Soil Dynamics and Earthquake Engineering 103 (2017) 141–150 Contents lists available at ScienceDirect Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn Vibration analysis of concrete foundations retroﬁt with NFRP layer resting on soil medium using sinusoidal shear deformation theory Abdollah Zamania, Mahmood Rabani Bidgolib, a b MARK ⁎ Department of Civil Engineering, Khomein Branch, Islamic Azad University, Khomein, Iran Department of Civil Engineering, Jasb Branch, Islamic Azad University, Jasb, Iran A R T I C L E I N F O A B S T R A C T Keywords: Vibration Concrete foundation NFRP layer CNTs Soil medium Sinusoidal shear deformation theory After decades of service, concrete foundations may be in need of strengthening and retroﬁtting due to continuous aging. However, the main objective of this work is presenting a mathematical model for vibration of concrete foundation with nano ﬁber reinforced polymer (NFRP) layer resting on soil medium. The nano ﬁbers of NFRP layer are made from carbon nanotubes (CNT) where the eﬀective material properties of nano-composite structure are obtained by Mori-Tanaka model considering agglomeration eﬀects. The soil medium is simulated with spring constants of Winker foundation model. Based on the sinusoidal shear deformation theory (SSDT), energy method and Hamilton's principle, the motion equations are derived. Applying Navier method, the frequency of the structure is calculated analytically so that the eﬀects of volume percent and agglomeration of CNTs, soil medium, structural damping and geometrical parameters of structure are shown on the frequency of system. Results show that considering NFRP layer leads to higher frequency for NFRP thickness to concrete foundation thickness ratio of higher than 0.005. 1. Introduction The concrete foundations resting on soil substrates are one of the most commonly used structural elements in civil engineering applications such as pavement of highways, footing of buildings and bases of machines. Since vibration of foundations cause many crucial problems in civil engineering ﬁeld especially for instruments with high-level safety and functionality, studying this topic is essential and therefore the vibrations may need to be reduced. There are several ways to reduce vibration of concrete foundations such as improving the properties of them with additive materials, retroﬁtting with ﬁber reinforced polymer (FRP) layers and so on. Persson et al. [1] done a numerical research on reducing building vibrations through foundation improvement. Investigations in recent years have clearly demonstrated that conﬁnement with a FRP cover leads to a substantial improvement in the strength, ductility and dynamical behavior of concrete elements. Wei and Wu [2] presented a uniﬁed stress–strain model of concrete for circular, square, and rectangular columns conﬁned by FRP jackets. Finite element analysis of masonry panels strengthened with FRPs was studied by Grande et al. [3]. They discussed some strategies, in the framework of the nonlinear ﬁnite element analysis, regarding the elements choice and the models to adopt for reliable nonlinear analyses of masonry structures reinforced ⁎ with FRP strips. Hemmatnezhad et al. [4] made an experimental, numerical and analytical investigation on free vibrational behavior of glass ﬁber reinforced polymer (GFRP)-stiﬀened composite cylindrical shells. Static and free vibration of reinforced concrete beams with carbon ﬁber reinforced polymer (CFRP) rectangular rods were analyzed by Capozucca and Bossoletti [5]. Pan and Wu [6] proposed an analytical modeling of bond behavior between FRP plate and concrete. CFRP composite retroﬁtting eﬀect on the dynamic characteristics of arch dams was investigated by Altunisik et al. [7]. The inﬂuence of using CFRP wraps on performance of buried steel pipelines under permanent ground deformations was studied by Mokhtari and Alai Nia [8]. Bond behavior of FRP carbon plates externally bonded over steel and concrete elements was considered experimentally and numerically by Ceroni et al. [9]. Seismic strengthening of inﬁlled reinforced concrete frames by CFRP was studied by Erol and Karadogan [10]. On the other hand, recent progresses in innovative and advanced engineering materials have given more emphasis on utilization of nano scale reinforcing elements. One of perfect candidates for using in the reinforcement phase of composite materials is CNTs. Because their Young's modulus and tensile strength are about 1 TPa and 150 GPa, respectively [11]. Many researchers studied the mechanical behaviors of CNTs reinforced composite plates. The nonlinear vibration behaviors of a reinforced composite plate with the CNTs under combined the Corresponding author. E-mail address: m.rabanibidgoli@gmail.com (M.R. Bidgoli). http://dx.doi.org/10.1016/j.soildyn.2017.09.018 Received 24 July 2017; Received in revised form 21 August 2017; Accepted 23 September 2017 0267-7261/ © 2017 Elsevier Ltd. All rights reserved. Soil Dynamics and Earthquake Engineering 103 (2017) 141–150 A. Zamani, M.R. Bidgoli graded plates using higher-order shear deformation theory was studied by Tran et al. [36]. Free vibration of functionally graded porous cylindrical shell using a sinusoidal shear deformation theory was studied by Wang and Wu [37]. Dynamic analyses of concrete structures improved by nanoparticles based on mathematical models are the novel topics in civil engineering. Research works undertaken in relation to mathematical modeling of concrete structures are very limited in the literature. One of these investigations is the research of Jafarian Arani and Kolahchi [38]. They studied buckling analysis of concrete columns retroﬁt with CNT-reinforced polymer layer by using Euler-Bernoulli and Timoshenko beam models. To the best of the authors’ knowledge, the mathematical modeling of concrete foundations strengthened with NFRP layer has not been reported yet. However, due to lack of this subject in the literature, vibration analysis of concrete foundations retroﬁtted by NFRP layer resting on soil medium is presented in this work. The soil medium is simulated by spring constants. In order to obtain the equivalent material properties of nano-composite structure, the Mori-Tanaka model is used considering agglomeration eﬀects of nano ﬁbers. Applying SSDT, the motion equations are obtained based on Hamilton's principal. Also, Navier method is applied for obtaining the frequency of the system. NFRP layers have the short and long-term eﬀects on the strength and lifetime of concrete structures. Hence, the aim of this research is studying the eﬀects of NFRP layer on the vibration response of the concrete foundations as well as showing the eﬀects of volume percent and agglomeration of CNTs, soil medium, structural damping and geometrical parameters of structure on the frequency of system. parametric and forcing excitations were studied by Guo and Zhang [12]. An element-free analysis of CNT-reinforced composite plates with column supports and elastically restrained edges under large deformation was considered by Zhang et al. [13]. Vibration analysis of CNTsreinforced thick laminated composite plates based on Reddy's higherorder shear deformation theory was done by Zhang and Selim [14]. Zhang et al. [15] conducted the ﬁrst-known vibration analysis of CNT reinforced functionally graded composite triangular plates subjected to in-plane stresses. Ahmadi et al. [11] analyzed the multi-scale bending, buckling and vibration of carbon ﬁber/carbon nanotube-reinforced polymer nanocomposite plates with various shapes. Shen and Wang [16] investigated the small- and large-amplitude vibrations of compressed and thermally postbuckled carbon nanotube-reinforced composite (CNTRC) plates resting on elastic foundations. Kumar and Srinivas [17] calculated the vibration, buckling and bending behavior of functionally graded multi-walled carbon nanotube reinforced polymer composite plates using the layer-wise formulation. Also, several works have been made in the ﬁeld of vibration analysis of plates on elastic foundations in recent years. Xiang et al. [18] studied analytically the vibration of rectangular Mindlin plates with simply supported boundary conditions on Pasternak foundation. The ﬁnite element method was used by Omurtag et al. [19] to study the free vibration of thin plates resting on Pasternak foundation. Lam et al. [20] employed the Green's functions to obtain canonical exact solutions of elastic bending, buckling and vibration for Levy plates resting on twoparameter elastic foundations. By employing the Rayleigh-Ritz method, the three dimensional vibration of rectangular thick plates on elastic foundations was analyzed by Zhou et al. [21]. By using the diﬀerential quadrature method (DQM), developing a three dimensional layerwiseﬁnite element method and coupling a three-dimensional (3D) discrete layer approach with DQM, Malekzadeh et al. studied the free vibration analysis of rectangular [22,23] plates resting on elastic foundation. Ferreira et al. [24] used the radial basis function collocation method to study the static deformation and free vibration of plates on Pasternak foundation. Kumar and Lal [25] studied the vibration analysis of nonhomogeneous orthotropic rectangular plates with bilinear thickness variation resting on Winkler foundation. Element Free Galerkin method was used by Bahmyari and Rahbar-Ranji [26] for free vibration analysis of orthotropic plate with variable thickness and resting on nonuniform elastic foundation. Bahmyari and khedmati [27] considered the vibration analysis of nonhomogeneous moderately thick plates with point supports resting on Pasternak elastic foundation using element free Galerkin method. Vibrational analysis of advanced composite plates resting on elastic foundation was studied by Mantari et al. [28]. They derived the governing equations of a type of functionally graded plates resting on elastic foundation by employing the Hamilton's principal. Nguyen-Thoi et al. [29] presented an edge-based smoothed three-node Mindlin plate element (ES-MIN3) for static and free vibration analyses of plates. Uğurlu [30] analyzed the vibration of elastic bottom plates of ﬂuid storage tanks resting on Pasternak foundation based on boundary element method. An original ﬁrst shear deformation theory to study advanced composites on the elastic foundation was presented by Mantari and Granados [31]. Recently, higher-order shear deformation theories (HSDTs) are used for mathematical modeling of structures. Because these theories can be employed for thick plates without any necessity to use shear correction factors. One of HSDTs is sinusoidal shear deformation theory. Neves et al. [32,33] investigated a quasi-3D sinusoidal shear deformation theory for the static and free vibration analysis of functionally graded plates. A simple quasi-3D sinusoidal shear deformation theory was considered for functionally graded plates by Thai and Kim [34]. Their studies showed that the obtained results with this theory are more accurate than those obtained by other higher-order shear deformation theories. A new sinusoidal shear deformation theory (SSDT) for bending, buckling, and vibration of functionally graded plates was investigated by Thai and Vo [35]. Isogeometric analysis of functionally 2. Mathematical modeling As shown in Fig. 1, a concrete foundation including NFRP layer resting on soil medium with length a, width b, concrete thickness 2h and NFRP layer thickness hf is considered. 2.1. Sinusoidal Shear Deformation Theory Based on SSDT, the displacement ﬁeld can be expressed as [39] U1 (x , y, z , t ) = U (x , y, t ) − z ∂Wb h πz ⎞ ∂Ws − ⎛z − ( sin ) , ∂x π h ⎠ ∂x ⎝ (1) U2 (x , y, z , t ) = V (x , y, t ) − z ∂Wb h πz ⎞ ∂Ws − ⎛z − ( sin , ) ∂y π h ⎠ ∂y ⎝ (2) U3 (x , y, z , t ) = Wb (x , y, t ) + Ws (x , y, t ), (3) where (U , V , Wb, Ws ) denote the displacement components. Based on above relations, the strain-displacement equations may be written as 0 b s ⎛ k xx ⎞ ⎛ εxx ⎞ ε ⎛ k xx ⎞ ⎛ xx ⎞ h πz ⎞ k s 0 b ⎟ εyy = ⎜ ε yy ⎟ + z ⎜ k yy ⎛ ⎜ yy ⎟, + z − sin ⎜ ⎟ π h ⎠⎜ s ⎟ ⎜⎜ b ⎟⎟ ⎝ ⎜ 0 ⎟⎟ γ k ⎝ xy ⎠ ⎜ γxy k xy ⎝ xy ⎠ ⎝ ⎠ ⎝ ⎠ (4) s γ ⎛⎜ γyz ⎞⎟ = cos πz ⎛ yz ⎞, ⎜γ s ⎟ γxz h ⎝ ⎠ ⎝ xz ⎠ Fig. 1. A schematic ﬁgure for concrete foundation with NFRP layer. 142 (5) Soil Dynamics and Earthquake Engineering 103 (2017) 141–150 A. Zamani, M.R. Bidgoli Vr = Vrinclusion + Vrm where ∂U 0 ⎛ ∂x ⎛ εxx ⎞ ⎜ ∂V 0 ⎜ ε yy ⎟ = ⎜ ∂y ⎜⎜ 0 ⎟⎟ γxy ⎜ ∂U + ⎝ ⎠ ⎝ ∂y ∂2Wb ∂x 2 ⎞ ⎛− b ⎛ k xx ⎞ ⎜ 2W ⎟ ∂ b b ⎜ k yy ⎟ = ⎜ − ⎟, ∂y 2 ⎜⎜ b ⎟⎟ ⎜ ⎟ 2 k ∂ W ⎝ xy ⎠ ⎜− 2 ∂x ∂yb ⎟ ⎠ ⎝ ⎞ ⎟ ⎟ , ∂V ⎟ ∂x ⎠ Vrinclusion ∂2Ws ∂x 2 ⎛− ⎞ s ⎛ k xx ⎞ ⎜ ⎟ ∂2Ws s ⎜ k yy ⎟ = ⎜ − 2 ⎟ ∂y ⎜k s ⎟ ⎜ 2 ⎟ ⎝ xy ⎠ ⎜− 2 ∂ Ws ⎟ ∂x ∂y ⎠ ⎝ and are the volumes of nanoﬁbers dispersed in the where spherical inclusions and in the matrix, respectively. Introduce two parameters ξ and ζ describe the agglomeration of nanoﬁbers (6a) ξ= Vinclusion , V (11) ζ= Vrinclusion . Vr (12) ∂W s s −4 ⎛ ∂y ⎞ ⎛ γyz ⎞ = ⎜ ⎟ ⎜γ s ⎟ h3 ⎜ ∂Ws ⎟ ⎝ xz ⎠ ⎝ ∂x ⎠ (6b) However, the average volume fraction Cr of nanoﬁbers in the composite is 2.2. Stress-strain relations Based on Hook's law, the stress-strain relation of the concrete foundation can be written as k+m l Q⏟ 12 n Q⏟ 22 k−m 0 0 0 0 0 0 0 Q13 ⎡ K = K out ⎢1 + ⎢ ⎣ (7) l Q⏟ 23 l k+m Q⏟ 32 Q 33 0 0 0 0 p Q⏟ 44 0 0 0 0 m Q⏟ 55 0 n= Em2 cm (1 + cr − cm νm) + 2cm cr (kr nr − lr2)(1 + νm)2 (1 − 2νm) (1 + νm)[Em (1 + cr − 2νm) + 2cm kr (1 − νm − 2νm2 )] Em [2cm2 kr (1 − νm) + cr nr (1 + cr − 2νm) − 4cm lr νm] + Em (1 + cr − 2νm) + 2cm kr (1 − νm − 2νm2 ) Em [Em cm + 2pr (1 + νm)(1 + cr )] p= 2(1 + νm)[Em (1 + cr ) + 2cm pr (1 + νm)] Em [Em cm + 2mr (1 + νm)(3 + cr − 4νm)] m= 2(1 + νm){Em [cm + 4cr (1 − νm)] + 2cm mr (3 − νm − 4νm2 )} (14) ) G ( ) (15) where (δr − 3Km χr ) Cr ζ , 3(ξ − Cr ζ + Cr ζχr ) Kin = Km + K out = Km + Gin = Gm + (8) Gout = Gm + Cr (δr − 3Km χr )(1 − ζ ) , 3[1 − ξ − Cr (1 − ζ ) + Cr χr (1 − ζ )] (ηr − 3Gm βr ) Cr ζ , 2(ξ − Cr ζ + Cr ζβr ) Cr (ηr − 3Gm βr )(1 − ζ ) , 2[1 − ξ − Cr (1 − ζ ) + Cr βr (1 − ζ )] (16) (17) (18) (19) where χr , βr , δr , ηr may be calculated as Em {Em cm + 2kr (1 + νm)[1 + cr (1 − 2νm)]} k= 2(1 + νm)[Em (1 + cr − 2νm) + 2cm kr (1 − νm − 2νm2 )] Em {cm νm [Em + 2kr (1 + νm)] + 2cr lr (1 − νm2 )]} (1 + νm)[Em (1 + cr − 2νm) + 2cm kr (1 − νm − 2νm2 )] ⎤ ⎥ , K 1 + α (1 − ξ ) ⎛ in − 1⎞ ⎥ ⎝ K out ⎠⎦ ( where σij , εij, γij, k , m , n, l, p are the stress components, the strain components and the stiﬀness coeﬃcients respectively. According to the Mori-Tanaka method, the stiﬀness coeﬃcients are given by [40] l= K ξ ⎛ in − 1⎞ ⎝ K out ⎠ ⎡ ⎤ ξ G in − 1 out ⎥, G = Gout ⎢1 + Gin ⎢ 1 + β (1 − ξ ) G − 1 ⎥ out ⎣ ⎦ 0 ⎤ ⎥ 0 ⎥ ε11 ⎧ ⎫ 0 ⎥ ⎪ ε22 ⎪ ⎥ ⎪ ε33 ⎪ 0 ⎥ ⎨ γ23 ⎬ ⎥ ⎪ γ13 ⎪ ⎪γ ⎪ 0 ⎥ ⎩ 12 ⎭ ⎥ p⎥ Q⏟ 66 ⎥ ⎦ (13) Assume that all the orientations of the nanoﬁbers are completely random. Hence, the eﬀective bulk modulus (K) and eﬀective shear modulus (G) may be written as where Cij are elastic constants. In addition, the stress-strain relation of NFRP layer is ⎡ Q ⎢ 11l ⎢ Q⏟12 σ ⎧ σ11 ⎫ ⎢ k − m ⎪ 22 ⎪ ⎢ ⎪ σ33 ⎪ Q31 = ⎨ σ23 ⎬ ⎢ 0 ⎪ σ13 ⎪ ⎢ ⎪ σ12 ⎪ ⎩ ⎭ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎣ Vr . V Cr = 0 0 ⎤ εxx C11 C12 C13 0 σ ⎡ ⎤ ⎡ σxx ⎤ ⎡ 0 0 0 ⎥ ⎢ εyy ⎥ C C C ⎢ yy 21 22 23 ⎢ ⎥ 0 0 ⎥ ⎢ εzz ⎥ ⎢ σzz ⎥ ⎢C31 C32 C33 0 ⎥ ⎢ γ ⎥, ⎢ σzy ⎥ = ⎢ 0 0 0 0 0 C 44 ⎥ ⎢ zy ⎥ ⎢σ ⎥ ⎢ xz ⎢ 0 0 0 0 C55 0 ⎥ ⎢ γxz ⎥ ⎢ ⎥ ⎥ γzy ⎥ ⎢ ⎣ σzy ⎦ ⎣ 0 0 0 0 0 C66 ⎦ ⎢ ⎣ ⎦ (10) Vrm χr = 3(Km + Gm) + kr − lr , 3(kr + Gm) βr = 1 ⎧ 4Gm + 2kr + lr 4Gm + k G p + 5⎨ 3( ) ( r m r + Gm ) ⎩ 2[Gm (3Km + Gm) + Gm (3Km + 7Gm)] ⎫ , Gm (3Km + Gm) + mr (3Km + 7Gm) ⎬ ⎭ (21) δr = (2kr − lr )(3Km + 2Gm − lr ) ⎤ 1⎡ n + 2lr + , ⎥ ⎢ r 3⎣ kr + Gm ⎦ (22) ηr = 4Gm pr 8Gm mr (3Km + 4Gm) 1 ⎡2 + (nr − lr ) + 5⎢ (pr + Gm) 3Km (mr + Gm) + Gm (7mr + Gm) ⎣3 + (9) + where the subscripts m and r stand for matrix and reinforcement respectively. Cm and Cr are the volume fractions of the matrix and the nanoﬁbers respectively and kr ، lr ، nr ، pr, mr are the Hills elastic modulus for the nanoﬁbers [40]. The experimental results show that the assumption of uniform dispersion for nanoﬁbers in the matrix is not correct and the most of nanoﬁbers are bent and centralized in one area of the matrix. These regions with concentrated nanoﬁbers are assumed to have spherical shapes, and are considered as ‘‘inclusions’’ with different elastic properties from the surrounding material. The total volume Vr of nanoﬁbers can be divided into the following two parts [41] (20) 2(kr − lr )(2Gm + lr ) ⎤ . ⎥ 3(kr + Gm) ⎦ (23) where, Km and Gm are the bulk and shear moduli of the matrix which can be written as Km = Em 3(1 − 2υm) Gm = Em . 2(1 + υm) , Furthermore, β , α can be obtained from 143 (24) (25) Soil Dynamics and Earthquake Engineering 103 (2017) 141–150 A. Zamani, M.R. Bidgoli α= (1 + υout ) , 3(1 − υout ) β= 2(4 − 5υout ) , 15(1 − υout ) υout = (26a) Qx = A55g ∂ ∂2 Ws + GA55g Ws, ∂x ∂x ∂t (39) Q y = A66g ∂2 ∂ Ws + GA66g Ws, ∂y ∂t ∂y (40) (26b) 3K out − 2Gout . 6K out + 2Gout ∂ ∂2 ∂2 ∂ ∂2 U − B11 2 Wb − A11zf 2 Ws + A12z V − B12 2 Wb ∂x ∂x ∂x ∂y ∂y ∂2 − A12zf 2 Ws, ∂y (41) MxxB = A11z (27) Finally, the elastic modulus (E) and poison's ratio (υ) can be calculated as E= 9KG 3K + G υ= 3K − 2G . 6K + 2G , ∂ ∂2 ∂2 ∂ ∂2 U − A11zf 2 Wb − E11 2 Ws + A12f V − A12zf 2 Wb ∂x ∂x ∂x ∂y ∂y ∂2 − E12 2 Ws, ∂y (42) MxxS = A11f (28) (29) ∂ ∂2 ∂2 ∂ ∂2 U − B21 2 Wb − A21zf 2 Ws + A22z V − B22 2 Wb ∂x ∂x ∂x ∂y ∂y ∂2 − A22zf 2 Ws, ∂y (43) MyyB = A21z 2.3. Energy method The potential energy can be written as U= 1 2 h ∫A ∫− h2 (σxx εxx + σyy εyy + σxy γxy + σxz γxz+σyz γyz ) dzdA ∂ ∂2 ∂2 ∂ ∂2 U − A21zf 2 Wb − E21 2 Ws + A22f V − A22zf 2 Wb ∂x ∂x ∂x ∂y ∂y ∂2 − E22 2 Ws, ∂y (44) MyyS = A21f (30) 2 Combining Eqs. (1)–(3) and (30) yields U= ∂U ∂U ∂V ∂V ∂Ws 1 + Nxy + Nxy + Nyy + Qx ∫ ⎜⎛Nxx ∂x ∂y ∂x ∂y ∂x 2 A⎝ ∂Ws ∂2Ws ∂2Ws ∂2Ws + Qy − MxxS − MyyS − 2MxyS ∂y ∂x 2 ∂y 2 ∂y ∂x − MxxB ∂2Wb ∂2Wb ∂2Wb ⎞ − MyyB − 2MxyB ⎟ dA , ∂x 2 ∂y 2 ∂y ∂x ⎠ h ∫−h ⎡ MxxB ⎤ ⎢ MyyB ⎥ = ⎢M ⎥ ⎣ xyB ⎦ σ ⎡ σxx ⎤ ⎢ yy ⎥ dz + ⎥ ⎢ σxy ⎦ ⎣ h ∫−h ∂ ∂ ∂2 ∂2 U + 2A 44z V − 2B44 Wb − 2A 44zf Ws, ∂y ∂x ∂x ∂y ∂x ∂y (45) MxyS = 2A 44f ∂ ∂ ∂2 ∂2 U + 2A 44f V − 2A 44zf Wb − 2E44 Ws, ∂y ∂x ∂x ∂y ∂x ∂y (46) where (31) h (A11 , A12 , A22 , A 44 ) = ∫−h (C11, C12, C22, C44 ) dz where the stress resultant-displacement relations can be written as ⎡ Nxx ⎤ ⎢ Nyy ⎥ = ⎢N ⎥ ⎣ xy ⎦ MxyB = 2A 44z h + hf + ∫h f ∫h σ ⎡ σxx ⎤ ⎢ yy ⎥ zdz + ⎢ ⎦ ⎣ σxy ⎥ h + hf ∫h ⎡ σxx ⎤ ⎢σ f ⎥ ⎢ yy ⎥ dz ⎢σ f ⎥ ⎣ xy ⎦ f ⎡ σxx ⎤ h + hf ⎢ f ⎥ ⎢ σyy ⎥ (Q11, Q12, Q22, Q44 ) dz , (47) h (A11z , A12z , A22z , A 44z ) = ∫−h (C11, C12, C22, C44 ) zdz (32) h + hf + ∫h (Q11, Q12, Q22, Q44 ) zdz , (48) h (A11f , A12f , A22f , A 44f ) = ∫−h (C11, C12, C22, C44 ) fdz zdz h + hf ⎢σ f ⎥ ⎣ xy ⎦ + ∫h (33) (Q11, Q12, Q22, Q44 ) fdz , (49) h ⎡ MxxS ⎤ ⎢ MyyS ⎥ = ⎢M ⎥ ⎣ xyS ⎦ ⎡Q x ⎤ = ⎢Q y ⎥ ⎣ ⎦ h ∫−h σ ⎡ σxx ⎤ ⎢ yy ⎥ fdz + ⎢ ⎦ ⎣ σxy ⎥ σ h ∫h h + hf h + hf ∫−h ⎡⎢ σxxyy ⎤⎥ pdz + ∫−h ⎣ ⎦ (A11zf , A12zf , A22zf , A 44zf ) = ∫−h (C11, C12, C22, C44 ) zfdz f ⎡ σxx ⎤ ⎢σ f ⎥ ⎢ yy ⎥ fdz ⎢σ f ⎥ ⎣ xy ⎦ h + hf + ∫h (34) h h + hf (35) ∂2 ∂ ∂2 ∂2 ∂ ∂2 Nyy = A21 U − A21z 2 Wb − A21f 2 Ws + A22 V − A22z 2 Wb ∂x ∂x ∂x ∂y ∂y ∂2 − A22f 2 Ws, ∂y Nxy = A 44 ∂ ∂ ∂2 ∂2 U + A 44 V − 2A 44z Wb − 2A 44f W, ∂y ∂x ∂x ∂y ∂x ∂y (51) (B11, B12, B22, B44 ) = ∫ (C11, C12, C22, C44 ) z 2dz h + hf + ∫h ∂2 ∂ ∂ U − A11z 2 Wb − A11f 2 Ws + A12 V − A12z 2 Wb ∂x ∂x ∂x ∂y ∂y ∂2 − A12f 2 Ws, ∂y Nxx = A11 (Q55, Q66) gdz , h −h Substituting stress-strain relation into Eqs. (32)–(35), the stress resultant-displacement relations can be obtained as follows ∂2 (50) (A55g , A66g ) = ∫−h (C55, C66) gdz + ∫h ⎡ σxx ⎤ pdz, ⎢ σyy ⎥ ⎣ ⎦ (Q11, Q12, Q22, Q44 ) zfdz , (Q11, Q12, Q22, Q44 ) z 2dz , (52) h (E11, E12, E22, E44 ) = ∫−h (C11, C12, C22, C44 ) f 2 dz h + hf + ∫h (36) (Q11, Q12, Q22, Q44 ) f 2 dz , (53) The kinetic energy of system may be written as K= 1 ρ 2 h 2 ∫A ∫− 2h ⎛ ⎛⎝ ∂∂ut ⎞⎠ ⎜ 2 ⎝ 2 2 ∂v ∂w + ⎛ ⎞ + ⎛ ⎞ ⎟⎞ dz dA . ⎝ ∂t ⎠ ⎝ ∂t ⎠ ⎠ (54) The external work due to soil medium can be written as [42] (37) We = (38) ∬ (−Kw w) wdA, (55) where Kw is Winkler's spring modulus. The governing equations can be 144 Soil Dynamics and Earthquake Engineering 103 (2017) 141–150 A. Zamani, M.R. Bidgoli derived by Hamilton's principal as follows ∂2 ⎛ ∂ ∂2 ∂2 ∂ ∂2 A11z U − B11 2 Wb − A11zf 2 Ws + A12z V − B12 2 Wb 2 ∂x ⎝ ∂x ∂x ∂x ∂y ∂y ⎜ ∫0 t (δU − δK − δWe ) dt = 0. (56) − A12zf Substituting Eqs. (30), (54) and (55) into Eq. (56) yields the following governing equations ∂ ∂ ∂ 2U ∂3Wb ∂3Ws Nxx + Nxy − I0 2 + I1 + J1 = 0, 2 ∂x ∂y ∂t ∂x ∂t ∂x ∂t 2 +2 + (57) ∂2 Ws ⎞ ∂y 2 ⎠ ∂2 ⎛ ∂ ∂ ∂2 ∂2 2A 44z U + 2A 44z V − 2B44 Wb − 2A 44zf Ws ⎞ ∂x ∂y ⎝ ∂y ∂x ∂x ∂y ∂x ∂y ⎠ ⎜ (58) ⎟ ∂2 ⎛ ∂ ∂2 ∂2 ∂ ∂2 A21z U − B21 2 Wb − A21zf 2 Ws + A22z V − B22 2 Wb 2 ∂y ⎝ ∂x ∂x ∂x ∂y ∂y ⎜ − A22zf ∂ ∂ ∂ 2V ∂3Wb ∂3Ws Nxy + Nyy − I0 2 + I1 + J1 = 0, 2 ∂x ∂y ∂t ∂y ∂t ∂y ∂t 2 ⎟ ∂2 Ws ⎞ ∂y 2 ⎠ ⎟ ∂ 2W ∂2Ws ⎞ ∂3U ∂3V ⎞ − Kw W − Kw Ws = I0 ⎛ 2 b + + I1 ⎛ + 2 2 t t x t y ∂t 2 ⎠ ∂ ∂ ∂ ∂ ∂ ⎝ ⎠ ⎝ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ∂ 4W ∂ 4W ∂ 4W ∂ 4W − I2 ⎛ 2 b2 + 2 b2 ⎞ − J2 ⎛ 2 s2 + 2 s2 ⎞ ⎤, ∂y ∂t ⎠ ⎥ ∂y ∂t ⎠ ⎝ ∂x ∂t ⎝ ∂x ∂t ⎦ ∂2 ∂2 ∂2 ∂ 2W ∂2Ws ⎞ MxxB + 2 MxyB + 2 MyyB − Kw w − I0 ⎛ 2 b + 2 ∂x ∂x ∂y ∂y ∂t 2 ⎠ ⎝ ∂t 3 3 4 4 4 ∂U ∂V ⎞ ∂W ∂W ∂W ∂ 4W − I1 ⎛ + + I2 ⎛ 2 b2 + 2 b2 ⎞ + J2 ⎛ 2 s2 + 2 s2 ⎞ 2 2 ∂y ∂t ⎠ ∂y ∂t ⎠ ∂y ∂t ⎠ ⎝ ∂x ∂t ⎝ ∂x ∂t ⎝ ∂x ∂t ⎜ ⎟ ⎟ ⎜ ⎟ (64) ⎟ ∂2 ⎛ ∂ ∂2 ∂2 ∂ ∂2 A11f U − A11zf 2 Wb − E11 2 Ws + A12f V − A12zf 2 Wb ∂x 2 ⎝ ∂x ∂x ∂x ∂y ∂y = 0, ⎜ (59) − E12 ∂2 ∂2 ∂2 ∂ ∂ MxxS + 2 MxyS + 2 MyyS + Qx + Q y − Kw w 2 ∂x ∂x ∂y ∂y ∂x ∂y +2 ∂ 2W ∂2Ws ⎞ − I0 ⎛ 2 b + ∂t 2 ⎠ ⎝ ∂t 3 3 ∂U ∂V ⎞ ∂ 4W ∂ 4W ∂ 4W ∂ 4W − J1 ⎛ + + J2 ⎛ 2 b2 + 2 b2 ⎞ + K2 ⎛ 2 s2 + 2 s2 ⎞ 2 2 ∂y ∂t ⎠ ∂y ∂t ⎠ ∂y ∂t ⎠ ⎝ ∂x ∂t ⎝ ∂x ∂t ⎝ ∂x ∂t = 0, ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ + ∂2 Ws ⎞ ∂y 2 ⎠ ⎟ ∂2 ⎛ ∂ ∂ ∂2 ∂2 Wb − 2E44 W⎞ 2A 44f U + 2A 44f V − 2A 44zf ∂x ∂y ⎝ ∂y ∂x ∂x ∂y ∂x ∂y ⎠ ⎜ ⎟ ∂2 ⎛ ∂ ∂2 ∂2 ∂ ∂2 A21f U − A21zf 2 Wb − E21 2 Ws + A22f V − A22zf 2 Wb 2 ∂y ⎝ ∂x ∂x ∂x ∂y ∂y ⎜ ⎟ − E22 ∂2 Ws+⎞ ∂y 2 ⎠ ⎟ ∂ ⎛ ∂ ∂ ⎛ ∂ A55g Wss ⎞ + A66g Ws ⎞+ ∂x ⎝ ∂x ∂y ⎝ ∂y ⎠ ⎠ ∂3V ⎞ ∂ 2W ∂2Ws ⎞ ∂3U + J1 ⎛ + − Kw W − Kw Ws = I0 ⎛ 2 b + 2 2 ∂y ∂t 2 ⎠ ∂t ⎠ ⎝ ∂t ⎝ ∂x ∂t (60) + where ⎜ ⎟ ⎜ h −h (I0, I1, I2, J1, J2 , K2) = ∫ ρ (1, z , f , zf , h + hf + ∫h z 2, f 2) ⎜ ⎟ ∂ 4W ∂ 4W ∂ 4W ∂ 4W − J2 ⎛ 2 b2 + 2 b2 ⎞ − K2 ⎛ 2 s2 + 2 s2 ⎞ ⎤, x t y t x t y ∂t ⎠ ⎥ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ⎠ ⎝ ⎝ ⎦ ⎜ ρf (1, z , f , zf , z 2, f 2 ) dz. ⎟ (61) ⎟ ⎜ ⎟ (65) Substituting Eqs. (36)–(46) into Eqs. (57)–(60), the governing equations can be written as follows 3. Solution procedure ∂ ⎛ ∂ ∂2 ∂2 ∂ ∂2 A11 U − A11z 2 Wb − A11f 2 Ws + A12 V − A12z 2 Wb ∂x ⎝ ∂x ∂x ∂x ∂y ∂y ⎜ − A12f + ∂2 ∂y 2 Steady state solutions to the governing equations of the system which relate to the simply supported boundary conditions can be assumed as Ws ⎞ ⎠ ⎟ ∂ ⎛ ∂ ∂ ∂2 ∂2 A 44 U + A 44 V − 2A 44z Wb − 2A 44f Ws ⎞ ∂y ⎝ ∂y ∂x ∂x ∂y ∂x ∂y ⎠ ⎜ ∂ 2U ∂3Wb ∂3Ws ⎤ = ⎡I0 2 − I1 − J1 , 2 ⎢ ∂t ⎥ ∂x ∂t ∂x ∂t 2 ⎦ ⎣ nπy iωt mπx )sin( )e , L b (66) nπy iωt mπx )cos( )e , L b (67) U (x , y, t ) = u 0 cos( ⎟ V (x , y, t ) = v0 sin( (62) nπy iωt mπx )sin( )e , L b (68) nπy iωt mπx )sin( )e , L b (69) Wb (x , y, t ) = wb0 sin( ∂2 ∂2 ∂ ⎛ ∂ ∂ A 44 U + A 44 V − 2A 44z Wb − 2A 44f Ws ⎞ ∂x ⎝ ∂y ∂x ∂x ∂y ∂x ∂y ⎠ ⎜ ⎟ Ws (x , y, t ) = ws0 sin( ∂ ⎛ ∂ ∂2 ∂2 ∂ ∂2 + A21 U − A21z 2 Wb − A21f 2 Ws + A22 V − A22z 2 Wb ∂y ⎝ ∂x ∂x ∂x ∂y ∂y ⎜ − A22f Substituting Eqs. (66)–(69) into Eqs. (62)–(65) yields ∂2 Ws ⎞ ∂y 2 ⎠ ⎡ K11 ⎢ K21 ⎢ K31 ⎢ ⎣ K 41 ⎟ ∂ 2V ∂3Wb ∂3Ws ⎤ = ⎡I0 2 − I1 − J1 , 2 ⎢ ∂y ∂t ∂y ∂t 2 ⎥ ⎣ ∂t ⎦ (63) K12 K22 K32 K 42 K13 K23 K33 K 43 K14 ⎤ u 0 ⎤ K24 ⎥ ⎡ ⎢ v0 ⎥ = 0, K34 ⎥ ⎢ wb0 ⎥ ⎥ ws0 ⎥ ⎦ ⎣ K 44 ⎦ ⎢ (70) where Kij are deﬁned in Appendix A. Finally, for calculating the frequency of the system (ω ), the determinant of matrix in Eq. (70) should be equal to zero. 145 Soil Dynamics and Earthquake Engineering 103 (2017) 141–150 A. Zamani, M.R. Bidgoli 4. Numerical results and discussion Here, a concrete foundation with Young's modulus of Ec = 20 GPa and Poisson's ratio of νc = 0.2 is considered. The NFRP layer is made from a polymer with Young's modulus of Em = 18 GPa and Poisson's ratio of νm = 0.3 which is reinforced by CNTs by Young's modulus of Er = 1 TPa and Poisson's ratio of νr = 0.3. The eﬀects of diﬀerent parameters are shown on the frequency (i.e. eigenfrequency) of the structure. The motivation for the choice of NFRP thickness, CNT volume percent and agglomeration of CNTs parameters is discussing about the NFRP layer eﬀect (main goal of this work) on the frequency of the concrete foundation. In addition, the eﬀects of diﬀerent soil mediums and geometrical parameters of concrete foundation are considered on the frequency of the structure. 4.1. Validation Fig. 2. Eﬀects of structural damping on the dimensionless frequency versus NFRP layer thickness to foundation thickness ratio. In this paper, to validate the results, the frequency (i.e. eigenfrequency) of the structure is obtained by neglecting the soil medium (Kw = 0 ) and NFRP layer. Therefore, all the mechanical properties and type of loading are the same as reference [43]. So the non-dimensional frequency is considered as Ω= ρhω2a4 D0 in which D0 = E1 h3/(12(1 − ν12 ν21)) . The results are compared with ﬁve references which have used diﬀerent solution methods. Whitney [43] is used exact solution while Secgin and Sarigul [44] applied discrete singular convolution approach. The numerical solution methods of Dai et al. [45], Chen et al. [46] and Chow et al. [47] are mesh-free, ﬁnite element and Ritz, respectively. As can be observed from Table 1, the results of present work are in accordance with the mentioned references. 4.2. Eﬀects of diﬀerent parameters Considering structural damping is crucial in most engineering problems. For this purpose, based on Kelvin-Voigt model [48], the stiﬀness coeﬃcient (Cij ) in Eq. (7) can be replaced by Cij (1 + g ∂/ ∂t ) where g is the structural damping parameter. Using the solution procedure outlined in Section 3, Eq. (70) can be written as where and [M ][d ]̈ + [C ][d]̇ + [K ][d] = [0] [M ], [C ], [K ] [d] = [u 0 , v0, wb0, ws0] are mass matrix, damp matrix, stiﬀness matrix and dynamic vector, respectively. Hence, utilizing the eigenvalue problem, the frequency of the structure may be calculated. In order to show the eﬀect of structural damping on the dimensionless frequency (Ω = ωL ρm / Em ) versus NFRP thickness to concrete foundation thickness ratio, Fig. 2 is plotted. As can be seen, considering structural damping leads to lower frequency. Since the exact value of the structural damping parameter (g) is unclear for concrete foundation with NFRP layer, we chose g = 0 in the following ﬁgures. Fig. 3 illustrates the eﬀect of the CNT volume fraction on the dimensionless frequency of structure versus NFRP thickness to concrete foundation thickness ratio. It can be seen that with increasing the values of CNT volume fraction, the frequency of the system is increased. This is due to the fact that the increase of CNT leads to a harder structure. In addition, increasing the NFRP thickness to concrete Fig. 3. Eﬀects of CNT volume percent on the dimensionless frequency versus NFRP layer thickness to foundation thickness ratio. foundation thickness ratio, the frequency is decreased for hf / h < 0.005 and increased for hf / h > 0.005. It is perhaps due to this fact that for hf / h < 0.005, the destructive eﬀect of CNTs agglomeration is higher than the eﬀect of NFRP layer while for hf / h > 0.005, the NFRP layer shows the positive eﬀect on the frequency of the structure. Fig. 4 presents the agglomeration of CNTs on the dimensionless Table 1 Validation of present work with the other references. Method Ref. [43] Ref. [44] Ref. [45] Ref. [46] Ref. [47] Present Mode number 1 2 3 4 15.171 15.171 15.17 15.18 15.19 15.169 33.248 33.248 33.32 33.34 33.31 33.241 44.387 44.387 44.51 44.51 44.52 44.382 60.682 60.682 60.78 60.78 60.79 60.674 Fig. 4. Eﬀects of CNT agglomeration on the dimensionless frequency versus the NFRP layer thickness to foundation thickness ratio. 146 Soil Dynamics and Earthquake Engineering 103 (2017) 141–150 A. Zamani, M.R. Bidgoli Fig. 5. Eﬀects of soil medium on the dimensionless frequencyversus NFRP layer thickness to foundation thickness ratio. Fig. 6. Eﬀects of length of concrete foundation on the dimensionless frequencyversus NFRP layer thickness to foundation thickness ratio. frequency versus the NFRP thickness to concrete foundation thickness ratio. It is shown that considering agglomeration of CNTs leads to lower frequency in the structure since in this state, the stiﬀness of the structure decreases. In addition, the eﬀect of agglomeration of CNTs on the frequency increases with enhancing the NFRP thickness to concrete foundation thickness ratio. The dimensionless frequency of the nano-composite concrete foundation versus NFRP thickness to concrete foundation thickness ratio is demonstrated in Fig. 5 for diﬀerent soil mediums. In this ﬁgure, four cases of loose sand, dense sand, Clayey medium dense sand and Clayey soil are considered with the spring constants stated in Table 2. As can be seen, considering soil medium increases the frequency of the structure. It is due to the fact that considering soil medium leads to stiﬀer structure. Furthermore, the frequency of the dense sand medium is higher than the other cases since the spring constant of this medium is maximum. The eﬀect of the length of concrete foundation on the dimensionless frequency of the system versus NFRP thickness to concrete foundation thickness ratio is depicted in Fig. 6. As can be seen, the frequency of the structure decreases with increasing the length of concrete foundation. It is because increasing the length leads to softer structure. In addition, the decreasing rate of frequency is reduced with increasing the concrete foundation length. Fig. 7 shows the dimensionless frequency of the structure versus NFRP thickness to concrete foundation thickness ratio for diﬀerent concrete foundation widths. It can be also found that the frequency of the structure decrease with increasing the width which is due to the higher stiﬀness of system with lower width of the concrete foundation. The eﬀect of length to thickness ratio of concrete foundation on the dimensionless frequency versus NFRP thickness to concrete foundation thickness ratio is shown in Fig. 8. It can be found that with increasing the length to thickness ratio, the frequency of the structure is decreased. It is because with rising the length to thickness ratio, the stiﬀness of the structure will be decreased. The eﬀect of three diﬀerent modes on the dimensionless frequency versus NFRP thickness to concrete foundation thickness ratio is plotted in Fig. 9. As can be seen, with increasing the vibrational mode, the Fig. 7. Eﬀects of width of concrete foundation on the dimensionless frequency versus NFRP layer thickness to foundation thickness ratio. Fig. 8. Eﬀects of length to thickness of concrete foundation on the dimensionless frequency versus NFRP layer thickness to foundation thickness ratio. Table 2 Spring constants of soil medium under concrete foundation. frequency increases. Soil Kw Loose sand Dense sand Clayey medium dense sand Clayey soil 4800–16,000 64,000–128,000 32,000–80,000 12,000–24,000 5. Conclusion Innovative strengthening methods based on FRP material for extending the lifetimes of concrete structures (old industrial buildings, 147 Soil Dynamics and Earthquake Engineering 103 (2017) 141–150 A. Zamani, M.R. Bidgoli SSDT, considering NFRP with agglomerated nano ﬁbers and simulating the soil medium by Winkler model were the main contributions of this work. The eﬀective material properties of the NFRP layer were obtained using the Mori-Tanaka model considering agglomeration eﬀects. Based on SSDT, the motion equations were derived using energy method and Hamilton's principle. Exact solution was applied for obtaining the frequency of system so that the eﬀects of the volume percent and agglomeration of CNTs, soil medium, structural damping and geometrical parameters of concrete foundation were considered. The results show that considering structural damping leads to lower frequency. Increasing the NFRP thickness to concrete foundation thickness ratio, the frequency is decreased for hf / h < 0.005 and increased for hf / h > 0.005. It was shown that considering agglomeration of CNTs leads to lower frequency in the structure. It can be seen that with increasing the values of CNT volume fraction, the frequency of the system was increased. Considering soil medium increases the frequency of the structure. The frequency of the structure decreases with increasing the length and width of the concrete foundation. In addition, the decreasing rate of frequency was reduced with rising the concrete foundation length. Present results were in good agreement with those reported by [43–47]. Finally, it is expected that the results presented in this paper would be helpful for further works in mathematical modeling of concrete foundations retroﬁtted with nano ﬁber reinforced polymers. Fig. 9. Eﬀects of three diﬀerent modes on the dimensionless frequency versus NFRP layer thickness to foundation thickness ratio. bridges, oﬀ-shore platforms, etc.) were gaining more and more interest in the practical applications. Hence, in this article, vibration response of the concrete foundations retroﬁtted with NFRP layer resting on soil medium was presented. Mathematical modeling of the structure using Appendix A K11 = − A11 m2π 2 A n2π 2 + I0 ω2 − 44 2 2 a b (A1) K12 = − A12 m π 2 n ab (A2) A12z m π 3 n2 A11z m3 π 3 I m π ω2 + − 1 2 3 ab a a 2A 44z m π 3 n2 + ab2 (A3) K13 = − 2A 44f m π 3n2 K14 = + ab2 A11f m3 π 3 a3 + A12f m π 3 n2 ab2 J1 m π ω2 − a (A4) K21 = − A 44 m π 2 n A m π2 n − 21 ab ab (A5) K22 = − A22 n2 π 2 + I0 ω2 b2 (A6) K23 = K24 = K31 = A22z n3 π 3 A21z m2 π 3 n + 3 b a2 b 2A 44z m2 π 3 n I n π ω2 + − 1 a2 b b A22f n3 π 3 + (A7) A21f m2 π 3 n b3 a2 b 2A 44f m2 π 3 n J n π ω2 + − 1 2 a b b (A8) A21z m π 3 n2 I m π ω2 − 1 2 ab a 4A 44z m π 3n2 A11z m3 π 3 + + a b2 a3 (A9) 148 Soil Dynamics and Earthquake Engineering 103 (2017) 141–150 A. Zamani, M.R. Bidgoli K32 = 4A 44z m2 π 3 n I n π ω2 − 1 2 a b b A12z m2 π 3n A22z n3 π 3 + + a2 b b3 (A10) B12 m2 π 4 n2 B n4 π 4 − 22 4 −K 2 2 a b b B m2 π 4 n2 4B44 m2 π 4 n2 − − 21 2 2 a2 b2 a b 2 π 2 ω2 2 π 2 ω2 m n ⎞ + I0 ω2 + I2 ⎛ + a2 b2 ⎠ ⎝ K33 = − ⎜ − ⎟ B11 m4 π 4 a4 (A11) m2 π 2 ω2 n2 π 2 ω2 ⎞ K34 = I0 ω2 + J2 ⎛ + 2 a b2 ⎠ ⎝ A22zf n4 π 4 A11zf m4 π 4 − − −K b4 a4 2 4 2 2 A21zf m π n A12zf m π 4 n2 − − 2 2 a b a2 b2 2 4 2 4A 44zf m π n − a2 b2 ⎜ K 41 = A11f m3 π 3 a3 + K 42 = − + ⎟ + 4A 44f m π 3 n2 a b2 − (A12) J1 m π ω2 a A21f m π 3 n2 a b2 (A13) A12f m2 π 3 n A22f n3 π 3 J1 n π ω2 + + b a2 b b3 4A 44f m2 π 3 n a2 b K 43 = J2 ⎛ ⎝ ⎜ (A14) m2 π 2 ω2 n2 π 2 ω2 ⎞ + a2 b2 ⎠ ⎟ n2 π 2 ω2 ⎞ m2 π 2 ω2 + K2 J ⎛ + a2 b2 ⎠ ⎝ 2 4 2 A21zf m π n A12zf m2 π 4 n2 − − 2 2 a b a2 b2 4 4 A11zf m π A22zf n4 π 4 − −K− 4 a b4 4A 44zf m2 π 4 n2 − + I0 ω2 a2 b2 ⎜ A66g n2 π 2 ⎟ (A15) E12 m2 π 4 n2 E m4 π 4 − 11 4 a2 b2 a E21 m2 π 4 n2 − − a2 a2 b2 4A 44zf m2 π 4 n2 E n4 π 4 − − 22 4 + I0 ω2 − K a2 b2 b K 44 = − b2 A55g m2 π 2 − (A16) [5] Capozucca R, Bossoletti S. 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