Plane and axisymmetric models in Mentat & MARC. Tutorial with some Background

Transcription

1 Plane and axisymmetric models in Mentat & MARC Tutorial with some Background Eindhoven University of Technology Department of Mechanical Engineering Piet J.G. Schreurs Lambèrt C.A. van Breemen March 6, 2014

2 Contents 1 Plane stress and plane strain Background : Theory and element formulation Plate with a central hole Modeling and analysis Results Edge load Axle support with radial load Modeling and analysis Results Quadratic elements Orthotropic plate Circular disc with prescribed radial edge displacement Modeling and analysis Results Rotating solid disc Modeling, analysis and results Axisymmetry Background : Axisymmetric modelling Axle support with axial load Modeling, analysis and results Circular discs

3 1 Plane stress and plane strain 1.1 Background : Theory and element formulation In the tutorial Truss and beam structures trusses and beams are described and used to model simple structures with MSC.Marc/Mentat. Although the nodes of the truss and beam elements can have a displacement (and rotation) in three directions, only one stress is relevant : the axial stress. For linear elastic material behavior, the stress in a truss is related to the axial strain ε = l l 0, according to σ = Eε, where E is the Young s modulus, which can be measured in a tensile experiment. Plane stress Many structures are build from flat plates, having a thickness, which is much smaller than the dimensions of the plate in its plane. The figure below shows such a plate, with its plane in the xy-coordinate plane and with a uniform thickness h 0. x z y h 0 In many cases it is allowed to assume that the plate is loaded in its plane, as shown in the figure. When the plate is isotropic, there will be no bending of the plate and it will stay flat after deformation. x z y P In a point P of the plate a column is cut out, with its sides parallel to the xz- and yz-coordinate planes, respectively, and with dimensions dx dy h 0. This small part of the plate is loaded with stresses, so forces per area. Because the thickness of the plate is very small, it can be assumed that these stresses are uniform over the thickness. The stress components working on a stress cube dx dy dz are the normal stresses σ xx and σ yy and the shear stresses σ xy and σ yx. It can be shown that σ xy = σ yx, so only three stress components are relevant. 3

4 x z y P σ yx σ yy σ xx σ xy Because there are no stresses working in z-direction, this stress state is referred to as plane stress. The deformation in point P is described by the normal strain components ε xx and ε yy and the shear strain ε xy. For linear isotropic elastic material behavior the next relation holds between stresses and strains : σ xx σ yy σ xy = E 1 ν 2 1 ν 0 ν ν ε xx ε yy ε xy Due to the stresses σ xx and σ yy the thickness of the plate will change. The strain in thickness direction, so in z-coordinate direction in our case, is : Plane strain ε zz = h h 0 = ν E (σ xx + σ yy ) The displacement in the z-direction, which is for our plate the direction perpendicular to its plane, is suppressed. The plate is for instance glued to two parallel rigid bodies, which stay at the same distance. In that case we have ε zz = 0, and call the deformation a state of plane strain. In a state of plane strain, there is generally a stress σ zz 0. The linear elastic material behavior is described by : σ xx σ yy σ xy = E (1 + ν)(1 2ν) 1 ν ν 0 ν 1 ν ν ε xx ε yy ε xy ; σ zz = ν(σ xx + σ yy ) It is noted that problems will occur when ν approaches 0.5. Stresses will become infinite. This is not strange, since in a linear elastic material the volume change will be zero for ν = 0.5. To prevent numerical problems in MSC.Marc, we use the option CONSTANT DILATATION in the menu GEOMETRY. Interpolation and integration : the element stiffness matrix When analyzing plane stress and plane strain problems with the finite element method we have to use plane stress or plane strain elements. These elements are in MSC.Marc always located in the 4

5 xy-plane. Also the deformation is in the xy-plane and is described by the x- and y-displacement of a number of nodal points, situated on the edges of the elements. The displacement of an internal point of the element is interpolated between the displacement of the nodes. For this interpolation, a local coordinate system is used with isoparametric coordinates ξ 1 and ξ 2, which have values between -1 and +1. In mathematical terms the interpolation of the displacement u and v in x- and y-direction, respectively, can be written as follows : u(ξ 1, ξ 2 ) = N 1 (ξ 1, ξ 2 )u 1 + N 2 (ξ 1, ξ 2 )u 2 + N 3 (ξ 1, ξ 2 )u 3 + = v(ξ 1, ξ 2 ) = N 1 (ξ 1, ξ 2 )v 1 + N 2 (ξ 1, ξ 2 )v 2 + N 3 (ξ 1, ξ 2 )v 3 + = n N i (ξ 1, ξ 2 )u i i=1 n N i (ξ 1, ξ 2 )v i where N i are the interpolation functions, associated to node i, and u i and v i the nodal displacement components. The number of element nodes is n. Calculation of the element stiffness matrix, which relates nodal displacements to nodal forces, implies integration of a function over the element volume. This integral can only be evaluated numerically : the integrand is calculated in a discrete number of internal integration points and these values are added after multiplication with a certain coefficient. The location of the integration points and the coefficients are fixed for a certain element type. Linear and quadratic elements The figure below shows a 4-node element with all nodes in the four corners of a quadrilateral with straight sides. Nodal points have a local counterclockwise numbering, which is in accordance with the MSC.Marc program. The figure also shows the element in the so-called isoparametric space, where points are identified with the local isoparametric coordinates ξ = [ ξ 1 ξ 2 ] T. i= η 3 ξ Because there are only two nodes on one element side, the displacement of an arbitrary point on the side can only vary linearly between the two nodal values. A 4-node quadrilateral is therefore called a linear element. The four interpolation functions, associated with the nodes are : N 1 = 1 4 (ξ 1 1)(ξ 2 1) ; N 2 = 1 4 (ξ 1 + 1)(ξ 2 1) N 3 = 1 4 (ξ 1 + 1)(ξ 2 + 1) ; N 4 = 1 4 (ξ 1 1)(ξ 2 + 1) The element has 4 integration points, which are shown in the figure below. Their local coordinates are fixed and determined such that the numerical integration is as accurate as possible η ξ 5

6 In a quadratic element, the displacement of an arbitrary point is interpolated using interpolation functions, which are quadratic in the local coordinates ξ 1 and ξ 2. This element has 8 nodes and is therefore also referred to as an 8-node element. Four of these nodes are situated in the corners of the element, the other four are in the middle of the element sides in the undeformed situation. The numbering of the local nodes is indicated in the figure below and is in accordance with the MSC.Marc program. The interpolation functions are : N 1 = 1 4 (ξ 1 1)(ξ 2 1)( ξ 1 ξ 2 1) ; N 2 = 1 4 (ξ 1 + 1)(ξ 2 1)( ξ 1 + ξ 2 + 1) N 3 = 1 4 (ξ 1 + 1)(ξ 2 + 1)(ξ 1 + ξ 2 1) ; N 4 = 1 4 (ξ 1 1)(ξ 2 + 1)(ξ 1 ξ 2 + 1) N 5 = 1 2 (ξ2 1 1)(ξ 2 1) ; N 6 = 1 2 ( ξ 1 1)(ξ 2 2 1) N 7 = 1 2 (ξ2 1 1)( ξ 2 1) ; N 8 = 1 2 (ξ 1 1)(ξ 2 2 1) The 8-node element has 9 integration points, the location of which is indicated in the figure ξ ξ ξ ξ

7 1.2 Plate with a central hole The figure below shows a square plate with a central hole. Relevant dimensions are indicated. Young s modulus = 210 [GPa] Poisson s ratio = 0.3 [-] thickness = 1 [cm] u [mm] y x 10 [cm] 16 [cm] The plate is loaded in its plane. Displacements of left and right edges (x = ±8 [cm]) are prescribed : u = ±0.1 [mm]. In y-direction the displacement of the edge points is free. Top and bottom edge (y = ±8 [cm]) are also free. This load leads to a plane stress state in the plate. The material of the plate is isotropic and can be assumed to be linearly elastic. Young s modulus and Poisson s ratio are known : E = 210 [GPa] and ν = 0.3 [-]. Deformation of and stresses in the plate will be determined using MSC.Marc and therefore a model must be made with MSC.Mentat. Considering symmetry and load learns that only a quarter of the plate has to be modeled. We choose the part with 0 (x, y) 8 [cm]. The correct symmetry conditions must of course be prescribed as boundary conditions. 7

8 1.2.1 Modeling and analysis GEOMETRY & MESH We start in the GEOMETRY & MESH. We will use the Cartesian coordinate system. This is shown on the computer screen as a two-dimensional grid in the xy-plane. In the grid points we can define (nodal) points and other things by clicking with the left mouse button. In the grid settings the grid dimensions and the spacing between the grid points can be specified. The program does not know of dimensions and units, so we have to choose them and stick to them during the modeling and analysis. Here, dimensions will be expressed in unit of meters. We choose the grid point spacing to be 0.01 meter. SET (Coordinate System) U DOMAIN : -0.1, 0.1 U SPACING : 0.01 V DOMAIN : -0.1, 0.1 V SPACING : 0.01 GRID (Coordinate Systen) (Fill view) In the tutorial for Truss and beam structures we used only (CURVES), but this is not enough any more; we have to do more. We want to define a (SURFACE) and subdivide this into elements. A simple method is used here, which can be applied in many cases. First two (CURVES) are defined, one POLYLINE and a quarter of a circle, an (ARC). The (POINTS) which define the (CURVES) can be located in grid points by clicking the left mouse button. After defining the (CURVES), a surface of the type RULED is made. The idea is that a stick is placed with its begin and end point on two separate (CURVES) and is subsequently rolled over them, thus describing a (SURFACE) in space. GEOMETRY & MESH (Basic Manipulation) (CURVES) POLYLINE (CURVES) ADD Define three points of the POLYLINE. Close the window with (End list). (CURVES) ARC CEN/PNT/PNT (CURVES) ADD Define the required (see Command-screen) points of the (ARC). When the (ARC) is drawn in the wrong direction, click latter two (ARC) points in swapped order. (Undo) and define the (SURFACES) RULED 8

9 (SURFACES) ADD Click on the two (CURVES). A strange surface may appear because of different orientation of the two curves. In that case we have to (Undo) the last action (definition of the surface) and flip one of the two curves in the CHECK (Operations) menu with FLIP CURVES. The surface must then be redefined. When this is done successfully, the surface is now going to be converted into elements. (ELEMENTS) QUAD(4) CONVERT (Operations) DIVISIONS : 8, 4 (Convert) SURFACES (To) ELEMENTS CONVERT Select (SURFACE). Close with (End list). After defining the element mesh, SWEEP (Operations) has to be used to remove coinciding nodes and elements. It is recommended to go to the CHECK (Operations) menu and check whether there are elements INSIDE OUT or UPSIDE DOWN. When this is the case these elements must be flipped. It is recommended to prevent future drawing of (CURVES) and (SURFACES) in the view plot control-menu. GEOMETRIC PROPERTIES NEW (STRUCTURAL) PLANAR PLANE STRESS PROPERTIES Thickness : 0.01 OK (ELEMENTS) ADD Enter geometry add element list : (All existing) MATERIAL PROPERTIES Prescribe the material parameters : E = 2.1e11 [Nm 2 ] and ν = 0.3 [-]. The initial yield stress does not have to be prescribed. Mentat uses a default value, which is very high (σ v0 = [Pa]). BOUNDARY CONDITIONS 9

10 Prescribe the NEW (STRUCTURAL) boundary conditions. Besides the prescribed boundary conditions symmetry conditions have to be prescribed. Selecting nodes after NODES ADD is done with the mouse. Hold the left mouse button and draw a box around the nodes to be selected. This is easy in this case, because the nodes are located on a straight line. Do some experiments with making boxes, by pressing the Control key during box drawing. Close the selection with (End list). Model definition is now completed. We are going to specify the analysis. First we choose the element type. Element 3 is a plane stress element with 4 nodes. ((QUAD(4))). As can be seen from the list, there are more elements with straight edges and four nodes. JOBS ELEMENT TYPES (Element Types) (ANALISYS DIMENSION) PLANAR SOLID (PLANE STRESS FULL INTEGRATION) 3 OK Enter element list : (All existing) After choosing the element type, the mechanical ((MECHANICAL)) analysis is specified further. JOBS NEW STRUCTURAL PROPERTIES OK - INITIAL LOADS : all applies - JOB RESULTS : stress, strain, von-mises - ANALYSIS DIMENSION : PLAIN STRESS file save as File name: plate1 SAVE 10

11 The model can now be analyzed. This is done by the finite element program MSC.Marc, by submitting the model in JOBS. JOBS RUN SUBMIT 1 MONITOR The program MSC.Marc is started and the model is analyzed. In the (RUN) menu we see some information about the analysis. In the status screen the word Running is seen. When the status indicates Ready the analysis is finished. When everything has worked well, we see in the (RUN) menu the exit number After completion of the analysis, three files are written by MSC.Marc : plate1 job1.log plate1 job1.out plate1 job1.t16 The results of the analysis can be visualized with Mentat. The file with extension.out contains the results in alpha-numerical format (ASCII). It is generally a rather long file and is mostly only opened when an error has occurred during the analysis. The file with extension.t16 contains the results which can be visualized and post-processed in Mentat. The file with the extension.log contains information about the analysis. 11

12 1.2.2 Results We take a look at the analysis results in Mentat. The.t16 file must be opened. This can be done directly from via file open default. It can also be opened from within the (MAIN)-menu in the submenu RESULTS. When MSC.Marc has been started by Mentat (via RUN), the Post file can be opened with OPEN DEFAULT. When Mentat has been closed and we want to open a new Post file, we have to use the OPEN-button. RESULTS file open default (Fill view) (DEFORMED SHAPE) DEFORMED & ORIGINAL Because the deformation is very small as it should be for a linear elastic analysis, we have to enlarge it in SETTINGS and when AUTOMATIC is used the scaling is selected so that a proper deformation is visible. Values of the variables selected in JOB RESULTS can be visualized. Selection is done with SCALAR, TENSOR or VECTOR. SCALAR Equivalent Von Mises Stress OK CONTOUR BANDS It is possible to make a plot of a variable along a path in the model, a PATH PLOT. Here we will plot the y-displacement of the top edge of the plate as a function of the x-distance along that edge. Try other possibilities. PATH PLOT NODE PATH Enter first node in Path-Plot node path : select (lm) node1 Enter next node in Path-Plot node path (1) : select (lm) node2 etc. etc. close with # ( (End list)) ADD CURVE Enter X-axis variable : Arc Length Enter Y-axis variable : Displacement y FIT Close the.t16 file with file close. It is very important to do this because problems may occur when loading a (new) model into Mentat. After closing the.t16 file, the model file is restored automatically. 12

13 1.2.3 Edge load Instead of prescribing the displacement of the right edge of the plate, we can also apply a distributed load. If necessary, we load the model file open plate1. The prescribed displacement is replaced with a prescribed edge load, which is a force per unit of area. In BOUNDARY CONDITIONS we make a new EDGE LOAD. To use it we have to select it in INITIAL LOADS. In that case we remove the prescribed displacement. file open plate1 BOUNDARY CONDITIONS NEW (STRUCTURAL) EDGE LOAD PRESSURE Enter value for p : -1e8 OK The given value must be negative for a tensile load. (EDGES) ADD Enter add apply element edge list : select (EDGES) Remove apply3 in INITIAL LOADS and select apply4. file save as File name: plate2 SAVE Run MSC.Marc and look at results. 13

14 1.3 Axle support with radial load An axle is fixed into a rubber ring in a rigid block as is shown in the figure below. Relevant dimensions are indicated. F R F R 0.05 [m] y x 0.15 [m] z 0.1 [m] Material properties of the axle material are known : Young s modulus : E = [Nm 2 ] Poisson s ratio : ν = 0.3 [-] The rubber material is assumed to be linearly elastic with the next material parameters given : Young s modulus : E = [Nm 2 ] Poisson s ratio : ν = 0.49 [-] The axle is loaded by two radial forces, which are equal : F R = [N]. Bending of the axle is not taken into account. It is assumed that a plane strain deformation state exists in the rubber material. Due to symmetry in the yz-plane w.r.t. the y-axis, only one half of the axle and rubber ring has to be modeled. Which half part are you going to model? 14

15 1.3.1 Modeling and analysis GEOMETRY & MESH The distance between the grid points is chosen in accordance with the dimensions of the model. In the grid points we can locate model (POINTS). SET (Coordinate System) U DOMAIN : -0.1, 0.1 U SPACING : V DOMAIN : -0.1, 0.1 V SPACING : GRID (Coordinate Systen) (Fill view) We define three (ARCS). Because we want to model a solid axle, we define the third arc of the type CENTER/POINT/POINT with radius zero by clicking three times on the central grid point (0, 0, 0). This virtual curve can be used to define a surface. (CURVES) ARC CEN/PNT/PNT (CURVES) ADD Define the half circle of the axle. Define the half circle of the rubber ring. Define the central arc with radius zero. Define the surface between the two half-circles. Define the surface between the axle-circle and the arc in the center point. The two defined (SURFACES) are converted to 4-node elements. The axle surface has some elements near the center point which are triangles. This is no problem : a quad4 element can have two points coinciding. (ELEMENTS) QUAD(4) CONVERT (Operations) DIVISIONS : 8, 8 (Convert) SURFACES (To) ELEMENTS CONVERT Select ring surface Select the axle surface After generating the element mesh we have to use SWEEP (Operations) and CHECK (Operations). 15

16 Geometry parameters, material properties, and boundary conditions (note the symmetry) must be prescribed. GEOMETRIC PROPERTIES NEW (STRUCTURAL) PLANAR PLANE STRAIN PROPERTIES Thickness : 0.1 OK (ELEMENTS) ADD Enter geometry add element list : (All existing) MATERIAL PROPERTIES Prescribe the elastic properties of the axle (material1) and the rubber ring (material2 strangely named material4 by Mentat). BOUNDARY CONDITIONS NEW (STRUCTURAL) Suppress all displacements of the edge of the rubber ring (apply1). Nodes on the symmetry axis can only have a displacement in y-direction (apply2). Prescribe the force F R (apply3). The question is where this POINT LOAD must be applied, i.e. in which node(s). Must we prescribe a total force of N or N in negative y-direction? In JOBS we select element 11, a 4-node linear plane strain element. Then we select the variables, which we want as output in JOB RESULTS and specify the load in INITIAL LOADS. Finally we indicate that the deformation state is plane strain. JOBS ELEMENT TYPES (Element Types) (ANALISYS DIMENSION) PLANAR SOLID (PLANE STRAIN FULL INTEGRATION) 11 OK Enter element list : (All existing) NEW STRUCTURAL PROPERTIES - INITIAL LOADS : apply1, apply2, apply3 - JOB RESULTS : stress, strain, von-mises - ANALYSIS DIMENSION : PLAIN STRAIN file save as File name: support1 SAVE and run MSC.Marc. The results, available in the.t16 files can then be loaded and the results can be visualized. 16

17 1.3.2 Results Looking at Von Mises contour plots it is immediately clear that the stresses in the rubber are much lower than those in the axle. We can visualize the stress state in the rubber in more detail by making the axle invisible in the SELECT menu. view visibility ELEMENTS (Make Invisible) Select the elements of the axle with a box Close with (End list) Push ELEMENTS (Make Visible) and (All existing) again to make everything visible. 17

18 1.3.3 Quadratic elements The analysis is done using linear elements with four element nodes (QUAD(4)). More accurate results, mostly with fewer elements, can be reached, using quadratic elements with eight element nodes. We can adapt the model support1. First file CLOSE the.t16 file. file open support1 GEOMETRY & MESH CHANGE CLASS (Operations) QUAD(8) ELEMENTS (All existing) SWEEP (Operations) BOUNDARY CONDITIONS Some boundary conditions must be adapted, as there are no boundary conditions defined in the new nodes. Adapt the boundary conditions. ELEMENT TYPES (Element Types) (ANALISYS DIMENSION) PLANAR SOLID (PLANE STRAIN FULL INTEGRATION) 27 OK Enter element list : (All existing) Save the model and run MSC.Marc. Results can be visualized in Mentat. 18

19 1.4 Orthotropic plate The figure below shows a square plate, which will be loaded in its plane. It can be assumed that a plane stress state exists in the plate : σ zz = σ xz = σ yz = 0. 2 y 1 α x The plate s material is a matrix in which long fibers are embedded, which all have the same orientation along the direction indicated as 1 in the material 1, 2-coordinate system. Both matrix and fibers are linearly elastic with Young s modulus and Poisson s ratio E m, E f, ν m and ν f, respectively. The volume fraction of the fibers is V. The angle between the 1-direction and the x-axis is α = 10 degrees. In the 1, 2-coordinate system the material behavior for plane stress is given by the next relation between stress and strain components : σ 11 σ 22 σ 12 1 = 1 ν 12 ν 21 E 1 ν 21 E 1 0 ν 12 E 2 E (1 ν 12 ν 21 )G 12 The Young s moduli, Poisson s ratios and shear modulus are defined as : ε 11 ε 22 γ 12 E 1 = σ 11 ε 11 ; E 2 = σ 22 ε 22 ; ν 12 = ε 22 ε 11 ; ν 21 = ε 11 ε 22 ; G 12 = σ 12 γ 12 The material stiffness matrix is symmetric : ν 21 E 1 = ν 12 E 2 which leaves us with four independent material parameters describing this orthotropic behavior. The material parameters can be calculated with the next formulas, which are based on the rule-ofmixtures : E 1 = V E f + (1 V )E m ; ν 21 = V ν f + (1 V )ν m ; 1 = V + 1 V E f E m E 2 = E 2 E f E m V E m + (1 V )E f 1 = V + 1 V G f G m G 12 = G 12 G f G m V G m + (1 V )G f The next table lists numerical values for material parameters where the above formulas are used to calculate E 1, E 2, ν 12 and G 12. E m = 70 GPa E f = 500 GPa E 1 = 242 GPa E 2 = GPa ν m = 0.4 ν f = 0.25 ν 21 = 0.34 G 12 = 38.5 GPa V = 0.4 ν 12 =

20 In Mentat we also have to give values for E 3, G 13, G 23, ν 31 and ν 23. These values are taken to be the same as the ones for the matrix material. E 3 = E 2 = GPa ; G 13 = G 12 = 38.5 GPa ; G 23 = G m = 25 GPa ν 31 = ν 21 = 0.34 ; ν 23 = ν m = 0.4 The sides of the plate have a length 1 m, and the thickness of the plate is 1 cm. The plate is modeled in plane stress with node elements of type 3. Input of orthotropic material parameters in the material coordinate system can be done with the following commands. MATERIAL PROPERTIES NEW STANDARD STRUCTURAL Type : ELASTIC-PLASTIC ORTHOTROPIC E1 E 1 E2 E 2 E3 E 3 NU12 ν 12 NU23 ν 23 NU31 ν 31 G12 G 12 G23 G 23 G31 G 31 (ELEMENTS) ADD Enter add material element list : (All existing) The orientation of the material coordinate system w.r.t. the global coordinate system must also be given. This is done per element in the submenu ORIENTATION. In this submenu we see that there are several TYPEs of orientations. The EDGE-types say that the 1-axis of the local coordinate system is defined by rotation over an ANGLE relative to a particular side of an element. Side 12 can be seen in the plot of the element mesh as a half-arrow on the element edges. It is the side from local node 1 to local node 2. We can also define the rotation of the local 1-axis relative to one of the global coordinate planes. NEW (Orientations) EDGE12 ANGLE 10 (ELEMENTS) ADD Enter add material element list : (All existing) Boundary conditions are defined for tensile loading in x- and y-direction and for simple shear loading in x-direction. In the latter case the y-displacement of the upper boundary is suppressed. Tensile loads and shear load are edge loads of 1 GPa. For all these loadcases, calculate the stress and strain components in the global and the material coordinate system. The calculated stress and strain components can be given in the global coordinate system (default) and in the preferred system. This preferred system coincides with the material coordinate system which is defined above in the NEW (Orientations) submenu. 20

21 1.5 Circular disc with prescribed radial edge displacement The next figure shows a circular disc with outer radius b having a central circular hole with radius a. a b For linear isotropic material behavior the analytical solution for radial displacement and radial and tangential stresses are given as a function of the radius r : u r = c 1 r + c 2 r ; σ rr = E 1 ν 2 [ (1 + ν)c 1 (1 ν) c 2 r 2 ] ; σ tt = E 1 ν 2 [ (1 + ν)c 1 + (1 ν) c 2 r 2 ] where E is Young s modulus and ν is Poisson s ratio. The integration constants c 1 and c 2 are determined by the boundary conditions. For u r (r = b) = u b ; σ rr (r = a) = 0 we have : c 1 = (1 ν)b (1 ν)b 2 + (1 + ν)a 2 u (1 + ν)a 2 b b ; c 2 = (1 ν)b 2 + (1 + ν)a 2 u b The next parameter values are given : a = 0.01 [m] b = 0.1 [m] d = 0.01 [m] E = [Pa] ν = 0.3 [-] u b = [m] For both a plane stress or a plane strain situation deformation and the stresses can be calculated with planar elements. Due to the axial symmetry of geometry and loading, a representative part of the disc can be modeled and analyzed. In this case we model and analyze a quarter of the plate as shown in the figure below. y y x z x 21

22 1.5.1 Modeling and analysis We start in the GEOMETRY & MESH. We will use the Cartesian coordinate system. Proceed as in the example Plate with a central hole and make the geometry using two arcs and a ruled surface between them. Use CONVERT (Operations) to make the mesh with QUAD(4) elements: GEOMETRY & MESH GEOMETRY & MESH (ELEMENTS) QUAD(4) CONVERT (Operations) DIVISIONS : 10, 10 (Convert) SURFACES (To) ELEMENTS Select (SURFACE). Close with (End list). After defining the element mesh, SWEEP (Operations) has to be used to remove coinciding nodes and elements. It is recommended to go to the CHECK (Operations) menu and check whether there are elements UPSIDE DOWN. When this is the case these elements must be flipped. It is recommended to prevent future drawing of (CURVES) and (SURFACES) in the view Plot Control-menu. In this example we want to analyze a plane stress situation. This has to be selected when we specify the thickness. GEOMETRIC PROPERTIES NEW (STRUCTURAL) PLANAR PLANE STRESS PROPERTIES Thickness : 0.01 OK (ELEMENTS) ADD Enter geometry add element list : (All existing) In BOUNDARY CONDITIONS we have to suppress the X-displacement on the vertical symmetry section and the Y -displacement on the horizontal symmetry section. To prescribe the radial displacement on the outer edge, a local coordinate system has to be used, which can be defined in TRANSFORMS. 22

23 TABLES & COORD. SYST. (Coordinate System) (New) CYLINDRICAL (R,Phi,Z) Give three points to define the axis of the cylindrical coordinate system, eg. [0 0 0], [0 0 1] and [1 0 0]. Now we need to transform the nodes on the outer edge to the cylindrical system. TOOLBOX TRANSFORMATIONS (General) NEW Coordinate System COORDINATE SYSTEM crdsyst1 NODES ADD select nodes at outer edge Insert the material properties as usual. Although the material is isotropic and there is no real need for a material coordinate system, we will define one. The reason is that we want MSC.Marc to calculate the radial and tangential stresses, which are the stresses in the preferred system. This coordinate system can be defined in all elements analogously, due to the fact that the elements are all oriented in the same way. In the submenu ORIENTATION we use EDGE12 for all elements with ANGLE zero. The preferred stress component-11 is then the tangential stress and the component-22 the radial stress. MATERIAL PROPERTIES (Orientations) EDGE12 ANGLE 0 ADD (All existing) (New) We are going to specify the analysis. First we choose the element type. Element 3 is a plane stress element with 4 nodes. ((QUAD(4))). As can be seen from the list, there are more elements with straight edges and four nodes. JOBS ELEMENT TYPES (Element Types) (ANALISYS DIMENSION) PLANAR SOLID (PLANE STRESS FULL INTEGRATION) 3 OK Enter element list : (All existing) After choosing the element type, the mechanical ((MECHANICAL)) analysis is specified further. 23

24 NEW STRUCTURAL PROPERTIES - INITIAL LOADS : all applies - JOB RESULTS : stress, preferred stress, strain, von-mises - ANALYSIS DIMENSION : PLANE STRESS file save as File name: Disc1 SAVE 24

25 1.5.2 Results After analysis the result can be visualized. The deformation can be shown as a deformed mesh. CONTOUR BANDS can be set to visualize displacements, strains, stresses and preferred stresses. Numerical values can also be viewed in nodal points. In a PATH PLOT the stresses can be plotted against the radius. The result is shown in the figure below. 6 x σ rr σ tt σ zz 4 σ [Pa] r [m] 25

26 1.6 Rotating solid disc A solid circular disc with radius b and uniform thickness, is rotated with a constant radial velocity ω [rad/s] about its central axis perpendicular to its plane. The material is isotropic and linearly elastic with Young s modulus E and Poisson s ratio ν. The density is ρ. r ω b z r The general solution is : σ rr = E 1 ν c 1 E 1 + ν c 2 r ν ρω 2 r 2 ; σ tt = E 8 1 ν c 1 + E 1 + ν c 2 r ν ρω 2 r 2 8 u = c 1 r + c 2 r where c 1 and c 2 are integration constants. For a solid disc the displacement at r = 0 must remain finite, while the stress boundary condition is σ rr (r = b) = 0. Integration constants can now be determined : c 1 = 3 + ν 1 + ν 1 ν 2 The next parameter values are given for this example : E 1 8 ρω2 b 2 ; c 2 = 0 b = 0.05 [m] f = 6 [c/s] E = 200 [GPa] ν = 0.3 [-] ρ = 7500 [kg/m 3 ] ( ) 1 ν 2 ρ E 26

27 1.6.1 Modeling, analysis and results In the menu MATERIAL PROPERTIES the density of the material has to be specified. Again the axial symmetry allows the analysis of the quarter plate, using proper boundary conditions on the section lines. The rotation is modeled as a CENTRIFUGAL LOAD in BOUNDARY CONDITONS. The rotation rate is given in rotations (= cycles) per second. The rotation axis has to be defined by two points in the three-dimensional space. BOUNDARY CONDITIONS NEW (STRUCTURAL) CENTRIFUGAL LOAD Angular Frequency (Cycles/Time) 6 Axis Of Rotation X1 0 Y1 0 Z1 0 X2 0 Y2 0 Z2 1 OK ELEMENTS ADD Enter element list : (All existing) Stress components for the analytical solution are plotted against the radius in the figure below. 12 x σ rr σ tt 8 σ [Pa] r [m] 27

28 2 Axisymmetry 2.1 Background : Axisymmetric modelling Many devices and components have a geometry which is symmetric w.r.t. an axis and are thus called axisymmetric. It is obvious that the position of material points of such an object can best be described in a cylindrical coordinate system. Coordinates are the distance z measured along the axis of symmetry, the distance r from and perpendicular to that axis and an angle θ, which indicates the position in circumferential direction, with respect to an arbitrary starting point (θ = 0 o : 0 θ 360 o (= 2π[rad]) (see figure below). z z P θ r r r When the load is independent of the angle θ, it is also called axisymmetric. With the additional assumption that there is no rotation around the z-axis, we only have to consider and model the crosssection of the object, when we want to analyze its mechanical behavior (see figure). Modeling such axisymmetric problems in MSC.Marc, the z-axis is oriented in the x-direction and the r-axis in the y-direction. Only one half of the geometry of the cross-section must be modeled and it must be located in the half-space y > 0, as is shown in the figure below. z y = r-as x = z-as r r The stress state in a material point is characterized by four stress components, which are indicated on the faces of a stress cube in the figure below. This stress cube is shown two times, once in the cross-section and once three-dimensional. 28

29 The deformation is described by four strain components, which are defined in accordance with the stress components. For isotropic linear elastic material behavior we have : σ rr 1 ν ν 0 ν ε rr σ tt σ zz = E ν 1 ν 0 ν ε tt (1 + ν)(1 2ν) ν 0 ε zz σ rz ν ν 0 1 2ν ε rz σ rr y = r-as P σ rr σ rz σ tt σ zr σ zz σ tt σ rz σ zr σ zz x = z-as Analyzing axisymmetric problems with the finite element method implies that axisymmetric elements must be used, which are defined in the cross-section. The deformation of such an element is defined by the displacement in r- and z-direction of the element nodal points. We must be aware that a nodal point is in fact a nodal ring as is indicated in the figure below. y = r-as x = z-as In the cross-section of the element the axial and radial displacement of a point is interpolated between the displacements of the element nodes. As with the planar elements, linear or quadratic interpolation can be used. Again we have 4-node and 8-node elements with 4 and 9 integration points respectively. In MSC.Marc these elements are indicated as element type 10 and 28. The cross-section of a 4-node element is shown in the figure below. 29

30 4 3 4 η 3 ξ η ξ In MSC.Marc/Mentat the strains in the integration points are defined as follows : Stress components are defined in the same way. strain 1 = global zz-strain = ε zz strain 2 = global rr-strain = ε rr strain 3 = global tt-strain = ε tt strain 4 = global rz-strain = γ rz 30

31 2.2 Axle support with axial load The load on the axle is assumed to be an axial force : F A = [N], as is indicated in the figure below. F A y x z The geometry and the load allow for an axisymmetric Modeling and analysis, as indicated in the figure below. y x manchet as F A NB. : The x-axis is the axial axis and the positive y-axis is the radial axis. This is the usual definition in MSC.Marc/Mentat. 31

32 2.2.1 Modeling, analysis and results GEOMETRY & MESH Define the Cartesian coordinate system with grid point spacing [m]. (CURVES) LINE (CURVES) ADD Define three (LINES) parallel to the x-axis between 0.05 < x < 0.05 for y = 0, y = and y = Define (SURFACES) between these (CURVES). Convert the surfaces in elements of type QUAD(4) with DIVISIONS [8, 4]. SWEEP and CHECK the mesh. BOUNDARY CONDITIONS NEW (STRUCTURAL) Fix the top edge of the rubber part (apply1). Nodes on the axial axis (y = 0) can only have a displacement in the x-direction (= axial) (apply2). Apply the axial force in one point of the axle (apply3). Material properties of axle and rubber are prescribed in the usual way. Geometric parameters are not relevant for this axisymmetric analysis. In JOBS we select element 10, a 4-node quadrilateral axisymmetric element. We indicate that the analysis is axisymmetric. JOBS ELEMENT TYPES (Element Types) (ANALASYS DIMENSION) AXISYMMETRIC SOLID (FULL INTEGRATION) 10 OK Enter element list : (All exisiting) NEW STRUCTURAL PROPERTIES - INITIAL LOADS : apply1, apply2, apply3 - JOB RESULTS : stress, strain, von-mises - ANALYSIS DIMENSION : AXISYMMETRIC Save the model as Support2 and run MSC.Marc. The results, available in the.t16 file can be loaded into Mentat and visualized. Change the model to use quadratic 8-node elements. element 28. For axisymmetric analyses in MARC this is 32

33 2.3 Circular discs A circular disc can be modeled with axi-symmetric elements. In fact these elements are ring -elements. What we model on the screen is the two-dimensional cross-section of the model. In MSC.Mentat the axial-direction always coincides with the horizontal x-axis. Only the part of the cross-section above this axis is modeled and converted in a finite element mesh. y = radial x = axial z = tangential For a disc with uniform thickness, this half-cross-section is just a rectangle, with the thickness as the dimension in the x-direction (= axial direction) and the inner ad outer radius as the dimensions in the y-direction (= radial direction). It would of course be no problem to model a disc with a nonuniform thickness. Boundary conditions have to be applied to prevent rigid body movement. It is obvious that there can not be a rigid body movement in radial direction, but in axial direction we have to prevent the rigid body displacement. When the disc is solid, it is advised to prescribe the radial displacements on the axis to be zero. Edge displacements and edge loads can be prescribed straightforwardly in BOUNDARY CONDI- TONS. Rotation leads to radial accelerations, which are modeled in MSC.Mentat as body forces in CENTRIFUGAL LOAD. We have to provide the rotational velocity as number of rotations (= cycles) per second. We also have to define the axis of rotation. This is a straight line and thus defined with two points in space. Below we have the x-axis as the rotation axis and the number of cycles/second is 6. BOUNDARY CONDITIONS NEW (STRUCTURAL) CENTRIFUGAL LOAD Angular Frequency (Cycles/Time) 6 Axis Of Rotation X1 0 Y1 0 Z1 0 X2 1 Y2 0 Z2 0 OK ELEMENTS ADD Enter element list : (All existing) 33