FEA convergence requirements. Dr. Nasrellah H A

FEA Element Types Elements fall into four major categories: 2D line elements, 2D planar elements, and 3D solid elements which are all used to define geometry. Special elements used to apply boundary conditions. For example special elements might include gap elements to specify a gap between two pieces of geometry. Spring elements are used to apply a specific spring constant at a specified node or set of nodes. Rigid elements are used to define a rigid connection to or in a model

Truss Element (2D Line) Truss elements are long and slender, have 2 nodes, and can be oriented anywhere in 3D space. Truss elements transmit force axially only and are 3 DOF elements which allow translation only and not rotation.

Beam Element (2D Line) Beam elements are long and slender, have two nodes, and can be oriented anywhere in 3D space. Beam elements are 6 DOF elements allowing both translation and rotation at each end node.

2D Element (2D Planar) 2D Elements are 3 or 4 node elements with only 2 DOF, Y and Z translation, and are normally created in the YZ plane. They are used for Plane Stress or Plane Strain analyses.

 Plane Stress typically occurs in thin flat plates that are acted upon only by load forces that are parallel to them. implies no stress normal to the cross section defined - strain is allowed - suitable to model the 2D cross section of a wall subjected to axial load.

Plane Strain implies no strain normal to the cross section defined - stress is allowed - suitable to model the 2D cross section of a long dam.

Membrane Element (2D Planar) Membrane Elements are 3 or 4 node 2D elements that can be oriented anywhere in 3D space. They can be used to model thin membrane like thin metal shells, etc. These elements support only translational DOF not rotational and in-plane loading. The thickness of the membrane must be small relative to its length or width

Plate Element (2D Planar) Plate elements are 3 or 4 node 2D planar elements that can be oriented anywhere in 3D space. They are typically used to model structures comprised of shells such as pressure vessels. Generally a thicker wall than for a membrane element but about 1/10 the length or width. All translational DOF are supported as well as rotational DOF that are not out of plane

3D Brick Element, 8 Nodes (3D Solid) Brick or tetrahedra elements may have 4, 5, 6, 7, 8, 15, or 20 nodes and support only translational DOF. They are normally used to model solid objects for which plate elements are not appropriate. You can usually specify either all tetrahedra, all bricks, or a mixture of both with some automatic mesh generators.

3D Tetrahedra Element, 4 Nodes (3D Solid) 3D Tetrahedra Element, 5 Nodes, Pyramid (3D Solid)

3D Tetrahedra Element, 6 Nodes, Wedge (3D Solid). 3D Tetrahedra Element with Midside Nodes, 15 Nodes, Wedge (3D Solid)

REQUIREMENTS FOR CONVERGENCE OF THE SOLUTION The finite element approximation must satisfy certain conditions which guarantee that as the mesh is refined the numerical solution converges to the exact values. Continuity condition: The displacement must be continuous within each element. This condition is automatically satisfied by using polynomial approximations for the displacement field.

Rigid body condition : The displacement ¯eld closed should not allow straining of an element to occur when the nodal displacements are caused by a rigid body motion. This physical condition is satisfied for a single element if the sum of the shape functions at any point is equal to one.

Constant strain condition: The displacement function has to be such that if nodal displacements are compatible with a constant strain field, such constant strain will in fact be obtained. Clearly, as elements get smaller, nearly constant strain conditions will prevail in them.

ASSESSMENT OF CONVERGENCE REQUIREMENTS. THE PATCH TEST The patch test provided a necessary and sufficient condition for convergence. The test is based on selecting an arbitrary patch of elements and imposing upon it nodal displacements corresponding to any state of constant strain. If nodal equilibrium is achieved without imposing external nodal forces, and if a state of constant stress is obtained, then clearly the constant strain criterion of the previous section is satisfied.

Example 3.5: Apply the patch test to the three element patch of 2-noded rod All elements have equal length and the same material properties.

FE MESHING The idea is that we are going to use a simple approximation method, but the errors in this approximation method become unnoticeable as the size of the subregion gets small.

COMPUTER CODE: truss.m MATLAB is a high-level language specially designed for dealing with matrices. This makes it particularly suited for programming the finite element method. In addition, MATLAB will allow the reader to focus on the finite element method rather than on the programming details. Programming the finite element method involves the following steps: • Step 1: Data preparation and input. • Step 2: Computation of element matrices.

Step 3: Assembly of elements’ stiffness matrices and elements’ force vectors in the global stiffness matrix and global force vector. Step 4: Imposition of boundary conditions such as prescribed displacements. This is usually carried out simultaneously with the assembly of the global matrix in Step 3. • Step 5: Solution of the global system of equations for the nodal unknowns. • Step 6: Computation of secondary variables such as stresses and strains from displacements. • Step 7: Print and/or plot desired results.

These steps are best illustrated by means of an example:

DATA PREPARATION: The input data for this structure consist of • nnd = 3; number of nodes • nel = 3; number of elements • nne = 2; number of nodes per element • nodof = 2; number of degrees of freedom per node

Nodes Coordinates The coordinates x and y of the nodes are given in the form of a matrix geom(nnd, 2): Element Connectivity The table of connectivity describes how the elements are connected to each other. given in the matrix connec(nel, 2):

Material and Geometrical Properties: The material and geometrical properties are given in the matrix prop(nel, 2): Boundary Conditions: A restrained degree of freedom is assigned the digit 0 A free degree of freedom is assigned the digit 1 are given in the matrix nf(nnd, nodof).

Loading: given in the form of a matrix, load(nnd, 2):

ELEMENT MATRICES: Stiffness Matrix in Local Coordinates:- node_1 = connec(i, 1) node_2 = connec(i, 2) Then using the values of the nodes, we retrieve their x and y coordinates from the geometry matrix: x1 = geom(node_1, 1); y1 = geom(node_1, 2) x2 = geom(node_2, 1); y2 = geom(node_2, 2)

Next, using Pythagoras theorem, we evaluate the length of the element: Finally, we retrieve the material and geometrical property of the section E = prop(i, 1); A = prop(i, 2). Transformation Matrix (C): it is depend on Stiffness Matrix in Global Coordinates:

“Steering” Vector: For example, for element 1, the vector g will look like

ASSEMBLY OF THE GLOBAL STIFFNESS MATRIX: The global stiffness matrix [KK] is assembled using a double loop over the components of the vector g:

ASSEMBLY OF THE GLOBAL FORCE VECTOR: F(nf(i, j)) = load(i, j) SOLUTION OF THE GLOBAL SYSTEM OF EQUATIONS: delta = KK\F