Eng. 6002 Ship Structures 1

1. Eng. 6002 Ship Structures 1 An introduction to the finite element method using MATLAB

2. Overview • This paper outlines an efficient approach to introducing the finite element method • This approach requires that the students have some prior experience with MATLAB and a fundamental understanding of solid mechanics. • Only two-dimensional beam element problems are considered, to simplify the development. • The approach emphasizes an orderly solution procedure and involves important finite element concepts, such as • the stiffness matrix, • element and global coordinates, • force equilibrium, and constraints.

3. Overview • The ability of MATLAB to manipulate matrices and solve matrix equations makes the computer solution concise and easy to follow. • The flexibility associated with the computer implementation allows example problems to be easily modified into design projects • and you may even feel motivated to modify things and tackle some other problems

4. Introduction • We will decompose a structure into a finite number of elements. • Once each element is completely described with appropriate geometric and material properties, the elements are properly combined or ‘assembled’

5. Introduction • Each element is modeled as an elastic member for which the displacement is linearly related to the applied loading. • where {f} is the element load vector, [k] is the element stiffness matrix, and {u} is the element displacement vector. • To indicate the specific element under consideration, a superscript will be used on the quantities in equation 1. The element or member loads can be a force, in which case the displacement is a linear translation, or a moment, in which case the displacement is an angular rotation.

6. Introduction • To model the structure, the individual beam elements are properly combined so that the entire structure is described by: • where {F} is the nodal load vector, [K] is the global stiffness matrix, and {U} is the nodal displacement vector. • The objective of the type of problems considered is determine the unknown nodal displacements.

7. Introduction • For the purposes of this paper, the FEM can be conveniently divided into five (or six) steps: • construction of the element stiffness matrix in local coordinates, • transformation of the element stiffness matrix into global coorindinates, • assembly to the global stiffness matrix using transformed element stiffness matrices, • application of the constraints to reduce the global stiffness matrix, • determination of unknown nodal displacements. • An additional step that might be implemented is the post-processing of results, i.e. the determination of unknown forces or stresses from the calculated displacements.

8. The Method • The first step in the formulation is to discretize the domain, i.e., to select the number and type of elements. Only two-dimensional beam elements are considered in this study. • Each beam element has two nodes. • Two-dimensional beam elements allow for an axial force fx, transverse force fy, and a bending moment fθ at each node and have three degrees of freedom (dof) per node, • axial displacement ux, transverse displacement uy, and angular rotation uθ. • The positive sign convention is shown.

9. The Method • The stiffness matrix for each element is

10. The Method • The element stiffness matrix is a square matrix whose size is given by: [number of nodes x dof per node] x [number of nodes x dof per node] • Thus, the stiffness matrix for a two-dimensional beam element is 6 x 6 because of the three degrees of freedom associated with each node.

11. The Method • this matrix describes the stiffness of an elastic element that relates the load vector to the displacement vector

12. The Method • To verify that this form of the stiffness matrix is correct and to illustrate several FEM concepts, consider the situation of a cantilevered beam such that • node 1 is constrained from movement, i.e., ux1 = uy1 = uθ1 = 0. • Next, consider a downward force applied at node 2, so that fy2 = -F and fx2 = fθ2 = 0.

13. The Method • With these considerations, the system of equations can be rewritten as:

14. The Method • The solution to this system of equations yields • ux2 = 0, • uy1 = -FL3/(3EI), • uθ2 =-FL2/(EI), • which agree with the expressions for the slope and deflection at the tip of a cantilevered beam as given in any mechanics of materials textbook. • Further verification can be achieved with consideration of an applied axial load and then an applied moment (on your own).

15. The Method • The second step in the development is to transform the element load vector {f} and the element displacement vector {u} into the global coordinate system • Consider a beam element at an arbitrary orientation described by an angle θ, measured counterclockwise from the horizontal, and a global coordinate system indicated with a bar

16. The Method • From simple trigonometry, the loads and displacements in the global coordinate system can be related to the loads and displacements in the element coordinate system

17. The Method • Substitution of equation 6 into equation 1 • yields: • Then, for an orthogonal matrix []-1 = []T, the element equation can be written in global coordinates as: • where the element stiffness matrix transformed to global coordinates is defined as:

18. The Method • The third step is to properly combine the individual transformed element stiffness matrices to construct the global stiffness matrix that describes the entire structure. • We only consider beam elements that are connected ‘end to end’. • We’ll look at a two-element example.

19. Example • Consider the case of two beam elements connected ‘end to end’. • The global stiffness matrix [K] for the two-element case is 9 x 9 because beam elements have three degrees of freedom per node and there are three nodes in the structure.

20. Example • The external loads at the nodes are given by: • And the displacements at the nodes:

21. Example • The external loads at the nodes are given by: • And the displacements at the nodes:

22. Example • Force and moment equilibrium at each node requires that the sum of the element loads must equal the nodal loads and • compatibility at each node requires that the element displacements must equal the nodal displacements

23. Example cont. • In order to easily combine the element stiffness matrices, each element stiffness matrix is stored in a matrix the size of the global stiffness matrix, with the extra spaces filled with zeros. • In this example, the element stiffness matrix for element 1 is stored in the portion of the global stiffness matrix that involves nodes 1 and 2, • i.e., the upper 6 x 6 portion of the matrix. Thus, the expanded stiffness matrix that describes element 1 is given by:

24. Example cont.

25. Example cont. • The element stiffness matrix for element 2 is stored in the portion of the global stiffness matrix that involves nodes 2 and 3, i.e., the lower 6 x 6 portion of the matrix. The expanded stiffness matrix that describes element 2 is given by:

26. Example cont. • The expanded individual stiffness matrices in equations 16 and 17 can now be added together so that the global stiffness matrix for the two-element structure is given as:

27. Example cont. • The fourth step is to apply the constraints and reduce the global stiffness matrix so that the specific problem of interest can be solved. • At the point of constraint, the displacement of the structure is known. • Because these displacements are known, matrix algebra allows the removal of the corresponding rows and columns. • The resulting system of equations can be written as:

28. Example cont. • [K]r is the reduced global stiffness matrix that contains information about the structure and the boundary conditions. • With MATLAB, rows and columns can be easily deleted, and a shift of the remaining elements in the matrix is performed automatically. • The final step is simply to solve the reduced system of equations for the unknown displacements. • MATLAB efficiently solves a system of equations with the backslash command. The backslash command uses • Cholesky factorization and Gaussian elimination to solve a system of equations

29. Next Class • In the next class we will be developing a matlab script to apply this methodology