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Nature of Finite Element Solutions FE Model-A mathematical model of the real structure,based on many approximations. Real Structures-Infinite number of nodes(physical points),thus infinite number of DOFs. FE Model-finite number of nodes,thus finite number DOFs. Displacement field is controlled by the shape functions and the values at a limited number of nodes. Stiffening Effect: FE model is stiffer than the real structure. In general,displacement results are smaller in magnitudes than the exact values

Nature of Finite Element Solutions • FE Model – A mathematical model of the real structure, based on many approximations. • Real Structures – Infinite number of nodes (physical points), thus infinite number of DOFs. • FE Model – finite number of nodes, thus finite number DOFs. • Displacement field is controlled by the shape functions and the values at a limited number of nodes. Stiffening Effect: • FE model is stiffer than the real structure. • In general, displacement results are smaller in magnitudes than the exact values

Common Sources of Error in FEA Controllable factors Domain approximation Element interpolation/Approximation Uncontrollable factors Numerical integration errors(Both spatial and time integration) Computer errors(Round-ff,etc.) Cause for incorrect results: ·Mistakes ·errors

Common Sources of Error in FEA - Controllable factors • Domain approximation • Element interpolation/Approximation - Uncontrollable factors • Numerical integration errors (Both spatial and time integration) • Computer errors(Round-ff, etc.) - Cause for incorrect results: • Mistakes • errors

Common Mistakes Mistakes that will cause a singular K matrix: -v=0.5 in a plain strain,axisymmetric or 3D solid element -E=0 in an element -Unconnected nodes -No supports,or insufficient supports -Part of the model is a mechanism -Large stiffness differences

Common Mistakes Mistakes that will cause a singular K matrix: – ν = 0.5 in a plain strain, axisymmetric or 3D solid element – E = 0 in an element – Unconnected nodes – No supports, or insufficient supports – Part of the model is a mechanism – Large stiffness differences

Insufficient supports will allow rigid body motion. Effect of wrong support types: (The stiffness matrix will be singular.) Each of these will result in different displacements,strains and stresses. Mistakes that may go unnoticed: -Wrong element type(e.g.axisymmetric instead of plane element) Supports wrong in location,type or direction Loads wrong in location,type,direction or magnitude Other data (e.g.units) -An element defined twice -Physically meaningless connections(e.g.axisymmetric element to plane element)

• Insufficient supports will allow rigid body motion. (The stiffness matrix will be singular.) • Mistakes that may go unnoticed: –Wrong element type (e.g. axisymmetric instead of plane element) – Supports wrong in location, type or direction – Loads wrong in location, type, direction or magnitude – Other data (e.g. units) – An element defined twice – Physically meaningless connections (e.g. axisymmetric element to plane element)

Structure and Element Behavior .To do a proper FE analysis,the analyst must understand how the structure is likely to behave and how elements are able to behave. E.g.If the analyst knows the stress varies linearly,Q4 elements will work,but if they vary quadratically,Q8 elements must be used

•To do a proper FE analysis, the analyst must understand how the structure is likely to behave and how elements are able to behave. • E.g. If the analyst knows the stress varies linearly, Q4 elements will work, but if they vary quadratically, Q8 elements must be used. Structure and Element Behavior

FEM Modeling-Element Attributes 1D 2D 2D 3D Intrinsic Dimensionality.Elements can have one,two or three space dimensions.There are also special elements with zero dimensionality,such as lumped springs or point masses. Nodal points.Each element possesses a set of distinguishing points called nodal points or nodes for short.Nodes serve a dual purpose:definition of element geometry,and home for degrees of freedom.They are usually located at the corners or end points of elements,as illustrated in Figure.In the so-called refined or higher-order elements nodes are also placed on sides or faces,as well as perhaps the interior of the element

Intrinsic Dimensionality. Elements can have one, two or three space dimensions. There are also special elements with zero dimensionality, such as lumped springs or point masses. Nodal points. Each element possesses a set of distinguishing points called nodal points or nodes for short. Nodes serve a dual purpose: definition of element geometry, and home for degrees of freedom. They are usually located at the corners or end points of elements, as illustrated in Figure . In the so-called refined or higher-order elements nodes are also placed on sides or faces, as well as perhaps the interior of the element. FEM Modeling - Element Attributes

Geometry.The geometry of the element is defined by the placement of the nodal points.Most elements used in practice have fairly simple geometries.In one-dimension,elements are usually straight lines or curved segments. In two dimensions they are of triangular or quadrilateral shape.In three dimensions the most common shapes are tetrahedra,pentahedra(also called wedges or prisms),and hexahedra(also called cuboids or"bricks"). Degrees of freedom.The degrees of freedom(DOF)specify the state of the element.They also function as "handles"through which adjacent elements are connected.DOFs are defined as the values (and possibly derivatives)of a primary field variable at nodal points.The actual selection depends on criteria studied at length in Part ll.Here we simply note that the key factor is the way in which the primary variable appears in the mathematical model.For mechanical elements,the primary variable is the displacement field and the DOF for many(but not all)elements are the displacement components at the nodes. Nodal forces.There is always a set of nodal forces in a one-to-one correspondence with degrees of freedom.In mechanical elements the correspondence is established through energy arguments. Constitutive properties.For a mechanical element these are relations that specify the material behavior.For example,in a linear elastic bar element it is sufficient to specify the elastic modulus E and the thermal coefficient of expansion a. Fabrication properties.For mechanical elements these are fabrication properties which have been integrated out from the element dimensionality.Examples are cross sectional properties of MoM elements such as bars,beams and shafts,as well as the thickness of a plate or shell element

Geometry. The geometry of the element is defined by the placement of the nodal points. Most elements used in practice have fairly simple geometries. In one-dimension, elements are usually straight lines or curved segments. In two dimensions they are of triangular or quadrilateral shape. In three dimensions the most common shapes are tetrahedra, pentahedra (also called wedges or prisms), and hexahedra (also called cuboids or “bricks”). Degrees of freedom. The degrees of freedom (DOF) specify the state of the element. They also function as “handles” through which adjacent elements are connected. DOFs are defined as the values (and possibly derivatives) of a primary field variable at nodal points. The actual selection depends on criteria studied at length in Part II. Here we simply note that the key factor is the way in which the primary variable appears in the mathematical model. For mechanical elements, the primary variable is the displacement field and the DOF for many (but not all) elements are the displacement components at the nodes. Nodal forces. There is always a set of nodal forces in a one-to-one correspondence with degrees of freedom. In mechanical elements the correspondence is established through energy arguments. Constitutive properties. For a mechanical element these are relations that specify the material behavior. For example, in a linear elastic bar element it is sufficient to specify the elastic modulus E and the thermal coefficient of expansion α. Fabrication properties. For mechanical elements these are fabrication properties which have been integrated out from the element dimensionality. Examples are cross sectional properties of MoM elements such as bars, beams and shafts, as well as the thickness of a plate or shell element

FEM Modeling-Classification of Mechanical Elements Physical Mathematical Finite Element Structural Model Name Discretization Component bar beam tube.pipe spar(web) shear panel (2D version of above) Primitive Structural Elements These elements are usually derived from Mechanics-of- Materials simplified theories and are better understood from a physical,rather than mathematical,standpoint

FEM Modeling - Classification of Mechanical Elements Primitive Structural Elements :These elements are usually derived from Mechanics-ofMaterials simplified theories and are better understood from a physical, rather than mathematical, standpoint

FEM Modeling-Classification of Mechanical Elements Physical Finite element Physical Finite element idealization idealization plates 3D solids Continuum Elements Unlike structural elements,continuum elements are better understood in terms of their mathematical interpretation.Examples:plates,slices, shells,axisymmetric solids,general solids

FEM Modeling - Classification of Mechanical Elements Continuum Elements : Unlike structural elements, continuum elements are better understood in terms of their mathematical interpretation. Examples: plates, slices, shells, axisymmetric solids, general solids

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