Dynamic behaviour of suction caissons

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14 Chapter 2 – Dynamic stiffness of suction caissons—vertical vibrations frequency domain is then given by ∂σ ij (x, ω) ∂x j + ρB i (x, ω) + ω 2 ρU i (x, ω) = 0, x ∈ Ω, (2.1) where summation is carried out over repeated indices. U i (x, ω) (i=1,2,3) and σ ij (x, ω) (j=1,2,3) are the complex amplitudes of the displacement field and the stresses, respectively. The latter may be computed from the displacements by the constitutive relation. Further, ρB i (x, ω) are the body forces. The boundary conditions on the surface Γ of the body Ω are: } U i (x, ω) = Ûi (x, ω) for x ∈ Γ U P i (x, ω) = ˆP , Γ = Γ U ∪ Γ P , Γ U ∩ Γ P = ∅, (2.2) i (x, ω) for x ∈ Γ P where the displacement amplitude U i (x, ω) is given on one part of the boundary, Γ U , and the surface traction P i (x, ω) = σ ij (x, ω)n j (x) is given on the remaining part of the boundary, Γ P . Here n j (x) are the components of the outward unit normal to the surface. To obtain the boundary element formulation of Equation (2.1), a second state Uil ∗ (x, ω; ξ) is identified as the fundamental solution to the equation of motion ∂σijl ∗ (x, ω; ξ) + ρδ (x − ξ)δ il + ω 2 ρUil ∗ (x, ω; ξ) = 0, (2.3) ∂x j where δ (x − ξ) is the Dirac delta function in vector form and δ il is the Kronecker delta. It should be noted that the Green’s function Uil ∗ (x, ω; ξ) represents a 3 × 3 matrix, i.e. there are three displacement components at the receiver point x for each direction l of the load applied at the source point ξ. In three dimensions, Uil ∗ (x, ω; ξ) has a singularity of the order 1/r, whereas the corresponding stress field σijl ∗ (x, ω; ξ) has a singularity of the order 1/r 2 . The fundamental solution is based on wave propagation in the full space and therefore only represents body waves emanating from the source, i.e. dilatation and shear waves with phase velocities c P and c S , respectively. The velocities c P and c S are given as √ √ λ + 2G G c P = , c S = ρ ρ , (2.4) where λ and G are the Lamé constants of the material, and ρ is the mass density. The Lamé constants λ and G can be written in terms of Young’s modulus E and Poisson’s ratio ν by the following relations: G = E 2 (1 + ν) , λ = νE (1 + ν) (1 − 2ν) (2.5) Material damping is introduced by a complex Young’s modulus E ∗ , resulting in complex Lamé constants. The complex Young’s modulus E ∗ is given by E ∗ = E (1 + iη), (2.6) Morten Liingaard
2.3 Boundary element/finite element formulation 15 where η is the loss factor of the material and i = √ −1 is the imaginary unit. Note that the loss factor is assumed to be constant for all frequencies, i.e. hysteretic damping is assumed. The fundamental solution is applied as a weight function in the weak formulation of the equation of motion (2.1) for the physical field and vice versa. After some manipulations, and disregarding body forces in the interior of the domain, Somigliana’s identity is derived: ∫ ∫ C il (x) U l (x, ω) + Pil ∗ (x, ω; ξ)U l (ξ, ω, )dΓ ξ = Uil ∗ (x, ω; ξ)P l (ξ, ω, )dΓ ξ (2.7) Γ Γ Here Pil ∗ (x, ω; ξ) is the surface traction related to the Green’s function U il ∗ (x, ω; ξ). C il (x) is a doubly indexed scalar that only depends on the geometry of the surface Γ. In particular, C il (x) = 1/2δ il on a smooth part of the boundary Γ and C il (x) = δ il inside the body Ω. A detailed derivation of (2.7) and properties of C il (x) are given in (Andersen 2002; Domínguez 1993). In order to evaluate the boundary integral equations in (2.7) for a point x on the boundary, the surface is discretized into a finite number of boundary elements. The boundary integral equation can then be solved numerically for any point x on the boundary. The boundary can be discretized by different types of elements with varying order of integration. In the present study, quadrilateral elements with quadratic interpolation are employed, due to the fact that nine-noded boundary elements are superior in performance and convergence compared to elements with constant or linear interpolation (Andersen 2002). To obtain the BE formulation, the state variable fields on the boundary are discretized. U j (x) and P j (x) be the vectors storing the displacements and tractions at the N j nodes in element j. The displacement and traction fields over the element surface Γ j then become U(x, ω) = Φ j (x)U j (ω), P(x, ω) = Φ j (x)P j (ω), (2.8) where Φ j (x) is a matrix storing the interpolation, or shape, functions for the element. This allows the unknown values of the state variables to be taken outside the integrals in Equation (2.7). Finally, the three-row matrices originating from Equation (2.7) for each of the observation points may be assembled into a single matrix equation for the entire BE domain, H (ω)U(ω) = G (ω)P(ω). (2.9) Component (i, k) of the matrices H (ω) and G (ω) stores the influence from degree-offreedom k to degree-of-freedom i for the traction and the displacement, respectively, i.e. the integral terms on the left- and right-hand side of Equation (2.7). The geometric constants C il (x) are absorbed into the diagonal of H (ω). 2.3.2 Coupling of FE and BE regions The finite element (FE) region of the model is formulated by the equation of motion in the frequency domain (Andersen and Jones 2002): ( −Mω 2 + iC + K ) U = K FE U = F, (2.10) December 4, 2006
Page 1 and 2: Aalborg University Department of Ci
Page 3: To my Father
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64 Chapter 4 - Prototype bucket fou
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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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86 Appendix A - Solution for an inf
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Dynamic behaviour of suction caissons

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