Undrained Load Capacity of Torpedo Anchors in ... - laceo - UFRJ

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Contact properties As pointed out before, surface to surface contact elements allow large relative displacement and separation between surfaces in contact. This task is performed with a contact detection algorithm based on the pinball technique (Belytschko and Neal [7]) and contact forces evaluated with the augmented Lagrangian method (Belytschko et al. [8]). The augmented Lagrangian method consists of calculating a series of penalty forces during the equilibrium iterations so that the final penetration between the two contact bodies is smaller than a value previously established. In the developed model, an allowable penetration of 0.1% of the smaller thickness between the two elements initially in contact is adopted. Penalty forces are calculated by placing fictitious springs along the contact boundaries of two bodies. When contact is established, these forces are applied to the nodal points of the contact elements placed along the contact bodies. The forces are proportional to the penetration and a chosen penalty parameter, which can be physically interpreted as the stiffness of the springs. The choice of the stiffness of the springs, called normal contact stiffness, f N , may be performed with relatively simplicity, but respecting some rules. On the one hand, very high values may lead to numerical instabilities and, on the other hand, very small values provoke violations of the contact conditions. Different estimates of contact normal stiffness proposed in Belytschko and Neal [7] and Benson and Hallquist [9], for instance, points to a value equivalent to the stiffness of the elements in contact. Therefore, an estimate of the normal contact stiffness may be given by: E( z) fN z) = ⋅ Ac tc ( (18) where t c is the smaller thickness of the two elements in contact and A c is the contact area between these elements. Another important aspect is the simulation of the relative sliding between the soil and the torpedo anchor. Here, the Mohr-Coulomb friction model is used and, according to API [10] may be expressed in the form: −0.5 () z , ψ () z −0.25 () z , ψ () z ⎪ ⎧0.5 ⋅ψ ≤ 1.0 α () z = ⎨ (21) ⎪⎩ 0.5 ⋅ψ > 1.0 where: ( z) () z Su ψ () z = (22) p o Finally, a stick stiffness, f T , must be defined to avoid large displacements between the two contact surfaces before the maximum shear stress is reached. In the proposed model, an expression similar to the one employed to the normal contact stiffness was adopted: G( z) fT ( z) = ⋅ Ac (23) t c If a purely cohesive soil is modeled, Eq. (9) reduces to: ( z) = α( z) ⋅ S ( z) τ (24) max u and only this initial adhesion is considered by the contact elements. Initial stress state of the soil An important aspect of the anchor analysis is the simulation of the initial stress state of the soil, i. e., the stresses in the soil prior to the imposition of any kind of structural load to the anchor. As the proposed FE model does not simulate the anchor penetration in the soil, it is assumed, by hypothesis, that the anchor is loaded with the soil stress state undisturbed by the installation process. The anchor, thus, is supposed to be “wished in place”. In order to induce this undisturbed stress state in the soil, three different FE meshes are present in the same FE model: the first mesh represents the soil that surrounds the anchor; the second is related to the torpedo anchor; and the third simulates the soil that occupies the anchor position before its installation. Figure 8 presents these meshes. τ max ( z) = α( z) ⋅ S ( z) + K ( z) ⋅ p ( z) tan( δ ) u 0 0 ⋅ (19) where τ max is the maximum allowable shear stress; α is the adhesion factor; p o is the effective overburden pressure; and δ is the friction angle between the soil and the anchor wall and may be stated as (API [10]): o δ = φ − 5 (20) The adhesion factor may be calculated by the formulas proposed by API [10]: Figure 8 – FE meshes for the analysis of a torpedo anchor: soil mesh, anchor mesh and soil mesh previous to the anchor installation. 6 Copyright © 2009 by ASME
The FE mesh of the anchor interacts with the surrounding soil with the contact elements previously described, as pointed in Fig. 9(a). The FE mesh of the soil that is at the same position of the anchor has the shape of the anchor and its outer nodes are coincident with the surrounding soil nodes. These nodes have their degrees of freedom coupled, as shown in Fig. 9(b). Moreover, each “slice” of the meshes presented in Fig. 9(b) has the same physical properties and, consequently, this mesh represents the media before the anchor installation. It is important to mention that there is no interaction between the meshes of the anchor or the soil that occupies its position. (a) (b) Figure 9 – Details of the FE mesh: (a) surrounding soil and torpedo anchor; (b) surrounding soil and soil within the torpedo anchor location. With this approach, in a first step, the anchor mesh is “deactivated” by multiplying the stiffness matrix of each of its elements by a small factor, e. g., 10 -6 , while the remaining soil finite elements maintain their stiffness matrices. In this step, only gravity acts and effective material properties are considered. Therefore, the soil mesh is deformed by its selfweight and the initial stress state of the soil is defined. The normal contact forces between the surrounding soil mesh and the soil mesh at the anchor position are transmitted to the contact elements that connect the surrounding soil mesh and the deactivated anchor mesh. In a second load step, the soil at the anchor position is deactivated and the anchor mesh is activated. The contact forces are transmitted to the anchor outer wall by the contact elements and friction forces, as defined in the previous section, arise when the anchor is loaded. If the self-weight of the anchor is to be addressed in the analysis, previous to the application of the load imposed by the mooring line, a load step in which only gravity acts on the anchor is included. By doing this, a small perturbation in the stress state of the soil is induced in the region around the anchor, which can be argued to be caused by the installation process. Solution procedure A typical FE mesh to predict the load capacity and the structural behavior of a torpedo anchor has between 100000 and 500000 degrees of freedom and, as previously mentioned, must account for contact, geometric and material nonlinearities. After testing several numerical methods, the sparse method was adopted since it provided accurate results with minimal computational effort. The analysis is divided in three different steps. In the first one, the initial stress state is generated by subjecting the soil to the action of gravity in a single load increment. In the second step, the analysis is restarted with the stress state from the first step and gravity acts on the anchor. This second load step is also performed with a single load increment. Finally, in the third step, the total load is imposed with adaptable increments since, as the surrounding soil progressively fails, the stability of the solution is affected and lower load increments are required. Typically, this last load step starts with an initial load increment of 5% of the total load applied and, as the analysis progresses, it can be reduced to 1% of the total load. At each load step, convergence is achieved if the Euclidian norm (L2) of the residual forces and moments is less than 0.1% of the absolute value of the total forces and moments applied. Model implementation The presented model was implemented in a program called ESTACAS. This software generates FE meshes to be analyzed with ANSYS® program, where the following finite elements are employed: SOLID185 in order to model the soil and the torpedo anchor; CONTA174 and TARGE170 are used to simulate the contact between the soil and the anchor. CASE STUDY Description In this paper, various FE analyses were performed in order to study the effect of four parameters on the hold capacity of the anchor. These parameters are the: 1. Undrained shear strength of the soil. 2. Number of flukes in the anchor. 3. Width of the flukes. 4. Angle between the applied load and the plane of the flukes. The geometry of the anchor is presented in Fig. 10 and its possible cross-sections are shown in Fig. 11. Initially, the anchor is assumed to have four flukes, but configurations with three or no flukes are also studied. The width of the flukes 7 Copyright © 2009 by ASME
Page 1 and 2: Proceedings of the ASME 2009 28th I
Page 3 and 4: E G = (7) 2⋅ K ( 1+υ) = 1000 ⋅
Page 5: Anchor modeling The torpedo anchor
Page 9 and 10: Load (kN) 14000 12000 10000 8000 60
Page 11 and 12: and dividing this value by the sine
Page 13: Comparisons with API [10] values sh

Undrained Load Capacity of Torpedo Anchors in ... - laceo - UFRJ

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