Advanced mechanical design analysis - Essay Example

ADVANCED MECHANICAL DESIGN ANALYSIS Part Introduction The structure given for analysis is simply supported using the rollers at both the ends.The beam element consists of three separate pieces. Two pieces of longer span both connected to the end portions while the middle piece, the smaller one, connects both the longer pieces using an assembly of pin joints. The loads to which the beam is subjected consist of three concentrated loads, one having a magnitude of 26 KN acting at the centre while the loads of magnitude 18 KN and 20 KN acting at 0.9 m and 1.1 m from both the ends. The connection provided in the members undertakes the roles of transfer of loads from one member to the other besides acting as means of support. As the connections are specified as pinned the members are given the freedom to rotate. The pinned connections are provided to facilitate the large deflection possible in the bridge element. In case stiff connection like welded connections are provided it would create additional stress in the body due to the induced rigidity of the connections. As the rigidity in the pinned connections are every low the members wouldn't experience any stress concentrations created by rotational effects or reaction forces. The objective of the problem is to determine the stresses in the four different beam cross sections and also the deflection caused when different beam cross sections are used. The beam theory is adopted whenever the structural analysis is required on a member characterized by significant span, L, comparable depth, D, and width, W. Thus the bending theory says that M/I = f /y= E/R. Where M is the maximum bending moment, I is the moment of Inertia, f is the maximum fibre stress, y is the depth to the outermost fibre from the neutral axis, E is the modulus of elasticity and R is the radius of gyration. The stresses and the deflection caused in the beam from the external loads depend on length of the beam or span and the depth of the beam. Also, the stresses and the deflections in the beam shall be expressed as the functions of shear force and bending moment. Thus it is very necessary to obtain the shear force and bending moment diagram. For the given problem the shear force and bending moment diagram shall be determined, it is shown separately as the hand calculations. The finite element analysis was undertaken using the MYSTRO AND LUSAS finite element software where the FEA models were formed using the command files than CAD interface systems. This approach even though was a little tedious and complex was described by separate set of commands. The success of FEA technique lies in the planning and preparation of suitable mesh arrangement for the problem to be addressed. The objective is to identify the most optimum mesh size arrangement that gives the most accurate stress distribution with in reasonable time for analysis. The region of high stress in the beam could be meshed very fine and those outside the region could be discretised using coarse mesh. The elements used for the discretisation is arrived based of variety of trial models. The elements that are used in the final analysis is three dimensional hexahedral elements which has eight nodes with each node having three degrees of freedom. These elements are geometrically linear and has an assumed internal strain field. They are sued to model the beam flanges and connecting plates. The beam webs are modelled using the three or four noded elements having 5 degrees of freedom and the connections are also modelled using hexahedral elements. The material used is assumed to behavior in the linear field. The modulus of elasticity and poisons ratio given in the problem could be used for the analysis. The beam is restrained to move in the vertical direction. The roller supports provided would give the model a free movement along it longitudinal direction. Three dimensional beam element refers to the element that have two nodes and each node having six degrees of freedom. Three translation motion and three rotational motion are the degrees of freedom for this element at each node. A typical sketch of a 3D beam element is as shown in the figure. Cross section Factor of safety Cross section 1 9.7 Cross section 2 4.2 Cross section 3 5.0 Cross section 4 4.6 Moment of Inertia (m4) Span (m) Deflection (mm) Cross section 1 0.00065 2.8 m 0.0022 Cross section 2 0.00029 2.8 m 0.0051 Cross section 3 0.00034 2.8 m 0.0043 Cross section 4 0.00031 2.8 m 0.0047 For crossection 1 For cross section 2 For Cross section 3 For cross section 4 The finite element solution is determined based on the above mentioned discretised elements The beam element chosen is two dimensional in nature. The characteristics of the beam elements are as follows The displacement analysis could be undertaken using a Finite Model body by appropriately dividing the system into an equivalent system of large number of smaller bodies or units (finite elements). The elements are connected between themselves at the common points across the elements. The method consists of arriving at a set of algebraic equations, which could be solved for unknown nodal quantity. The stresses and strains are expressed in relation with the nodal values of primary quantity. A typical FEA software consists of three different separate set of operations like preprocessing, FEA Solver and Postprocessing. The section chosen is the cross-section no 1 Part 2 A typical two dimensional beam element is shown in the figure below. Each node is expected to have three degrees of freedom. Two translational and one rotational degrees of freedom are assigned to the nodes. I and J represents the nods of the element and 1,2 and 3 represents the degrees of freedom assigned to the node I while 4,5, and 6 represents the degrees of freedom assigned to node J. among them 1,2 , 5 and 6 are the translation components and 3 and 6 are the rotational components. These models are often used in building linear elastic model of the beam element. The elementary beam theory states that And The deflection in the beam is represented as = The curvature of the beam is represented as Where The deflection computed is = 0.0037 mm In order to limit the deflection to 2.5 to 3 mm the options available is the increase the moment of inertia of the cross section. This could be achieved by providing additional plates on the top and bottom surface of the beam. Increasing the moment of inertia by 30 percent would help to reduce the deflection within the desired limits. Part 3 Stress analysis, using finite element method, has been widely used in many engineering disciplines. Discuss the following points: a) The method of simulation construct during the analysis. All finite element computations would result in the element stiffness matrix in all cases would be of the form Thus all the simulation problems in the stress analysis would lead to the solution to large number of algebraic equations. The number of equations would depend in the number of nodes or the extend of discretisation carried out on the system. The solution of equation gives the nodal displacements. From the known value of displacements, the strain in each member is estimated. The known values of the material properties like modulus of elasticity, the stresses are computed by multiplying the strains with these constants. The results obtained in this manner is compared with the results obtained under different loading conditions by changing the boundary conditions that are incorporated in the column matrix on the right hand side. b) Possible mistakes that could be made when conducting simulation. The mistakes that could be conducted while undertaking any simulation studies on stress analysis are as follows Poor Idealization of the system. The system being analyzed must be well understood towards choosing the right element for the simulation exercise. Thus this error would give poor stress distribution, which could have been obtained by selection of most acceptable elements. Incorporating the boundary conditions - Every system would have particular boundary conditions and the results obtained in the simulation would be based on this articular boundary conditions. Clearly depicting the boundary conditions relevant to the problem is a very vital step in the simulation programme. Error in the selection of material constants - The material properties allocated during the simulation process shall be undertaken carefully inorder to obtain more reliable results. c) The use of mesh size and types. Finite element analysis is an approximate method that involves discretising the entire problem domain into finite elements and later combining the element properties to obtain a global solution. The global system is analyzed by incorporating the boundary conditions to get the required solution (Cooks et al, 1989). Finite element analysis uses a network of grids obtained from connecting several points called nodes specified on the problem body or surface. These grids are called as meshes. The characteristics about the materials and its properties are programmed in the meshes which will be used to determine the system behavior under externally imposed conditions. The distribution of meshes is based on the type of the problem. If a location requires very refined analysis this region could have very fine meshes (Widas, 1997). A typical finite element application is shown as below. Here, the finite element model of the problem is shown in figure 1. The material is divided into small triangle shaped elements and hence it can be said that the body is meshed using triangular elements. The FEA software consists of different type of elements that could be used in the analysis. The element refers to the geometrical shapes that are used to mesh the system. Some of the types of elements used usually used are rod elements, beam elements, plate/shell/composite elements and Solid elements Figure 1 (http://www.sv.vt.edu/classes/MSE2094_NoteBook/97ClassProj/num/widas/history.html) The FEA software consists of different type of elements that could be used in the analysis. The element refers to the geometrical shapes that are used to mesh the system. Some of the types of elements used usually used are rod elements, beam elements, plate/shell/composite elements and Solid elements Elements ; the different type of elements used for the finite element analysis is (i) Linear bar element : These elements have only linear dimension and is used to represent with nodes representing one unknown. Quadratic bar element: These elements are used to represent more complex curvatures. (ii) Rectangular element / Bilinear element (iii) Triangular element (iv) Solid element Meshing The process of dividing the problem element into the fine grids is known as meshing. The mesh would consist of similar elements or combination of different elements. Like for a irregular body the using square or rectangular elements might be useful in proper division of the domain space into small elements. This disadvantage shall be overcome by different type of elements like triangular elements to wards the boundary or irregular side for proper geometrical matching between the finite element model and the actual physical body. Another important aspect that need to be considered is the size of meshes that are used for the process of discretisation. The size of meshes could be course , large and well spread, or fine , small size and closer placed. The choice of the size depends on variery of factors. The type of the element chosen for a particular problem, nature of the variation of the unknowns, computational efficiency etc. For example, the choice between linear bar element and quadratic bar element is based on the experience of the analyst. Problems might require few number of complex elements or larger number of simpler element no strict guideline exist in this aspect. The experience gained from analysis would be the prime factor that would help in the proper choice of elements for a particular type of problem (Chandrupatla and Belegundu, 2007) (d) The possible different between the hand calculation and simulation. The finite element model helps in the creation of construct the element level matrices for the particular problem. The size of the matrix would depend on the type of the element chosen. For linear bar element with one unknown at each node the size of the matrix is 2 2. While for a bilinear element like rectangle with two unknowns at each node the size of the matrix would be 8 8. The element matrices hence formed shall be combined to get the global equations with appropriate boundary conditions incorporated in it. The equations are hence solved using any standard solver. The results obtained as presented using any post processing software attached to the finite element packages for better appreciation of the results (Cooks et al, 1989). Thus the simulation exercise undertaken using Finite Element Approach helps to obtain the stress distribution in the chosen system considering the actual material parameters without any approximations in the materials properties. On the other hand, the hand calculation helps us to obtain the results subjecting the given system to considerable level of simplification only. Thus the simulated system also help to get the behavior of the system under wide range of scenarios very efficiently than the hand calculations. References Chandrupatla, T R and Belegundu, A D (2007) , Introduction to finite elements in engineering, Pearson Prentice hall, Cooks, R D, Malkus, D S and Plesha, M E, (1989) Concepts and applications of finite element analysis, John Wiley and Sons. 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