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.