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A BEM–FEM model for the dynamic analysis of building structures founded on viscoelastic or poroelastic soils

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Abstract

This paper presents a time-harmonic boundary element–finite element three–dimensional model for the dynamic analysis of building structures founded on viscoelastic or poroelastic soils. The building foundation and soil domains are modelled as homogeneous, isotropic, viscoelastic or poroelastic media using boundary elements. The foundation can also be modelled as a perfectly rigid body coupled to soil and structure. The buildings are modelled using Timoshenko beam finite elements that include the torsional eccentricity of non-symmetrical buildings. The excitation model includes far-field plane seismic waves of P, S or Rayleigh type for viscoelastic soils and P1 and S type for poroelastic soils. Modelling foundation and structure as rigid body and Timoshenko beam respectively conveys important benefits such as a significant reduction in the number of degrees of freedom in the problem, which allows to study problems involving several building structures and the interactions between them with acceptable computational effort. Results are presented for validation purposes first, and for studying the influence of modelling the soil as a viscoelastic or poroelastic region afterwards. Results involving structure–soil–structure interaction are also presented for illustration purposes.

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Acknowledgments

This work was supported by the Subdirección General de Proyectos de Investigación of the Ministerio de Economía y Competitividad (MINECO) of Spain and FEDER through research project BIA2010-21399-C02-01 and also by the Agencia Canaria de Investigación, Innovación y Sociedad de la Información (ACIISI) of the Government of the Canary Islands and FEDER through research project ProID20100224. A. Santana is recipient of the FPI research fellowship BES-2009-029161 from the MINECO. The authors are grateful for this support. The authors also want to thank the reviewers for their valuable comments and suggestions.

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Appendix: Plane harmonic waves in viscoelastic and poroelastic halfspace

Appendix: Plane harmonic waves in viscoelastic and poroelastic halfspace

1.1 P, S and Rayleigh waves in viscoelastic halfspace

For a viscoelastic media, the vector of the displacement field of vertical generic incident wave (P and S) can be written as follows

$$\begin{aligned} {\mathbf {u}} _{\text {I}}=\left( \text {A}\,\text {e}^{-\text {i} \,k\,z}+\text {B}\,\text {e}^{\text {i}\,k\,z}\right) \mathbf d \end{aligned}$$
(21)

where \(\text {A}\) and \(\text {B}\) are respectively the amplitudes of the incident and reflected waves, \(\mathbf d\) is the vector containing the direction cosines of the displacement and \(k=\omega /c\) is the wave number, being c (\(c_{\text {s}}\text {\, or\, }c_{\text {p}}\)) the wave velocity. The boundary conditions on the free-surface in terms of unitary displacement and zero stresses allow to compute the values of the amplitudes \(\text {A}=\text {B}=0.5\).

In the case of Rayleigh’s wave propagating along the x-direction (Fig. 13a), the three components of the vector of the displacement field are written as follows

$$\begin{aligned}u_{\text {I}}=({\text {A}}_1\,{\text {e}}^{{\text {b}}_1\,z}+\text{A}_2 \,{\text {e}}^{{\text {b}}_2\,z})\,{\text {e}}^{-{\text{i}}\,k\, x};\quad v_{\text {I}}=0; \quad w_{\text{I}}=\left(-\frac{{\text {i}}\,k}{{\text {b}}_1}{\text {A}}_1 \,{\text{e}}^{{\text {b}}_1\,z}+\frac{{\text {b}}_2}{{\text {i}}\, k}\text{A} _2\,{\text {e}}^{{\text {b}}_2\,z}\right) \, {\text{e}}^{-{\text {i}}\,k\,x} \end{aligned}$$
(22)

being \(k=\omega /c_{\text {R}}\) the wave number, \(\text {b}_1=k\,\sqrt{1-c_{\text {R}}^2/c_{\text {s}}^2}\) and \(\text {b}_2=k\,\sqrt{1-c_{\text {R}}^2/c_{\text {p}}^2}\). The amplitudes \(\text {A}_1,\,\text {A}_2\) and the Rayleigh wave velocity \(c_{\text {R}}\) are computed on the basis of zero stresses at the free–surface and depends on the material properties of the soil. See Achenbach (1973) for more details.

In any case, taking into account the kinematic relations and the constitutive law (Hooke’s law), the strain \((\varepsilon _{ij})_{\text {I}}=\frac{1}{2}((u_{i,j})_{\text {I}} +(u_{j,i})_{\text {I}})\) and the stress \((\sigma _{ij})_{\text {I}}=\lambda \,\delta _{ij}\,e+2\,\mu \,(\varepsilon _{ij})_{\text {I}}\) tensors are respectively obtained \((i,j=x,y,z)\), being e the volumetric dilatation and \(\lambda\) and \(\mu\) the Lame’s constants. Finally the incident traction field is computed as \({\mathbf {p}} _{\text {I}}=(\sigma _{ij})_{\text {I}} \,{\mathbf {n}}\) where \({\mathbf {n}}\) is the normal vector.

1.2 P1 and S waves in poroelastic halfspace

For a vertical incident wave, the vectors of the displacement field for the solid skeleton and the fluid phase can be respectively written as follows

$$\begin{aligned} {\mathbf {u}} _{\text {I}}=\left({ \text {A}}\,{\text{e}}^{-{\text {i}} \,k\,z}+{\text {B}}\,{\text{e}}^{{\text {i}}\,k\,z}\right) {\mathbf {d}} ;\quad {\mathbf {U}} _{\text{I}}=\beta \left( {\text {A}}\, {\text {e}}^{-{\text {i}}\,k\,z}+{\text {B}}\,{\text {e}}^{{\text {i}}\,k\,z}\right) {\mathbf{d}} \end{aligned}$$
(23)

As before, \(\text {A}\) and \(\text {B}\) are respectively the amplitudes of the incident and reflected waves and \(\mathbf d\) is the vector containing the direction cosines of the displacement and \(k=\omega /c\) is the complex-value wave number. For a shear incident S-wave \(c=c_{\text {s}}\,(k=k_{\text {s}})\) and for the longitudinal P1-wave \(c=c_{p1}\,(k=k_{p1})\). The expressions of \(\beta\) regarding the type of wave are respectively expressed as follows

$$\begin{aligned} \beta _\text {s}=-\frac{\hat{\rho }_{12}}{\hat{\rho }_{22}} ;\quad \beta _{p1}=\frac{k_{p1}^2\,(\lambda +2\mu +\frac{Q^2}{R}) -\omega ^2\,\hat{\rho }_{11}}{\omega ^2\,\hat{\rho }_{12}-k_{p1}^2\,Q} \end{aligned}$$

where \(Q,\,R\) are the Biot’s constants and \(\hat{\rho }_{11}=\rho _{1}-\rho _{12}-{\text {i}}\,{\text{b}}/\omega ,\, \hat{\rho }_{12}=\rho _{12}+{\text {i}}\,{\text{b}}/\omega ,\,\hat{\rho }_{22} =\rho _{2}-\rho _{12}-\text {i}\,b/\omega\) (see Norris 1985) are parameters that include the dissipation constant b and the densities \(\rho _1=\rho _s\,(1-\phi ),\,\rho _2=\rho _f\,\phi ,\,\rho _{12}=-\rho _a\), being \(\phi\) the soil porosity, \(\rho _{s}\) the density of the solid skeleton, \(\rho _{f}\) the density of the fluid phase and \(\rho _{a}\) the apparent added density.

For both type of waves, taking into account the kinematic relations and the constitutive law (see Biot 1956), the strain tensor \((\varepsilon _{ij})_{I}=\frac{1}{2}((u_{i,j})_{I}+(u_{j,i})_{I})\), the stress tensor in the solid skeleton \((\tau _{ij})_{I}=(\lambda +\frac{Q^2}{R}) \,e\,\delta _{ij}+2\,\mu \,(\varepsilon _{ij})_{I}+Q \,\varepsilon \,\delta _{ij}\) and the fluid equivalent stress \((\tau )_{I}=Q\,e+R\,\varepsilon\) are obtained \((i,j=x,y,z)\), being e and \(\varepsilon\) the solid and fluid dilatations, respectively. The boundary conditions on the free–surface of the solid skeleton in terms of unitary displacement and zero stresses allow to compute the values of the amplitudes \(\text {A}=\text {B}=0.5\). Finally the incident traction field is computed as \({\mathbf {p}} _{I}=(\tau _{ij})_{I}\,{\mathbf {n}}\) where \({\mathbf {n}}\) is the normal vector.

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Santana, A., Aznárez, J.J., Padrón, L.A. et al. A BEM–FEM model for the dynamic analysis of building structures founded on viscoelastic or poroelastic soils. Bull Earthquake Eng 14, 115–138 (2016). https://doi.org/10.1007/s10518-015-9817-z

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