership of not exceed 40% A finite-element model fortyvalue elements equal length 0.5 exceed m is used for the analysis Using of membership with respect to theofmain at thewith membership of not 40% membershipmodel levels,ofthe estimated membership of m theistop deflection the maximum Afive finite-element forty elements with equalfunctions length 0.5 used for the and analysis Using five A finite-element model of forty elements with equal length 0.5 m is used for the analysis Using five membership levels, membership ofFig the top deflection bending bending moment theestimated pile are shown in functions Fig.functions 5(a) of and respectively Themaximum corresponding membership levels, theinthe estimated membership the top 5(b), deflection andand the the maximum bending moment in the pile are shown in Fig 5(a) and Fig 5(b), respectively The corresponding membership moment in the pile are shown in Fig Fig 5(b), respectively The corresponding membership functions obtained by 5(a) directand optimization using differential evolution (DE)membership [18] are also functions obtained by direct optimization using differential evolution (DE) [18] are also plotted in Fig functions obtained by direct optimization using differential evolution (DE) [18] are also plotted in Fig 5memfor5 for plotted in Fig for comparison Moreover, the values functions of these membership functionslevel at each comparison Moreover, the values of these membership at each membership are listed comparison Moreover, the values of these membership functions at each membership level are listed in in bership level are listed in Table Table Table 1 Membership level Membership level 0.6 0.4 0.2 0.6 0.4 0.2 0.8 Membership level 0.8 0.8 DE DE Proposed Proposed Membership level 1 0.6 0.4 0.2 5 6 7 u [m] u [m] 9 x 10 10 10 -3 -3 x 10 DE DE Proposed Proposed 0.8 0.6 0.4 0.2 0 180 190 200 210 220 230 180 190 200 210 220 230 M [kNm] M [kNm] (a)(a)(a) (b)(b) (b) Figure Membership function: (a) Top displacement; (b) Maximum bending moment Figure Membership function: (a) (a) Top displacement; (b)(b) Maximum bending moment Figure Membership function: Top displacement; Maximum bending moment is seen results obtained by the proposed approach those provided by direct optimization It isItseen thatthat the the results obtained by the proposed approach andand those provided by direct optimization are are almost identical In this example, the membership functions of the pile responses are approximated almost identical In this example, the membership functions of the pile responses are approximated withwith membership levels To obtain sufficient good DE requires 1000 analyses, while Table Results ofresults theresults fuzzy formore themore pile fivefive membership levels To obtain sufficient good DEanalysis requires thanthan 1000 FE FE analyses, while the proposed approach needs only 2(8+5)+1=27 FE analyses to produce exact results This clearly the proposed approach needs only 2(8+5)+1=27 FE analyses to produce exact results This clearly demonstrates computational efficiency of the proposed approach demonstrates the the computational efficiency of the proposed approach µY (y) Top displacement (min;max) [m] DE Max bending moment (min;max) [kNm] Table Results of the fuzzy analysis for the Table Results of the fuzzy analysis for the pile pile Proposed DE Top displacement (min;max) Top displacement (min;max) [m] [m] 0.0058 0.0058 Proposed Max bending moment (min;max) [kNm] Max bending moment (min;max) [kNm] 199.8863 199.8863 1.0(y ) 0.8 0.6 1 0.4 0.8 0.80.2 0.0 0.6 0.6 DE (0.0055; DE0.0062) (0.0052;0.0058 0.0067) 0.0058 (0.0049; 0.0072) (0.0055; 0.0062) (0.0047; 0.0079) (0.0055; 0.0062) (0.0045; 0.0087) (0.0052; 0.0067) 0.4 0.4 (0.0049; 0.0072) (0.0049; (0.0049; 0.0072) (188.7972; (188.7972; 213.5837)(188.7972; (188.7972; 213.5838) (0.0049; 0.0072) 0.0072) 213.5837) 213.5838) Y (yY) (0.0052; 0.0067) Proposed DE Proposed (0.0055; 0.0062) (195.9505; (195.9505; 204.1065) Proposed DE204.1065) Proposed (0.0052; 0.0058 0.0067) (192.2638;199.8863 208.6543) (192.2638; 208.6544) 199.8863 0.0058 199.8863 199.8863 (0.0049; 0.0072) (188.7972; 213.5837) (188.7972; 213.5838) (0.0055; 0.0062) (185.5262; (195.9505; 204.1065)(195.9505; (195.9505; 204.1065) (0.0047; 0.0079) 218.9637) (185.5261; 218.9637) (0.0055; 0.0062) (195.9505; 204.1065) 204.1065) (0.0045; 0.0087) 227.6920) 227.6922) (0.0052; 0.0067) (182.4303; (192.2638; 208.6543) (182.4300; (192.2638; 208.6544) (0.0052; 0.0067) (192.2638; 208.6543) (192.2638; 208.6544) It is seen that the results obtained by the proposed approach and those provided by direct opti0.2 are (0.0047; 0.0079) In (0.0047; 0.0079) (185.5262; 218.9637)of (185.5261; 218.9637) 0.2 (0.0047; 0.0079) (0.0047; 0.0079) 218.9637) (185.5261; 218.9637) mization almost identical this example, the (185.5262; membership functions the pile responses are approximated with five membership levels To obtain sufficient good results DE requires more than JOURNAL SCIENCE TECHNOLOGY IN CIVIL ENGINEERING JOURNAL OF OF SCIENCE ANDAND TECHNOLOGY IN CIVIL 7ENGINEERING xxxxxx 7 Anh, P H / Journal of Science and Technology in Civil Engineering 1000 FE analyses, while the proposed approach needs only 2(8 + 5) + = 27 FE analyses to produce exact results This clearly demonstrates the computational efficiency of the proposed approach Conclusion This paper presents a fuzzy finite element analysis approach for the laterally-loaded pile in multilayered soils The pile is idealized as a one-dimensional beam and the soil as two-parameter elastic foundation model A fast α-level optimization procedure is developed using a response surface methodology based on the first order Taylor’s expansion of the pile response The procedure is validated by an example of a pile in 4-layer soil with fuzziness in soil parameters Numerical results show that the obtained fuzzy pile responses agree well with those obtained by direct optimization The advantage of the approach is that it does not require a large number of finite-element analyses as often found in direct optimization strategy Acknowledgment This study was carried out within the project supported by National University of Civil Engineering, Vietnam; grant number: 82-2016/KHXD References [1] Fan, H and Liang, R (2012) Application of Monte Carlo simulation to laterally loaded piles In GeoCongress 2012: State of the Art and Practice in Geotechnical Engineering, 376–384 [2] Fan, H and Liang, R (2013) Performance-based reliability analysis of laterally loaded drilled shafts Journal of Geotechnical and Geoenvironmental Engineering, 139(12):2020–2027 [3] Fan, H and Liang, R (2013) Reliability-based design of laterally loaded piles considering soil spatial variability In Foundation Engineering in the Face of Uncertainty: Honoring Fred H Kulhawy, 475–486 [4] Chan, C L and Low, B K (2009) Reliability analysis of laterally loaded piles involving nonlinear soil and pile behavior Journal of Geotechnical and Geoenvironmental Engineering, 135(3):431–443 [5] Tandjiria, V., Teh, C I., and Low, B K (2000) Reliability analysis of laterally loaded piles using response surface methods Structural Safety, 22(4):335–355 [6] Chan, C L and Low, B K (2012) Probabilistic analysis of laterally loaded piles using response surface and neural network approaches Computers and Geotechnics, 43:101–110 [7] Zadeh, L A (1965) Fuzzy set Information Control, 8(1):338–353 [8] Moens, D and Hanss, M (2011) Non-probabilistic finite element analysis for parametric uncertainty treatment in applied mechanics: Recent advances Finite Elements in Analysis and Design, 47(1):4–16 [9] Huynh, L X and Duy, L C (2013) An algorithm for solving basic equations of the finite element method with membership functions of fuzzy input parameters Journal of Science and Technology in Civil Engineering, 6(3):45–53 [10] Tuan, N H., Huynh, L X., and Anh, P H (2015) A fuzzy finite element algorithm based on response surface method for free vibration analysis of structure Vietnam Journal of Mechanics, 37(1):17–27 [11] Tuan, N H., Huynh, L X., and Anh, P H (2014) Fuzzy structural analysis subjected to harmonic forces using the improvement response surface method and assessing the safety level Journal of Science and Technology in Civil Engineering (STCE)-NUCE, 8(3):12–21 [12] Măoller, B., Graf, W., and Beer, M (2000) Fuzzy structural analysis using α-level optimization Computational Mechanics, 26(6):547–565 [13] Basu, D and Salgado, R (2007) Elastic analysis of laterally loaded pile in multi-layered soil Geomechanics and Geoengineering: An International Journal, 2(3):183–196 Anh, P H / Journal of Science and Technology in Civil Engineering [14] Vlasov, V Z and Leont’ev, N N (1966) Beams, plates and shells on elastic foundations Israel Program for Scientific Translations, Jerusalem [15] Reese, L C and Van Impe, W F (2010) Single piles and pile groups under lateral loading A.A Balkema: Rotterdam, Netherlands [16] Anh, P H (2014) Fuzzy analysis of laterally-loaded pile in layered soil Vietnam Journal of Mechanics, 36(3):173–183 [17] Basu, D., Salgado, R., and Prezzi, M (2008) Analysis of laterally loaded piles in multilayered soil deposits Joint Transportation Research Program, Department of Transportation and Purdue University, West Lafayette, Indiana [18] Anh, P H., Thanh, N X., and Hung, N V (2014) Fuzzy structural analysis using improved differential evolutionary optimization In Proceedings of the International Conference on Engineering Mechanics and Automation (ICEMA 3), Hanoi, 492–498 ... finite element analysis approach for the laterally- loaded pile in multilayered soils The pile is idealized as a one-dimensional beam and the soil as two-parameter elastic foundation model A fast. .. laterally- loaded pile a layered soil; discretization; laterally- loaded pile in in asoil; layered (b)(b) FEFE discretization; AFigure laterally- loaded pile in a layered (b) soil; FE discretization;... on a response Fuzzy finite element analysis of laterally loadedanalysis pile surface method, which is applicable for the fuzzy of laterally loaded piles with uncertain soil parameters The methodology
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