1. Introduction to Finite Element Method (FEM) In order to analyze an engineering system, a mathematical model is developed to describe the system. While developing the mathematical model, some assumptions are made for simplification. Finally, the governing mathematical expression is developed to describe the behaviour of the system. The mathematical expression usually consists of differential equations and given conditions. The best way to solve any physical problem governed by a differential equation is to obtain the analytical solution. There are many situations, however, where the analytical solution is difficult to obtain. The region under consideration may be so irregular that it is mathematically impossible to describe the boundary. The configuration may be composed off several different materials whose regions are mathematically difficult to describe. Problems involving anisotropic materials are usually difficult to solve analytically, as are equations having non-linear terms. A numerical method can be used to obtain an approximate solution when an analytical solution cannot be developed. Especially, the finite element method has been one of the major numerical solution techniques. One of the major advantages of the finite element method is that a general purpose computer program can be developed easily to analyze various kinds of problems. In particular, any complex shape of problem domain with prescribed conditions can be handled with ease using the finite element method. The underlying premise of the method states that a complicated domain can be sub-divided into a series of smaller regions in which the differential equations are approximately solved. By assembling the set of equations for each region, the behaviour over the entire problem domain is determined. Each region is referred to as an element and the process of subdividing a domain into a finite number of elements is referred to as discretization. Elements are connected at specific points, called nodes, and the assembly process requires that the solution be continuous along common boundaries of adjacent elements.
When modeling a problem using a finite element program, it is very important to check whether the solution has converged. The word convergence is used because the output from the finite element program is converging on a single correct solution. In order to check the convergence, more than one solution to the same problem are required. If the solution is dramatically different from the original solution, then solution of the problem is not converged. However, if the solution does not change much (less than a few percent difference) then solution of the problem is considered converged. Currently, two types method are used to demonstrate the numerical convergence of the solution : 1). h – method 2). p – method The h- and p- versions of the finite element method are different ways of adding degrees of freedom (dof) to the model.
h-method –> The h-method improves results by using a finer mesh of the same type of element. This method refers to decreasing the characteristic length (h) of elements, dividing each existing element into two or more elements without changing the type of elements used. p-method –> The p-method improves results by using the same mesh but increasing the displacement field accuracy in each element. This method refers to increasing the degree of the highest complete polynomial (p) within an element without changing the number of elements used. The difference between the two methods lies in how these elements are treated. The h-method uses many simple elements, whereas the p-method uses few complex elements.
In general, a finite element solution may be classified into the following three stages. This is a general guideline that can be used for setting up any finite element analysis. 1. Preprocessing: defining the problem The major steps in preprocessing are given below: • Define key points/lines/areas/volumes • Define element types and material/geometrical properties • Mesh lines/areas/volumes as required 2. Solution: assigning loads, constraints and solving In the solution level, the loading conditions such as point load or pressure and constraints or boundary conditions are specified and finally the resulting sets of equations are solved. 3. Post processing: further processing and viewing of the results This stage provides different tools to view the results including: • Lists of nodal displacements • Element forces and moments • Deflection plots • Stress contour diagrams
In this level our job as an engineer is to interpret the output results and verify their accuracy. Typically with increasing number of elements (using finer mesh), we should expect to more accurate results.
Introduction to ANSYS ANSYS is a powerful general purpose finite element modeling package to numerically solve a wide variety of mechanical, structural and non-structural problems. These problems include: static/dynamic structural analysis (both linear and non-linear), heat transfer and fluid problems, as well as acoustic and electro-magnetic problems Ansys, as most of other FEA packages, has two major interfaces: an old one (Classical GUI, now named “Mechanical APDL”) and a new one (Workbench, also named “Ansys Mechanical”). Mechanical APDL is not so convenient to use, but is very useful when you want to solve some specific problem which is not ordinary. Mechanical APDL could be driven via text commands, thus it’s liked by experienced users. Workbench is more user friendly, and allows the easy usage of the geometry from modern CAD packages, maintaining the associativity through the whole FEA process. However, Workbench has few limitations; it is not as flexible as APDL. In APDL we can program everything that can imagine, for instance we can implement our own finite elements. But for ordinary problems, Workbench is quicker and simpler to use.
1. Introduction to ANSYS Mechanical APDL ANSYS Mechanical APDL is a very powerful tool that allows the user to explore the versatile capabilities of the software fully. APDL stands for ANSYS Parametric Design Language, a scripting language that we can use to automate common tasks or even build our model in terms of parameters (variables). In the mechanical structure mechanics analysis, there are generally two kinds of analysis methods: (1) Command flow mode. (2) GUI mode. In two kinds of modes, the command flow mode is provided by ANSYS APDL analysis method. ANSYS Mechanical APDL Graphical User Interface (GUI) is split into four main areas. The graphics area, the utility menu, the main menu and the ANSYS toolbar.
Pre-processor Within the pre-processor the model is set up. It includes a number of steps and usually in the following order: Build geometry. Depending on whether the problem geometry is one, two or three dimensional, the geometry consists of creating lines, areas or volumes. These geometries can then, if necessary, be used to create other geometries by the use of boolean operations. The key idea when building the geometry like this is to simplify the generation of the element mesh. Hence, this step is optional but most often used. Nodes and elements can however be created from coordinates only. Define materials. A material is defined by its material constants. Every element has to be assigned a particular material. Generate element mesh. The problem is discretized with nodal points. The nodes are connected to form finite elements, which together form the material volume. Depending on the problem and the assumptions that are made, the element type has to be determined. Common element types are truss, beam, plate, shell and solid elements. Each element type may contain several subtypes, e.g. 2D 4-noded solid, 3D 20-noded solid elements. Therefore, care has to be taken when the element type is chosen.
The element mesh can in ANSYS be created in several ways. The most common way is that it is automatically created, however more or less controlled. For example you can specify a certain number of elements in a specific area, or you can force the mesh generator to maintain a specific element size within an area. Certain element shapes or sizes are not recommended and if these limits are violated, a warning will be generated in ANSYS. It is up to the user to create a mesh which is able to generate results with a sufficient degree of accuracy. PREP7 pre-processor is used to define the element types, element real constants, material properties, and the model geometry. Solution processor Here you solve the problem by gathering all specified information about the problem: Apply loads: Boundary conditions are usually applied on nodes or elements. The prescribed quantity can for example be force, traction, displacement, moment, rotation. The loads may in ANSYS also be edited from the pre-processor. Obtain solution: The solution to the problem can be obtained if the whole problem is defined. Postprocessor Within this part of the analysis you can for example: Visualize the results: For example plot the deformed shape of the geometry or stresses. List the results: If you prefer tabular listings or file printouts, it is possible Two postprocessors are available: (1) POST1: The general postprocessor is used to review results at one substep (time step) over the entire model or selected portion of the model. (2) POST26: The time history postprocessor is used to review results at specific points in the model over all time steps. There are two methods to use ANSYS; Graphical Interface and Command File Coding. The graphical user interface or GUI follows the conventions of Windows based programs. This method is probably the best approach for new users. The command approach is used by professional users. It has the advantage that an entire analysis can be described in a small text file, typically in less than 50 lines of commands. This approach enables easy model modifications and minimal file space requirements. Types of Analysis with ANSYS Structural Analysis Thermal Analysis CFD Analysis Electromagnetic Analysis Field and Coupled-Field Analysis
STRUCTURAL ANALYSIS Structural analysis is probably the most common application of the finite element method. The term structural (or structure) implies not only civil engineering structures such as bridges and buildings, but also naval, aeronautical, and mechanical structures such as ship hulls, aircraft bodies, and machine housings, as well as mechanical components such as pistons, machine parts, and tools Types of Structural Analysis We can perform the following types of structural analyses: • Static Analysis -- Used to determine displacements, stresses, etc. under static loading conditions. Both linear and nonlinear static analyses. Nonlinearities can include plasticity, stress stiffening, large deflection, large strain, hyper elasticity, contact surfaces, and creep. • Modal Analysis -- Used to calculate the natural frequencies and mode shapes of a structure. Several mode extraction methods are available. • Harmonic Analysis -- Used to determine the response of a structure to harmonically time-varying loads. • Transient Dynamic Analysis -- Used to determine the response of a structure to arbitrarily time-varying loads. All nonlinearities mentioned under Static Analysis above are allowed. • Spectrum Analysis -- An extension of the modal analysis, used to calculate stresses and strains due to a response spectrum or a PSD input (random vibrations). • Buckling Analysis -- Used to calculate the buckling loads and determine the buckling mode shape. Both linear (Eigen value) buckling and nonlinear buckling analyses are possible. • Explicit Dynamic Analysis -- This type of structural analysis is available via the ANSYS LS-DYNA product, which provides an interface to the LS-DYNA explicit finite element program. Explicit dynamic analysis calculates fast solutions for large deformation dynamics and complex contact problems. Several special-purpose structural analysis capabilities are available: • Fracture mechanics • Composites • Fatigue • Beam analyses and cross sections Type of Elements in Structural Analysis
a)Static Analysis 1. Axially loaded stepped bars
The structure consists of two stepped bars. An axially load, P= 15 kN is loaded as shown in figure. Take, Esteel = 20 x 106 N/cm2, Eal = 7.6 x 106 N/cm2. Poisson, ratio for steel & aluminum are 0.27 and 0.3 respectively. Determine the following. i. Nodal displacements ii. Stress in each bar iii. Reaction forces
2. 2DTruss Determine the axial force in each member of the truss loaded as shown in the figure given below. Also find the support reactions and nodal deflections. Take E = 200GPa, ν = 0.3, Area = 5000 mm 2
3. Beams
Construct the Shear force and bending moment diagrams for the beam shown and find the maximum deflection. Assume rectangular c/s area of 0.2 m x 0.3 m, Young’s modulus of 210 GPa, Poisson’s ratio 0.27. What is the maximum bending moment? Locate the points of contraflexure, if any. Also find the magnitude & location of the maximum deflection of the beam.
4. Frames In the frame shown in figure below members 1, 2, 3 ,4, and 5 are aluminum beams and have a hollow square cross-section with outer wall size 2 cm and wall thickness 2 mm, and member 6 is a steel wire with diameter 2 mm. The applied force is 100 N. Find the loads and stresses in the members of the frame.
5. Plain Stress Analysis Plane Stress: Plane stress assumes zero stress in the Z direction. Valid for components in which the Z dimension is smaller than the X and Y dimensions. Z-strain is non-zero. Optional thickness (Z direction) allowed. Used for structures such as flat plates subjected to in-plane loading, or thin disks under pressure or centrifugal loading. The corner angle bracket is shown below. The upper left hand pin-hole is constrained around its entire circumference and a tapered pressure load is applied to the bottom of lower right hand pin-hole. Compute Maximum displacement, Von-Mises stress.
6. Plain Strain Analysis Plane Strain: Plane strain assumes zero strain in the Z direction. Valid for components in which the Z dimension is much larger than the X and Y dimensions. Z-stress is non-zero. Used for long, constant cross-section structures such as structural beams. Examine the expansion of a pressure vessel due to an internal pressure. Determine the principal stresses in the pressure vessel due to the applied loading and boundary conditions. Use a two-dimensional plane strain element for this analysis. The vessel is made from steel (E = 207 Gpa, v = 0.27) and the internal pressure is 10,000 Pa.
A plane strain analysis assumes that all strains occur in the X-Y and there are no strains in the Z-direction. This does allow for stresses in the Z-direction as stresses may be required to prevent displacement in the Z-direction. We can use a plane strain analysis in this case by taking a slice through the pressure vessel at
its midpoint. We are assuming that the mid point doesn't move and we are modeling this non-vertically moving mid plane.
Figure 2 shows an overview of the plane strain model of the pressure vessel. The model on the left hand side is a full plane strain model of a slice through the pressure vessel. By recognising the symmetry in the problem we can reduce this model to a 1/4 symmetry plane strain model as shown on the right hand side of figure 2. In this tutorial, we will build the 1/4 symmetry plane strain model as it easily allows for the application of boundary conditions (which are not so easily applied to the full model).
7. Axisymmetric Analysis Axisymmetric: Axisymmetry assumes that the 3-D model and its loading can be generated by revolving a 2-D section 360° about the Y axis. Axis of symmetry must coincide with the global Y axis. Negative X coordinates are not permitted. Y direction is axial, X direction is radial, and Z direction is circumferential (hoop) direction. Hoop displacement is zero; hoop strains and stresses are usually very significant. Used for pressure vessels, straight pipes, shafts, etc. Consider the problem of finding the stresses in a thick open-ended cylinder with an internal pressure (such as a pipe discharging to the atmosphere). The steel cylinder below has an inner radius of 5 inches and an outer radius of 11 inches.
A cross section is shown below
The length of the object is arbitrary and represents a segment of a long, open-ended cylinder. The Y axis is the axis of symmetry. The cylinder can be generated by revolving a rectangle 6 inches wide and of arbitrary height 360 degrees about the Y axis. Since the height of the segment considered is arbitrary, we will use a segment 1 inch in height for the finite element model. The geometry is shown below.
8. Analysis of thin walled channels ( Use of Shell type elements) Thin plate or shell elements find applications in many practical situations such as structural angle brackets, automotive hoods or trunk lid attachments and similar thin components with fairly complex shape. Find the deformations and stress in a cantilevered thin walled channel subject to an end load. The 6 x 2 x 0.2 inch thick steel channel shown below is 30 inches long and has a total load of 1 lbf divided among the four corner points on the right end as shown.
9. Analysis of a bicycle frame. ( Use of Pipe Elements) The problem to be modeled is a simple bicycle frame shown in the following figure. The frame is to be built of hollow aluminum tubing having an outside diameter of 25mm and a wall thickness of 2mm.
10. Static Analysis with 3D elements - Pillow Block
11. Model the object using tetrahedral 10 node element. Assume the structure as made of steel with modulus of elasticity E = 200GPa. The object is fixed around the inner surface of the hole. The object is loaded uniformly (1000 N/cm2) along the top surface of the extended beam. Plot the deformed shape. Determine the principal stress and Von-mises stress.
b)Dynamic Analysis
DYNAMIC ANALYSIS: MODAL AND TRANSIENT ANALYSES
Modal Analysis of Cantilever beam for natural frequency determination. Modulus of elasticity = 200GPa, Density = 7800 Kg/m3
Free Vibration: Solid Structure (Modal Analysis) Obtain natural frequencies and mode shapes of free squared isotropic plate using ANSYS Modal Analysis
Length (a) Width (b) Thickness (h) Material: Aluminium Young’s modulus Density Poisson’s ratio
0.3m 0.3m 6.35*10-3m 73.1 GPa 2780 Kg/m3 0.3
Modal or Dynamic Analysis of an Airplane Wing The wing is of uniform configuration along its length and its cross-sectional area is defined to be a straight line and a spline as shown. It is held fixed to the body of the airplane on one end and hangs freely at the other. Find the wing's natural frequencies and mode shapes. The dimensions of the wing are as shown above. The wing is made of low density polyethylene with a Young's modulus of 38x103 psi, Poisson's ration of 0.3, and a density of 1.033E-3 slugs/in3 .
Determine transient response of mass 2 of the system shown in Figure
1. Introduction to Ansys Workbench ANSYS Workbench is a simulation platform that enables and solves a wide range of engineering problems using the FEA. It provides access to the ANSYS family of design and analysis modules in an integrated simulation environment. The workbench interface is composed primarily of a Toolbox region and a Project Schematic region.
