<p>The purpose of this work has been to investigate the ultimate capacities of aluminium plates. The aluminium plates are supposed to be part of a marine structure. Their behaviour has been explored by taking advantage of non-linear finite element simulations. The loading conditions included axial compression, transverse compression, pure shear loading, axial compression in combination with transverse compression and axial compression in combination with shear loading.</p><p>Different patterns of heat affected zones have been studied with particular attention on conventionally welded plates and extruded, welded plates. The plates have been given realistic boundary conditions in addition to boundary conditions believed to represent the extremities in plate collapse behaviour. The initial deflection patterns chosen, were generally conservative, and the variations in collapse capacities for different initial deflection patterns have been investigated. The initial deflection amplitudes were systematically varied to cover all values of the initial deflection amplitudes likely to occur.</p><p>In addition to studying different patterns of the heat affected zones, systematic variations of the breadths of the heat affected zones have been performed. Systematic variations in the magnitudes of the residual stresses and systematic variations of the reductions of the 0.2 percent tensile proof stresses in the heat affected zones have also been carried out. Three different aluminium alloys were studied. They were believed to represent a high, medium and low utilised aluminium alloy. The aspect ratio of the plates was varied to cover both plates that were completely square to plates with a length equal to five times the breadth of the plates.</p><p>It was found that axially loaded plates with heat affected zones along loaded edges had lower ultimate capacities than plates with heat affected zones along unloaded edges, and if the plates had heat affected zones along loaded edges, having additional heat affected zones along unloaded edges did not alter the buckling capacities. The reduction in ultimate capacity could be modelled to vary linearly with the breadth of the heat affected zones, and the reduction in ultimate capacity could be modelled to vary linearly with the relationship between the value of the 0.2 percent tensile proof stress in the heat affected zones and the 0.2 percent tensile proof stress of the base material.</p><p>The ultimate capacities of transversally loaded plates could be found as the sum of a square plate and a plate strip, even if the plates contained heat affected zones. Square plates and plate strips with relevant patterns of heat affected zones had to be employed. Biaxial interaction curves showed a considerable slenderness dependence for all values of the aspect ratio investigated. The biaxial interaction curves were only slightly affected by introduction of different patterns of heat affected zones. The stress corresponding to equal elastic and plastic strain was better suited as the normalising stress than the more commonly used 0.2 percent tensile proof stress. One design curve could be used to represent all the different aluminium alloys investigated. Changing the aluminium alloy did not change the shape of the biaxial interaction curves.</p><p>Based on the detailed study of the collapse behaviour of aluminium plates carried out, and established practice, new ULS design formulas for axial compression, transverse compression, pure shear loading, axial compression in combination with transverse compression and axial compression in combination with shear loading were proposed. The new design formulas accounted for reductions in ultimate capacities due to heat affected zones, including heat affected zones along loaded edges. They took full advantage of non-linear residual strength for slender plates; and the slenderness dependence and aspect ratio dependence for biaxial interaction curves were systematically incorporated.</p>

<p>The present thesis deals with ultimate strength and collapse analysis of ship structures. Within this are, the attention is directed towards simple and efficient nonlinear finite element models for stiffened plate panels in ship structures. Several types of Morley elements are investigated and two are selected for further development. These triangular elements assume constant stress distribution over the element area. </p><p>The first element considered is a nonlinear facet shell element which is valid within moderate rotations. Its elastic formulation has been proposed earlier. In this work, the element matrices are extended to account for material plasticity. The second element, is a displacement-based curved element which undergoes arbitrary large displacements and rotations. The discrete equilibrium equations for this element are re-derived so as to make them more efficient with standard Newton-Raphson solution procedures. </p><p>Material plasticity formulation using through-the thickness integration as well as resultants plasticity is presented. The evolution laws are derived from the natural laws of thermodynamics, and a return mapping algorithm with a backward Euler difference scheme is used for a solution of the evolution equations. The plasticity computations involve a solution of a single scalar yield surface for the plasticity multiplier. By performing the matrix algebra analytically, simple and explicit expressions are derived. These equations reduce the computational costs remarkably.</p><p>Numerical examples, mostly selected from well-know benchmark problems, are presented to demonstrate the performance of the proposed formulations. Very good agreement is obtained when compared with published results. In addition, typical problems for ultimate strength and collapse analysis of ship hull-girder are analyzed. These include plate girders, stiffened plate panels, as well as a cruciform element. The results show good agreement not only with those obtained from commercial finite element programs, but also from the experimental observations. For stiffened plate panels, comparison is made with DNV design rules, which is found to give non-conservative estimates for some load conditions.</p><p>Finally, a study on multi-span stiffened panels is performed so as to compare the estimates provided by the conventional single span model. It is observed that the conventional model provides conservative estimates, and the effect of transverse frames is especially significant on the finite element model of stiffened panels. </p>

The purpose of this work has been to investigate the ultimate capacities of aluminium plates. The aluminium plates are supposed to be part of a marine structure. Their behaviour has been explored by taking advantage of non-linear finite element simulations. The loading conditions included axial compression, transverse compression, pure shear loading, axial compression in combination with transverse compression and axial compression in combination with shear loading. Different patterns of heat affected zones have been studied with particular attention on conventionally welded plates and extruded, welded plates. The plates have been given realistic boundary conditions in addition to boundary conditions believed to represent the extremities in plate collapse behaviour. The initial deflection patterns chosen, were generally conservative, and the variations in collapse capacities for different initial deflection patterns have been investigated. The initial deflection amplitudes were systematically varied to cover all values of the initial deflection amplitudes likely to occur. In addition to studying different patterns of the heat affected zones, systematic variations of the breadths of the heat affected zones have been performed. Systematic variations in the magnitudes of the residual stresses and systematic variations of the reductions of the 0.2 percent tensile proof stresses in the heat affected zones have also been carried out. Three different aluminium alloys were studied. They were believed to represent a high, medium and low utilised aluminium alloy. The aspect ratio of the plates was varied to cover both plates that were completely square to plates with a length equal to five times the breadth of the plates. It was found that axially loaded plates with heat affected zones along loaded edges had lower ultimate capacities than plates with heat affected zones along unloaded edges, and if the plates had heat affected zones along loaded edges, having additional heat affected zones along unloaded edges did not alter the buckling capacities. The reduction in ultimate capacity could be modelled to vary linearly with the breadth of the heat affected zones, and the reduction in ultimate capacity could be modelled to vary linearly with the relationship between the value of the 0.2 percent tensile proof stress in the heat affected zones and the 0.2 percent tensile proof stress of the base material. The ultimate capacities of transversally loaded plates could be found as the sum of a square plate and a plate strip, even if the plates contained heat affected zones. Square plates and plate strips with relevant patterns of heat affected zones had to be employed. Biaxial interaction curves showed a considerable slenderness dependence for all values of the aspect ratio investigated. The biaxial interaction curves were only slightly affected by introduction of different patterns of heat affected zones. The stress corresponding to equal elastic and plastic strain was better suited as the normalising stress than the more commonly used 0.2 percent tensile proof stress. One design curve could be used to represent all the different aluminium alloys investigated. Changing the aluminium alloy did not change the shape of the biaxial interaction curves. Based on the detailed study of the collapse behaviour of aluminium plates carried out, and established practice, new ULS design formulas for axial compression, transverse compression, pure shear loading, axial compression in combination with transverse compression and axial compression in combination with shear loading were proposed. The new design formulas accounted for reductions in ultimate capacities due to heat affected zones, including heat affected zones along loaded edges. They took full advantage of non-linear residual strength for slender plates; and the slenderness dependence and aspect ratio dependence for biaxial interaction curves were systematically incorporated.

The present thesis deals with ultimate strength and collapse analysis of ship structures. Within this are, the attention is directed towards simple and efficient nonlinear finite element models for stiffened plate panels in ship structures. Several types of Morley elements are investigated and two are selected for further development. These triangular elements assume constant stress distribution over the element area. The first element considered is a nonlinear facet shell element which is valid within moderate rotations. Its elastic formulation has been proposed earlier. In this work, the element matrices are extended to account for material plasticity. The second element, is a displacement-based curved element which undergoes arbitrary large displacements and rotations. The discrete equilibrium equations for this element are re-derived so as to make them more efficient with standard Newton-Raphson solution procedures. Material plasticity formulation using through-the thickness integration as well as resultants plasticity is presented. The evolution laws are derived from the natural laws of thermodynamics, and a return mapping algorithm with a backward Euler difference scheme is used for a solution of the evolution equations. The plasticity computations involve a solution of a single scalar yield surface for the plasticity multiplier. By performing the matrix algebra analytically, simple and explicit expressions are derived. These equations reduce the computational costs remarkably. Numerical examples, mostly selected from well-know benchmark problems, are presented to demonstrate the performance of the proposed formulations. Very good agreement is obtained when compared with published results. In addition, typical problems for ultimate strength and collapse analysis of ship hull-girder are analyzed. These include plate girders, stiffened plate panels, as well as a cruciform element. The results show good agreement not only with those obtained from commercial finite element programs, but also from the experimental observations. For stiffened plate panels, comparison is made with DNV design rules, which is found to give non-conservative estimates for some load conditions. Finally, a study on multi-span stiffened panels is performed so as to compare the estimates provided by the conventional single span model. It is observed that the conventional model provides conservative estimates, and the effect of transverse frames is especially significant on the finite element model of stiffened panels.

<p>This study is centered on assessment of the fatigue strength of titanium fusion welds for deep-water riser’s applications. Deep-water risers are subjected to significant fatigue loading.</p><p>Relevant fatigue data for titanium fusion welds are very scarce. Hence there is a need for fatigue data and life prediction models for such weldments.</p><p>The study has covered three topics: Fatigue testing, Fractography and defect assessment, and Fracture Mechanics modelling of fatigue crack growth.</p><p>Two series of welded grade of titanium consisting of 14 specimens in each series were fatigue tested under constant amplitude loading. Prior to fatigue testing, strain gauge measurements of some specimens was conducted to enable the definition of stress range in the fatigue assessment procedure. The results were compared with finite solid element analysis and related to fatigue stresses in a riser pipe wall.</p><p>Distribution and geometry of internal and surface defects both in the aswelded and in the post-welded machined conditions were assessed using fractography. This served as a tool to determine the fatigue initiation point in the welds.</p><p>Fracture mechanics was applied to model fatigue strength of titanium welds with initiation from weld defects. Two different stress intensity factor formulations for embedded eccentrically placed cracks were used for analysis of elliptical cracks with the major axis parallel and close to one of the free surface. The methods were combined to give a satisfactory model for crack growth analysis.</p><p>The model analyses crack growth of elliptical and semi-elliptical cracks in two directions, with updating of the crack geometry. Fatigue strength assessment was conducted using two crack growth models, the Paris-Erdogan relation with no threshold and the Donahue et al. relation with an implied threshold.</p><p>The model was validated against experimental data, with a discussion on the choice of crack growth model.</p>

<p>The contributions of this thesis fall naturally into two main categories: Part I: Feedback control of fluid flows, and; Part II: Modelling and control of marine structures.</p><p><b>Main Con tributions of Part I</b></p><p>Part I presents new results on stabilization (for the purpose of drag reduction or vortex shedding suppression) and destabilization (for the purpose of mixing) of channel, pipe and cylinder flows. In order to provide a stand-alone reference on this topic, the thesis also contains a comprehensive review of the research carried out in this field over the last decade or so, along with introductory chapters on fluid mechanics and control theory. The review also serves the purpose of placing the contributions by the author into the wider context of the field. The contributions by the author are the following:</p><p><b>Section 4.3.1:</b> A new boundary feedback control law for stabilization of the parabolic equilibrium flow in 2D channel flow is derived using Lyapunov stability theory. The controller uses pressure measurements taken on the channel wall, and applies actuation in the form of wall transpiration, that is, suction and blowing of fluid across the wall. Although the analysis is valid for small Reynolds numbers, only, simulations indicate that the control is very effcient in stabilizing the flow at Reynolds numbers several orders of magnitude higher. The pressure-based control law performed much better than other Lyapunov-based control laws studied.</p><p><b>Section 4.3.3: </b>The simple pressure-based control law derived in Section 4.3.1 is generalized to the 3D pipe flow. As for the 2D channel flow, the analysis is valid for small Reynolds numbers, only.</p><p><b>Section 4.3.4: </b>The pressure-based feedback control law derived in Section 4.3.1 for the 2D channel flow results in flow transients with instantaneous drag far lower than that of the corresponding laminar flow. In fact, for the first time, instantaneous total drag in a constant-mass- flow 2D channel flow is driven to negative levels. The physical mechanisms by which this phenomenon occur is explained, and the possibility of achieving sustained drag reductions to below the laminar level by initiating such low-drag transients on a periodic basis is explored. The results add to the evidence that the laminar ow represents a fundamental limit to the drag reduction achievable by wall transpiration.</p><p><b>Section 4.4:</b> A state feedback controller that achieves global asymptotic stabilization of a nonlinear Ginzburg-Landau model of vortex shedding from bluff bodies is designed using backstepping. Stabilization is obtained in two steps. First, the upstream and downstream parts of the system are shown to exhibit the inputto- state stability property with respect to certain boundary input terms governed by the core flow in the vicinity of the bluff body. Second, a finite difference approximation of arbitrary order of the core flow is stabilized using the backstepping method. Consequently, all the states in the core flow are driven to zero, including the boundary input terms of the upstream and downstream subsystems. The control design is valid for any Reynolds number, and simulations demonstrate its performance.</p><p><b>Section 5.2:</b> For thefirst time, active feedback control is used to enhance mixing by exploiting the natural tendency in the flow to mix. By applying the pressurebased feedback control law derived for stabilizing the 2D channel flow in Section 4.3.1, with the sign of the input reversed, enhanced instability of the parabolic equilibrium flow is obtained, which leads rapidly to highly complex flow patterns. The mixing enhancement is quantified using various diagnostic tools.</p><p><b>Section 5.3: </b>A Lyapunov based boundary feedback controller for achieving mixing in a 3D pipe flow governed by the Navier-Stokes equation is designed. It is shown that the control law maximizes a measure of mixing that incorporates stretching and folding of material elemen ts, while at the same time minim izing the control effort and the sensing effort. The penalty on sensing results in a static output- feedback control law (rather than full-state feedback). A lower bound on the gain from the control effort to the mixing measure is also deriv ed. For the openloop system, input/output-to-state stability properties are established, which show a form of detectability of mixing in the interior of the pipe from the chosen outputs on the wall. The effectiveness of the optimal control in achieving mixing enhancement is demonstrated in numerical sim ulations. Simulation results also show that the spatial changes in the control velocity are smooth and small, promising that a low number of actuators will suffice in practice.</p><p><b>Section 5.4: </b>Motivated by the mixing results for channels and pipes in Sections 5.2 and 5.3, a simulation study that investigates the feasibility of enhancing particle dispersion in the wake of a circular cylinder is carried out. For a subcritical case, vortex shedding is successfully provoked using feedback.</p><p><b>Main Contributions of Part II</b></p><p>Part II deals with modelling and control of slender marine structures and marine vessels.</p><p><b>Chapter 8:</b> A new finite element model for a cable suspended in water is developed. Global existence and uniqueness of solutions of the truncated system is shown for a slightly simplified equation describing the motion of a cable with negligible added mass and supported by fixed end-points. Based on this, along with well known results on local existence and uniqueness of solutions for symmetrizable hyperbolic systems, a global result for the initial-boundary value problem is conjectured. The FEM model for the cable is assembled to give a model of a multi-cable mooring system, whic h, in turn, is coupled to a rigid body model of the floating vessel. The result is a coupled dynamical model of a moored v essel, which can be applied to applications such as turret-based moored ships, or tension leg platforms. As a simple application of the sim ulator, controlling the line tensions dynamically as an additional means of station keeping is explored.</p><p><b>Chapter 9: </b>Output feedback tracking control laws for a class of Euler-Lagrange systems subject to nonlinear dissipative loads are designed. By imposing a monotone damping condition on the nonlinearities of the unmeasured states, the common restriction that the nonlinearities be globally Lipschitz is removed. The proposed observer-controller scheme renders the origin of the error dynamics uniformly globally asymptotically stable, in the general case. Under certain additional assumptions, the result continue to hold for a simplified control law that is less sensitive to noise and unmodeled phenomena.</p>

This study is centered on assessment of the fatigue strength of titanium fusion welds for deep-water riser’s applications. Deep-water risers are subjected to significant fatigue loading. Relevant fatigue data for titanium fusion welds are very scarce. Hence there is a need for fatigue data and life prediction models for such weldments. The study has covered three topics: Fatigue testing, Fractography and defect assessment, and Fracture Mechanics modelling of fatigue crack growth. Two series of welded grade of titanium consisting of 14 specimens in each series were fatigue tested under constant amplitude loading. Prior to fatigue testing, strain gauge measurements of some specimens was conducted to enable the definition of stress range in the fatigue assessment procedure. The results were compared with finite solid element analysis and related to fatigue stresses in a riser pipe wall. Distribution and geometry of internal and surface defects both in the aswelded and in the post-welded machined conditions were assessed using fractography. This served as a tool to determine the fatigue initiation point in the welds. Fracture mechanics was applied to model fatigue strength of titanium welds with initiation from weld defects. Two different stress intensity factor formulations for embedded eccentrically placed cracks were used for analysis of elliptical cracks with the major axis parallel and close to one of the free surface. The methods were combined to give a satisfactory model for crack growth analysis. The model analyses crack growth of elliptical and semi-elliptical cracks in two directions, with updating of the crack geometry. Fatigue strength assessment was conducted using two crack growth models, the Paris-Erdogan relation with no threshold and the Donahue et al. relation with an implied threshold. The model was validated against experimental data, with a discussion on the choice of crack growth model.

The contributions of this thesis fall naturally into two main categories: Part I: Feedback control of fluid flows, and; Part II: Modelling and control of marine structures. <b>Main Con tributions of Part I</b> Part I presents new results on stabilization (for the purpose of drag reduction or vortex shedding suppression) and destabilization (for the purpose of mixing) of channel, pipe and cylinder flows. In order to provide a stand-alone reference on this topic, the thesis also contains a comprehensive review of the research carried out in this field over the last decade or so, along with introductory chapters on fluid mechanics and control theory. The review also serves the purpose of placing the contributions by the author into the wider context of the field. The contributions by the author are the following: <b>Section 4.3.1:</b> A new boundary feedback control law for stabilization of the parabolic equilibrium flow in 2D channel flow is derived using Lyapunov stability theory. The controller uses pressure measurements taken on the channel wall, and applies actuation in the form of wall transpiration, that is, suction and blowing of fluid across the wall. Although the analysis is valid for small Reynolds numbers, only, simulations indicate that the control is very effcient in stabilizing the flow at Reynolds numbers several orders of magnitude higher. The pressure-based control law performed much better than other Lyapunov-based control laws studied. <b>Section 4.3.3: </b>The simple pressure-based control law derived in Section 4.3.1 is generalized to the 3D pipe flow. As for the 2D channel flow, the analysis is valid for small Reynolds numbers, only. <b>Section 4.3.4: </b>The pressure-based feedback control law derived in Section 4.3.1 for the 2D channel flow results in flow transients with instantaneous drag far lower than that of the corresponding laminar flow. In fact, for the first time, instantaneous total drag in a constant-mass- flow 2D channel flow is driven to negative levels. The physical mechanisms by which this phenomenon occur is explained, and the possibility of achieving sustained drag reductions to below the laminar level by initiating such low-drag transients on a periodic basis is explored. The results add to the evidence that the laminar ow represents a fundamental limit to the drag reduction achievable by wall transpiration. <b>Section 4.4:</b> A state feedback controller that achieves global asymptotic stabilization of a nonlinear Ginzburg-Landau model of vortex shedding from bluff bodies is designed using backstepping. Stabilization is obtained in two steps. First, the upstream and downstream parts of the system are shown to exhibit the inputto- state stability property with respect to certain boundary input terms governed by the core flow in the vicinity of the bluff body. Second, a finite difference approximation of arbitrary order of the core flow is stabilized using the backstepping method. Consequently, all the states in the core flow are driven to zero, including the boundary input terms of the upstream and downstream subsystems. The control design is valid for any Reynolds number, and simulations demonstrate its performance. <b>Section 5.2:</b> For thefirst time, active feedback control is used to enhance mixing by exploiting the natural tendency in the flow to mix. By applying the pressurebased feedback control law derived for stabilizing the 2D channel flow in Section 4.3.1, with the sign of the input reversed, enhanced instability of the parabolic equilibrium flow is obtained, which leads rapidly to highly complex flow patterns. The mixing enhancement is quantified using various diagnostic tools. <b>Section 5.3: </b>A Lyapunov based boundary feedback controller for achieving mixing in a 3D pipe flow governed by the Navier-Stokes equation is designed. It is shown that the control law maximizes a measure of mixing that incorporates stretching and folding of material elemen ts, while at the same time minim izing the control effort and the sensing effort. The penalty on sensing results in a static output- feedback control law (rather than full-state feedback). A lower bound on the gain from the control effort to the mixing measure is also deriv ed. For the openloop system, input/output-to-state stability properties are established, which show a form of detectability of mixing in the interior of the pipe from the chosen outputs on the wall. The effectiveness of the optimal control in achieving mixing enhancement is demonstrated in numerical sim ulations. Simulation results also show that the spatial changes in the control velocity are smooth and small, promising that a low number of actuators will suffice in practice. <b>Section 5.4: </b>Motivated by the mixing results for channels and pipes in Sections 5.2 and 5.3, a simulation study that investigates the feasibility of enhancing particle dispersion in the wake of a circular cylinder is carried out. For a subcritical case, vortex shedding is successfully provoked using feedback. <b>Main Contributions of Part II</b> Part II deals with modelling and control of slender marine structures and marine vessels. <b>Chapter 8:</b> A new finite element model for a cable suspended in water is developed. Global existence and uniqueness of solutions of the truncated system is shown for a slightly simplified equation describing the motion of a cable with negligible added mass and supported by fixed end-points. Based on this, along with well known results on local existence and uniqueness of solutions for symmetrizable hyperbolic systems, a global result for the initial-boundary value problem is conjectured. The FEM model for the cable is assembled to give a model of a multi-cable mooring system, whic h, in turn, is coupled to a rigid body model of the floating vessel. The result is a coupled dynamical model of a moored v essel, which can be applied to applications such as turret-based moored ships, or tension leg platforms. As a simple application of the sim ulator, controlling the line tensions dynamically as an additional means of station keeping is explored. <b>Chapter 9: </b>Output feedback tracking control laws for a class of Euler-Lagrange systems subject to nonlinear dissipative loads are designed. By imposing a monotone damping condition on the nonlinearities of the unmeasured states, the common restriction that the nonlinearities be globally Lipschitz is removed. The proposed observer-controller scheme renders the origin of the error dynamics uniformly globally asymptotically stable, in the general case. Under certain additional assumptions, the result continue to hold for a simplified control law that is less sensitive to noise and unmodeled phenomena.

This Master thesis details the development of a mathematical model of ROV SF30k, and its implementation into a DP control system developed by Espen Tolpinrud.The project was performed as a part of the Applied Underwater Robotics atNTNU.A 6DOF model of the ROV was developed. The parameters of the model werefound using both the 3D model of the ROV provided by Sperre AS, and based onthe parameters found previously for ROV Minerva. Both ROVs were developedby Sperre AS and share many similarities. Thrust configuration was developed forthe ROV based on the geometric positions and orientations of its thrusters, whilethrust coefficients were found using propeller data provided by Sperre AS. The DPcontrol system was configured to operate SF 30k by using a configuration file basedon the found parameters.A number of modules were ported from the old DP system, which was tailormadefor ROV Minerva, to the new one. This includes the Kalman Filter, controllersand thrust allocation. An adaptive passive nonlinear observer was developedand implemented.The existing simulator model for ROV Minerva was modified to be able tosimulate ROV SF 30k. This was done by changing the parameters of the processplant model and thrust dynamics.The parameters in the configuration file for the new control system were verifiedduring sea trials onboard R/V Gunnerus on May 29-30th 2012. The thrust allocationparameters were found to be satisfactory during ROV operation, however, themathematical model of the ROV could not be verified due to the small window ofoperation during the sea trials.