The purpose of this course is to broaden the basis for use of the finite element method for calculating static linear and non-linear behavior of complex structures and dynamic response of marine structures. The course is lectured in two parallel modules, respectively non-linear static and dynamic analysis. In the case of static analysis an introduction to the modeling of complex structures is given - exemplified by ship structures. Element formulations for linear analysis of prismatic and curved shell structures are shown, and an introduction to non-linear structural analysis caused by geometry, material and contact behavior is given. There is also an introduction to practical modeling and analysis of structures with finite element method with appropriate software. The basis for the dynamic analysis is the static finite element method and d'Alemberts principle. The dynamic equilibrium equation is discussed in terms of virtual work, kinetic and potential energy and Rayleigh and Rayleigh-Ritz methods. The properties of the mass and damping matrices are discussed. Methods to reduce the number of degrees of freedom in a model for static analysis for use in dynamic analysis are given. Methods for calculating the dynamic response like the mode superposition technique, the frequency-response method and numerical time integration are outlined. Modeling of non-linear drag forces and vortex-induced vibrations are lectured as special topics of large importance for slender marine structures. An important part of the course is the use of Matlab for making simple finite element programs for dynamic analysis.

## Prerequisites

Basic understanding of the finite element method as applied on static cases including beam and plate elements, and experience with static analysis of marine structures like jacket platforms and ships.

Basic understanding of dynamic response of systems with one degree of freedom and simple continous beams, and how analysis models can be established for realistic structures by use of generalized coordinates and ode superposition.

##### Specific conditions

Admission to a programme of study is required:

- Marine Technology (MIMART)
- Marine Technology (MSN1)
- Marine Technology (MTMART)
- Maritime Engineering (MSNMME)
- Wind Energy (MSWIND)

##### Recommended previous knowledge

TMR4167 Marine Technology Structures, TMR4182 Marine Dynamics, TMR4190 Finite Element Methods, TMR4195 Design of Offshore Structures.

## Learning outcomes

After completing the course the student should be able to perform deformation and stress analyses of complex structures using the finite element method, taking into account nonlinear and dynamic behavior, and conduct a critical assessment of the results. Students will be able to:

- understand the theoretical basis of the finite element method applied on non-linear and dynamic analysis
- explain the derivation of the stiffness relationship of elements for the analysis of stiffened plate and shell structures - based on energy principles and interpolation of the displacement pattern
- understand the mathematical basis for the description of the behavior of stiffened plates (prismatic shell) and curved shell structures (equilibrium, kinematic compatibility, stress - strain - relation)
- understand the available methods for describing contact between structural elements.
- explain the derivation of the incremental stiffness relation for rod and beam elements taking into account the nonlinear behavior due to large deformation and material behavior
- explain the criteria that must be fulfilled by a finite element model for certain structure types in order to converge to the exact solution when the element size is reduced
- understand how the prismatic and curved shell structures behave and what element types and boundary conditions must be selected in order to achieve a given accuracy
- understand how a ship - considered as a prismatic shell - behave and how various elements models and corresponding boundary conditions should be selected to determine the deformations and stresses with a given accuracy. Practical models include the ship as a whole, a number of tanks, transverse frame, and a randomly defined part ("submodel") of the ship could carry out simple element analysis of prismatic shell and frame structures using ABAQUS, or similar software
- understand methods for eigenvalue computation and dynamic response analysis and hence be able to make simple computer programs, and thereby develope a good basis for the use of commercial computer program
- understand the importance of reducing the number frihetsgradet in relation to the element models for static analysis, and how the reduction can be implemented without significant loss of accuracy
- define damping effects in a dynamic response model, and describe how different damping mechanisms can be formulated in a element model
- be able to select an element model and calculation method for the analysis of practical problems so that the results are sufficiently accurate
- know how hydrodynamic effects influence the mass, damping and driving forces of slender marine structures.

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