Dynamic response of foundations on threedimensional layered soil using the scaled boundary finite element method

Transcription

1 IOP Conference Series: Materials Science and ngineering Dynamic response of foundations on threedimensional layered soil using the scaled oundary finite element method To cite this article: Carolin irk and Ronny ehnke IOP Conf. Ser.: Mater. Sci. ng. 8 View the article online for updates and enhancements. Related content - Development of polygon elements ased on the scaled oundary finite element method Irene Chiong and Chongmin Song - A coupled F and scaled oundary Fapproach for the earthquake response analysis of arch dam-reservoir-foundation system Yi Wang, Gao Lin and Zhiqiang Hu - A Hamiltonian-ased derivation of Scaled oundary Finite lement Method for elasticity prolems Zhiqiang Hu, Gao Lin, Yi Wang et al. This content was downloaded from IP address 6... on 9//7 at :

2 WCCM/APCOM IOP Conf. Series: Materials Science and ngineering () 8 doi:.88/ x///8 Dynamic response of foundations on three-dimensional layered soil using the scaled oundary finite element method Carolin irk and Ronny ehnke Institut für Statik und Dynamik der Tragwerke, Fakultät auingenieurwesen, Technische Universität Dresden, 6 Dresden, Germany -mail: Carolin.irk@tu-dresden.de, Ronny.ehnke@tu-dresden.de Astract. This paper is devoted to the dynamic analysis of aritrarily shaped threedimensional foundations on layered ground using a coupled FM-SFM approach. A novel scaled oundary finite element method for the analysis of three-dimensional layered continua over rigid edrock is derived. The accuracy of the new method is demonstrated using rigid circular foundations resting on or emedded in nonhomogeneous soil layers as examples.. Introduction This paper is devoted to the dynamic analysis of three-dimensional foundations resting on or emedded in inhomogeneous layered ground. Here, the major challenge is the accurate description of radiation damping in the unounded soil domain. No analytical solutions of prolems of this type exist. First numerical models have een proposed more than years ago ]. They led to the development of the thin-layer method,, ], which is closely related to the scaled oundary finite element method. Alternative approaches are ased on the oundary element method 5] or on simple cone models 6, 7]. For a comprehensive overview of existing alternative methods and their limitations and advantages the reader is referred to Ref. 8]. The general approach taken here is to couple a finite element model of the foundation and a certain irregular part of the surrounding soil to a scaled oundary finite element model of the unounded layered soil domain. The scaled oundary finite element method (SFM) 9] is a semi-analytical technique which is particularly suitale for wave propagation analyses in unounded domains. Originally, it is ased on descriing the geometry of a three-dimensional domain y scaling the geometry of the near field / far field interface using a radial coordinate. A scaling centre is introduced from where the total oundary must e visile. A modified form of the SFM for scalar or vector waves in a two-dimensional layer is also availale, ]. In this case, the scaling centre is located at infinity. oth derivations can, however, not e used for the description of a three-dimensional layered medium in their present form. In this paper, a modified scaled oundary finite element method for the analysis of dynamic prolems in three-dimensional layered continua is derived. ased on the use of a scaling line, rather than a scaling centre, a modified scaled oundary transformation is proposed. The derivation of the corresponding scaled oundary finite element equations in displacements and stiffness is presented. The latter is a nonlinear differential equation with respect to the radial c Pulished under licence y Ltd

3 WCCM/APCOM IOP Conf. Series: Materials Science and ngineering () 8 doi:.88/ x///8 PLAN PLAN A A A A near field far field SCTION A - A foundation SCTION A - A interface (a) soil layers () far field near field far field Figure. D foundation of aritrary shape emedded in layered soil deposit. (a) prolem setup, () sustructure method coordinate, which has to e solved numerically for each excitation frequency of interest. This is facilitated using a high-asymptotic expansion of the stiffness as a function of the radial coordinate. Rigid circular and square foundations emedded in or resting on layered homogeneous or inhomogeneous three-dimensional soil deposits over rigid edrock are examined as numerical examples. The accuracy of the novel method is demonstrated y comparing the dynamic stiffness coefficients calculated using the proposed method to numerical reference solutions.. Prolem statement and assumptions Consider an elastic foundation of aritrary shape which is emedded in a layered soil deposit as shown in Fig. (a). The interfaces etween two consecutive layers, the interface to the rigid ground as well as the free surface are regarded as horizontal. The individual layers with shear modulus G j, Poisson s ratio ν j, mass density ρ j and thickness d j are unounded in horizontal direction. The sustructure method is used to analyse the coupled soil-structure system, as indicated in Fig. (). The near field, which may contain irregularities and non-linearities, is modelled using the finite element method. The latter is well estalished and not addressed in this paper. The far field is analysed using a modified scaled oundary finite element method which is derived in the next section. The corresponding derivation is ased on the following assumptions. The near field / far field oundary is vertical to the horizontal rigid ase and is denoted as interface S. The Cartesian coordinate system is chosen such that the ẑ-axis is vertical and the ˆx ŷ plane is parallel to the rigid ground, respectively, as illustrated in Fig. (a). In order to facilitate the derivation of the modified SFM, special restrictions with respect to the choice of the scaled oundary finite element mesh apply. The upper and lower oundaries of all elements must e horizontal, where as the vertical oundaries must e perpendicular to the ˆx-ŷ-plane. From a practical point of view this means the following: The interfaces of all layers are horizontal. If the material parameters vary within one layer, the resulting oundaries are assumed to e vertical and perpendicular to the ˆx-ŷ-plane.

4 WCCM/APCOM IOP Conf. Series: Materials Science and ngineering () 8 doi:.88/ x///8 z^ ζ x^ O y^ S e V e η ξ ξ = S e x^ ^ z O y^ ξ = (a) () Scaling Line Figure. Scaled oundary finite element model of the far field. (a) Cartesian coordinates, () scaled oundary coordinates. Scaled oundary finite element method for D layered soil ased on the aove assumptions, a modified scaled oundary finite element method for threedimensional layered media is derived analogously to that of the original SFM 9]... Scaled oundary transformation of the geometry The scaled oundary coordinates ξ, η and ζ are introduced to descrie the geometry of the near field / far field interface S. As shown in Fig. (), the scaling centre is replaced y a straight line - the scaling line - which is identical to the ẑ-axis. The radial direction ξ is descried y a series of parallel rays which are perpendicular to the scaling line. ξ is equal to and at the intersection with the ẑ-axis and the interface S, respectively. The geometry of the interface S is descried y the circumferential coordinates η and ζ, where η and ζ are parallel and perpendicular to the rigid ground, respectively. The corresponding scaled oundary transformation is formulated as ˆx(ξ, η, ζ) = ξ N(η, ζ)] {x}, () ŷ(ξ, η, ζ) = ξ N(η, ζ)] {y}, () ẑ(ξ, η, ζ) = N(η, ζ)] {z}, () where the symols {x}, {y}, {z} and N(η, ζ)] denote the nodal coordinates of a scaled oundary finite element on the interface S and the row of shape functions used to interpolate them. The assumptions formulated in section with respect to the mesh geometry imply that the scaled oundary finite elements are singly curved rectangles, where the curvature occurs around the ζ-axis. It can e shown that in this case the partial derivatives x,ζ, y,ζ and z,η vanish, x(η, ζ),ζ = y(η, ζ),ζ = z(η, ζ),η =. () Using qs. ()-() the Jacoian matrix is formulated as ] Ĵ(ξ, η, ζ) = x y ξ x,η ξ y,η = ξ z,ζ J(η, ζ)], (5) with J(η, ζ)] = x,η y,η x y z,ζ. (6)

5 WCCM/APCOM IOP Conf. Series: Materials Science and ngineering () 8 doi:.88/ x///8 Note that the third row and column of J], which corresponds to z, is decoupled. The upper left lock is identical to the matrix J] of the original scaled oundary finite element method for two-dimensional prolems. Using qs. (5) and (6), the partial derivatives with respect to the Cartesian coordinates are expressed as /ˆx /ŷ /ẑ = y,ηz,ζ x,η z,ζ ξ + The infinitesimal volume dv is otained as yz,ζ xz,ζ ξ η + xy,η yx,η ζ. (7) dv = ξdξdηdζ. (8).. Scaled oundary finite element equation in displacement The governing differential equation for time-harmonic three-dimensional elastodynamics is L] T {σ} + {p} + ω ρ {u} =, (9) with the displacement vector {u} = {u(ˆx, ŷ, ẑ)} = u x u y u z ] T, the stresses {σ}, the volume forces {p} = p x p y p z ] T and the differential operator L], L] T = ˆx ŷ ẑ ẑ ŷ ẑ ˆx ŷ ˆx Using Hooke s law, the stress-strain relationship is expressed as. () {σ} = σ xx σ yy σ zz σ yz σ xz σ xy ] T = D] {ε}, () with the elasticity matrix D] and the strain vector {ε}, {ε} = ε xx ε yy ε zz γ yz γ xz γ xy ] T = L] {u}. () Using q. (7), the differential operator L] is expressed in terms of the scaled oundary finite element coordinates as L] = ] ξ + ] ξ η + ] ζ, () with ] = y,η z,ζ x,η z,ζ x,η z,ζ y,η z,ζ x,η z,ζ y,η z,ζ ] =, ] = xy,η yx,η xy,η yx,η xy,η yx,η yz,ζ xz,ζ xz,ζ yz,ζ xz,ζ yz,ζ, (). (5)

6 WCCM/APCOM IOP Conf. Series: Materials Science and ngineering () 8 doi:.88/ x///8 The stress vectors { t ξ}, {t η } and { t ζ} of the surfaces (η, ζ), (ζ, ξ) and (ξ, η), respectively, are expressed as { t ξ} = ] T {σ}, g ξ g ξ = (y,η z,ζ ) + ( x,η z,ζ ), (6) { t η} = ] T {σ}, g η g η = ( yz,ζ ) + (xz,ζ ), (7) { t ζ} = g ζ ] T {σ}, g ζ = xy,η yx,η. (8) The method of weighted residuals with respect to the circumferential directions is applied to the governing differential equation (9) formulated in terms of the scaled oundary coordinates. {w} T ] T {σ,ξ } dv + {w} T ] T {σ,η } dv + {w} T ] T {σ,ζ } dv ξ V V V + {w} T {p} dv + ω {w} T ρ {u} dv =. (9) V Integrating the second and third term in q. (9) y parts and using qs. (7) and (8), the weighted residual statement is transformed into T ξ {w} ] ( { T T {σ,ξ } dηdζ + ξ {w} t ζ} g ζ dη + ) ξ {tη } g η dζ Γ ξ ( {w} T ] T + {w,η } T ] T + ξ {w,ζ } T ] T ) {σ} dηdζ () V +ξ {w} T {p} dηdζ + ω ξ {w} T ρ {u} dηdζ =. The displacement field {u} and the weighting functions {w} are discretized analogously to the geometry, {u(ξ, η, ζ)} = N u (η, ζ)] {u(ξ)}, {w} = N u (η, ζ)] {w(ξ)}. () Using qs. (), (), () and (), the stresses {σ} are expressed in terms of the nodal displacements {u(ξ)} as ( {σ} = {σ(ξ, η, ζ)} = D] ] ( {u(ξ)} +,ξ ] + ]) ) {u(ξ)}, () ξ with ] = ] N u (η, ζ)], ] = ] N u (η, ζ)],η, ] = ] N u (η, ζ)],ζ. () Sustituting q. () in q. () and evaluating further manipulations, the scaled oundary finite element equation in displacement for a three-dimensional layered medium () is otained. ] ( ξ {u(ξ)},ξξ + ] ] + ] T + ξ ] ξ ] ) T {u(ξ)},ξ ( + ] + ] ] ] T ξ 5 ]) {u(ξ)} + ω M ] ξ {u(ξ)} + {F (ξ)} =. () ξ 5

7 WCCM/APCOM IOP Conf. Series: Materials Science and ngineering () 8 doi:.88/ x///8 The coefficient matrices ] - 5], M ] and the term {F (ξ)} are defined as ] = ] T D] ] dηdζ, ] = ] T D] ] dηdζ, (5) ] = ] T D] ] dηdζ, ] = ] T D] ] dηdζ, (6) ] = ] T D] ] dηdζ, 5 ] = ] T D] ] dηdζ, (7) { F t (ξ) } = Γ ξ M ] = N u (η, ζ)] T ρ N u (η, ζ)] dηdζ, (8) {F (ξ)} = ξ { F t (ξ) } + ξ { } F (ξ), (9) ( { T N(η, ζ)] t ζ} (xy,η yx,η ) dη + ) ξ {tη } ( yz,ζ ) + (xz,ζ ) dζ, () { } F (ξ) = N(η, ζ)] T {p} dηdζ. ().. Scaled oundary finite element equation in dynamic stiffness The linear second order differential equation () in displacement can e transformed into an equivalent nonlinear first order differential equation in dynamic stiffness. Using the principle of virtual work, the internal nodal forces can e expressed as {Q(ξ)} = ] ξ {u(ξ)},ξ + ( ] T + ξ ]) {u(ξ)}. () In the asence of ody forces and surface tractions, the dynamic stiffness S (ω, ξ)] of an unounded domain is defined as {Q(ξ)} = S (ω, ξ)] {u(ξ)]. () quating q. () to q. () and evaluating a numer of sustitutions, the scaled oundary finite element equation in dynamic stiffness () is derived, (S (ω, ξ)] + ] + ξ ] T ) ] ( S (ω, ξ)] + ] T + ξ ]) ξ S (ω, ξ)],ξ ] ξ ( ] + ] T ) ξ 5] + ω ξ M ] =. () Note that q. () varies consideraly from the original scaled oundary finite element equation in dynamic stiffness derived in Ref. 9]. Differences occur oth with respect to the numer of coefficient matrices and the order of ξ. As a result, it is not possile to transform q. () into a differential equation with respect to the excitation frequency ω. The first order differential equation with respect to ξ is solved numerically using a Runge-Kutta scheme for a given frequency ω instead. An initial value is required to start the numerical integration scheme. It is constructed using an asymptotic expansion of the dynamic stiffness S(ω, ξ)] with respect to the radial coordinate ξ. 6

8 WCCM/APCOM IOP Conf. Series: Materials Science and ngineering () 8 doi:.88/ x///8.. High-asymptotic expansion of dynamic stiffness The eigenvalue prolem (5), M ] Φ] = ] Φ] m ], Φ] T ] Φ] = I], Φ] T M ] Φ] = m ], (5) is used to transform q. () into with ( s (ξ)] + e ] + ξ e ] T ) ( s (ξ)] + e ] T + ξ e ]) ξ s (ξ)],ξ The power series s (ξ)] = Φ] T S (ξ)] Φ], e ] ξ ( e ] + e ] T ) ξ e 5] + ω ξ m ] =, (6) s (ξ)] (ξ) c ] + (ξ) k ] + e i ] = Φ] T i] Φ], i =,, 5. (7) m j= (ξ) j a j], (8) is sustituted in q. (6). The coefficients c ], k ] and a j ] are calculated equating terms corresponding to decreasing powers of ξ to zero. A Riccati equation, c ] c ] + e ] T c ] + c ] e ] + e ] T e ] e 5] + ω m ] =. (9) is otained for c ]. The Lyapunov equations ()-() for the calculation of the coefficients k ] and a ] follow. ( c ] + e ] ) T k ] + k ] ( c ] + e ]) = (c ] + e ] ) T e ] T e ] ( c ] + e ]) + c ] + ( e ] + e ] T ). () ( c ] + e ] T ) a ] + a ] ( c ] + e ]) = ( k ] + e ]) ( k ] + e ] T ) + e ], () quations for higher order terms a j ], j >, can e derived in an analogous manner. The initial value of the dynamic stiffness is constructed evaluating q. () for a high ut finite value ξ h, S (ω, ξ h )] (Φ] ) T m ξ h c ] + k ] + (ξ h ) j a j] Φ]. ().5. Material damping So far, material damping has not een considered in the aove derivation. Linear hysteretic material damping can, however, easily e incorporated into the proposed method y using a complex-valued shear modulus G instead of the real-valued quanity G, j= G = ( + i D) G. () In q. () the symols i and D denote the imaginary unit and the damping ratio, respectively. Using q. (), the complex-valued coefficient matrices ] - 5 ] follow from those given in qs. (5)-(7) as i ] = ( + i D) i], i =,,, 5. () The mass matrix M ] remains unchanged. 7

9 WCCM/APCOM IOP Conf. Series: Materials Science and ngineering () 8 doi:.88/ x///8 r G ρ v = / D = % r r x. r DISCRTIZATION nxe = nye = nze = Figure. Rigid circular foundation resting on homogeneous soil layer over edrock r.5 r.5 r G ρ v =.5 t =.5 r r.5 r.5 r.5g ρ v =. r.g.89 ρ v = / D = 5%.5 r r DISCRTIZATION nxe = nye = nze = 6.5 r.5 r Figure. Rigid cylindrical foundation emedded in nonhomogeneous soil layer. Numerical examples A rigid massless circular foundation of radius r as shown in Fig. is considered. It is resting on a homogeneous soil layer of thickness d = r. Only one quarter of the system is analysed due to symmetry. The unounded soil is modelled using 8-node scaled oundary finite elements, as shown in Fig.. The near field is modelled using -node finite elements. The dynamic stiffness matrix of the three-dimensional unounded layered soil is calculated using the proposed method. The asymptotic expansion of S (ω, ξ)] is evaluated at ξ h = to otain an initial value for the numerical solution of q. (). The frequency-dependent dynamic stiffness coefficients of the rigid circular foundation are calculated and evaluated in the form S(a ) = K stat (k(a ) + ia c(a )), a = ωr c s. (5) Here, the symols k(a ), c(a ), c s and a denote the frequency-dependent stiffness and damping coefficients, the shear wave velocity c s = G/ρ and the dimensionless frequency, respectively. The dynamic stiffness coefficients otained for vertical, horizontal, rocking and torsional motion are shown in Fig. 5 (a)-(d), respectively. The results are compared to a reference solution otained using the thin-layer method (see Ref. K] in Ref. ]) with a very fine vertical discretization. As a second example, a rigid cylindrical foundation of radius r, which is emedded with depth t =.5r in a nonhomogeneous soil layer as shown in Fig. is considered. One quarter of the system is analysed using 8-node scaled oundary finite elements and -node finite elements to model the far and near field, respectively. The dynamic stiffness matrix of the soil layer is calculated evaluating the asymptotic expansion of S (ω, ξ)] at ξ h = 7 to otain an initial value for the numerical solution of q. (). The resulting dynamic stiffness coefficients of the rigid emedded foundation for vertical and rocking motion are shown in Fig. 6 (a)- (), respectively. The corresponding dynamic stiffness coefficients otained using the thin layer method (see Ref. ] in Ref. ]) are also shown for comparison. oth examples demonstrate the accuracy and the correct implementation of the proposed method. The cutoff frequency and higher eigenfrequencies of the three-dimensional soil layer are accurately represented. 8

10 WCCM/APCOM IOP Conf. Series: Materials Science and ngineering () 8 doi:.88/ x///8 (a) (c) SPRING COFFICINT k v (a ) -] DAMPING COFFICINT c v (a ) -] SPRING COFFICINT k r (a ) -] DAMPING COFFICINT c r (a ) -] SFM-FM XACT TLM K] in 5] DIMNSIONLSS FRQUNCY a = r /cs a =.57 a = 6.8 a = DIMNSIONLSS FRQUNCY a = r /cs DIMNSIONLSS FRQUNCY a = r /cs a =.57 SFM-FM XACT TLM K] in 5] DIMNSIONLSS FRQUNCY a = r /cs a = 9. a) c) () (d) SPRING COFFICINT k h (a ) -] DAMPING COFFICINT c h (a ) -] SPRING COFFICINT k t (a ) -] DAMPING COFFICINT c t (a ) -] SFM-FM XACT TLM K] in 5] DIMNSIONLSS FRQUNCY a = r /cs a =.57 a =.7 5 DIMNSIONLSS FRQUNCY a = r /cs DIMNSIONLSS FRQUNCY a = r /cs a =.57 a =.7 5 DIMNSIONLSS FRQUNCY a = r /cs ) d) SFM-FM XACT TLM K] in 5] Figure 5. Dynamic stiffness coefficients of circular rigid foundation resting on homogeneous soil layer over rigid edrock. (a) vertical, () horizontal, (c) rocking, (d) torsional 9

11 WCCM/APCOM IOP Conf. Series: Materials Science and ngineering () 8 doi:.88/ x///8 (a) () SPRING COFFICINT k v (a ) -] SPRING COFFICINT k r (a ) -] DIMNSIONLSS FRQUNCY a = r /cs 5 6 DIMNSIONLSS FRQUNCY a = r /cs DAMPING COFFICINT c v (a ) -] DAMPING COFFICINT c r (a ) -] 5 6 DIMNSIONLSS FRQUNCY a = r /cs 5 6 DIMNSIONLSS FRQUNCY a = r /cs a) c) Figure 6. Dynamic stiffness coefficients of cylindrical rigid foundation emedded in nonhomogeneous soil layer over rigid edrock. (a) vertical, () rocking 5. Conclusions The proposed modified formulation significantly roadens the area of application of the scaled oundary finite element method. It can e used for the frequency-domain analysis of D elastic foundations of aritrary shape emedded in or resting on layered inhomogeneous soil. Its extension to the time-domain is the suject of current research. Acknowledgments Carolin irk performed some of the research reported herein during her visit to The University of New South Wales within the scope of a Marie Curie International Outgoing Fellowship for Career Development of the uropean Community. The financial support provided y the uropean Community and her appointment as a Visiting Fellow are gratefully acknowledged. References ] Lysmer J and Waas G 97 Journal of the ngineering Mechanics Division ] Kausel and Peek R 98 ulletin of the Seismological Society of America ] Kausel 986 International Journal for Numerical Methods in ngineering ] Kausel 99 International Journal for Numerical Methods in ngineering ] Karaalis D L and Mohammadi M 998 Soil Dynamics and arthquake ngineering ] Meek J W and Wolf J P 99 Journal of Geotechnical ngineering ] Pradhan P K, aidya D K and Ghosh D P Soil Dynamics and arthquake ngineering 5 8] Gazetas G 98 Soil Dynamics and arthquake ngineering 9] Wolf J P The Scaled oundary Finite lement Method (Chichester: Wiley & Sons) ] Wolf J P and Song C 996 Finite-lement Modelling of Unounded Media (Chichester: Wiley & Sons) ] Li, Cheng L, Deeks A J and Teng 5 Applied Ocean Research 7 6 ] Wolf J P 99 Foundation Viration Analysis Using Simple Physical Models (Prentice Hall) ] Wolf J P and Preisig M arthquake ngineering and Structural Dynamics 75 98