Axisymmetric Solid - Linear Quadrilateral

Transcription

1 Axisymmetric Solid - Linear Quadrilateral Axisymmetric elements model structures which are symmetric about an axis of rotation and are subjected to axisymmetric boundary conditions. Linear quadrilateral axisymmetric solid elements are supported by the following types of analyses: Structural Analyses (Linear Statics, Normal Mode Dynamics, Superelement Creation, Linear Buckling, Nonlinear Statics, Response Dynamics) Heat Transfer Analysis Potential Flow Analysis A diagram of the element is shown below. Note that the cumulative mass of an axisymmetric element model as computed in a verification run is the mass of the rotated model - not the mass of a 1-radian sector. FE Descriptor ID The FE descriptor ID for this element is 84 in the universal file. Topology nodes = 4 edges = 4

2 Assumptions The Z axis is assumed to be the axis of symmetry. The X axis is in the radial direction. All nodes must lie in the X-Z plane of the part coordinate system, and must have a nonnegative X (radial) coordinate. All loads must be in the X-Z plane. The Y axis of all nodal displacement coordinate systems must be parallel to the part's axis. XY and YZ shear stresses and strains are zero. All gravity vectors must be parallel to the axis of symmetry. All angular velocity vectors must lie on the axis of symmetry. Features (for Structural Analyses) This section lists and describes the features of linear quadrilateral axisymmetric solid elements for linear and nonlinear structural analyses. Summary of Features Number of nodes: 4 Nodal DOF: 2 translational degrees of freedom assigned to each node Boundary conditions: Load Type Units Loads: Mechanical....Nodal forces force / radian...in-plane loads force / area...shear loads force / area Loads: Acceleration....Load set gravity vector force / mass...load set angular velocity vector cycles / time...load set translational acceleration vector length / (time * time)...load set angular acceleration vector cycles / (time * time) Temperature set....load set reference temperature temperature

3 ...Load set ambient temperature temperature...nodal temperatures temperature Displacement restraints....nodal displacements. Physical properties: Formulation option. Plasticity model (nonlinear structural only) Creep model (nonlinear structural only) Plastic yield function Plastic hardening rule Creep equation option Creep hardening rule Material types: Isotropic Orthotropic Associated data: Material orientation vector (for orthotropic materials) Output: Displacements Strains Elastic strain energy Plastic strains (nonlinear structural) Creep strains (nonlinear structural) Reaction forces Element forces Stresses Element Formulation The default and recommended formulation for this element uses serendipity interpolation functions with enhanced strain field interpolation (Simo et al, 1990). This element exhibits good performance in bending-dominated structures, and also does not lock for nearly incompressible materials.

4 An element formulated with the standard serendipity bilinear interpolation functions is also available. Although computationally inexpensive, this formulation is too stiff in bending and locks when used with nearly incompressible materials. Details for this element may be found in Zienkiewicz and Taylor, 1989, or in Cook et al, A mean dilatational formulation may also be selected. This formulation is useful for eliminating element locking which occurs when modeling nearly incompressible materials (Poisson's ratio greater than.49). The standard displacement interpolation functions are used, but the strain-displacement matrix is split into dilatational and deviatoric parts. The dilatational part is replaced with the mean dilatational strain-displacement matrix for the element. This technique is called the strain projection or B technique. See Hughes, 1987 for details. With the mean dilatational formulation, a single value for the mean stress or pressure is obtained across the whole element. Therefore, this formulation is not recommended unless locking due to incompressibility is a problem. Even under these circumstances, the default formulation is usually a better choice. References: Cook, R. D., Malkus, D. S. and Plesha, M. E., Concepts and Applications of Finite Element Analysis, 3rd Edition, John Wiley & Sons, New York, 1989, pp , Hughes, T. J. R., The Finite Element Method, Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1987, pp Simo, J. C. and Rifai, M. S., "A Class of Mixed Assumed Strain Methods and the Method of Incompatible Modes," International Journal for Numerical Methods In Engineering, Vol 29, 1990, PP Zienkiewicz, O. C. and Taylor, R. L., The Finite Element Method, Volume 1, Basic Formulation and Linear Problems, 4th Edition, McGraw-Hill, London, 1989, pp 72-80, Element Integration Points The software numerically integrates element stiffness matrices, mass matrices, and internal force matrices. It also recovers stresses and strains, and performs plastic and creep strain calculations at these element integration points. The parametric coordinate definitions and integration points for this element are listed below. Parametric coordinate definition Parametric Coordinate Axis Definition -1.0 along side along side along side 1-2

5 +1.0 along side 4-3 Order and parametric coordinates of element integration points Integration Point PC1 PC If you request integration point stress and strain list output, the software reports it in the order of the integration points listed above. The software calculates nodal stresses and strains by extrapolating the integration point values to the nodes on an element-by-element basis. When error analysis is requested in Post Processing, the software back-calculates the integration point results using the solver's element assumptions. These data are then used for the error analysis calculations. Nodal DOF Two translational degrees of freedom are assigned to each node. Translations are in the directions of the nodal displacement coordinate system X and Z axes and have the dimensions of length. Boundary Conditions Boundary conditions for the element include loads, temperature set, and displacement restraints. Loads: Mechanical nodal forces in-plane loads shear loads. In this figure: A = edge start B = edge end C = positive shear load, linear variation along the edge D = positive in-plane load, linear variation along the edge

6 Nodal forces are in the directions of nodal displacement coordinate system X and Z axes and represent force per unit radian. For a geometric nonlinear analysis: nodal forces are fixed in global direction and magnitude element-based in-plane and shear loads change global direction as the element deforms and displaces Loads: Acceleration See Also Common Features of the Elements Temperature Set The software uses the temperature set to interpolate material properties if temperature dependent material properties are defined in the I-DEAS Material Data System software. The software also uses the temperature set to compute thermal strain. load set reference temperature load set ambient temperature nodal temperatures See Also For thermal effects, nodal temperature data can be created to define a bilinear temperature variation through the element volume. However, thermal stresses are exact only for the case of constant temperature throughout the element. Common Features of the Elements The linear quadrilateral axisymmetric solid element in Heat Transfer Analysis uses the same temperature field as in structural thermal loading. This allows Heat Transfer Analysis to automatically create a structural temperature set for use in Linear Statics Analysis. Physical Properties formulation option

7 Recommend (recommended - equivalent to full integration using internal shape functions) Full-Int (full integration using internal shape functions) Full-NoInt (full integration without using internal shape functions) Mean-Dil (mean dilatational formulation) plasticity model (nonlinear structural analysis only) creep model (nonlinear structural analysis only) See Also Common Features of the Elements Associated Data material orientation vector When defining material orientation vectors for orthotropic materials, the material X and Y axes must lie in the XZ plane of the element. The material Z axis is perpendicular to or normal to the plane of the element. Therefore, you only need to define the orientation of the material X axis in the elemental XZ plane. In this figure, (A) is the part coordinate system, and (B) is the material orientation coordinate system. Output If the material orientation vector is not defined, the material X axis is aligned with the part's X axis; the material Y axis is aligned with the part's Z axis; and the material Z axis is aligned with the part's Y axis. displacements strains

8 elastic strain energy plastic strains (nonlinear structural analysis only) creep strains (nonlinear structural analysis only) reaction forces element forces stresses Stress, strain, strain energy, element forces, and elastic strain energy for Normal Mode Dynamics and Linear Buckling are based on normalized modal displacements. When performing a Buckling or Normal Mode Dynamic Analysis, only the axisymmetric modes are obtained. These may not be all the modes of interest in the physical structure being modeled. Out-of-plane modes are not captured with this type of element. Features (for Heat Transfer Analysis) This section lists and describes the features of linear quadrilateral axisymmetric solid elements for Heat Transfer Analysis. Summary of Features Number of nodes: 4 Nodal DOF: 1 temperature assigned to each node Boundary conditions: Load Type Units...Nodal heat source energy / (degree radian * time)...element heat generation energy / (volume * time)...edge flux energy / (area * time)...edge convection energy / (area * temp. * time) Temperature restraints....nodal temperatures. Physical properties: Null property table Material types: Isotropic Orthotropic

9 Associated data: Material orientation vector (for orthotropic materials) Output: Temperature Flux Reaction heat sources Note: All units are per time, and time units must be the same for all quantities. The finite element modeling software does not process any time units. Element Formulation for Heat Transfer Analysis This element uses bilinear interpolation functions for temperature within the element. Details for this element may be found in Zienkiewicz and Taylor, 1989, or in Cook et al, References: Cook, R. D., Malkus, D. S. and Plesha, M. E., Concepts and Applications of Finite Element Analysis, 3rd Edition, John Wiley & Sons, New York,1989, pp , , Zienkiewicz, O. C. and Taylor, R. L., The Finite Element Method, Volume 1, Basic Formulation and Linear Problems, 4th Edition, McGraw-Hill, London, 1989, pp 72-80, , Boundary Conditions Boundary conditions for the element include loads and temperature restraints. Loads nodal heat source loads element heat generation loads The positive sense of nodal heat sources and element heat generation represents the production of thermal energy. Nodal heat sources represent energy / (degree radian * time). edge flux loads Positive edge flux is inward and may vary linearly along the edge. In this figure, (A) is the edge start, and (B) is the end.

10 edge convection loads Edge convection loads are defined by a convective film coefficient and a temperature representing distant conditions. The convective boundary condition is given by: outward flow of heat = h * ( temperature - T ) Where h = convective film coefficient - energy/(area*t*time) T = temperature representing distant conditions Edge convection film coefficients are dimensioned as energy/(area*temperature*time). Note: Default units are the units that are current for the model, and time is in seconds. Associated Data material orientation vector For orthotropic materials, the position of the material X axis is determined by projecting the material orientation vector onto the element at each point on the element. If no material orientation vector exists, the material axes are taken to be parallel to the part's axes. Since no flow of heat in the direction of the part's Y axis is possible, the Y thermal conductivity is ignored. In this figure, (A) is the part coordinate system, and (B) is the material orientation coordinate system.

11 Features (for Potential Flow Analysis) This section lists and describes the features of linear quadrilateral axisymmetric solid elements for Potential Flow Analysis. Summary of Features Number of nodes: 4 Nodal DOF: 1 velocity potential assigned to each node Boundary conditions: Load Type Units...Nodal source loads volume / time...edge flux loads volume / (area * time) Output: Velocity Pressure Coefficient of pressure Element Formulation for Potential Flow Analysis The formulation for this element uses bilinear interpolation functions for velocity potential within the element. Details for this element may be found in Zienkiewicz and Taylor, 1989, or in Cook et al, References: Cook, R. D., Malkus, D. S. and Plesha, M. E., Concepts and Applications of Finite Element Analysis, 3rd Edition, John Wiley & Sons, New York, 1989, pp , , Zienkiewicz, O. C. and Taylor, R. L., The Finite Element Method, Volume 1, Basic Formulation and Linear Problems, 4th Edition, McGraw-Hill, London, 1989, pp 72-80, , 260, 285. Boundary Conditions nodal source loads The positive sense of nodal sources represents the production of fluid per unit radian. The X component of structural nodal force is used to define nodal source data.

12 edge flux loads Positive edge flux is inward and may vary linearly along the edge. The membrane component of structural edge pressure is used to define edge flux. In this figure, (A) is the edge start, and (B) is the end. The X component of structural nodal displacement restraint is used to define velocity potential restraint. Copyright (c) 2002 Unigraphics Solutions Inc. All Rights Reserved.