Lusas Modeller User Manual

Fixing Mesh Problems

Chapter 5 Model Attributes Attribute datasets are used to describe the properties of the model. Attributes are assigned on a feature basis and hence are not lost when the geometry is edited, or the feature is re-meshed at a different density. Attribute assignments are inherited when features are copied and are retained when features are moved. The LUSAS attribute types are: General q Mesh describes the element type and discretisation on the geometry. See page 92. q Geometric specifies any relevant geometrical information that is not inherent in the feature geometry, for example section properties or thickness. See page 117. q Material defines the behaviour of the element material, including linear, plasticity, creep and damage effects. See page 118. q Support specifies how the structure is restrained. Applicable to structural, pore water and thermal analyses. See page 145. q Loading specifies how the structure is loaded. See page 148. Specific q Local Coordinate provides a transformation for loads and supports, and an alternative to the global coordinate system. See page 167. q Composite defines the lay-up properties of composite materials in the model. See page 171. q Slideline slidelines control the interaction of disconnected meshes. See page 173. q Constraint Equations provides the ability to constrain the mesh to deform in certain pre-defined ways. See page 179.

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Chapter 5 Model Attributes q Thermal Surface defines thermal surfaces, which are required for modelling thermal effects. See page 183. q Retained Freedoms specifies the master nodes used in a Guyan reduction or superelement analysis. See page 187. q Damping defines the damping properties for use in dynamic analyses. See page 188. q Birth and Death allows elements to be added (birth) and removed (death) throughout an analysis, e.g. in a tunnelling process or a staged construction. See page 189. q Equivalencing allows nodes which are close to each other but on different features to be merged into one according to defined tolerances. See page 193. q Search Area restricts discrete (point and patch) loads to only apply over certain areas of the model. See page 195.

Manipulating Attributes Attributes are defined from the Attributes menu. Defined attribute datasets are and can be assigned to selected arranged in the Attribute panel of the Treeview features by dragging them onto the model, by using the shortcut menu, (RH mouse button), or by setting them as a default.

1. Attributes are defined using the Attributes Menu.

2. Defined attributes are displayed in the Treeview. 3. To assign an attribute to features, select the features with the mouse, then drag the attribute onto the model.

4. Manipulate the attributes using the shortcut menu (right mouse button).

Attributes are manipulated using the shortcut menu in the Treeview following commands:

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, with the

Visualising Attributes q Rename Attributes can be given meaningful names, for example, 'Steel' to describe a material, or 'Plate - Four Divisions' to describe a Line mesh. q Delete Existing datasets may be deleted, provided they are not assigned to features. q Edit Attribute dataset may be edited. If the name is changed by editing a dataset a new dataset is created and the original dataset is left unchanged. q Select users Selects the features that have the current attribute assignment. q Visualise users Visualises the current attribute assignment. See Visualising Attributes below. q Assign Assigns the current attribute to any features selected on the model. LUSAS only assigns attributes to features that are valid. Some attributes require further information in order to be assigned, in these cases, a dialog is displayed. Assigning an attribute to a feature overwrites any previous assignment of that attribute type. q Deassign Deassigns the current attribute. Choose from all assignments or the selected features only.

Set default Assignment Certain attributes, (mesh, geometric properties and material properties), can be assigned automatically to all newly created features. For this to happen they must first be set as default, by right-clicking the attribute dataset in the Treeview , then choosing Set default from the shortcut menu. This is useful for models with similar materials or thickness throughout, or where the same element is to be applied to all features. Attributes that are set as default are displayed with a red box around them in the Treeview.

Visualising Attributes Attribute assignments can be visualised using three methods: q Attributes layer The Attributes layer is a window layer in the Treeview and is normally added in the initial start-up. The Attributes layer properties define the styles by which assigned attributes are visualised. The attributes layer properties may be edited directly, by double-clicking the , or attributes can be visualised individually by layer in the Treeview selecting an attribute dataset in the Treeview , clicking the right mouse button to choose Visualise Users from the shortcut menu. This is the easiest way to visualise a single attribute. q Contour layer (materials/geometry/loading only) Allows the model to be contoured with material, geometric or loading attribute assignments. 91

Chapter 5 Model Attributes (With nothing selected), click the right mouse button in the graphics area. Choose Contours from the shortcut menu. Double-click the contour layer in the Treeview to display the properties and select either Loading (model), Geometry (model) or Materials (model). ). Colours the geometry q Colour by attribute (From the Geometry layer according to which attributes are assigned to which features. A key is generated to identify the colours. See also Composites for visualising composites and materials.

Drawing Attribute Labels Labels are a window layer in the Treeview

. To display attribute labels:

1. With nothing selected, click the right mouse button in the graphics area. Choose Labels from the shortcut menu. 2. Labels are not displayed for attributes by default. Double-click the Labels layer in the Treeview to display the Properties. 3. Switch on labels for the attribute type. Scroll through the labels list if necessary.

Meshing a Model What is Meshing? LUSAS models are defined in terms of geometric features which must be sub-divided into finite elements for solution. This process is called meshing, and mesh datasets contain information about: q Element Type Specifies the element type to be used in a Line, Surface or Volume mesh dataset may be selected either by describing the generic element type, or naming the specific LUSAS element. q Element Discretisation Controls the density of the mesh, by specifying the element length or the number of mesh divisions, spacing values and ratios. q Mesh type Controls the mesh type e.g. regular or irregular, transition or grid. Mesh datasets are defined from the Attributes menu for a particular geometry type i.e. Line, Surface or Volume. They are then assigned to the required features. Various techniques exist for meshing different types of models, these are described below.

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Meshing a Model

Mesh Types There are various mesh patterns which can be achieved using LUSAS. These are: q Regular meshes only used on regular or analytical Surfaces, and regular Volumes. Regular meshing uses two discretisation techniques, grid and transition. Any element shape may be selected for regular meshing. LUSAS will automatically insert triangular elements in the appropriate positions of a triangular surface for a regular or a transition mesh. Regular grid mesh

Non-uniform Line spacing

q Transition or Grid if the number of mesh divisions on opposite sides of a surface are equal a grid will be generated, otherwise transition patterns (or an irregular mesh) will have to be used. Transition meshes do not produce results which are as good as those from grid meshes or irregular meshes, therefore transition meshes will only be used if specified in the mesh dataset. q Irregular used for Surfaces and irregular swept Volumes. A Surface mesh consists of triangular elements or a quadrilateral and triangular mixture following no set pattern, and may be used on regular or irregular surfaces. A Volume mesh consists of pentahedral or a hexahedral and pentahedral mixture of elements following no set pattern. Regular transition mesh

Irregular mesh

q Extruded irregular mesh used for Volumes which have been swept from an irregular Surface. See below. q Interface Meshes Applicable to joint and composite interface elements only.

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Chapter 5 Model Attributes

Techniques for Meshing a Model The simplest way to mesh a model might be to define a dataset for a particular feature type containing the element type and discretisation, for example a Surface mesh could be used for a 3D shell model, or a Line mesh for a frame model. Two other methods exist allowing greater control over the mesh distribution. These are: q Boundary and Surface discretisation Volume meshing.

Useful for Surface and

q Background grid method Useful for creating a graduated element mesh. q Meshing Volumes Default Number of Mesh Divisions If the discretisation has not been specified in the mesh dataset, or using a Line mesh of element type ‘none’, then the feature will be sub-divided according to the default number of mesh divisions. This may be specified in File > Model Properties > Meshing tab. Tip. Fixing Mesh Problems A group can be created containing all the features that failed to mesh.

Boundary discretisation In the case of Surface or Volume meshing, the boundary divisions may either be specified in the Surface or Volume mesh dataset, or they can by defined using Line meshes of element type ‘None’. In many realistic problems, where several Surfaces (or Volumes) exist, using Line meshes may be the most convenient way to define the mesh. The spacing can be specified using either element length or number of divisions. LUSAS provides several Line meshes of type ‘None’ by default, with different numbers of divisions. Regular Surface Meshing

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Techniques for Meshing a Model

Irregular Surface Meshing

The applied boundary discretisation (top) produces the irregular mesh pattern on the Surface (bottom).

Surface Discretisation Applicable to Volume meshes. The Volume discretisation is specified in the Surface meshes defining the Volume.

Using a Point Mesh and a Background Grid A background grid is a collection of triangular or tetrahedral shapes which are used to specify the element edge length when meshing surfaces automatically. A Line or Surface mesh is used to define the element type and the background grid defines the discretisation, (applicable to Lines and Surfaces). Can be used to create a graded element edge length, but mostly used for an adaptive analysis (remeshing). A background grid is defined from the Utilities menu. When a background grid has been defined, the background grid layer is added to the current window. Graded Element Mesh on Surface Background Grid Point Mesh Spacing a Point mesh, defining the element edge length can be assigned to any Point used to define a Background Grid. The element edge lengths in the vicinity of these Point mesh assignments can then be controlled. Finer control is available using more Points in the Background Grid definition. If a graded element edge length is required on a Surface when meshed, then this can be specified using a Background Grid. The procedure is as follows:

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Chapter 5 Model Attributes Define the background grid as a series of triangular or tetrahedral shapes completely encompassing the Surfaces to be meshed. This may be specified explicitly by specifying Point numbers at each vertex or generated automatically. If the background grid is generated automatically, tetrahedral shapes will always be used. The example on the right shows an irregular Surface bounded by a background grid. The Background Grid is used when the Surface mesh dataset containing the element information is assigned. The element edge length may then be graded by assigning different spacing parameters to various Points in the Background Grid definition. In addition, the Background Grid may be used to control the overall Surface mesh discretisation and additional Line mesh assignments can be used to control the mesh on specific edges. Define a Point mesh dataset setting the spacing parameter to the required element edge length. Any mesh distortion required may be entered as stretching parameters. Note the Attributes > Mesh > Define/Edit by Description method must be used to define Point meshes. Assign the point mesh dataset to the points defining the background grid using Attributes > Mesh > Assign to Features. Repeat this process with different spacing parameters to grade the mesh with as much control as required. Constant Mesh Spacing

Same spacing parameters (Point meshes) are assigned to all Points in background grid.

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Meshing Volumes

Varied Mesh Spacing

Different spacing parameters (Point meshes) are assigned to the top Points (spacing=7) and the bottom Points (spacing=1) in the background grid.

Meshing Volumes Volumes are meshed using regular and limited transition mesh patterns. Only regular volumes, defined by 4, 5 or 6 Surfaces forming tetrahedral, pentahedral or hexahedral bodies, or certain irregular swept Volumes, can be meshed in LUSAS. Tetrahedral Volumes

Tetrahedral Elements

Pentahedral/Tetrahedral Elements

Pentahedral Volumes

Pentahedral Elements

Hexahedral/Pentahedral Elements

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Chapter 5 Model Attributes

Hexahedral Volumes

Pentahedral Elements

Hexahedral Elements

Hexahedral/Pentahedral Elements

Mesh Discretisation The mesh density may be controlled by: q Boundary Discretisation taking the Volume mesh density from the mesh discretisation on the Lines defining the Surfaces of the Volume. q Surface Discretisation taking the Volume mesh density from the specified Surface discretisation defining the Volume. q Volume Discretisation specifying the Volume discretisation explicitly in the Volume mesh dataset. Regular, Irregular or Transition Grid In order to generate a regular grid mesh pattern the number of mesh divisions on opposite faces of the volume must match. If they do not match then transition patterns will be used. A transition mesh may only be used in one direction through the volume. LUSAS will automatically insert pentahedral/tetrahedral elements in the appropriate positions of a transition mesh. Tip. The Volume mesh may be graduated by using non-uniform spacing in the Line mesh assignments on the boundary Lines.

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Meshing Volumes

Extruded Irregular Mesh Volumes defined by sweeping an irregular Surface may now be meshed by extruding the irregular Surface mesh. The interconnecting lines between the irregular end Surfaces must all be straight Lines, or all minor or major arcs with a common axis of rotation. The side Surfaces must all be defined by 4 Lines and LUSAS meshes them with a regular grid of quadrilateral faces. The irregular end Surfaces must not share any common boundary lines therefore wedge-shaped Volumes cannot be meshed as extruded irregular Volumes.

Composite Material Assignment When a Volume feature with a composite material assignment is meshed LUSAS will move the nodes so that they lie on the composite layer boundaries. This ensures an exact number of layers in each element.

MESH ADJUSTED AUTOMATICALLY TO LAY ON INTERLAMINA BOUNDARIES WHEN > 1 THROUGH DEPTH

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Chapter 5 Model Attributes Case Study. Meshing Volumes by Extruding Irregular Surfaces It is possible to mesh an irregular volume if it has been formed by extruding an irregular surface i.e. by sweeping the irregular surface. 1. Define an irregular Surface with more than 4 sides. 2. Define a Volume by sweeping the irregular surface. 3. Define a Volume mesh and leave the number of divisions blank; this will ensure an equal number of divisions on the swept edges. 4. Assign the Volume mesh to the Volume. 5. Draw the mesh. Case Study. Connecting Shells and Solids Solid and shell elements may be connected but the procedure is not as straightforward as it as first appears. Solids and shells have different sets of nodal freedoms and the rotational freedom present in the shells can only be passed through to the solid elements by extending the shell around the side of the solid, thus passing through the rotation via combined translational effects. This form of connection stops rotation relative to a solid which only has translational degrees of freedom. The following case study outlines the general method of fixing shells to solids. 1. Define the Surfaces and Volumes. 2. Assign suitable meshes, for example HX8 elements for the solid and QSI4 elements for the shells. 3. Mesh a Surface that forms part of the solid with shell elements. The surface should share a common edge with the shell Surface that is being fixed to the solid part of the model. Do not forget to assign material and geometric properties to the surface attached to the solid. The properties can be relatively weak in comparison to the main shell properties, or indeed to the solid as the shell is present purely to pass forces and moments through to the underlying solid elements. It is advisable to make a connection such as this reasonably distant from the main area of interest as it may affect the quality of the results locally.

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Meshing Surfaces

Beam Shell Connectivity

Beam Solid Connectivity

Shell Solid Connectivity

Extend the beams along the edge of the shell indicated by thick lines.

Extend the beams along the edge of the solid elements indicated by thick lines. Torsion is restrained using out of plane beams.

Extend the shells over a portion of the solids indicated by dark shaded area.

Overlapping beams

Overlapping shells

Overlapping beams

Beams to be attached Shells to be attached

Beams to be attached

End Releases for Beam Elements Rotational freedoms at the ends of a Line can be made free to rotate by using an element with moment release end conditions. See the Element Reference Manual for more information on these element types.

x 2 y

When defining a Line mesh dataset, with a valid 1 element selected click on the End Release button. Options are available to free rotations about element local y and z axes (θy and θz). Releasing beam element end rotational freedoms could be used as an alternative to using a joint element between beam elements, for example when defining a Pin which is free to rotate.

Meshing Surfaces Mesh Discretisation Regular meshing is used to generate a set pattern of elements on Surfaces and Volumes. Only surfaces which are regular (defined by 3 or 4 lines) can be meshed using a regular mesh pattern. The meshing pattern may be chosen as:

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Chapter 5 Model Attributes q Grid only in order to generate a regular grid mesh pattern the number of mesh divisions on opposite sides of the Surface must match. The examples shown here mesh triangular and quadrilateral Surfaces using both triangular and quadrilateral elements. Note that for a triangular Surface the apex is defined opposite the first Line in the Surface definition and therefore, the second and third Lines in the definition must have the same number of divisions assigned. The Surface mesh may be graded using mesh spacing parameters in null element Line meshes assigned to the boundary Lines. In the examples shown here mesh spacing has been used to zoom the same number of elements as above into the apex of the triangle and one corner of the rectangle. The mesh discretisation for regular meshing can be controlled using one of two methods. These are: q Surface Mesh Dataset specifying the surface discretisation explicitly in the Surface mesh dataset. This is useful only for small problems with few Surfaces and is not recommended.

Irregular Surface Meshing Irregular meshing is used to generate elements on any arbitrary surface. The mesh density can be controlled in a number of ways. These are: q Element Length specifying the required approximate element edge length. Two Surface mesh parameters are used in irregular meshing only. An optional default mesh size, will be applied to a whole Surface if a Surface has no background grid assignment. A mesh quality parameter determines the greatest deviation allowed in element size from either the size interpolated from the background grid or from the default mesh size. This method has a tendency to produce a more uniform mesh as the element size within a Surface is controlled more closely than when just the boundary sizes are specified.

Element Selection About LUSAS Elements The LUSAS Element Library contains over 100 element types. The elements are classified into groups according to their function. The LUSAS element groups are 102

Line Element Selection listed below. Refer to the Element Reference Manual for further details. For full details of the element formulations refer to the LUSAS Theory Manual. q q q q q

Bar Elements Beam Elements 2D Continuum Elements 3D Continuum Elements Plate Elements

q q q q

Shell Elements Membrane Elements Joint Elements Thermal Elements

Line Element Selection The following table lists the elements available for Line meshing by type and by name. The first column matches the option list in the Line mesh dialog box. Element Types q ’None’ Element - One of the generic element types is ‘none’. This type generates no structural elements on the line, the mesh dataset will be used purely to control the line discretisation. q Bar - Bar elements transfer axial force but have no bending or rotational stiffness. q Thin Beam - Thin beam elements behave as a bar, but will also support moment transfer. This formulation of beam neglects shear deformations. q Thick Beam - Thick beam elements support shear effects. q Thick Beam (Nonlinear) q Engineering Grillage - Engineering grillage beam elements in 2D only, with constant shear force along the length, constant torsion and linear bending moment variation. Shear deformations are included. q Ribbed Plate Beam - Ribbed plate beam elements are straight, eccentric beam elements with shear effects. q Cross-section Beam - Cross-section beams are curved, thin beam elements with user-specified quadrilateral cross-section. Shear deformations are neglected. q Semiloof Beam - Semiloof thin beam elements are for use with semiloof shell elements. q Axisymmetric Membrane - The axisymmetric membrane element is the axisymmetric equivalent of a shell element. Modelled as a line over a unit radian segment. q Joint - Joint elements are used to connect two or more nodes by springs, with translational and rotational stiffness. They may have an associated mass and damping.

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Chapter 5 Model Attributes q Thermal Bar - Thermal bar elements are isoparametric bars for use in a field analysis. q Axisymmetric Thermal Membrane - The axisymmetric thermal membrane element has a formulation that applies over a unit radian segment. q Thermal Link - The thermal link element is a straight conductive, convective or radiative link element for field analyses.

Generic Element Types

’None’ Element Bar Thin Beam Thick Beam Thick Beam (Nonlinear) Engineering Grillage Ribbed Plate Beam Cross-section Beam Semiloof Beam Axisymmetric Membrane Joint (no rotational stiffness) Joint (for beams) Joint (for grillages) Joint (for ribbed plates) Joint (for axisymmetric solids) Joint (for axisymmetric shells) Thermal Bar Axisymmetric Thermal Membrane Thermal Link

2D 2 noded

2D 3 noded

3D 2 noded

3D 3 noded

BAR2 BEAM GRIL BRP2 BXM2 JNT3 JPH3 JF3 JRP3 JAX3 JXS3 BFD2 BFX2 LFD2

BAR3 BM3 BMX3 BXM3 BFD3 BFX3 -

BRS2 BMS3 BTS3 JNT4 JSH4 BFD2 BFX2 LFS2

BRS3 BS4 BSX4 BSL4 BFD3 BFX3 -

Notes

q Elements in italic text are only available with the LUSAS +Plus option. q Quadratic elements are curved with a mid-side node. q Rotational freedoms at the ends of a Line can be made free to rotate by using an element with moment release end conditions. q No check is made at this stage as to whether the element type is valid for the analysis being performed, however the LUSAS Solver will stop the analysis if the element is unsuitable. q This list is a guide as to which elements to use. Not all elements are listed here. See the Element Library for full details. 104

Surface Element Selection

Surface Element Selection The following table lists the elements available for surface meshing by type and by name. The first column matches the option list in the Surface mesh dialog box. Triangle 3 noded

Generic Element Types

Plane Stress Plane Strain Axisymmetric Solid Thin Plate Thick Plate Ribbed Plate Thin Shell Thick Shell Membrane Fourier Plane Field (Thermal) Axisymmetric Solid Field Explicit Dynamic - Plane Stress Explicit Dynamic - Plane Strain Explicit Dynamic Axisymmetric

TPM3 TPN3 TAX3 TF3 TTF6 TRP3 TS3 TTS3 TSM3 TAX3F TFD3 TXF3 TPM3E TPN3E TAX3E

Quadrilateral 4 noded

QPM4M QPN4M QAX4M QF4 QSC4 RPI4 QSI4 QTS4 SMI4 QAX4F QFD4 QXF4 QPM4E QPN4E QAX4E

Triangle 6 noded

TPM6 TPN6 TAX6 TTF6 TSL6 TTS6 TAX6F TFD6 TXF3 -

Quadrilateral 8 noded

QPM8 QPN8 QAX8 QTF8 QSL8 QTS8 QAX8F QFD8 QXF8 -

Notes

q Elements in italic text are only available with the LUSAS +Plus option. q No check is made at this stage as to whether the element type is valid for the analysis being performed, however the LUSAS Solver will stop the analysis if the element is unsuitable. q This list is a guide as to which elements to use. Not all elements are listed here. See the Element Library for full details.

Volume Element Selection The following table lists the elements available for volume meshing by type and by name. The first column matches the option list in the Line mesh dialog box. Tetrahedral

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Pentahedral

Chapter 5 Model Attributes Generic Element Types

4 noded

Stress Thermal Explicit Dynamic Composite

TH4 TF4 TH4E -

Generic Element Types

8 noded

Stress Thermal Explicit Dynamic Composite

HX8M HF8 HX8E -

10 noded

TH10 TF10 -

6 noded

PN6 PF6 PN6E -

12 noded

15 noded

PN15 PF15 PNC12C

Hexahedral 16 noded 20 noded

HX20 HF20 HX16C

Notes q Elements in italic text are only available with the LUSAS +Plus option. q No check is made at this stage as to whether the element type is valid for the analysis being performed, however the LUSAS Solver will stop the analysis if the element is unsuitable.

Joint/Interface Element Meshes Joint elements are used to connect two or more nodes by springs, with translational and rotational stiffness. They may have initial gaps, contact properties, an associated mass and damping, and other nonlinear behaviour. Interface elements are used for modelling interface delamination in composite materials. Joint and Interface elements may be inserted between corresponding nodes and features by using interface meshes.

Using Joint Meshes Joint elements are defined in either a Line or Surface mesh dataset. In a 2D analysis the Line joint mesh is assigned to Line features and in a 3D analysis the Surface joint mesh is assigned to Surface features. There are two methods of assigning joint mesh datasets: q Single Joint (Lines only) Joint meshes are assigned directly to the Line(s) that join two structures. This method is more suitable for defining one or two joints since a Line feature must be defined for every joint required.

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Joint/Interface Element Meshes q Joint Mesh Interface (Lines and Surfaces) Uses a master and slave connection to tie two Lines (2D) or two Surfaces (3D) together with a joint mesh.

Modelling a single Joint Element To model a single joint element between two Points. 1. Create a Line joining the two Points. 2. Define a Line mesh dataset with the chosen joint element. 3. Assign the Line mesh to the Line between the two Points.

Modelling a Joint Mesh Interface To define a joint mesh interface between two Lines or two Surfaces: 1. Add the slave feature to selection memory, 2. Assign the joint mesh to the master feature. As the same mesh dataset is assigned to both master and slave features, a mesh pattern is created between the two, with the number of divisions in the mesh determining the divisions along the interface edge. Joint elements are automatically created joining all nodes on the master and slave features. Joint elements cannot be created between two Points using the interface mesh technique.

Joint Local Axis Direction The joint local axis direction is defined when the Line mesh is assigned. Three options are available: q Global axis (default). q A Point in selection memory. q Local coordinate dataset, if at least one has been defined. Example. Interface Mesh (2D) In this example a Line joint mesh with 6 divisions is assigned to Line 1 with Line 2 as the slave. Joints are created automatically to tie the Lines together with an interface joint mesh.

S1 L1 L2

S2

Note. The LUSAS Unmerge facility allows coincident features to be created from a single feature and also allows a feature to be set as Unmergable, so it will not be 107

Chapter 5 Model Attributes accidentally merged back with another coincident feature. See Merging and Unmerging for more details. Example. Cylindrical Interface Mesh (3D) P1 P2 In this example, a Master Surface joint mesh q r is assigned to Slave Surfaces between Joint two concentric Cylindrical Local Cordinate Set (P1,P2,P3) cylinders. used on Assignment to Cylindrical axes align Joint Properties are defined for the P3 joint properties using a local coordinate set. Joint local x axes will then coincide with the cylinder radial direction.

Joint Material and Geometric Properties Joint properties are assigned to the master feature. q Joint Geometric Properties For joints with rotational degrees of freedom an eccentricity must be specified using the Attributes > Geometric menu. q Joint Material Properties Joint meshes also require joint properties to be assigned to them, defined from the Attributes > Material menu.

Composite Delamination using Interface Elements Interface elements may be used at planes of potential delamination to model interlaminar failure, and crack initiation and propagation. If the strength exceeds the strength threshold value in the opening or shearing directions the material properties of the interface element are reduced linearly as defined by the material parameters and complete failure is assumed to have occurred when the fracture energy is exceeded. No initial crack is inserted so the interface elements can be placed in the model at potential delamination areas where they lie dormant until failure occurs. Fracture Modes Three fracture modes exist: open, shear, and orthogonal shear for 3D models. The number of fracture modes corresponds to dimension of the model. (INT6 = 2, INT16 = 3). The diagram below illustrates the three modes.

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Composite Delamination using Interface Elements

Mode 1 - Open

Mode 2 - Shear

Mode 3 - Shear (orthogonal to mode 2)

Interface Elements The interface elements, INT6 and INT16, are used to model composite delamination in an incremental nonlinear analysis. These elements have no geometric properties and are assumed to have no thickness. Interface elements are selected from within Line or Surface mesh datasets using the Attribute, Mesh menu. The mesh datasets are then assigned to the required geometry.

Interface Material Properties The interface material properties are defined from the Attribute, Material, Specialised menu then assigned to the same geometry. Strength

Initial failure strength

Softening Area = Fracture energy (G) Elastic

Failure

Relative displacement

Opening distance

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Chapter 5 Model Attributes Material Parameters q Fracture energy Measured values for each fracture mode depending on the material being used, i.e. carbon fibre, glass fibre. q Initiation Stress The tension threshold /interface strength is the stress at which delamination is initiated. This should be a good estimate of the actual delamination tensile strength but, for many problems the precise value has little effect on the computed response. If convergence difficulties arise it may be necessary to reduce the threshold values to obtain a solution. q Relative displacement The maximum relative displacement is used to define the stiffness of the interface before failure. Provided it is sufficiently small to simulate an initially very stiff interface it will have little effect. Coupling Model q Coupled/mixed interface damage Recommended method. q Uncoupled /reversible Unloading is reversible along the loading path. q Uncoupled /origin Unloading is directly towards the origin ignoring the loading path.

Notes on Delamination Analyses q It is recommended that the arc length procedure is adopted with the option to select the root with the lowest residual norm, when defining the transient control [option 261]. q It is recommended that fine integration [option 18] is selected for the parent elements from the Model Properties, Solution tab. q The nonlinear convergence criteria should be selected to converge on the residual norm. q Continue Solution if more than one Negative Pivot Occurs [option 62] should be selected (from the Model properties, Solution tab) to continue if more than one negative pivot is encountered and option 252 should be used to suppress pivot warning messages from the solution process. q The non symmetric solver is selected automatically when mixed mode delamination is specified. q Although the solution is largely independent of the mesh discretisation, to avoid convergence difficulties it is recommended that a least 2 elements are placed in the process zone.

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Adaptive Analysis (Remeshing)

Adaptive Analysis (Remeshing) Adaptive analysis allows a LUSAS model to be remeshed based on the solution from a previous analysis. The remeshing procedure uses the background grid meshing approach and bases the mesh spacing on nodal error values. The procedure involves creating a background grid from the current mesh using a specified results entity as a measure of the error at each node. The error is calculated from the difference in the nodal values common to a node from the average value at the same node. The discontinuity in the chosen results parameter at the nodes, quantified by the error, is then used to define the required mesh spacing at the Points defining the background grid. The existing mesh is adjusted by increasing or decreasing the mesh spacing at a Point. If the discontinuity error at a node is less than a user specified value then the elements in the region of that node will become larger, conversely at nodes with errors above the acceptable limit the elements will become smaller. Performing an Adaptive Analysis The adaptive process is not automatic and should be controlled using a parametric command file. The model must be tabulated and the LUSAS analysis run to obtain the next set of results and hence the next model mesh. At this stage adaptivity is limited to a Surface mesh implementation only. If a valid results type is active, a command dialog is opened allowing the results column to be selected using an options list. Having created a model and solved it, the results must be read into LUSAS on top of the model. The adaptive process is controlled using three steps, which must be executed in the following order: 1. From Surface Mesh generates a background grid from the mesh on the Surfaces specified. If specified Surfaces lie in a common plane a background grid of triangles is created, otherwise tetrahedral features enclosing the Surfaces are created. A new Point feature for the background grid is created at each corner node of elements meshed in the specified Surfaces. 2. Mesh Spacing from Results specifies the required mesh spacing by generating Point mesh datasets that define the mesh spacing for the background grid created from the Surface mesh in 1 above. Mesh spacing parameters are established by examining the discontinuity of the chosen results parameter (including calculator column results) between elements sharing a common node. This value is termed the Discontinuity Error. Point mesh datasets are then automatically assigned to the correct Point features defining the background grid. This command causes adaptive error results to be created in a results column. See

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Chapter 5 Model Attributes the section titled Discontinuity Error later for more details. The following command options are available: • Background Grid Number specifies the number of the background grid created using this command. • Results Column to Process specifies the results column to be used as an error estimate for calculating new mesh spacing parameters. Shear and principal stresses have been excluded from the adaptivity results column selection as these stress types cannot be used to calculate a sensible mean nodal value. • Element Size Reducing Scaling Factor specifies a scaling factor to control the amount by which an element size is reduced during the adaptive process. See the section titled Discontinuity Error below for more details. • Element Size Increasing Scaling Factor specifies a scaling factor to control the amount by which an element size is increased during the adaptive process. See the section titled Discontinuity Error below for more details. 3. Assign to Defining Features assigns the new background grid or grids specified to the Surfaces from which each was created. Any command that redraws the mesh will cause an updated mesh to be created using the new spacing parameters. As the adaptive process proceeds and the number of remeshing cycles increases, redundant Point features, and background grid and Point mesh datasets will start to be accumulated. This data may be deleted with the existing commands using the ALL entry for the parameter list, allowing LUSAS to check for redundancy.

Discontinuity Error The discontinuity error at a node is given by: ( MaxNode - Mean ) / Mean * 100% Where MaxNode represents the maximum result from any element defined by a node, and Mean represents the mean result from all the elements contributing to that node. The discontinuity error may be contoured using PostView > Nodal Results > Plot Contours and printed using PostView > Nodal Results > Print Adaptive in a similar manner to existing results columns. The results column header is Eadp. Element re-sizing is controlled using two relationships. For values of discontinuity error greater than the specified acceptable limit: SizeNew = SizeOld / [ 1 + (DErr * ScaleRed) ] For values of discontinuity error less than the specified acceptable limit: 112

Adaptive Analysis (Remeshing) SizeNew = SizeOld [ 1 + ScaleInc * (AErr - DErr) ] Where DErr is the calculated discontinuity error, AErr is the acceptable error set in the Utilities > Background Grid > Options dialog (default=1%), ScaleRed is the element size reducing scaling factor and ScaleInc is the element size increasing scaling factor.

Excluding Features Using the menu commands Geometry > Feature type > Exclude/Include features may be excluded from the adaptive process so that the mesh on those features remains fixed. Features may be excluded from the discontinuity error calculations to stop the elements gathering around point loads where there can be artificially high stress concentrations. Nodes excluded from the adaptive process can be printed to the text window using MeshView > Show Nodes Excluded from Adaptivity.

Element Size Control From the Meshing tab (Adaptivity button) of the model properties global element parameters can be set to stop elements becoming too large or too small or to stop the elements changing size by a large percentage between solutions. Options are as follows: q Minimum and Maximum Element Size sets the minimum and maximum allowable element size in model units (initially unset). See note below. q Maximum Element % Change sets the maximum allowable percentage change in element size (default=50%). q Error % Cut-off sets the acceptable limit of percentage discontinuity error, below which the element size is not changed (default=1%). The adaptivity error results are stored in the Eadp column. q Minimum Result sets the percentage of the absolute maximum results value below which there is no change in mesh size (default = 0%). This value is useful to remove large areas of fairly constant stress from the solution, where the stress level is well below the peak values in the areas of interest. The maximum and minimum element sizes are initially unset. LUSAS sets the parameters when the model is scaled, or when the adaptivity convergence parameters are calculated. In both cases the overall model dimensions are used to calculate maximum and minimum element sizes, but only if these parameters are unset. Once set these values will remain unchanged. LUSAS sets the parameters using the following formulae: 113

Chapter 5 Model Attributes Elem(max) = DMINSZ * Current model size Elem(min) = DMAXSZ * Current model size Where Elem(max) and Elem(min) represent maximum and minimum element sizes respectively, and DMINSZ and DMAXSZ are system variables with default sizes 0.010 and 0.50 respectively.

Convergence Control A check of the current level of convergence of the adaptive process can be made using the command line command show adaptive change. These parameters are designed to provide a measure of the converged state of the adaptive process and are ideal for using adaptive meshing in a parametric command file. Having retrieved the convergence values, the parametric file can stop or continue with another remesh/solution cycle. Valid convergence parameters are as follows: q Maximum Change in Element Size (chgesz) Maximum percentage change in mesh spacing. This value is initialised to 100%. q Change in Strain Energy (chgstr) Percentage change in strain energy between solutions of the problem. This value is initialised to 100%. Within a parametric these real number parameters can be accessed via their internal LUSAS names using the function m$mysvbs. The function usage is shown in the example below: real csiz, cnrg ... csiz = m$mysvbs(chgesz) cnrg = m$mysvbs(chgstr) printf("Change in element size: %s", csiz) printf("Change in strain energy: %s", cnrg) if ( csiz < 2.0

&&

cnrg < 5.0 ) then

{ goto exit } ... exit: Files. The adaptive process lends itself to execution via the command file facility. An example set of command files that can be used to analyse a simple plate with hole have been included with the release kit. Command files csadapt1.cmd,

114

Adaptive Analysis (Remeshing) csadapt2.cmd and csadapt3.cmd are included in the tutorial directory. They should be used with reference to the following case study. Case Study. Plate with a Hole Adaptive Mesh Improvement This case study makes reference to the command files csadapt1.cmd, csadapt2.cmd and csadapt3.cmd. 1. Using Files > Command File > Open, run the command file csadapt1.cmd. This will define a single irregular Surface model and tabulate a LUSAS data file plate_1.dat and save a model database plate_1.mdl. 2. Run LUSAS using the generated data file plate_1.dat. A results database is then created. 3. Run the command file csadapt2.cmd. This file uses the model file and results file from the analysis to regenerate an improved mesh based on the errors calculated in the x direction direct stress results. After remeshing, the model is saved as plate_2.mdl and a second data file plate_2.dat is tabulated. 4. After running the Solver a second time to create results file plate_2.mys, run the command file csadapt3.cmd. This creates a further level of mesh refinement. The meshes for each level of adaptive meshing are shown in the accompanying diagrams following this case study.

115

Chapter 5 Model Attributes

Plate 1

Plate 2

Plate 3

Original mesh.

1st mesh iteration.

2nd mesh iteration.

Mesh

Original mesh refined in 2 stages. Contour Plot

showing x direction stress used for adaptivity. Error Plot

As mesh is refined, errors reduce and localise. Tip. The post-processing command files in the case study above can be merged by including parametric variables to refer to the model and results files read in and the data file and model names saved. In this way a single post-processing command file could be used instead.

116

Geometric Properties

Geometric Properties General Geometric properties are used to describe geometric attributes which have not been defined by the feature geometry. For example below. The properties required are element dependent and are defined for an element family. The dataset is then assigned to the required Line, Surface or Joint feature. If a geometric property of a type incompatible with the mesh is assigned to a feature a warning will be issued when the model is tabulated. Beam Centre-Line

Bar/Link Grillage Thin/Thick Beam Elements Structures modelled using bar or beam elements require section properties to be defined, for example, thickness, shear area and eccentricity.

Plate/Membrane/Shell Elements Structures modelled using shell elements require their thickness to be defined, and eccentric behaviour can also be specified for certain element types.

Eccentricity (e) y x z

Nodal Line Beam Local (Element) Axes

Thickness (T) Nodal Line

Local Surface (Element) Axes Thickness (T) Eccentricity (e) Nodal Line Plate Centre-Line

Using Geometric Properties Geometric properties are defined as attribute datasets from the Attribute Menu. A geometric property dataset may be nominated as the default assignment, which is then automatically assigned to all Lines and Surfaces subsequently created. Notes on Use

q Geometric properties are not required for plane strain, axisymmetric or solid elements. q The geometric properties are specified in generic form for all elements and only the properties required for the intended element need be specified. For

117

Chapter 5 Model Attributes example eccentricity is ignored by semi-loof shells which do not use it and so it may be entered as zero in the property dialog. q Geometric properties can be varied over a given feature by using a Variation dataset. See Variations for more details. q For more details on the properties required for specific elements refer to the LUSAS Element Library.

Material Properties Every part of an FE model must be assigned a material property dataset. LUSAS material datasets are defined from the Attributes > Materials menu. Note that not all elements accept all material property types. Refer to the LUSAS Element Library for full details of valid element/material combinations.

Linear and Nonlinear Material Properties q Isotropic/Orthotropic Defines linear elastic or nonlinear material properties with options for plasticity, hardening, creep, damage, viscosity and two-phase materials. q Anisotropic Different material properties are specified in arbitrary (nonorthotropic) directions by direct specification of the modulus matrix. q Rigidities Allows direct specification of the material rigidity matrix.

Specialised Material Properties q Thermal Applicable to thermal elements only. Whenever thermal elements have been used in a model thermal material properties should be defined and assigned to the relevant parts of the model. Thermal material properties include thermal conductivity, specific heat, enthalpy. Sub-types are Isotropic and Orthotropic. q Joint Linear and nonlinear joint material models for contact and impact analyses using joint elements. q Interface Material models for use with the composite delamination interface elements. These elements enable composite delaminations to be modelled using an incremental nonlinear analysis. q Rubber Defines materials with hyper-elastic or rubber-like mechanical behaviour. q Crushing A volumetric crushing model such as would be used for crushable foam-filled composite structures. Notes

q Material property datasets can be formed into a composite lay-up using the composite attribute facility.

118

Isotropic/Orthotropic Material Definition q Once assigned to geometry material directions can be visualised using the Attributes layer . q Rubber, crushing, and plastic material datasets cannot be combined.

Isotropic/Orthotropic Material Definition Isotropic and orthotropic material datasets can be used to specify the following material properties: q Elasticity Linear elastic material properties including Young’s modulus, Poisson’s ratio, mass density, (orthotropic angle). Optional thermal and dynamic properties. Note that not all elements accept all the orthotropic models. Refer to the LUSAS Element Library for full details of valid element/material combinations. Orthotropic models are Plane stress, Plane strain, Thick, Sheet, Axisymmetric, Solid. q Plasticity Used to model ductile yielding of nonlinear elasto-plastic materials such as metals, concrete, soils/rocks/sand. q Hardening Used to model a nonlinear hardening curve data. Hardening is defined as part of the plastic properties. Isotropic, Kinematic and Granular sub-types are available. Isotropic hardening can be input in three ways. q Creep Used to model the inelastic behaviour that occurs when the relationship between stress and strain is time dependent. q Damage Used to model the initiation and growth of cavities and microcracks. q Viscosity Used to model viscoelastic behaviour. Coupling of the viscoelastic with nonlinear elasto-plastic materials enables hysteresis effects to be modelled. q Two-phase Required when performing an analysis in which two-phase elements are used to define the drained and undrained state for soil.

Plastic Material Models - Isotropic The following are Isotropic models available from the Attributes > Material > Isotropic dialog, after clicking the Plastic check box. q Stress Resultant (Model 29) May be used for certain beams and shells. The model is formulated directly with the beam or shell stress resultants plus geometric properties, therefore it is computationally cheaper. q Tresca (Model 61) Represents ductile behaviour of materials which exhibit little volumetric strain (for example, metals). Incorporates isotropic hardening.

119

Chapter 5 Model Attributes q Optimised implicit von Mises (Model 75) Represents ductile behaviour of materials which exhibit little volumetric strain (for example, metals). Especially for explicit dynamics. q Stress Potential (von Mises model) Nonlinear material properties applicable to a general multi-axial stress state requiring the specification of yield stresses in each direction of the stress space. Incorporates hardening, yield stress and Heat fraction. q Mohr-Coulomb (Model 63) Represents ductile behaviour of materials which exhibit volumetric plastic strain (for example, granular materials such as concrete, rock and soils). Incorporates isotropic hardening. q Drucker-Prager (Model 64) Represents ductile behaviour of materials which exhibit volumetric plastic strain (for example, granular materials such as concrete, rock and soils). Incorporates isotropic hardening. q Concrete Cracking Represents the nonlinear material effects associated with the three dimensional cracking of concrete.

Plastic Material Models - Orthotropic The Stress Potential Hill and Hoffman models are available from the Attributes > Material > Orthotropic dialog, click the Plastic check box. The stress potential model defines nonlinear material properties applicable to a general multi-axial stress state requiring the specification of yield stresses in each direction of the stress space. Incorporates hardening, yield stress and Heat fraction. Hoffmann is a pressure dependent material model allowing for different properties in tension and compression.

Stress Resultant Material Model The model is formulated directly with the beam or shell stress resultants plus geometric properties, therefore it is computationally cheaper. Consult the LUSAS Element Library the check which elements are valid for this material model. Material Parameters q Yield stress The level of stress at which a material is said to start unrecoverable or plastic behaviour. q Section shape Match the section type to the element being used. Notes 1. The yield criteria, when used with beam elements, includes the effects of nonlinear torsion. Note that the effect of torsion is to uniformly shrink the yield surface. 2. The stress-strain curve is elastic/perfectly plastic.

120

Isotropic/Orthotropic Material Definition 3. The fully plastic torsional moment is constant. 4. Transverse shear distortions are neglected. 5. Plastification is an abrupt process with the whole cross-section transformed from an elastic to fully plastic stress state. 6. Updated Lagrangian (Option 54) and Eulerian (Option 167) geometric nonlinearities are not applicable with this model. The model, however, does support the total strain approach given by Total Lagrangian and Co-rotational geometric nonlinearities, Option 87 and Option 229, respectively. Geometric nonlinearity options are set from the Model properties.

Tresca Material Model (Model 61) Material Parameters Uniaxial Yield Stress

α σ yo

α

=tan -1C 1

L1

Equivalent Plastic Strain, ε p

Hardening Curve Definition for the Tresca Yield Model Yield stress The level of stress at which a material is said to start unrecoverable or plastic behaviour. Heat fraction The fraction of plastic work that is converted into heat energy. Only applicable to temperature dependent materials and coupled analyses where the heat produced due to the rate of generation of plastic work is of interest. The value should be between 0 and 1.

121

Chapter 5 Model Attributes

Optimised implicit von Mises Material Model Represents ductile behaviour of materials which exhibit little volumetric strain (for example, metals). Especially for explicit dynamics. Material Parameters q Yield stress The level of stress at which a material is said to start unrecoverable or plastic behaviour. q Heat fraction The fraction of plastic work that is converted into heat energy. Only applicable to temperature dependent materials and coupled analyses where the heat produced due to the rate of generation of plastic work is of interest. The value should be between 0 and 1. Hardening (von Mises)

q Kinematic hardening Plasticity hardening formulation associated with translation, as opposed to expansion, of the yield surface. In the optimised implicit model the direction of plastic flow is evaluated from the stress return path. The implicit method allows the proper definition of a tangent stiffness matrix which maintains the quadratic convergence of the Newton-Raphson iteration scheme otherwise lost with the explicit method. This allows larger load steps to be taken with faster convergence. For most applications, the implicit method should be preferred to the explicit method. The model incorporates linear isotropic and kinematic hardening. Uniaxial Yield Stress

α = tan-1C α1 σ yo L1

Equivalent Plastic Strain, ε p

Nonlinear Hardening Curve for the von Mises Yield Model (Model 75)

122

Isotropic/Orthotropic Material Definition

Stress Potential Stress Potential The use of nonlinear material properties applicable to a general multi-axial stress state requires the specification of yield stresses in each direction of the stress space when defining the yield surface (see the LUSAS Theory Manual). Notes

q The yield surface must be defined in full, irrespective of the type of analysis undertaken. This means that none of the stresses defining the yield surface can be set to zero. For example, in a plane stress analysis, the out of plane direct stress σzz, must be given a value which physically represents the model to be analysed. q The stresses defining the yield surface in both tension and compression for the Hoffman potential must be positive. Material Properties q Yield stress The level of stress at which a material is said to start unrecoverable or plastic behaviour. q Heat fraction The fraction of plastic work that is converted into heat energy. Only applicable to temperature dependent materials and coupled analyses where the heat produced due to the rate of generation of plastic work is of interest. The value should be between 0 and 1. Hardening Properties There are three methods for defining nonlinear hardening. Hardening curves can be defined in terms of either the hardening gradient, the plastic strain or the total strain as follows: q Hardening gradient vs. Effective plastic strain Requires specification of gradient and limiting strain values for successive straight line approximations to the stress vs. effective plastic strain curve. In this case hardening gradient data will be input as (C1, ep1), (C2, ep2) for each straight line segment. LUSAS extrapolates the curve past the last specified point.

123

Chapter 5 Model Attributes Gradient

Stress

C1 = s1-sy/ep1 S2 S1

Sy

Gradient C2 = s2-s1/ep2-ep1

ep1

ep2

Plastic Strain

q Uniaxial yield stress vs. Effective plastic strain Requires input of coordinate points at the ends of straight line approximations to the uniaxial yield stress vs. effective plastic strain curve. For the curve shown here the plastic properties will contain the yield stress (sy) and the hardening data will be input as (s1, ep1), (s2, ep2), etc. LUSAS extrapolates the curve past the last specified point. Elastic/Plastic Strain Boundary

Stress S2 S1

ee2

ep3

Curve Extrapolation

ee1

Sy

ep2

ep1

ep3

Young's ep2 Modulus

Effective Plastic Strain

q Uniaxial yield stress vs. Total Strain Requires input of coordinate points at the ends of straight line approximations to the stress strain curve. Linear properties specify the slope of the stress strain curve up to yield in terms of a Young's modulus. Plastic properties specify the yield stress (sy) and the hardening data is input as a series of coordinates, for example (s1, e1), (s2, e2), etc. LUSAS extrapolates the curve past the last specified point.

124

Isotropic/Orthotropic Material Definition Stress S2

Curve Extrapolation

S1

Sy

Total Strain Young's Modulus

e2

e3

Mohr-Coulomb Material Model The Mohr-Coulomb elasto-plastic model may be used to represent the ductile behaviour of materials which exhibit volumetric plastic strain (for example, granular materials such as concrete, rock and soils). The model incorporates isotropic hardening. Material Properties q Initial Cohesion A material property of granular materials, such as soils or rocks, describing the degree of granular bond and a measure of the shear strength. q Initial Friction angle A material property of granular materials, such as cohesive soils and rocks. q Heat fraction The fraction of plastic work that is converted into heat energy. Only applicable to temperature dependent materials and coupled analyses where the heat produced due to the rate of generation of plastic work is of interest. The value should be between 0 and 1. Notes

q The heat fraction coefficient represents the fraction of plastic work which is converted to heat and takes a value between 0 and 1. For compatibility with pre LUSAS 12 data files specify Option -235. q Setting the initial cohesion (C) to zero is not recommended as this could cause numerical instability under certain loading conditions.

125

Chapter 5 Model Attributes

Cohesion

α1 Co

α 1=tan-1C11

L1 Equivalent Plastic Strain, ε p Cohesion Definition for the Mohr-Coulomb and Drucker-Prager Yield Models (Models 63 and 64)

φo

α2 α 2=tan-1C21

L1 Equivalent Plastic Strain, ε p Friction Angle Definition for the Mohr-Coulomb and Drucker-Prager Yield Models (Models 63 and 64)

Drucker-Prager Material Model The Drucker-Prager elasto-plastic model (see figures on page) may be used to represent the ductile behaviour of materials which exhibit volumetric plastic strain

126

Creep Material Properties (for example, granular materials such as concrete, rock and soils). The model incorporates isotropic hardening. Material Properties q Initial Cohesion A material property of granular materials, such as soils or rocks, describing the degree of granular bond and a measure of the shear strength. Setting the initial cohesion to zero is not recommended as this could cause numerical instability under certain loading conditions. q Initial Friction angle A material property of granular materials, such as cohesive soils and rocks. q Heat fraction The fraction of plastic work that is converted into heat energy. Only applicable to temperature dependent materials and coupled analyses where the heat produced due to the rate of generation of plastic work is of interest. The value should be between 0 and 1.

Concrete Cracking Model The multi-crack model assumes that, at any one point in the material, there are a defined number of permissible cracking directions. The model assumes that the material can soften and eventually loose strength in positive loading. q The softening follows an exponential curve defined by the tensile strength and the strain at end of softening curve. To ensure the softening function is a valid shape the following restriction should be used: strain at end of softening curve > 1.5 * tensile strength Young’s modulus q Fracture energy per unit area (to fully open the crack) should be specified (instead of strain at end of softening curve) when defining a localised fracture rather than a distributed fracture. Note. Either the Fracture energy per unit area or the Strain at end of softening curve should be defined. If both are specified then the Fracture energy per unit area is ignored.

Creep Material Properties Creep is the inelastic behaviour that occurs when the relationship between stress and strain is time dependent. The creep response is usually a function of the stress, strain, time and temperature history. Unlike time independent plasticity where a limited set of yield criteria may be applied to many materials, the creep response differs greatly for different materials.

127

Chapter 5 Model Attributes

Creep Properties There are three uniaxial creep laws available in LUSAS and a time hardening form is available for all laws. The power creep law is also available in a strain hardening form. Fully 3D creep strains are computed using the differential of the von Mises or Hill stress potential. A user-definable creep interface is also available which allows a programmable uniaxial creep law. The required creep properties for each law are: q Power law (time dependent / strain hardening) ε c = f1q f2 t f3

LM N

q Exponential law ε c = f1e f2 q 1 − − f3tq

−f 4

OP + f te Q 5

f6 q

q Eight parameter law ε c = f1q f2 t f3 + f4 t f5 + f6 t f7 e − f8 / T

b

& = f q , t, T q User-supplied εc

g

where: ε c = uniaxial creep strain ε& c = rate of uniaxial equivalent creep strain

q = (von Mises or Hill) equivalent deviatoric stress t = current time T = temperature (Kelvin) Stress Potential The definition of creep properties requires that the shape of the yield surface is defined. The stresses defining the yield surface are specified using the Stress Potential material model. If a Stress Potential model is used in the Plastic definition then this will override the Creep stress potential and will apply to both the plastic properties and the creep properties. The Creep stress potential is only required when defining linear materials. If a stress potential type is not specified then von Mises is set as default. None of the stresses defining the stress potential may be set to zero. For example, in a plane stress analysis, the out of plane direct stress must be given a value which physically represents the model to be analysed.

User Supplied Creep Properties The User creep property facility allows user supplied creep law routines to be used from within LUSAS. This facility provides completely general access to the LUSAS property data input and provides controlled access to the pre- and post-solution constitutive processing and nonlinear state variable output.

128

Creep Material Properties Source code access is available to interface routines and object library access is available to the remainder of the LUSAS code to enable this facility to be utilised. Contact FEA for full details of this facility. Since user specification of a creep law involves the external development of source FORTRAN code, as well as access to LUSAS code, this facility is aimed at the advanced LUSAS user. Notes

q The user-supplied routine must return the increment in creep strain. Further, if implicit integration is to be used then the variation of the creep strain increment with respect to the equivalent stress and also with respect to the creep strain increment, must also be defined. q If the function involves time dependent state variables they must be integrated in the user-supplied routine. q If both plasticity and creep are defined for a material, the creep strains will be processed during the plastic strain update. Stresses in the user routine may therefore exceed the yield stress. q User-supplied creep laws may be used as part of a composite element material assembly.

Creep Data in Rate Form Creep data is sometimes provided for the creep law in rate form. The time component of the law must be integrated so that the law takes a total form before data input. For example the rate form of the Power law ε c = Aq n t m

integrates to

a

f

ε c = A / m + 1 q n t m +1

The properties specified as input data then become

a

f

f1 = A / m + 1 f2 = n f3 = m + 1

where A, n and m are temperature dependent constants.

129

Chapter 5 Model Attributes

Damage Material Properties Damage is assumed to occur in a material by the initiation and growth of cavities and micro-cracks. The damage model allows parameters to be defined which control the initiation of damage and post damage behaviour. In LUSAS a scalar damage variable is used in the degradation of the elastic modulus matrix. This means that the effect of damage is considered to be non-directional or isotropic. Two LUSAS damage models are available (Simo and Oliver) together with a facility for a user-supplied model. A damage analysis can be carried out using any of the elastic material models and the following nonlinear models: q von Mises (models 72 and 77) q Hill (model 76) q Hoffman (model 78) Creep material properties may be included in a damage analysis.

Damage Properties The initial damage threshold, r0 , can be considered to carry out a similar function to the initial yield stress in an analysis involving an elasto-plastic material. However, in a damage analysis, the value of the damage threshold influences the degradation of the elastic modulus matrix. A value for r0 may be obtained from: r0 =

σtd

bE g

1/ 2

0

where σ t d is the uniaxial tensile stress at which damage commences and E 0 is the undamaged Young’s modulus. The damage criterion is enforced by computing the elastic complementary energy function as damage progresses:

e

β σ T De σ

j

1/ 2

− rt ≤ 0

where σ is the vector of stress components, De the elastic modulus matrix and rt the current damage norm. The factor β is taken as 1 for the Simo damage model, while for the Oliver model takes the value:

FG H

β= θ+

1− θ η

IJ K

where θ=

< σ1 > + < σ 2 > + < σ 3 > | σ1|+| σ 2 |+| σ 3 |

η=

130

σcd σtd

Viscous Material Properties Only positive values are considered for , any negative components are set to zero. The values σ c d and σtd represent the stresses that cause initial damage in compression and tension respectively (note that if σ c d = σ t d , β=1). The damage accumulation functions for each model are given by:

a

f

bg

r0 1 − A − A exp B r0 − rt rt

bg

r0 r exp A 1 − 0 rt rt

Simo:

G rt = 1 −

Oliver:

G rt = 1 −

LM F MN GH

b

g

I OP JK PQ

For no damage, G(rt)=0. The characteristic material parameters, A and B, would generally be obtained from experimental data. However, a means of computing A has been postulated for the Oliver model:

LM G E A=M MN I eσ j

f o t 2 d ch

O 1P − P 2 PQ

−1

where G f is the fracture energy per unit area, I ch is a characteristic length of the finite element which can be approximated by the square root of the element area. These damage models are explained in greater detail in the LUSAS Theory Manual. Damage ratio (Oliver model only) The Damage ratio is the ratio of the stresses that cause initial damage in tension and compression = σ c d / σ t d . It is invoked if different stress levels cause initial damage in tension and compression.

Viscous Material Properties Viscoelasticity can be coupled with the linear elastic and non-linear plasticity, (isotropic or orthotropic), creep and damage models available in LUSAS. The model restricts the viscoelastic effects to the deviatoric component of the material response. This enables the viscoelastic material behaviour to be represented by a shear modulus Gv and a decay constant β. Viscoelasticity imposed in this way acts like a springdamper in parallel with the elastic-plastic, damage and creep response. Coupling of the viscoelastic and the existing nonlinear material behaviour enables hysteresis effects to be modelled.

131

Chapter 5 Model Attributes Notes

q It is assumed that the viscoelastic effects are restricted to the deviatoric component of the material response. The deviatoric viscoelastic components of stress are obtained using a stress relaxation function G(t), which is assumed to be dependent on the viscoelastic shear modulus and the decay constant.

af

σ' v t =

z

t

0

a f ∂ε∂s ∂s

2G t − s

'

af

G t = G v e − tβ

q The viscoelastic shear modulus G v can be related to the instantaneous shear modulus, G 0 , and long term shear modulus, G ∞ , using G v = G 0 − G ∞ . q When viscoelastic properties are combined with isotropic elastic properties, the elastic modulus and Poisson’s ratio relate to the long term behaviour of the material, that is, E ∞ and υ∞ . q At each iteration, the current deviatoric viscoelastic stresses are added to the current elastic stresses. The deviatoric viscoelastic stresses are updated using;

a

f

af

σ' v t + ∆t = σ' v t e −β∆t + 2G v

e1 − e j ∆ε − β∆t

β

'

∆t

where • σ' v • Gv

= deviatoric viscoelastic stresses = viscoelastic shear modulus

• β • ∆t • ∆ε'

= viscoelastic decay constant = current time step increment = incremental deviatoric strains

q When viscoelastic properties are coupled with a nonlinear material model it is assumed that the resulting viscoelastic stresses play no part in causing the material to yield and no part in any damage or creep calculations. Consequently the viscoelastic stresses are stored separately and deducted from the total stress vector at each iteration prior to any plasticity, creep or damage computations. Note that this applies to both implicit and explicit integration of the creep equations. q Nonlinear Control must always be specified when viscoelastic properties are assigned. In addition Dynamic Control must also be specified to provide a time step increment for use in the viscoelastic constitutive equations. If no time control is used the viscoelastic properties will be ignored.

132

Two-Phase Material Properties

User Supplied Visco Elastic Properties The user supplied visco elastic properties facility enables routines for implementing a user supplied viscoelastic model to be invoked from within LUSAS. This facility provides completely general access to the LUSAS property data input via this data section and provides controlled access to the pre- and post-solution constitutive processing and nonlinear state variable output via these user supplied routines. Source code access is available to interface routines and object library access is available to the remainder of the LUSAS code to enable this facility to be utilised. Contact FEA for full details of this facility. Since user specification of a viscoelastic model involves the external development of a FORTRAN source code, as well as access to the LUSAS code, this facility is aimed at the advanced LUSAS user. Notes

q Option 179 can be set for argument verification within the user routines. q The current viscoelastic stresses must be evaluated at each iteration and added to the current Gauss point stresses. These viscoelastic stresses are subsequently subtracted at the next iteration, internally within LUSAS, before any plasticity, creep or damage calculations are performed.

Two-Phase Material Properties Two-phase material properties are required when performing an analysis in which two-phase elements are used to define a drained and undrained state for soil. Notes

q Usually, the value of Bulk modulus of solid phase is quite large compared to Bulk modulus of fluid phase and not readily available to the user. If Bulk modulus of solid phase is input as 0, LUSAS assumes an incompressible solid phase. Bulk modulus of fluid phase is more obtainable, e.g. for water Bulk modulus of fluid phase = 2200 Mpa [N1] q Two-phase material properties can only be assigned to geotechnical elements, that is, TPN6P and QPN8P. q When performing a linear consolidation analysis TRANSIENT CONTROL must be specified. DYNAMIC or NONLINEAR CONTROL cannot be used. q In an un-drained analysis two-phase material properties may be combined with any other material properties, and creep, damage and viscoelastic properties. In a drained analysis only linear material properties may be used.

133

Chapter 5 Model Attributes

Rigidity Material Definition The linear rigidity model is used to define the in-plane and bending rigidities from prior explicit integration through the element thickness. Note q Angle of orthotropy is relative to the reference axis (degrees). q The element reference axes may be local or global (see Local Axes in the LUSAS Element Library for the proposed element type). If the angle of orthotropy is set to zero, the anisotropy coincides with the reference axes. Example 1. Membrane Behaviour

R|N S|N TN

x y xy

U| LD V| = MMD W MND

1

D2

D4

2

D3

D5

4

D5

D6

OPR|R|ε PPS|S|ε Q|TTε

x y xy

U| R|ε V| − S|ε W Tε

xo yo xyo

U|U| R|N V|V| + S|N W|W TN

xo yo xyo

U| V| W

where: N D e

are the membrane stress resultants (force per unit width). membrane rigidities. membrane strains.

and for isotropic behaviour, where t is the thickness: D1 = D 3 =

Εt 1−ν

D2 =

2

νΕt 1−ν

D6 =

2

Εt 2 1+ν

b g

D4 = D5 = 0

The initial strains due to a temperature rise T are: ε ot

R|ε = Sε |Tε

xo yo xyo

U| Rα V| + T|S|α W Tα

1 2 3

U| V| W

Example 2. Thin Plate Flexural Behaviour

R|M S|M TM

x y xy

U| LD V| = MMD W MND

1

D2

D4

2

D3

D5

4

D5

D6

OPR|R|Ψ PPS|S|Ψ Q|TTΨ

x y xy

U| R|Ψ V| − S|Ψ W TΨ

xo yo xyo

U|U| R|M V|V| + S|M W|W TM

xo yo xyo

U| V| W

where: M D Ψ

are the flexural stress resultants (moments per unit width). flexural rigidities. flexural strains given by: 134

Rigidity Material Definition

R|- ∂ w U| U| || ∂∂x || V| = |S|- ∂yw |V| W | 2∂ w | ||- ∂x∂y || T W 2

R|Ψ S|Ψ TΨ

2

x

2

y

2

xy

2

and for an isotropic plate for example, where t is the thickness: D1 = D 3 =

Et

3

D2 =

12(1 − ν ) 2

νEt 3

D6 =

12(1 − ν ) 2

Et

3

24 (1 + ν )

D4 = D5 = 0

The initial strains due to a temperature rise T are:

R|Ψ = SΨ |TΨ

Ψ ot

xo yo xyo

U| ∂T Rα V| = ∂z |S|α W Tα

1 2 3

U| V| W

Example 3. Thick Plate Flexural Behaviour

R|M ||M S|M ||SS T

x y xy

xz yz

U| LD || MMMD V| = MD || MMDD W N

1

D2

D4

D7

D11

2

D3

D5

D8

D12

4

D5

D6

D9

D13

7

D8

D9

D10

D14

11

D12

D13

D14

D15

OPR|R|Ψ PP||||ΨΨ PPS|S|Γ PQ||T||TΓ

x y xy xz yz

U| R|Ψ || ||Ψ V| − S|Ψ || ||ΓΓ W T

xo yo xyo xzo yzo

U|U| R|M |||| ||M V|V| + S|M |||| ||SS WW T

xo yo xyo

xzo

yzo

U| || V| || W

where: M S D Ψ

are the flexural stress resultants (moments per unit width). shear stress resultants (shear force per unit width). flexural and shear rigidities. flexural strains given by:

R|− ∂ w U| U| || ∂∂x || V| = |S|− ∂yw |V| W | 2∂ w | ||− ∂x∂y || T W 2

R|Ψ S|Ψ TΨ Γ

2

x y xy

2

2

2

shear strains given by:

135

Chapter 5 Model Attributes

R ∂w ∂u U RSΓ UV = |S ∂x + ∂z |V TΓ W || ∂∂wy + ∂∂vz || T W xz yz

and for an isotropic plate for example: Et

D10 = D15 =

2 (1 + ν )k

D 7 = D 8 = D 9 = D11 = D12 = D13 = D14 = 0

where t is the plate thickness and k is a factor taken as 1.2 which provides the correct shear strain energy when the shear strain is assumed constant through the plate thickness. D1 to D6 are the same as defined for the thin plate flexural behaviour (see Example 2. Thin Plate Flexural Behaviour). The initial strains due to a temperature rise T are:

R|Ψ RSΨ UV = ||SΨΨ Tε W |Γ ||Γ T ot

ot

xo yo xyo xzo yzo

U| Rα || ∂T |||α V| = ∂z S|α || ||00 W T

1 2 3

U| R|0 || ||00 V| + TS| || ||αα W T

4 5

U| || V| || W

Example 4. Shell Behaviour

R|N ||N N S|M ||M |TM

x y xy x y xy

U| LMD || MMD V| = MMDD || MMD |W MND

1

D2

D4

D7

D11

D16

2

D3

D5

D8

D12

D17

4

D5

D6

D9

D13

D18

7

D8

D9

D10

D14

D19

11

D12

D13

D14

D15

D 20

16

D17

D18

D19

D 20

D 21

OPR|R|ε PP||||εΓ PP|SSΨ PP||||Ψ PQ||T||TΨ

x y xy x y xy

U| R|ε || ||ε Γ V| − |SΨ || ||Ψ |W |TΨ

xo yo xyo xo yo xyo

U|U| R|N |||| ||N N V||V| + S|M |||| ||M |W|W |TM

xo yo xyo xo yo xyo

where: N M D ε Γ

are the membrane stress resultants (forces per unit width). are the flexural stress resultants (moments per unit width). flexural and shear rigidities. membrane strains. flexural strains.

The initial strains due to a temperature rise T are:

136

U| || V| || |W

Joint Material Properties

R|ε |ε RSΨ UV = |SΓ Tε W |Ψ ||Ψ |TΨ

xo yo

ot

xyo

ot

xo yo xyo

U| Rα || ||α | V| − TS|α0 || ||0 |W |T0

1 2 3

U| R|0 || ||0 V| + ∂∂Tz S|α0 || ||α |W |Tα

4 5 6

U| || V| || |W

Joint Material Properties Joint material models are used in conjunction with joint elements to define the material properties for linear and nonlinear joint models. See Joint Element Meshes for information about using joints. Six joint models are available: Linear Joint Models q Spring stiffness only corresponding to each local freedom. These local directions are defined for each joint element in the LUSAS Element Library. q General Properties full joint properties of spring stiffness, mass, coefficient of linear expansion and damping factor. Non-Linear Joint Models q Elasto-Plastic uniform tension and compression with isotropic hardening. Equal tension and compression yield conditions are assumed. q Elasto-Plastic General with isotropic hardening. Unequal tension and compression yield conditions are assumed. q Smooth Contact with an initial gap. See note below. q Frictional Contact with an initial gap. See note below.

137

Chapter 5 Model Attributes + : tension - : compression

F +κ strain hardening stiffness K - elastic spring stiffness

+Yield force

ε = δ2 - δ1 -Yield force

-κ strain hardening stiffness

Elasto-Plastic Joint Models F

κ - lift-off stiffness

Lift-off force ε = δ2 - δ1 ψ = θ2 - θ1

Gap

K - contact spring stiffness

Smooth Contact

138

Joint Material Properties Fy or Fz Fo

Fx

Kcy or Kcz - contact spring stiffness

Gap

γ xy = δ y2 - δ y1 or

γ xz = δ z2 - δ z1

ε xx = δ x2 - δ x1 K - contact spring stiffness

-Fo

Fo µ - coeff. of friction

Fx

Frictional Contact Notes q Initial gaps are measured in units of length for translational freedoms and in radians for rotational freedoms. q Smooth Contact: If an initial gap is used in a spring, then the positive local axis for this spring must go from node 1 to 2. If nodes 1 and 2 are coincident the relative displacement of the nodes in a local direction (δ2- δ1) must be negative to close an initial gap in that direction. q Frictional Contact: If an initial gap is used in a spring, then the positive local x axis for this spring must go from node 1 to 2. If nodes 1 and 2 are coincident the relative displacement of the nodes in the local x direction (δx2δx1) must be negative to close an initial gap.

139

Chapter 5 Model Attributes

Rubber Material Rubber materials maintain a σ Hard Rubber linear relationship between stress and strain up to very large strains (typically 0.1 0.2). The behaviour after the proportional limit is exceeded depends on the Soft Rubber type of rubber (see diagram below). Some kinds of soft rubber continue to stretch enormously without failure. ε The material eventually offers increasing resistance to the load, however, and the stress-strain curve turns markedly upward prior to failure. Rubber is, therefore, an exceptional material in that it remains elastic far beyond the proportional limit. Rubber materials are also practically incompressible, that is, they retain their original volume under deformation. This is equivalent to specifying a Poisson's ratio approaching 0.5.

Rubber Material Models The strain measure used in LUSAS to model rubber deformation is termed a stretch and is measured in general terms as: λ = dnew/dold where: • dnew is the current length of a fibre • dold is the original length of a fibre Several representations of the mechanical behaviour for hyper-elastic or rubber-like materials can be used for practical applications. Within LUSAS, the usual way of defining hyper-elasticity, i.e. to associate the hyper-elastic material to the existence of a strain energy function that represents this material, is employed. There are currently four rubber material models available: q q q q

Ogden Mooney-Rivlin Neo-Hookean Hencky

140

Rubber Material The rubber constants (used for Ogden, Mooney-Rivlin and Neo-Hookean) are obtained from experimental testing or may be estimated from a stress-strain curve for the material together with a subsequent curve fitting exercise. The Neo-Hookean and Mooney-Rivlin material models can be regarded as special cases of the more general Ogden material model. In LUSAS these models can be reformulated in terms of the Ogden model. The strain energy functions used in these models includes both the deviatoric and volumetric parts and are, therefore, suitable to analyse rubber materials where some degree of compressibility is allowed. To enforce strict incompressibility (where the volumetric ratio equals unity), the bulk modulus tends to infinity and the resulting strain energy function only represents the deviatoric portion. This is particularly useful when the material is applied in plane stress problems where full incompressibility is assumed. However, such an assumption cannot be used in plane strain or 3D analyses because numerical difficulties can occur if a very high bulk modulus is used. In these cases, a small compressibility is mandatory but this should not cause concern since only near-incompressibility needs to be ensured for most of the rubber-like materials. Using Rubber Material Rubber is applicable for use with the following element types at present: q 2D Continuum q 3D Continuum q 2D Membrane

QPM4M, QPN4M HX8M BXM2

Notes

1. For membrane and plane stress analyses, the bulk modulus is ignored because the formulation assumes full incompressibility. The bulk modulus has to be specified if any other 2D or 3D continuum element is used. 2. Ogden, Mooney-Rivlin and Neo-Hookean material models must be run with geometric nonlinearity using either the total Lagrangian formulation (for membrane elements) or the co-rotational formulation (for continuum elements). The Hencky material model is only available for continuum elements and must be run using the co-rotational formulation. The large strain formulation is required in order to include the incompressibility constraints into the material definition. 3. Option 39 can be specified for smoothing of stresses. This is particularly useful when the rubber model is used to analyse highly compressed plane strain or 3D continuum problems where oscillatory stresses may result in a "patchwork quilt" stress pattern. This option averages the Gauss point stresses to obtain a mean value for the element.

141

Chapter 5 Model Attributes 4. When rubber materials are utilised, the value of det F or J (the volume ratio) is output at each Gauss point. The closeness of this value to 1.0 indicates the degree of incompressibility of the rubber model used. For totally incompressible materials J=1.0. However, this is difficult to obtain due to numerical problems when a very high bulk modulus is introduced for plane strain and 3D analyses. Subsequent selection of state variables for displaying will include the variable PL1 which corresponds to the volume ratio. 5. Rubber material models are not applicable for use with the axisymmetric solid element QAX4M since this element does not support the co-rotational geometric nonlinear formulation. The use of total Lagrangian would not be advised as an alternative. 6. There are no associated triangular, tetrahedral or pentahedral elements for use with the rubber material models at present. 7. The rubber material models are inherently nonlinear and, hence, must be used in conjunction with the NONLINEAR CONTROL command. 8. The rubber material models may be used in conjunction with any of the other LUSAS material models. However, it is not possible to combine rubber with any other nonlinear material model within the same material dataset.

Volumetric Crushing Material Material behaviour can generally be described in terms of deviatoric and volumetric behaviour which combine to give the overall material response. The crushable foam material model accounts for both of these responses. The model defines the volumetric behaviour of the material by means of a piece-wise linear curve of pressure versus the logarithm of relative volume. An example of such a curve is shown in the diagram below, where relative volume is denoted by V/V0.

142

Volumetric Crushing Material

Compression

pressure

K - Bulk modulus

-ln (V/V0)

K - Bulk modulus Tension

cut-off pressure Pressure - Logarithm of Relative Volume Curve From this figure, it can also be seen that the material model permits two different unloading characteristics volumetrically. q Unloading may be in a nonlinear elastic manner in which loading and unloading take place along the same nonlinear curve. q Volumetric crushing may be included (by clicking in the Volumetric crushing check box) in which case unloading takes place along a straight line defined by the unloading/tensile bulk modulus K which is, in general, different from the initial compressive bulk modulus defined by the initial slope of the curve. In both cases, however, there is a maximum (or cut-off) tensile stress, (cut-off pressure), that is employed to limit the amount of stress the material may sustain in tension. The deviatoric behaviour of the material is assumed to be elastic-perfectly plastic. The plasticity is governed by a yield criterion that is dependent upon the volumetric pressure (compared with the classical von Mises yield stress dependency on equivalent plastic strain) and is defined as: τ 2 = a 0 − a 1p + a 2 p 2 where p is the volumetric pressure, τ is the deviatoric stress and a0, a1, a2 are pressure dependent yield stress constants. Note that, if a1 = a2 = 0 and a0 = (syld2)/3, then classical von Mises yield criterion is obtained. 143

Chapter 5 Model Attributes

τ hyperbolic a2>0

parabolic a2=0

elliptic a2 Loading > Options dialog. If the multiple intersection check is suppressed, then the first intersection found will be used.

157

Chapter 5 Model Attributes Note. The distribution of load to the nodes follows the shape functions of the particular element. In quadratic elements, this distribution can appear at first unlikely. For example, a unit positive load at the centre of an 8-noded quadratic element, results in negative 0.25 loads at the corners and positive 0.5 loads at the mid-side nodes.

Discrete Load Types A discrete load consists of coordinates defining the vertices (different to geometric Points defining the model), load intensity and local x, y and z position. Any geometric Points selected when the Discrete loading dialog is initiated are entered as coordinates. Discrete load types available are Point load and Patch load: q Point Load Defines a general set of discrete y point loads in 3D space. Each individual point load can have a x P1 P1 separate load value. This example uses 16 distinct load values. The loads are applied to the model as distinct point loads. q Patch Load 2 or 3 points define a continuous line load in 3D space. 2 points give a straight line load and 3 points give a curved line load. A set of 4 or 8 points define a general patch load in 3D space. A patch consisting of 4 points will give a straight-sided patch, while 8 points define a patch with curved sides. The following examples show patch loads assigned to Point 1: Example 1. A line load defined using a 2 point line. The local origin of the line is assigned to Point 1.

Example 2. A curved line load. The curved load is defined using a 3 point line. The local origin of the line is assigned to Point 1.

p2

p3 p2

p1

p1 x3,y3

x2,y2 x2,y2 x1,y1

x1,y1

P1 y

P1

x1,y1,p1

x2,y2,p2

y P1

P1

x

158

x1,y1,p1 x

x3,y3,p3 x2,y2,p2

Defining Discrete Loads Example 3. A standard 4 point patch load. The local origin of the patch is assigned to Point 1.

Example 4. A fully curved 8 point patch load. The local origin of the patch is assigned to Point 1.

P3 P4

P5 P4

P6

P7

P2

P7 P1

P3

P2

P1 x2,y2

x3,y3 x7,y7 x8,y8

x4,y4 x1,y1

x1,y1 P1

P1 y

x2,y2

x4,y4,p4

x7,y7,p7

x3,y3,p3 y

x1,y1,p1

x2,y2,p2

x1,y1,p1

x

x6,y6,p6

x5,y5,p5

x2,y2,p2

x3,y3,p3

x8,y8,p8

x4,y4,p4

x

P1

P1

Defining Discrete Loads Coordinates and magnitude Point coordinates The coordinates defining the discrete loads. Any Points selected before initiating the Discrete load dialog, are entered as point coordinates in the discrete load dialog. q For a point load multiple points can be defined on one dataset. q For patch loads (2, 3, 4 or 8) points define the vertices for a single patch. Load magnitude Specified at each given coordinate set of the general point or patch load, therefore the load may be varied over features.

159

Chapter 5 Model Attributes

Load Projection Load Projection Vector (untransformed) defines the direction in which the load is projected onto the model. The vector is expressed by inputting an X Y and Z component. For patch loads this direction is always perpendicular to the patch. Discrete loads are not restricted to elements lying in the XY plane. For Point and Patch Load types, options are available for the application of the load in the global X, Y or Z axes directions, or normal to the surface onto which the load is projected. This example shows a typical 3D patch load where the patch is defined in space and projected onto the model. Load Direction (untransformed) defines the direction of the loads in the patch before any transformation is carried out at the assignment stage. Options are: Global X, Y and Z and Surface Normal.

Untransformed Load Global Z or Surface Normal

Untransformed Load Global X

In this example, loads Z are projected onto a Y model normal to the X patch definition. The patch normal direction is denoted by a double-headed arrow on a visualised patch. The direction of the load applied to the model is defined using the Untransformed Load Direction options form on the define load dialog.

160

Assigning Discrete Loads

Assigning Discrete Loads Discrete loads are independent of features therefore their application can be more flexible. The load assignment parameters are explained below: Patch Transformation Changes the patch orientation. For example, a patch load may be skewed by applying a rotation transformation dataset when assigning the load. In the example shown right the Point load defined about local xy axes is assigned to Point 1 subject to a patch direction transformation using a 30 degree xy rotation about the global origin. Note that the local origin of the patch load is rotated and repositioned as well as the patch itself.

Y X P1 y

x

To rotate a patch about its centre, define the patch with its local origin at its centre. Load Transformation Changes the load orientation from the (untransformed) direction given in the load definition. The transformation applies to the direction of the individual load components rather that to the patch as a whole. For example, it can be used to model breaking loads on a 3D model that have horizontal and vertical components by specifying a transformation that will rotate the loads out of the vertical direction and into an inclined plane in the direction of vehicle travel. Search Area A search area restricts loading to a specified portion of the model. If a search area is not specified, the load is projected onto the active model. For 2D models it is usually acceptable to default to the whole model, but for 3D models where multiple intersections of the load projection onto the model may occur it is safer to restrict the loading to the required face using a search area. In either case the time taken to assemble the loads is significantly improved by using a search area to restrict the number of elements tested for intersection with the load. Options for Processing Loads Outside Search Area The load outside the search area can be moved into the search area along the paths defined by the local x and y axes of the loading patch. The patch load is transferred into the path along the projected directions and added to the first loading positions found inside the patch in the projected direction. Options are: q Exclude All Load (default) q Include Local X Projected Load

161

Chapter 5 Model Attributes q Include Local Y Projected Load q Include Local X and Y Projected Loads q Include Non-Projected Load q Include Full Local X Load q Include Full Local Y Load q Include Full Load The load cannot be moved if the entire patch load lies outside the search area. Loads inside the search area are not moved. See Processing Loads Outside the Search Area below for explanations of each option. Number of Divisions in Discrete Extrapolated Load Local X and Y Direction Patch Concentrated Extrapolation Loads Nodal Loads Path specifies the numbers of divisions in the local x and y directions of the patch being assigned. The divisions are used to split the applied patch into individual component loads before they are in turn used to calculate equivalent nodal loads on the model. By default, 10 divisions are used in the local x direction and the aspect ratio of the patch is used to calculate the divisions in the y direction. At least one division should be used per element division. The more individual loads a patch is split into, the more accurate the solution obtained. Equivalent weighting values are used to calculate the portion of each discrete load that is applied to each corner of the element that it lies within. The load is then applied as Concentrated Loads. These weighting values are based on element shape functions and may vary with element type. Note. Discrete patch loads are not work equivalent as the discrete points are simply lumped at the nearest node. Load Case specifies which loadcase the loading is to be applied. Load cases can themselves be manipulated. See Load Case Management for more details. Load Factor specifies a factor by which the loading is multiplied before the equivalent nodal loads are calculated.

162

Assigning Discrete Loads Case Study. Hydrostatic Loading In this example, a Patch Load will be used to apply a hydrostatic load to the sidewall of an underground box culvert. 1.

Assuming the box culvert wall is defined using a Surface in the global XZ plane with corners at coordinates (0,0,0), (5,0,0), (5,0,3) and (0,0,3). Define a Surface using the New Surface button at the specified corner positions.

2.

Rotate the view until the Surface can be visualised using the Dynamic Rotate button. 3. Using Attributes > Mesh > Surface, define a mesh using Thick Shell, Quadrilateral, Linear elements. Specify the spacing as 15 divisions in the local x and 9 divisions in the local y directions. Since only one Surface is present in the model, the divisions for the mesh can be entered directly onto the Surface mesh dialog. 4. 5.

6.

7.

8.

With the cursor in normal mode assign the mesh to the Surface by dragging the attribute from the Treeview onto the selected Surface. To define a patch load that is coincident with the side-wall Surface, first select the four Points defining the Surface in the order they were defined. Choose the menu command Attributes > Loading > Discrete, click on the Patch tab, then specify a 4 node patch. Notice that LUSAS has filled the Point coordinates into the patch coordinates. The load direction coincides with the global Y axis direction so select Y from the Untransformed Load Direction. Specify patch corner load intensity values of -3, -3, -1, -1 respectively. The defined patch uses a local coordinate system that is coincident to the global Cartesian axis system, so it can be assigned to the Point at the origin (Point 1). Assign the load to Point 1 (0,0,0). The Assign Leave all dialog entries as default. Press OK to assign the load. Visualise the loading by selecting the loading attribute in the Treeview, clicking the right mouse button, then choosing Visualise from the shortcut menu. Note that the patch is drawn as discrete point loads. This is because the patch load is automatically split into point loads by LUSAS. . The number of discrete loads in each direction is dependent on the numbers of divisions entered in the Assign Loading dialog. In this case, the default number of divisions (4) is insufficient as there are insufficient loads to apply at least one load per element along the culvert. To improve the load application accuracy, deassign the load from the Point, and reassign using 15 divisions in the local X direction. Leave the Y divisions field blank. Draw the load again. Note that

163

Chapter 5 Model Attributes LUSAS has automatically used the aspect ratio of the patch load to calculate a suitable number of divisions in local Y. Hydrostatic Patch Load

Hydrostatic Patch Load

Default number of divisions showing insufficient discrete point loads.

Increased number of divisions on assignment. The double arrow vector indicating patch orientation.

Processing Loads Outside the Search Area Patch loads outside the search area are lumped onto the nearest edge of the search area

164

Assigning Discrete Loads

Patch Load Divisions

Patch Load Local Coordinates

Number of divisions in local x (div x) and y (div y) are specified at load assignment. The load intensity is then split into individual load components with an associated area of application.

The local coordinate set is dependent on the order in which the coordinates of the patch vertices are defined.

7

6

4

5

3 Y

Search Area Boundary

8

X

4

1

2

7 1

Area of individual Load Application

2

3

5

Y

8

Patch Load divs x = 6 divs y = 5

6

4 X

1

2

3

Local X Projected Load

Local Y Projected Load

Loads in the local y projected region (dark area) are lumped at nearest loading positions within the search area (light area).

Loads in the local y projected region (dark area) are lumped at nearest loading positions within the search area (light area).

Search Area

Search Area

Patch load

Patch load

165

Chapter 5 Model Attributes

Local X and Y Projected Loads

Non-Projected Load

Loads in the local x and y projected regions (dark area) are lumped at nearest loading positions within the search area (light area).

Loads not in the local x and y projected regions (dark area) are lumped at nearest loading positions within the search area (light area).

Search Area

Search Area

Patch load

Patch load

Full Local X Load

Full Local Y Load

Loads in the full local x region of the patch (dark area) are lumped at nearest loading positions within the search area (light area).

Loads in the full local y region of the patch (dark area) are lumped at nearest loading positions within the search area (light area).

Search Area

Search Area

Patch load

Patch load

166

Local Coordinate Systems

Full Load

All patch loads lying outside the search area (dark area) are lumped at nearest loading positions within the search area (light area). Search Area

Patch load

Local Coordinate Systems Local Coordinate datasets define coordinate systems that differ from the default global Cartesian system. Local coordinate datasets are attributes and are defined from the Attributes menu. Local coordinate datasets have several uses in LUSAS: q Feature Definition They may be defined and activated in order to enter coordinates in place of the global coordinate system. When a local coordinate P3 (19,10.5,0) system is active, all Local Coordinate 1 Global Origin at P2, dialog entries Local X at P3 relating to global X, P4 (5,0,0) Local 1 y Y and Z coordinate Y x input use the z Z X transformed axis set P2 (10,3,0) P1 (0,0,0) Global as a basis for input. Global

q Transforming Nodal Freedoms When assigned to features the effect is to transform the degrees of freedom on the mesh on those features. This has the effect of transforming the directions of applied global load and support conditions. In this example, global freedoms are transformed to radial directions by assigning a cylindrical coordinate set to the Lines around the hole. This method of transforming nodal freedoms is only valid for small deflections, since the freedom 167

Chapter 5 Model Attributes directions are not updated during analysis. q Material assignments A local coordinate set may be used to align orthotropic and anisotropic materials. q Variations Thermal variations may be defined using functions in terms of the coordinate variations. The function may use alternative coordinates if a local coordinate set is specified in the variation. q Composites A local coordinate set may be used to align material directions in composite layups. q Element Orientation May be used at the mesh assignment stage to orient beam and joint elements. q Results Transformation When activated during post-processing results can be output relative to the local coordinate set. For example, this is useful when looking at local results for triangular elements, where axes are not consistent.

Local Coordinate System Types Three main types of local coordinate systems can be defined. All three types are defined in LUSAS by indicating three positions in space defining a local Cartesian xy plane (origin, x axis, xy plane). The type of coordinate set selected will dictate how features are defined based on the specified plane. q Cartesian Based on standard X, Y and Z coordinates arbitrarily oriented in space. Y3 q Cylindrical Based on the axes of a cylinder - radius, angle subtended and distance along the cylinder t axis. 1 P1 Coordinates of a Point are specified as (r, t, z), where 2 r is the radius perpendicular to the X local z axis; Z z r

t is the angle in degrees measured from the positive x direction of the local xz plane, clockwise about the local z axis when looking in the positive z direction; z is the distance along the z axis.

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Local Coordinate Systems q Spherical Based on the axes of a sphere defined by a radius, subtended tangential angle and subtended angle around a meridian. Note that there is no equivalent spherical set in the LUSAS Solver so nodal freedoms cannot be transformed using this method. Coordinates of a Point are specified as (r, t, c), where:

Y

3 t

c 1 P1

2 X

Z

r

r is the radius of the sphere on which the Point lies from the local origin; t is the angle in degrees measured from the positive x direction of the local xz plane, clockwise about the local z axis when looking in the positive z direction; c is the angle in degrees measured from the positive z axis to the radius line

Defining Local Coordinate Systems Local coordinate datasets are defined from the Attributes Menu. Local coordinate sets are defined by specifying an origin and either a rotation about a global plane, rotation matrix or a scale factor (Cartesian only). They may also be defined by first selecting 3 Points (to specify 3 required positions), then selecting Attributes > Local Coordinate. Note. Defining a new coordinate set does not automatically make it the active set, see Using Local Coordinate Sets below. q Rotation about a global plane specifying angular rotations about the global planes, XY, YZ or XZ. When defining coordinate systems using this method, the local x axis is oriented parallel to the global X axis and rotated into position using the specified angle in the specified plane. q Scale Factors specifying scale factors in all three global directions about a specified origin. This method is only used to define local Cartesian coordinate sets. Scale factor local coordinate sets can be used to scale a model or portions of a model. See the case study below for more details. q Rotation matrix specifying a direction cosine matrix. q Rotation matrix from selected Points using a Cartesian set generated from selected Points. Three Points define a cartesian set anywhere in space, two Points define a Cartesian set with the XY plane parallel to the global XY plane. 1st Point defines the origin, 2nd Point defines the positive direction of

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Chapter 5 Model Attributes the local x axis, 3rd Point defines the local xy plane. Note that the Cartesian set generated from Points can be changed to a cylindrical or spherical set.

Using Local Coordinate Sets There are three ways of using a local coordinate set: q By setting it as the active coordinate set. In order to use local coordinate systems for entering coordinate values, the active coordinate set must be changed from global Cartesian by selecting a local coordinate dataset from the geometry tab of the model Properties box. q From the geometry by coordinates dialogs. When defining Points, Lines, Surfaces or Volumes by coordinates a different coordinate set to the active one may be used. An option also exists to set the active set from the by coordinates dialogs. q By assigning a local coordinate set to geometry. To transform degrees of freedom, (as described above), the local coordinate set must be assigned to underlying geometry by dragging a saved dataset from the Treeview onto the geometric features. Warning. There is no equivalent spherical set in the LUSAS Solver, therefore freedoms cannot be transformed using this type of local coordinate system.

Visualising Coordinate sets The active coordinate set is visualised on the graphics area by default, this can be switched off from the Window properties. To display the Window properties doubleclick in the current window away from any features (i.e. in the space around the model). Click on the View Axes tab to alter change the view axes settings. Local coordinate sets assigned to features may be visualised in the same way as all attributes. Case Study. Scaling a Model Local coordinate systems defined by scale factors can be used to shrink or enlarge a model. This case study will scale a model defined in mm units to metre units. Follow the procedure outlined below: 1. Load the model to be scaled. Next, create a command file using Files > Save as. Choose Command file from the Save as type, specify a suitable file name, for example fullscale.cmd. 2. Start a new model using File > New.

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Composites 3. Define a local Cartesian coordinate set using Attributes > Local Coordinate. Select the Scale option. Enter scale factors in X, Y and Z of 1e-3 to scale from mm to metres. A scaling origin can be entered if any position other than the global Cartesian origin is required. OK the dialog. 4. Set the scaling local coordinate set as the active coordinate set from the Geometry tab of the model properties. (From the Treeview select the top level branch in the layers tab, click the RH mouse button and select Properties from the shortcut menu. Click on the Geometry tab.) OK the dialog. 5. Finally, read in the command file using File > Command File > Open, specifying the file in which the model was saved in step 1 above, for example fullscale.cmd. As the command file replays, the model features are redefined at scaled coordinates. Caution. The method outlined in the case study above should not be used if a local coordinate system is already active in the model. If this is the case, it is better to define a scale transformation and then move all the Points in the model using the move command. Higher order features will be updated also.

Composites Composite datasets are used to represent the material characteristics that are formed by applying layers of differing materials in varying orientations and thicknesses.

Using Composite Datasets Composite lay-up properties are defined from the Attributes Menu. Previously defined material datasets are combined in a composite property dataset, specifying the layer name, the relative thickness and the relative angle for each layer. 171

Chapter 5 Model Attributes The composite dataset must then be assigned to Surfaces or Volumes, specifying the overall composite orientation. Options for orientation are: Local Coordinate Set, Local Element Axes, Axes From Support Surface. Note that an angle of 0° aligns with the appropriate x axis and an angle of 90° with the y axis. Tips q Material directions may be used when viewing the results to display stresses on or off axis. q Only Isotropic and Orthotropic materials can be used in composite lay-ups. q Orthotropic plane stress and orthotropic solid materials can be used for shell composite lay-ups, but solid composites must use the orthotropic or anisotropic solid material model. Isotropic materials may additionally be used in either. q For shell elements an appropriate plane stress material model must be used while for solid elements a 3D continuum model should be used (see the LUSAS Element Library). q The lay-up sequence is from bottom to top. In the case of a shell this will be in the direction of the Surface normal. In the case of a solid this will be in the direction of the local z. q In the case of composites assigned to Volumes, nodal positions may be moved by LUSAS to correspond with layer positioning/thicknesses. q In the case of composites assigned to Volumes, the number of layers must correspond with the number of elements through the Volume. q Composite datasets can only be used with Surfaces and Volumes that use composite elements. q Composite datasets may not include materials that contain variations. q The layer thicknesses are relative and determine the proportion of the total thickness (specified in geometric properties) apportioned to each layer. Composite properties for Surface models can also be defined using the CACE-Drape system.

Composite Material Visualisation 1. When defining a composite layup click on the Visualise button to display a layered representation of the composite with annotations of layer orientation angle and material dataset number automatically included. This representation may also be annotated to the screen if desired, to do so click on the Create Annotation button. 2. Once assigned to Surface or Volume features which have a mesh assigned, the composite material definition can be displayed in a number of ways using the Attributes layer . 172

Slidelines • Material Axes Material directions can be plotted in the form of an axes set at any layer within a composite. For solids the axes set is placed at the top/bottom or middle of the selected layer, for shells the axes set is placed on the centroid of the shell element. If no layer is specified (i.e. layer 0) the zero degrees fibre direction of the composite is drawn at the element centroid. If a layer is specified the axes will only be drawn for elements with orthotropic or anisotropic material property types. If an element has a material property in which material directions are not valid, a warning message is output. • Layer Element/layer material directions and element layers can be drawn in both pre- and post-processing models. 3.

Note. Good use can be made of the LUSAS cycling facility, where feature local axes can be cycled relative to a reference feature to ensure a consistent set of composite material axes.

Composite Post-Processing It is often useful to change Results orientation to material directions to view the results from a composite analysis. For shells, plates, membrane and solid elements stresses and stress resultants can be displayed and printed in the element local and element material directions. See Local and Global Results for more details. When setting a results layer, LUSAS accepts the layer name in addition to the layer number. q Layer stresses are output by requesting output at Gauss points. All layers will then be output. q To get stresses for a shell layer, the layer must be set and results set to ‘stresses’. If the results are set to Top/Middle/Bottom, then the results for the T/M/B of the element will be obtained, and the layer is not relevant. q A layer of a solid composite element will act like a shell. Thus both stress resultants and T/M/B stresses can be obtained.

Slidelines Slidelines are attributes which are used to model contact and impact problems, or to tie dissimilar meshes together. They can be an alternative to joint elements or constraint equations, and have advantages when there is no exact prior knowledge of the contact process. Their applications range from projectile impact, vehicle crash worthiness, the containment of failed components such as turbine blades, to interference fits, rock joints and bolt/plate connections. Note. The slideline facility is inherently nonlinear, except for tied slidelines used in an implicit dynamic or static analysis where the solution may be linear.

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Chapter 5 Model Attributes Slidelines allow the definition of properties such as the stiffness scale factor, friction coefficient and pre-contact. They are assigned to corresponding pairs or groups of features, known as master and slave. This diagram shows a contact application using a slideline with friction between two bodies and a tied slideline to join dissimilar meshes without the need for stepped mesh refinement.

Tied slideline

Friction slideline Tied slideline

Slideline properties are defined from the Attributes > Slideline menu entry.

Slideline Types There are several different types of slide: q Null used to perform a straightforward linear analysis ignoring the slide definition. Useful for performing a preliminary check on the model. q No Friction used for contact analyses but ignores friction between the two surfaces. q Friction used for constant or intermittent contact or impact. q Tied used to tie together two dissimilar meshes. The tied slideline option eliminates the requirement of a transition zone in mesh discretisation comprising differing degrees of refinement and is extremely useful in creating a highly localised mesh in the region of high stress gradients. q Sliding used for problems where surfaces are kept in contact, allowing sliding without friction. The general slideline options may be utilised for modelling finite relative deformations of colliding solids in two or three dimensions which involve sliding (with or without friction) and constant or intermittent contact conditions. The sliding only option is similar to the general sliding options but does not permit intermittent contact conditions. The figure below shows the use of both the general and the tied slideline options.

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Slidelines

Slideline Properties q Master/Slave interface stiffness scale factor Controls the amount of interpenetration between the two slidelines. Increasing the stiffness scales will decrease the amount of penetration between the slides, but may cause illconditioning. Recommended values: Implicit/static solution 1.0 Explicit solution 0.1 Tied slidelines 100 to 1000 The scale factor should be increased slightly for slidelines involving rigid wall contact. Note. Scaling of the slideline stiffnesses is automatically invoked at the beginning of each analysis if the ratio of the average stiffness values for each constituent slideline differ by a factor greater than 100. In this manner account is made for bodies having significantly different material properties. Option 185 will suppress this facility. q Coulomb friction coefficient Only applicable for friction slidelines. q Zonal contact detection parameter Controls the extent of the contact detection test. The zonal detection distance for a slideline is taken as the product of the detection parameter and the longest segment found on both slidelines. If the zonal contact detection parameter is less than 0.5, undetected material interpenetration may occur. To apply the refined contact detection test for every contact node, the parameter should be set to a large number. For further information refer to the LUSAS Theory Manual. Explicit solution schemes (default = 5/9) Implicit/static solution schemes (default = 10/9) q Slideline extension parameter Slideline extensions are continuations of a slideline segment beyond that of the original definition. The extension eliminates interpenetration for slidelines which are significantly irregular. q Close contact detection parameter The close contact detection parameter is used to check if a node is threatening to contact a slideline. The surface tolerance used is the product of the detection parameter and the length of the surface segment where the node is threatening to penetrate. If the distance between the surfaces is less than the surface tolerance a spring is included at that point just prior to contact. By default, the stiffness of the spring is taken as 1/1000 of the surface stiffness. The inclusion of springs in this manner helps to stabilise the solution algorithm when surface nodes come in and out of contact during the iteration process. The close contact detection facility is not available for explicit dynamics.

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Chapter 5 Model Attributes Temperature Dependency Selecting the Temperature dependent check box enables a reference temperature to be attached to each slideline property allowing multiple properties to be used in a slide table to provide temperature dependence. If temperature dependent properties are specified linear interpolation is applied to the interface stiffness scale factors and the friction coefficient. The zonal contact detection parameter and the slideline extension distance are unchanged when interpolation of temperature dependent properties is carried out. Pre-contact Parameter Pre-contact is used to overcome problems encountered when applying an initial load to a discrete body which would be subjected to unrestrained rigid body motion. This procedure is only applicable to static analyses and it is required when an initial gap exists between the slidelines and a loading is to be applied (other than Prescribed Displacement). 1 The surfaces of a slideline are initially P brought into contact under the action of the applied loading and interface forces between the surfaces. This 2 allows the surfaces of a slideline to be defined with a gap between them and an automatic procedure is invoked to P bring the bodies into contact to avoid unrestrained rigid body motion. The interface forces which bring the bodies together act at right angles to each surface. One of the surfaces must be free to move as a rigid body and the direction of movement is dictated by the interface forces, applied loading and support conditions. In the example above pre-contact is defined for slideline 1 but not for slideline 2.

Warning. Incorrect use of this procedure could lead to initial straining in the bodies or to an undesirable starting configuration. By selecting specific slidelines for the pre-contact process (i.e. slidelines where initial contact is expected) minimum initial straining will occur and more control over the direction of rigid body movement can be exercised. Loadcase Title Specification of the loadcase allows the slide properties and type to be changed between load increments. The slideline properties need only be defined for loadcases where there is a change in the definition. For example, if the slideline stiffness changes on loadcase 4 then slidelines need only be defined for loadcases 1 and 4, loadcases 2 and 3 taking the properties of loadcase 1.

176

Slidelines

Assigning Slidelines Slides are defined by assigning the slideline dataset to the required Lines or Surfaces and specifying whether they are to be treated as Master or Slave slides. In general, the smaller of the two contact areas should be the slave. Slideline Modelling Considerations q Only the expected region of contact should be defined as a slideline surface for tied slideline analyses. q Coarse mesh discretisation in the region of contact should be avoided q Slides must be continuous and should not subtend an angle greater than 90 degrees. Sharp corners are best described by two separate slides. q Rigid target surfaces may be modelled by fully restraining the slideline surface q The surface scale factors and friction coefficient are not utilised in explicit tied slideline analyses q The nodal constraint slideline (explicit tied slideline) treatment is more robust if the mesh with the greatest contact node density is designated the slave surface. q The use of a larger value of Young's modulus to simulate a rigid surface in a dynamic contact analysis is not advisable since this will increase the wave speed in that part of the model and give rise to a reduced time step. This practice significantly increases the computing time required. q Do not converge on residual norm with PDSP loading. This norm uses external forces to normalise which do not exist with PDSP loading q The use of tied slidelines to eliminate transition meshes is recommended for areas removed from the point of interest in the structure. q Slidelines may be used with automatic solution procedures (constant load level and arc-length methods). The line search and the step reduction algorithms are also applicable to analyses that contain slidelines. Using Slidelines with LUSAS Elements The slideline facility may only be used with the following elements: Element Type

Element Name

Plane Stress

TPM3, QPM4,

QPM4M,

TPM3E, QPM4E

Plane Strain

TPN3, QPN4,

QPN4M,

TPN3E, QPN4E

Axisymmetric

TAX3, QAX4,

QAX4M,

TAX3E, QAX4E

Shell

TTS3, QTS4

Solid

TH4, PN6, HX8, HX8M,

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TH4E, PN6E, HX8E

Chapter 5 Model Attributes q When defining slidelines for use in implicit dynamics or static analyses linear continuum elements are recommended. q Slidelines may be utilised with higher order elements (quadratic variation of displacements). However, this is generally not recommended because it is necessary to constrain the displacements of the slideline nodes so that they behave in a linear manner (LUSAS Modeller will do this automatically).

Slideline Options Options relating to slidelines are set from the Attributes tab of the Model Properties, File menu. Case Study. Metal Forming Analysis Initial configuration.

Deformed configuration.

Processing Slideline Results The results from analyses involving slidelines may be processed in the same manner as other problems. However, extra information is available concerning the performance of the slide and the results on the interface.

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Constraint Equations Viewing Slideline Results LUSAS automatically creates a group of every slideline used in the analysis. To view results on the slideline alone make the whole model invisible (top level group) then make the slideline group visible. Use the right-click shortcut menu to make groups visible or invisible . Graph datasets can be generated containing the variation of slide variables through an analysis (time or load increments) using the Graph Wizard. Note. When looking at a deformed mesh plot of contact analysis results, the exaggeration factor should be set to unity to avoid a misleading visualisation.

Constraint Equations A constraint can be defined to constrain the movement of a geometric or nodal freedom. Constraint equations allow linear relationships between nodal freedoms to be set up. This facility allows the user to constrain plane surfaces to remain plane while they may translate and/or rotate in space. Similarly straight lines can be constrained to remain straight, and different parts of a model can be connected so as to behave as if connected by rigid links. These geometric constraints are only valid for small displacements. Constraint equations can also be used to model cyclic symmetry, for example a single blade from a complete rotor may be modelled and then constrained to behave as if it were part of the complete model. As constraint equations in LUSAS refer to transformed nodal freedoms, any local coordinate datasets assigned to the features are taken into account during tabulation when the constraint equations are assembled. Several different types of Constraint Equations can be defined from the Attributes menu, grouped under the following types (described below): q Displacement Control q Geometric q Cyclic q Tied Mesh

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Chapter 5 Model Attributes

Displacement Control q Specified Variable a nodal freedom takes a specified value across all the nodes in the assigned features. In this example, a specified variable constraint of Displacement in X direction with value 1.0 is assigned to Point 1. The underlying node is then allowed to displace only by the specified distance in the specified X direction. q Constant Variable used where a nodal freedom value is constant but unknown across all the nodes in the assigned features. In this example, a constant variable constraint of displacement in the X direction is assigned to Line 1. The underlying nodes move a constant amount in that direction. q Vector Path The nodes in the assigned features may be constrained to move along a specified vector defined by 2 Points or by 2 sets of X, Y and Z coordinates. In this example, vertical and horizontal vectors are used to restrict movement in those directions. Note that the vectors are used purely to define a direction. Nodes can travel along a vector in either direction.

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u=1.0

P1

u=C

L1

Constraint Equations

Geometric q Rigid Displacements The nodes in the assigned features may be constrained to be rigid, the group of nodes may translate and/or rotate but their positions relative to one another remain constant. Only translational displacements can be constrained using this type of constraint. This type of constraint is only valid for small displacements. Assigning a constraint of this type to Lines on either side of a gap, as in the example shown, maintains the underlying undeflected node positions relative to each other as if a rigid block were in place between the structures. q Rigid Links A rigid link will create a rigid fixity between two features. It is similar to the Rigid Displacements constraint type, except that rotational freedoms are also constrained to be rigid. In the example shown here, the end of a beam is rigidly linked to the shell edges around a cylinder. The plane containing beam and cylinder end will remain plane throughout the analysis. q Planar Surface A surface may be constrained to remain plane, the surface may translate and/or rotate but remains plane. Nodal positions may vary relative to other nodes on the surface. This type of constraint is only valid for small displacements. In this example, a planar Surface constraint is assigned to the top Surface to force the underlying nodes to remain planar during loading. Constrained nodes may move relative to each other as long as they remain in plane.

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Gap=δ

Gap=δ

Chapter 5 Model Attributes q Straight Line A straight line may be constrained to remain straight, the line may translate and/or rotate but will remain straight. Nodal positions may vary relative to other nodes along the line. This constraint type is only valid for small displacements. In the example shown, a straight Line constraint is assigned to Line 3 to force underlying nodes to remain in a straight line relative to each other during loading. Constrained nodes may move together or apart as long as they remain in a straight line.

L3

Cyclic q Cyclic Rotation Cyclic rotational symmetry may be used to model a section from a continuous ring. The mesh on the two planes of symmetry may be different. In the example shown, the radial Lines are defined as a Master and Slave pair maintaining cyclic symmetry around the structure. Meshes on the Master and Slave Lines need not match. q Cyclic Translation Cyclic translational symmetry may be used to model a section from a continuous strip. The mesh on the two planes of symmetry may be different. In the example shown here, Master and Slave Surfaces define start and finish positions of repeating sections. Meshes on Master and Slave need not match.

Master

Slave

Master

Slave

Tied Mesh q Tied Mesh Specified Tied meshes may be used to force two sets of assigned features to move together in a similar manner to tied slidelines. The meshes are tied along Master and Slave Lines to restrict relative movement. The mesh on the two sets of features need not match. A search direction vector is defined to limit the mesh to which

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Vector Slave

Master

Thermal Surfaces and Heat Transfer it is tied. A vector defines the direction in which the constraint is applied. q Tied Mesh Normal Meshes tied along Master and Slave Lines to restrict relative movement. The underlying nodes maintain their original relative positions under loading. Meshes on Master/Slave need not match. This form of tied mesh constraint uses a search direction normal to the Master/Slave surfaces to detect the mesh to which it is tied.

Slave

Master

Case Study. Using Constraint Equations Differing meshes may be constrained to displace together in a similar way to a tied slideline. 1. Define two Surfaces separated by a small gap using Geometry > Surface > Coordinates. 2. Mesh the Surfaces with Linear Plane Strain elements using different mesh spacing on each Surface using Attributes > Mesh > Surface. 3. Define and assign a valid Material to the Surfaces and define and assign Supports and Load datasets so that the Surfaces are being forced towards each other. 4. Define a normal tied mesh Constraint using Attributes > Constraint Equation > Tied Mesh. Assign it to the Lines on either side of the gap. One Line must be selected as a Master and the opposing Surface as a slave. If meshes on tied Lines have different spacing, choose the Line containing the finer mesh as the master. 5. Run the model through LUSAS and use the plot file to view the deformed mesh. The constraint equations will have prevented one surface from passing through the other.

Thermal Surfaces and Heat Transfer The thermal surface facility in LUSAS allows thermal gaps, contact and diffuse radiation to be modelled. Thermal surfaces are used to model the thermal interaction of two distinct bodies, or two different parts of the same body through a fluid medium. q Thermal Gaps are used to model gaps between structures that are relatively close together. q Contact is used in a thermo-mechanical coupled analysis where contact takes place and the contact pressure effects are then included in the analysis. 183

Chapter 5 Model Attributes q Diffuse Radiation is the process of heat transfer from a radiation surface to the environment or to another thermal surface defining the same radiation surface. Radiation is modelled by specifying radiative properties for thermal surfaces. Thermal Surfaces are the thermal equivalent of structural slidelines. They are defined from the Attributes menu, and are assigned and manipulated in the same way as all attributes, see Manipulating Attributes.

Heat Transfer Thermal Surfaces work in conjunction with Thermal Gaps, Radiation Surfaces and Surface Properties, defined from the Utilities > Heat Transfer menu. They are utilities meaning they can be used in the definition of thermal surfaces but not assigned to features. q Surface Properties Gap conductance, Contact conductance, Environment and Radiation. q Thermal Gaps Thermal gaps are used to model heat transfer across a gap and heat transfer by contact when a gap is deemed to have closed. If these effects are required, a thermal gap dataset must be specified during definition of a thermal surface. q Radiation Surfaces Diffuse radiation exchange may be modelled with a radiation surface that is defined by any number of thermal surfaces. Planes of symmetry that cut through the radiation enclosures may be defined so that it is not necessary to model the whole structure. Radiation surfaces allow for the calculation of diffuse view factors. These view factors may be output to a print file.

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Thermal Surfaces and Heat Transfer Radiation Surface Surface Properties Utilities > Heat Transfer > Surface Properties ...

Utilities > Heat Transfer > Radiation Surface... Controls heat transfer from body to body over large distances (where radiation is dominant).

Environment Heat transfer to the environment (convection and conduction).

Thermal Surface Attributes > Thermal Surface...

Radiation Heat transfer by radiation exchange.

Used to associated heat transfer properties with parts of the model.

Assign to features

Gap Heat transfer across a gap.

Contact Heat transfer between two contacting Surfaces (includes slideline contact pressure effects, so can only be used in a coupled analysis).

Thermal Gap Utilities > Heat Transfer > Thermal Gap... Controls heat transfer from body to body in close proximity (where conduction and convection are dominant).

The thermal surface definition process is shown in the above table. Environment and Radiation thermal properties are referred to directly in the thermal surface definition. Gap and contact thermal properties are used to define a thermal gap dataset which is then referred to in the thermal surface definition. A radiation surface pair, can be referenced from the thermal surface definition to control heat transfer from one body to another over large distances where radiation is dominant.

Choosing Thermal Properties The following flowchart guides the decision making process for choosing thermal properties. The process is simplified if the analysis only considers a single body, when only environmental thermal properties are required. For analyses where discrete (multiple) bodies are considered, factors such as body proximity and whether the bodies are touching, or are likely to touch during the analysis, become important and the choice of thermal properties changes. Follow a route through the flowchart below and define your thermal surfaces using the properties given in the shaded box.

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Chapter 5 Model Attributes Start Here

Is the structure made up of discrete bodies?

No

Thermal Surface + Thermal Properties (Environment) or Environmental Loading

No

Thermal Surface + Thermal Properties (Radiation)+ Radiation Surfaces

No

Thermal Surface + Thermal Properties (Gap) + Thermal Gap

Yes

Are the bodies close together (conduction and convection effects dominant)? Yes

Are the bodies touching (include pressure effects)? Yes

Thermal Surface + Thermal Properties (Gap + Contact) + Thermal Gap + Coupled Analysis

Environmental Nodes (LUSAS datafile) Environmental nodes may be used to represent the medium which separates the thermal surfaces between which heat is flowing. As the length of a link directly connecting two surfaces increases, the validity of the assumed flow becomes more tenuous. Alternatively, instead of forming a link, heat could flow directly to the surroundings, but in this case, the heat is lost from the solution. This, in some cases, is a poor approximation to reality, particularly when the thermal surfaces form an enclosure. In this instance an environmental node can be used to model the intervening medium, with all nodal areas which are not directly linked to other areas linked to the environmental node. The environmental node then re-distributes heat from the hotter surfaces of the enclosure to the cooler ones without defining the exact process of the transfer. 186

Processing Thermal Surface Results Note. Environmental nodes cannot be defined in LUSAS Modeller, and must be edited directly into the LUSAS data file if required. See the Data Syntax Manual, Thermal Surfaces for further information.

Radiation Options Radiation options are set from the Model Properties dialog. Available options are: q Temperatures Input and Output in Degrees Celsius [242] (Model properties, Attributes tab). Changes the temperature units from the default of Kelvin to degrees Celsius. q Suppress Recalculation of View Factors in Coupled Analysis [256] (Model properties, Solution tab, Thermal options). Turns on/off the view factor recalculation. The option should be turned off when the radiation surface geometry is unchanged by the structural analysis. This stops recalculation of the view factors.

Processing Thermal Surface Results The results from analyses involving thermal surfaces may be processed in the same manner as other problems. However, extra information is available concerning the performance of the thermal surface and the results on the interface. Viewing Thermal surface Results LUSAS automatically creates a group of every thermal surface used in the analysis. To view results on the thermal surface alone make the whole model invisible (top level group) then make the thermal surface group visible. Use the right-click shortcut menu to make groups visible or invisible . Graph datasets can be generated containing the variation of slide variables through an analysis (time or load increments) using the Graph Wizard.

Retained Freedoms Retained Freedoms are used to manually define the master freedoms for use in the following analyses: q Guyan reduction eigenvalue analysis q Superelement analysis Retained Freedom datasets are defined from the Attributes menu. They contain the definition of the master and slave degrees of freedom and are then assigned to the features with the master nodes. 187

Chapter 5 Model Attributes

Full Subspace Iteration

20 Masters

15 Masters

10 Masters

5 Masters

Damping Properties This facility is used to define the frequency dependent Rayleigh damping parameters for elements which contribute to the damping of the structure. Viscous (modal) and structural (hysteretic) damping can be specified. If no damping datasets are specified the properties are taken from the material properties (click on dynamic properties). Damping properties are usually required when distributed viscous and/or structural damping factors are required for Modal Damping control. A Modal damping analysis is performed as part of an eigenvalue analysis. Defining Damping Properties LUSAS structural or viscous damping properties are defined from the Attributes menu and assigned to features in the usual way. Mass and stiffness Rayleigh damping parameters are linked with the corresponding reference circular frequency value at which they apply in a damping properties dataset. If more than one set of damping values is defined in a damping table, linear interpolation is used to calculate damping values at intervening frequencies.

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Birth and Death

Birth and Death Birth and death enables the modelling of a staged construction process (e.g. tunnelling), whereby selected elements are activated and deactivated as the simulation process requires. Birth and death datasets are defined from the Attributes menu, and are assigned and manipulated in the same way as other attributes. If an element is deactivated, all stresses/strains are set to zero and the magnitude of the stiffness matrix is reduced so that the element has negligible effect on the behaviour of the residual structure. The main difference between deactivating an element and assigning very weak material properties to an element is the way in which the internal forces that may exist in that element are processed. Of course, if an element is deactivated from the outset there will be no difference but if deactivation occurs as the analysis progresses there will probably be internal stresses associated with the element. Percent to Redistribute The deactivate command provides control over the way in which these internal forces are processed by specifying how much of the internal forces should be redistributed: q Zero Redistribution 0% of the internal forces in a deactivated element may be redistributed in the system (if this is prescribed in a static analysis, and the load remains constant, the stress contours, displacements etc. in the other elements will remain unchanged). q Full Redistribution 100% of the internal forces in a deactivated element may be redistributed in the system (this has the same effect as reassigning very weak material properties to the element). q Fractional Redistribution A percentage of the internal force to be redistributed is specified. Provides a solution which is part way between the two extremes. Any remaining internal equilibrating force associated with a deactivated element is maintained in the system until the element is subsequently activated. When an element is activated it is assumed that the element has just been introduced to the model (although all elements must be defined at the outset). The current (deformed) geometry for that element is taken as the initial geometry and the element is assumed to be in a stress/strain free state (unless initial stresses or strains are defined). All internal forces that exist in the element are redistributed and the computed strains are incremented from the time at which the element becomes active.

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Chapter 5 Model Attributes

Tunnel Excavation Stage 1

Tunnel Excavation Stage 2

Top layer of soil deactivated and lining activated. Lining and soil elements duplicated in the model.

Second layer deactivated as soil excavated. Surrounding lining elements activated.

Lining constructed

Lining constructed

Soil excavated

Soil excavated

Tunnel Excavation Stage 3

Tunnel Excavation Stage 4

Remaining second layer soil elements deactivated.

Supporting soil pillar removed and top lining activated. Lining construction Soil excavated

Soil excavated

Tunnel Excavation Stage 5

Final central soil column removed.

Soil excavated

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Birth and Death

Equivalent Stresses in Surrounding Material

Equivalent Stresses in Tunnel Lining

Using Birth and Death Datasets Activate and deactivate datasets are defined from the Attributes menu. The datasets are assigned on a feature basis to control the history of the underlying elements throughout the analysis. The load case is specified during assignment to indicate at what point the elements are added or removed. Notes on use q Elements cannot be activated and deactivated in the following circumstances: • Field analyses • Explicit dynamics analyses 191

Chapter 5 Model Attributes

q q

q

q q

q

q

q

• Fourier analyses • When using updated Lagrangian or Eulerian geometric nonlinearity • When they are adjacent to slidelines Deactivation and activation can take place over several increments if convergence difficulties are encountered. Deactivated elements remain in the solution but with a scaled down stiffness so that they have little effect on the residual structure. The stiffness is scaled down by a parameter which can be changed by the user. In a dynamic analysis the mass and damping matrices are also scaled down by the same factor. When an element is deactivated, all loads associated with that element are removed from the system and will not be re-applied if an element is subsequently re-activated. This includes concentrated nodal loads unless the load is applied at a boundary with an active element. The only exception to this rule is a prescribed displacement which may be applied to a node on deactivated elements. Accelerations and velocities may also be prescribed in a dynamic analysis but this is not recommended. If required, initial stresses/strains and residual stresses may be defined for an element at the re-activation stage. The activation of an element which is currently active results in an initialisation of stresses/strains to zero, an update of the initial geometry to the current geometry and the element is considered to have just become active. The internal equilibrating forces which currently exist in the element will immediately be redistributed throughout the mesh. This provides a simplified approach in some cases. The direction of local element axes can change during an analysis when elements are deactivated and reactivated. In particular, 3-noded beam elements that use the central node to define the local axes should be avoided as this can lead to confusion. For such elements the sign convention for bending moments for a particular element may change after re-activation (e.g. it is recommended that BSL4 should be preferred to BSL3 so that the 4th node is used to define the local axes and not the initial element curvature). Care should be taken when deactivating elements in a geometrically nonlinear analysis, especially if large displacements are present. It may be necessary to apply prescribed displacements to deactivated elements in order to attain a required configuration for reactivation. It should be noted that the internal forces in the elements will not balance the applied loading until all residual forces in activated/deactivated elements have been redistributed.

192

Nodal Equivalencing

Nodal Equivalencing The equivalence facility is used to merge coincident nodes on otherwise unconnected features. If an equivalence dataset is assigned to any features LUSAS will automatically equivalence the required nodes after meshing has been carried out. There are several ways equivalencing can be set up to work: q By assigning equivalence tolerances to certain features - only these features will be equivalenced, all other are ignored. q By switching on the Automatic tolerancing, and accepting the default tolerance - all features are equivalenced according to the default tolerance. q By switching on the Automatic tolerancing, and assigning other equivalence tolerances to certain features - all features are equivalenced according to either the an assigned tolerance or the default tolerance. In this example, Surfaces 1 and 2 do not share a common boundary Line, therefore the nodes created on their common boundaries will not be joined and must be equivalenced. Node 2 becomes node 1 if both δu and δv are less than the equivalence tolerance.

For a grillage, created with Lines spanning boundary to boundary, as shown in the above example, internal nodes on the Lines will not be common to both longitudinal and transverse members even when mesh divisions cause nodes to have identical coordinates. All Lines must therefore be equivalenced.

L1 L2 S1

S2

2 1

δz

δv δu

Using Nodal Equivalencing Equivalence datasets are defined from the Attributes menu, See About Attributes. They are defined as a tolerance, which is used to determine whether nodes are considered to be coincident.

193

Chapter 5 Model Attributes The equivalence dataset is assigned to the features that are to be checked for coincident nodes. When an equivalence dataset is assigned to a lower order feature it will search through all higher order features for nodes to be checked. For example, in order to equivalence two Volumes at their boundaries, it is more efficient to assign the equivalence to the Surfaces on the boundaries, as a smaller number of nodes need to be checked. Automatic Equivalencing Automatic equivalencing can be activated from the Meshing tab of the model Property box. This will equivalence all features in the model on meshing if they are within the default equivalence tolerance, or within an assigned tolerance. Note. Remeshing occurs each time a new command is issued, but a forced remesh is possible from the command line. Automatic equivalencing can be time consuming for models with a large number of nodes.

Visualising Nodal Equivalences Displays the features which have a specified equivalence dataset assigned to them in a chosen colour and line style. In this example different equivalence tolerances are assigned to different parts of a model to merge more coarsely or finely as required. Using visualisation, the lines to which the equivalence dataset is assigned can be highlighted. Equivalenced nodes can also be displayed as they are removed (see below). In this diagram they are shown using the square symbol.

EQV1

EQV2

Summary q Nodal equivalencing is useful when creating grillage structures where unconnected longitudinal and orthogonal lines result in duplicate nodes. q More than one equivalence dataset may be defined in order to rationalise more than one section of the model independently. q More than one equivalence dataset can be assigned to a feature to equivalence it within a different subset of the model. q A check for unconnected elements and nodes can be performed using an outline mesh plot, (Mesh layer properties), or by checking for duplicate node numbers, (Label layer properties), . q Nodal equivalencing may be used to position a point load or support at a node (which is not a defining feature Point). A Point must be created, the load or support assigned, and the Point and meshed feature equivalenced.

194

Search Areas q Equivalencing may be used to merge nodes on the constituent Lines of combined Lines i.e. the entire combined Line may be equivalenced, including the Lines forming it. Case Study. Removing Duplicate Nodes on Grillage Members When a grillage is defined by a network of lines, two nodes will lie at each point at which the lines cross. In order to model the grillage correctly these pairs of nodes must be merged into one node so that the grillage elements are connected. 1. Define a grillage using a series of criss-crossing Lines and mesh the Lines using the Engineering Grillage element. Take care to specify your mesh divisions so that internal nodal positions on longitudinal and transverse members are coincident. 2. Define an equivalence dataset with a suitable tolerance using Attributes > Equivalence. Nodes whose X, Y and Z coordinate lies within this tolerance of each other will be joined as one. 3. Assign the equivalence dataset to all the Lines in the model. 4. Re-set the mesh using the syntax for the Force Remesh of Model command. 5. Draw the mesh and node labels. If duplicates still exist, adjust the equivalence tolerance and re-mesh the model.

Search Areas Search areas may be used to restrict the area of application of discrete loads (point and patch). This is useful for several reasons: q Improved Control of Load Application the search area will effectively limit the area over which the load is applied so that the effect of loads on certain features may be removed from the analysis. For 3D models, where general loading is applied, it is possible that a chosen projected direction will cross a model in several locations. A search area is therefore used to limit the application of load to one of these multiple intersections. Restricting the area of application of discrete loads allows the same load datasets to be used to apply loads to different parts of the model. q Speed Improvement the speed of calculation of equivalent nodal loads will be increased by cutting down the number of features considered in the calculation.

195

Chapter 5 Model Attributes In the example shown, a multiple span grillage structure is defined with Span 1 as the search area. A discrete Patch load, indicated by the grey shaded region in the upper diagram, is applied across the whole structure, Span 1 and 2. The area of the structure coinciding with both the Search Area and the patch load will take the load as shown in the lower diagram.

Span 1 Loaded

Span 2 Unloaded

HA Load

Search Area

Tip. Search areas should be used if the model is three dimensional and discrete loads are applied, for example boxsection or cellular construction decks.

Defining and Assigning Search Areas Search areas are defined from the Attributes Menu then assigned to the required features, (Lines or Surfaces only). Control of application of load lying outside the search area is available when the load is assigned, see Assigning Discrete Loads. If a search area dataset is not specified when the load is assigned, all of the highest order features, excluding volumes, in the model will be used as a default search area. Valid search area configurations are shown below. Rules for Creating Search Areas The following general guidelines should be noted when defining a search area on a mesh. Overhanging elements defined such that only one side of a cell is missing are included in the search area, as shown in the left side diagram.

Valid Search Areas

Invalid: 2 edges overhanging from same node

Elements cannot be included in the search area when they overhang from the same node, as shown in the right side diagram. In this case, dummy Lines can be added with ‘none’ type element mesh assignments to close the bay to make the search area valid.

196

Varying Attributes over Features Cells of more than 4 edges are subdivided into triangles (left diagram), but overhanging elements are only included if divided by no more than one edge Invalid: Valid (invalid left, valid right). There is overhanging Search edges separated no limit to the number of edges Area by 2 edges that may hang over the main body if the overhanging members are only separated by one edge (right).

Valid Search Areas

Varying Attributes over Features The variation dataset allows values in Material, Geometry, Loading and Support datasets to be varied over a feature in a number of ways by defining the manner in which the chosen entity will vary. If a variation is not used the individual components within an attribute, such as Young's modulus within a material dataset, are considered to be constant over a feature, which results in the definition of a large number of features to accurately describe the required variation. The variation facility allows these analysis requirements, and many more, to be easily defined. Depending on the type of variation defined, both continuous and discontinuous variations can be modelled. Three different types of variation dataset can be defined. These are: q Field allowing variation in terms of the global Cartesian coordinate system variables. This form of variation would be used for hydrostatic and wind loading and is applicable to all feature types except Points. Variations on volumes are limited to field variations. q Interpolation variations may be applied to Lines and Surfaces. The variation is defined by interpolating between values at specified feature distances. The order of the interpolation may be specified as constant, linear, quadratic and cubic in either actual (local) or parametric distance. q Function variations are expressed as symbolic functions in terms of the parametric coordinates of a feature. They can be applied to Lines and Surfaces. For Lines, the parametric distance is the distance along the Line (u), and for a Surface the distances are the local x and y coordinates expressed as u and v.

Using Variation datasets Variations are defined from the Utilities menu. Variation datasets are stored in the Utilities Treeview .

197

Chapter 5 Model Attributes Once defined, a variation can be used in a Loading, Material, Geometry, Support or Damping dataset by clicking on the variation button in an edit box of any parameter on the dialog. This way, different parameters can be varied in different ways by applying different variation datasets to them. Notes on Varying Attributes It is possible to apply variations to Loading, Geometry, Material, Support and Damping datasets. A variation dataset will apply to a particular parameter, for example WZ in a uniformly distributed load or Young's modulus in an elastic isotropic material dataset. q Loading It is possible to vary all load types except General Point and Patch loads and Element Point and Distributed loads, which incorporate variable loading implicitly in their definition. Values of loads which are applied to elements will be evaluated at the element centroid. Contouring is carried out on an element by element basis and is available in pre-processing only. q Geometry Datasets containing a variation are tabulated as multiple datasets. An additional parameter is added to the assignment command to relate to the original defining dataset number for use in post-processing. Contouring is carried out on an element by element basis and is available in pre-processing only. q Material Variations in materials are limited to elastic material values and certain joint properties. Datasets containing a variation are tabulated as multiple datasets containing the material value calculated at the element centroid. An additional parameter is tabulated to the assignment data chapter in the LUSAS data file to relate to the original defining dataset number for use in post-processing. Contouring is carried out on an element by element basis and is available in pre-processing only. q Supports Only spring stiffness values can be varied. In post-processing spring stiffness values are not scaled when drawn. q Damping Variations of the Rayleigh parameters cannot be contoured as they are calculated at element centroid positions.

Field Variations Field variations allow a variation according to a mathematical expression in terms of coordinate variables in either the global Cartesian or a specified local coordinate system. Coordinates may be Cartesian, cylindrical or spherical. Field variations are applicable to Lines, Surfaces and Volumes and the value of the variation at any position on the structure will be calculated by substituting the values of the coordinate variables at that position. 198

Varying Attributes over Features A field variation is defined by specifying a field expression and an optional local coordinate dataset number, which will be used to specify a coordinate system other than the global Cartesian set. Specifying a local coordinate number of zero will default to the global Cartesian set. These examples (right) show field variations expressed in terms of the global X coordinate displayed along a Line parallel to the global X axis. The typical field expressions used are shown in the boxes next to each diagram. For example, a field expression in Cartesian coordinates would typically be:

F=Sin(x)

F=A+Bx

F=x**n

F=Ax

-9.81*y and in cylindrical coordinates: 10+r*tan(thetax) Coordinate Systems in Field Variations Functions and variables used in field expressions are limited to those used in the parametric language plus the Cartesian, cylindrical and spherical coordinate variable names. The coordinate variable names that should be used in a field expression are dependent on the type of coordinate systems in use. Definitions are given in the tables starting below. See F=A+Bx x=0.75 Local coordinate systems for more information. In this example (right), a field expression referring to the global axis coordinates (XY), is also used with a local coordinate axis set (indicated by xy) to create a variation relative to a rotated system. Cylindrical and spherical axis sets can also be used.

x=0.25

Field Variation in Local Axis System

y x Y

X=0.25

X=0.75 F=A+BX

X Field Variation in Global Axis System

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Chapter 5 Model Attributes

Cartesian (global/local)

X Cylindrical (local)

X Spherical (local)

x

X coordinate

x

Distance along cylinder longitudinal axis

r

Radial distance

y

Y coordinate

r

Radial distance

thetax

Angle about x axis

x

Z coordinate

thetax

Angle about axis of cylinder

thetac

Second angle

Z Cylindrical (local)

Z Spherical (local)

r

Radial distance

r

Radial distance

thetaz

Angle about axis of cylinder

thetaz

Angle about z axis

z

Distance along cylinder longitudinal axis

thetac

Second angle

Cylindrical and spherical field variation expressions can use radians (default) or degrees to specify angles. If trigonometric functions are used in a field expression, they will dictate what angular measure is used. For example, a function will use degrees if degree-based trigonometric functions, such as sind, cosd, tand and atan2d are used. An expression may not mix radian and degree functions. Any angle cut-off values will use the same units as the expression. Maximum and Minimum Cut-Off Values Maximum and minimum [0.25] [0.75] F=Sin(x) cut-off values may be specified for the chosen coordinate system. This F=A+Bx allows the range of application of load to be limited, such as would be F=x**n necessary to model a structure not wholly submerged in water. These F=Ax examples (right) show field variations in terms of the global X coordinate displayed along a Line parallel to the global X axis. The typical field expressions used are shown in the boxes next to each diagram. All expressions are subject to a cut-off in minimum and maximum X at parametric distances of 0.25 and 0.75 respectively. 200

Varying Attributes over Features This example shows a variation in terms of the global Z axis coordinate with minimum and maximum cut-offs at specified Z coordinate values. Field Variation in terms of global Z coordinate Maximum Z cut-off

Minimum Z cut-off

Z

X

Variation subject to cut-offs in global Z

Case Study. Applying Hydrostatic Loading A hydrostatic loading may be modelled using a combination of a field variation and a Structural Face Loading. The loading can be considered to be dependent on the depth varying as: water density*g*(h-y) where g is the acceleration due to gravity, h is the height of the water above the structure origin and y is the height of the structure. Use the following procedure: 1. Define a simple 100 unit square Surface using Geometry > Surface > By Coordinates and entering the following coordinate (0,0,0), (100,0,0), (100,100,0) and (0,100,0). 2. Define a simple thin shell mesh using Attributes > Mesh > Surface. Assign the mesh to the Surface. 3. Define a field variation using Utilities > Variation > Field and specify a function of density*g*(h-y), where density is the water density (1000), g is acceleration due to gravity (9.81), h is the maximum height of the water above the structure origin (80) and y is the global Cartesian y coordinate. This function will apply a hydrostatic loading down the depth of the Surface (global y axis). Enter 1000*9.81*(80-y) on the dialog. To model a water depth of 80 (and to avoid negative loading above the surface of the water), select a Cut-off in Maximum y at 80. Click on the Advanced button and set the maximum second coordinate to 80. Click the OK button. 201

Chapter 5 Model Attributes Give the dataset a suitable name. Type Hydrostatic variation into the dataset box. Click the OK button. 4. Using Attributes > Loading > Structural, define a Local Distributed load entering the Z component as 1, notice that in doing so the variation button appears. Click on the button and select the Hydrostatic variation dataset. This will factor a negative unit load using the variation defined in 1. Type Water load into the dataset box. Click the OK button. Assign the loading to the Surface. 5. The applied loading with the variation is visualised as arrows on the model. Use the dynamic rotate to get a 3D perspective of the surface. If the load is not visualised, select the load dataset in the Treeview , right-click and choose Visualise from the shortcut menu. Note. Visualising attribute assignment requires that the model is meshed.

Line Interpolation Variations Line variations by interpolating are defined by a series of pairs of distances, and the values to be interpolated between at those distances. The specified interpolation values may in turn reference field variation datasets. Discontinuous variations can be defined in this way. q By Equal Distances defines a set of interpolation values at a specified number of equal divisions along a Line. If n divisions are specified, n+1 interpolation values are required. In the examples shown right, a number of equal divisions are specified along with

4.0

4.0 3.0 1.0

2.0

Linear 4 Divisions 4;1;3;2;4 Quadratic 2 Divisions 4;3;4 Multiple Quadratic 4 Divisions 4;1;3;2;4

Cubic 3 Divisions 4;1;3;4

202

Varying Attributes over Features interpolation values for the end and internal interpolation positions. For n divisions, n+1 interpolation values must be specified. Discontinuities cannot be defined using this method apart from those indicated at the junction of separate quadratic variations. q By Unequal 200 [1.0] Actual [parametric] 140 [0.7] Distances distances 100 [0.5] defines a set of 60 [0.3] interpolation values 4.0 4.0 3.0 at a specified set of 2.0 1.0 distances along a Line. Distances can Linear be entered as actual [0;0.3;0.5;1] 4;1;3;4 or parametric values. Quadratic [0;0.3;0.5] The examples 4;3;4 shown right use Discontinuous distances specified Linear [0;0.3; by actual or 0.5;0.5;0.5;0.7;1] 4;1;3;2;4 parametric values (indicated in square Cubic brackets) with a [0;0.3;0.5;1] corresponding 4;1;3;4 interpolation value at each position. Quadratic variations require a minimum of 3 interpolation values with multiple quadratic variations being defined for 5, 7, 9 sets of interpolation values. Repeating a coordinate and specifying an additional associated interpolation value will allow a discontinuity in the variation to be modelled. See the discontinuous linear variation example above. The discontinuity tolerance is used to separate adjacent points within a variation at a discontinuity so as to avoid multiple values being present at an underlying node.

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Chapter 5 Model Attributes

Line Function Variations A Line function variation is defined by a series of pairs of distances, and the symbolic functions at those distances and beyond. The function is a mathematical function in terms of the parametric or actual coordinate along the Line. For a Line function variation, the value of the variation at any point on the Line is calculated by finding the interval in which the point occurs and then substituting the parametric or local distance into the function. q By Equal Distances [in u] defines a number of equal divisions and a set of functions (one for each division) in terms of u, the parametric distance along the Line. In this example, F=2 F=0 F=u**2 the Line is split into a specified u=0.92 u=0 u=0.33 number of distances, each with an associated function. q By Unequal Distances [in u] defines a series of parametric or actual distances, and a set of functions. The distance specified is the starting point for the function associated with it. Each distance must have an associated function specified. To enter a maximum cut-off point, associate a zero function with it. In this example, a F=-1.0+2*u F=0 F=-1.0+2*u F=0 distance of 0 is associated with function 0, 0.33 is associated with u**2 and 0.92 is associated with 2.

Surface Interpolation Variations On Surfaces, interpolation may be defined by either a grid of values or a set of line variations which are to be applied to the boundary Lines and interpolated between. For interpolation by grid the order may be constant, linear, quadratic or cubic and the grid values supplied are interpolated between. q Surface By Grid defines a full grid of interpolation values by specifying numbers of equal divisions in Surface local X and Y directions and interpolation values at each resulting grid position. Surface grid interpolation can only be used on 3 and 4 sided Surfaces. For irregular Surfaces, the variation can be supplied by a support Surface.

204

Varying Attributes over Features Care must be taken to supply the correct number of interpolation values for the type of interpolation chosen (see table below). Values are specified along the local X axis first, then along each line of grid positions parallel to the x axis in turn. Interpolation order can be different in each direction. Order

Number of Divisions (n)

Interpolation Points

Constant

Automatically set to 1

1

Linear

No restriction

n+1

Quadratic

Minimum of 2 (or 4 or 6 ...)

n+1

Cubic

Minimum of 3 (or 4 or 5 ...)

n+1

This example (right) shows simple Quadratic vs. Linear Surface 3 grid variation. The 2 quadratic variation Local Y Constant 1 in the local X 1 Division (1 Point) direction is Local X Quadratic 2 Divisions (3 Points) specified with 2 divisions (3 interpolation points). The constant variation in the local Y direction requires no additional points. A total of 3 interpolation points are required. This second example shows a 12 cubic vs. linear 11 Surface grid 10 8 7 9 variation. The 6 4 local X direction 5 3 takes a cubic 2 Y (Linear) variation defined Local 1 2 Divisions (3 Points) with 3 divisions (4 Local X (Cubic) 3 Divisions (4 Points) interpolation points) and the local Y direction takes a linear variation using 2 divisions and 3 interpolation points. A total of 12 interpolation points are required. q Surface By Boundary defines a set of interpolation values by specifying variation datasets around the Surface boundary Lines. A variation must be specified for each Line in the Surface definition. If no variation is required along a Line, a constant order variation must be specified. Care must be taken to ensure that interpolation values at corner points are common to both variation datasets meeting there otherwise strange results may occur. 205

Chapter 5 Model Attributes Variations are defined in the same direction on opposite sides of the Surface (see the following example) and use the Line order in the Surface definition on which to base the variation direction. Individual Line directions have no effect on variation directions. Same Variation As This example Line First Line (1) shows a Interpolation Constant (2) Surface Value = 3 boundary interpolation using three Line Line interpolation Interpolation Line Interpolation datasets. A (3) Constant Discontinuous Linear discontinuous Value = 4 (1) Values = 4-5, 3-2, 4-3 Line interpolation (1) is specified for first and third Lines. Note that the Surface axes drawn here dictate the variation directions and that opposite sides of a Surface will vary in the same direction whatever the underlying Line directions indicate. The second and fourth Lines in the Surface definition use constant interpolation datasets. The variation sense is denoted by double and single arrows shown on boundary Lines. The variation along the local X axis (signified by the double arrow) is specified first. The Surface variation in this case is 1;3;1;2.

Note. When variations are defined, distances may be defined by Points on a Line, as a number of divisions, or as a series of parametric or local distances. However, a variation is only stored in terms of parametric distances. Therefore, the values used to define a variation can only be filled into the dialog when editing an existing dataset by using the unequal distance method.

206

Varying Attributes over Features

Surface Function Variations A Surface function variation consists of a single function in terms of the parametric coordinates of the Surface u and v. The value of the variation at any point on the Surface is given by finding the parametric coordinates of the point within the Surface and substituting them into the specified function. Surface function variations are only allowed for 3 and 4 sided Surfaces.

Variation Max(4,10*u)

v

u

The example shown here defines a variation using the function max(4,10*u) in terms of the local Surface x direction parametric distance. The max function takes two arguments and returns the maximum of both arguments. In this case, 4 is the maximum value until u exceeds 0.4.

Plotting Graphs of Variation Datasets Variation datasets can be evaluated as datasets and displayed using the LUSAS Graph Wizard from the Utilities menu. The variations can be evaluated along a specified Line by 5.000 creating a pair of datasets (distance and 4.000 variation) which can subsequently be plotted. A number of points at which to 3.000 sample the variation can be specified. 2.000 For variations that have more than one 1.000 interval, the variation is evaluated at the specified number of points on each 0.0000.0 0.2 0.4 0.6 0.8 1.0 interval. A factor may be applied to the variation value before the dataset coordinates are calculated. This example demonstrates the graphical visualisation of a discontinuous Line interpolation variation.

207

Chapter 5 Model Attributes

Influence Lines An influence line is a physical break in the finite element mesh at a node. Additional parameters define the behaviour of the structure around the influence node. The influence type may be a force, reaction or displacement.

Defining Influence Lines Influence lines may be defined from the Utilities menu. A single node must be selected for which the influence curve is required. Note. The Force type influence is only available if a node and an element have been selected with the mouse, and an element may also be added to selection memory to define element axes. Influence Line Parameters q Freedom type defines the direction of the influence line: U, V, W, THX, THY, THZ. q Displacement direction defines the direction of the influence node deflection. A negative displacement direction will produce a negative displacement in the chosen axes direction. Influence axes can be specified as: q local (via a previously defined coordinate dataset) q global q the axes of an element in memory selection

208

Load Cases The influence type may be specified as a force (moment), reaction or displacement: q Force (Moment) breaks the mesh and defines constraint equations to fix the displacements and rotations of the nodes at the break, apart from the displacement in the force direction at which a unit relative displacement is imposed. A moment influence curve is produced by breaking the mesh at the specified node and then defining constraint equations to fix the displacements and rotations of the nodes at the break, apart from the rotation in the moment direction at which a unit relative rotation is imposed. If the specified node is attached to more than two elements, then the elements defined by the node at which the break is required must also be selected before using the command. q Reaction defines constraint equations to impose a unit displacement or rotation in the specified direction. q Displacement for unit forces or moments applied at nodes.

Manipulating Influence Lines Defined Influence lines are stored in the Utilities Treeview manipulated from here using the right-click shortcut menu.

and may be

Writing a Datafile with Influence Lines Once the correct influence datasets have been defined, they are tabulated to a LUSAS data file using the Files > LUSAS DataFile, then specifying the datafile type as a Influence Line analysis. Data file names are generated from the specified file name and the influence dataset number. For example, if the specified file name is root, then files root1.dat and root2.dat will be created for influence line datasets 1 and 2 respectively.

Load Cases Load cases are used differently during the modelling stage and the results processing stage. q Modelling load cases (pre processing) contain the load assignments and the analysis control. The order of the load cases in the Treeview defines the order in which the load case are solved by the LUSAS Solver. A model may be specified as a load curve problem from the Solution tab of the model properties. q Results load cases (post processing) contain the results from an analysis. They exist in the Treeview but cannot be edited. Results load cases may be manipulated using combinations and envelopes, fatigue loadcases and IMD loadcases. All of which are added as new load cases in the Treeview. 209

Chapter 5 Model Attributes

Manipulating Load Cases Load cases are contained in the Treeview ,. At least one load case always exists. New load cases may be added to the Treeview using the Utilities menu. Double-clicking on a load case will display the properties for editing, except results load cases. General editing commands are available by right-clicking a load case. The following commands are available for all (results load cases have reduced commands): q Delete Attribute assignments must be deassigned before a load case can be deleted. A load case cannot be deleted if it is the first load case or the active load case. q Rename Sets the Load case title. Note. Load cases are tabulated in the order laid out in the Treeview. q Remove Deassigns attributes from the load case, by choosing from a list of attributes. q Set active Sets the active load case for the current window. See below. q Properties Displays the properties relating the a load case. Double-clicking the load case also displays the properties. See below. Setting the Active Load Case An important concept in LUSAS is the active load case in the current window. The active load case a window property, and is the load case that all results will come from when visualising results. This speeds up the process of comparing results by using different windows for each load case to be compared. The active load case is set from the Treeview using the shortcut menu. A black dot next to a load case indicates the active load case.

Load Curves Load curves can be used to describe the variation of the applied loading in nonlinear, transient, dynamic and Fourier analyses. For example, in a transient thermal problem the loading changes with time, and in a nonlinear problem the loading level may vary with load increment. Load curves are used to simplify the input of load data in situations where the variation of load is known with respect to a certain parameter. An example of this 210

Load Curves could be the dynamic response of a pipe to an increase of pressure over a given period. The data input would consist of the definition of the load and its variation with time.

Using Load Curves 1. Specify a Load Curve Problem

A model is specified as a load curve problem by clicking on Load Curves in the Solution tab of the Model properties, then click on the Set Control button to define the analysis control. Load curves scale all loads in the specified load case. Therefore, to scale loads using different load curves they must be assigned to different load cases. Further load curves may be added using the Utilities, Load case menu. 2. Define the Load Curve Properties

The load curve is made active by setting the load case properties, right-click on a load case and choose Properties from the shortcut menu. A load curve is defined either using a sine, cosine or square wave input, or using a defined Variation. q Sine, cosine, square wave input Values for peak (amplitude), frequency and phase angle must be defined along with activation and termination points. q Variation dataset curve A line interpolation variation must be defined using the Utilities, Variation, Line menu command, and referenced on the load case properties. The dependent variable in the variation dataset will represent time or increment number depending on the type of analysis, and the value of the variation will be the factor by which to scale the values in the loading dataset. Notes

q When load curves are used, load cases cease to be analysed sequentially, instead they are used merely to allow load curves to reference different load datasets. q Load curves are only applicable to nonlinear, transient, dynamic and Fourier analyses. q For Fourier analysis the load must only be applied over an angular range of 0 to 360 degrees.

Evaluating a Load Curve Load curves (and variation datasets) may be viewed using the Graph Wizard.

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Chapter 5 Model Attributes Case Study. Pressurisation of Two Tanks with Multiple Load Curves Two tanks are to be pressurised at different stages of a nonlinear dynamic analysis. This will be achieved using two different load cases and two load curves to vary the loads individually. The following procedure outlines the steps required: 1. Define 2 variation datasets which give the correct pressure variation with time. Use the Utilities, Variation, Line menu entry. Specify an Interpolation variation, Actual distance type and enter coordinate pairs for load factor and time values on each graph. 2. Click on File, Model Properties and go to the Solution tab. Set model to a Load Curve Problem (checkbox), then click on the Set Control button. From the analysis control dialog click on the nonlinear checkbox. Click on the transient checkbox and choose dynamic from the drop-down list. OK the dialogs. 3. Using Utilities > Load Case add a second load case. Go to the load case Treeview and double-click on the first load case. Click on the Active checkbox, then specify the first variation dataset from the dataset list. Enter suitable activation, termination and increment values which will be used at the tabulation phase to define the curve. These curves can be evaluated using Utilities, Graph Wizard. 4. Using Attributes, Loading, Structural, define 2 Face Load datasets containing the pressure loads for each tank. Note that the pressure values entered in the load definition will be multiplied by the load factors used on the load curve associated with it. 5. Finally, assign these loads to the features in the model, specifying different load cases for each tank. For example assign load 1 to tank 1 using load case 1, and load 2 to tank 2 using load case 2. The accompanying diagram shows a schematic of the tanks under internal pressure with their corresponding force versus time graphs.

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