Dynamic behaviour of suction caissons

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20 Chapter 2 – Dynamic stiffness of suction caissons—vertical vibrations Boundary Element/Finite Element models Due to symmetry only half the foundation is included. In the finite element region only half the model needs to be analysed when a plane of symmetric exists. The degrees of freedom in the plane of symmetry are simply eliminated in the system of equations in order to satisfy the conditions at the interface between the modelled and non-modelled part. The procedure for introducing a plane of symmetry in the BE region is more complex, and will not be given here. The procedure for BE analysis of problems with geometrical symmetry is discussed in details by Andersen and Jones (2001b). The BE/FE model of the surface footing contains a massless circular foundation with the radius R = 5 m. The foundation is modelled by 40 quadrilateral finite elements employing quadratic interpolation. The thickness (height) of the foundation is one meter. The soil is discretized into a total of 152 boundary elements with quadratic interpolation. The model is illustrated in Figure 2.3a. The BE/FE model of the suction caisson consists of four sections: a massless finite element section that forms the top of the foundation where the load is applied, a finite element section of the skirts, a boundary element domain inside the skirts and, finally, a boundary element domain outside the skirts that also forms the free surface. The skirt of the suction caisson is considered flexible, and the lid is assumed to be rigid. The lid is modelled as a solid finite element section with a thickness of one meter. Again, quadratic interpolation is employed. The models of the suction caisson and the subsoil contain approx. 100 finite elements and 350 boundary elements. The model is illustrated in Figure 2.3b. For both numerical models, the soil is linear elastic with G s = 1 MPa, ν s = 1/3 and ρ s = 1000 kg/m 3 . The material of the surface foundation and the lid of the suction caisson is linear elastic with G s = 10 6 MPa, ν s = 1/3 and ρ s = 0 kg/m 3 . The connection between soil and foundation corresponds to the condition of ‘rough’ contact since the foundation and the surrounding soil have common degrees of freedom. The mesh of the free surface for the surface foundation has been truncated at a distance of 15 m (3 times radius R) from the centre of the footing, based on convergence studies. Regarding the suction caisson (H/D = 1), the mesh of the free surface is truncated at a distance of 30 m (6 times radius R) from the centre of the foundation. The truncation distance for the models of the suction caisson depends on the skirt embedment. Convergence studies for the worst case (H/D = 2) suggested a truncation distance of 30 m from the centre of the foundation. This length has been used for all the BE/FE analyses of the suction caisson, regardless of embedment depth of the skirt. Adaptive meshing could possibly improve the accuracy versus the number of degrees of freedom, but this facility is currently not available in the BE/FE software. For a given excitation frequency a vertical load equal to 1 N is applied in the centre on top of the foundations and the complex displacements are computed. The complex vertical dynamic stiffness is then determined from the load and the displacement response. Note that load control has been used to generate the stiffness values. Displacement control would be more appropriate, but this feature is currently not available in the BE/FE software. The models contains approximately 1000 (surface model) to 3000 (suction caisson model) degrees of freedom and the runtime is approximately 5 to 30 minutes for each excitation frequency on a 2.0 GHz P4 laptop computer. Morten Liingaard
2.5 Benchmark tests 21 |SV V |/K 0 V V 6 4 2 Veletsos and Tang (1987) Surface foundation model Suction caisson model 0 0 1 2 3 4 5 6 Dimensionless frequency a 0 π 2 φV V (rad) π 4 0 0 1 2 3 4 5 6 Dimensionless frequency a 0 Figure 2.4: Vertical dynamic stiffness for a surface foundation calculated by two different BE/FE models. The numerical results are compared with a known analytical solution. Results The BE/FE models have been utilized for 13 excitation frequencies in the range a 0 ∈ ]0;6]. The results obtained from the numerical models are given in Figure 2.4 together with the known analytical solution reported by Veletsos and Tang (1987). The upper plot shows the normalized magnitude |S V V |/K 0 V V , and the phase angle φ V V is shown in the lower plot. First of all, the numerical models are able to reproduce the overall pattern of the frequency dependent stiffness of the analytical solution, when it is considered that the analytical solution by Veletsos and Tang (1987) is based on relaxed boundary conditions and the boundary element solutions corresponds to welded, or rough, contact. The same type of results have been reported by Alarcon et al. (1989). The results from the suction caisson model with ‘soil skirts’ match the results of the surface foundation model quite well, and it is concluded that the model of the suction caisson is able to reproduce the frequency dependent behaviour of a surface foundation without introducing errors due to the complexity of the model (two boundary element domains separated by a thin finite element structure). December 4, 2006
Page 1 and 2: Aalborg University Department of Ci
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Page 14 and 15: x Contents 3 Dynamic stiffness of s
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76 BIBLIOGRAPHY Brincker, R. and P.
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78 BIBLIOGRAPHY Kitahara, M. (1984)
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80 BIBLIOGRAPHY Morten Liingaard
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82 BIBLIOGRAPHY Morten Liingaard
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84 Appendix A - Solution for an inf
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Dynamic behaviour of suction caissons

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