Model for Deflection Analysis in Cantilever Beam

Exposure to Finite Element Method is beneficial to undergraduate engineering students; and educators have an obligation to introduce students to modern engineering tools. However, teaching of the course is computational intensive and existing propriety software are very expensive. Different approaches to introducing students to the FEM have been proposed. Existing approaches make use of learning modules of commercial Finite Element Analysis (FEA) packages such as ANSYS, ABAQUS, COMSOL Multiphysics, ALGOR, JL AutoFEA Analyzer and PRO/MECHANICA. This paper presents an in-house developed finite-element-based computer model via the virtual work principle using Linear Strain Triangular (LST) elements for deflection analysis on cantilever beam. The validation and capability characteristics was demonstrated by applying it to a cantilever beam, subjected to a point load, using both coarse (4-element) and fine (10-element) meshes. The model gave results very close to those obtained analytically; the 10-element mesh gave better results than the 4-element mesh. The model has made the analysis more flexible, and also made visualization and presentation of results easier for better judgment. It facilitates the presentation of the basic rules which govern FEA and helps in the learning of different aspects of the numerical technique; hence the model will serve as reliable tool in undergraduate engineering programme


I. INTRODUCTION
Finite Element Method (FEM) is a numerical technique for finding approximate solutions of Partial Differential Equations (PDE) as well as integral equations.The solution approach is based either on eliminating the differential equation completely, or rendering the PDE into an approximately system of ordinary differential equations, which are then numerically integrated using standard technique such as Euler's method, Runge-Kutta, etc. Plane elasticity problems involve continua that are loaded in their plane.Where the continuum is loaded by forces normal to the plane, out-of-plane displacements, (for instance bending of a plate) are induced.Plane elasticity problems may be separated into two classes, namely plane stress problems and plane strain problems.In a plane stress problem, the continuum (such as a plate) is thin relative to other dimensions, and stresses normal to the plane are neglected.Problems in this category include diaphragm plates in box sections and plate girder webs where the applied loads are in the plane of the member.In a plane strain problem, the strain normal to the plane of loading is assumed to be zero.Cantilever beam with a point load at the extreme end is a case of plane elasticity problems that is subjected to finite element analysis in this study.
Finite Element Analysis (FEA) is a computer simulation technique using the FEM to carry out engineering analysis.There are many propriety FEA software packages.However, they are very expensive to purchase.Development of the finite element method in structural mechanics is usually based on an energy principle such as the virtual work principle or the minimum total potential energy principle.The finite element analysis from the mathematical side was first developed in 1943 by Richard Courant, who used the Ritz method of numerical analysis and minimization of variational calculus to obtain approximate solutions to vibration systems [1].From the engineering side, the finite element analysis originated as the displacement method of the matrix structural analysis, which emerged over the course of several decades mainly in British aerospace research as a variant suitable for computers.The FEM has been successfully applied to the various practical problems of civil engineering (bridge deck analysis, plane truss, space frame and multi-storey building) and mechanical engineering (aircraft, motor vehicle, turbine blade and housing).Maximov et al. [2] employed isotropic plastic hardening and Von Mises yielding criterion for finite element simulation of spherical mandrelling process of holes with cracks.Yorgun et al. [3] investigated, by experimental and finite element modelling, the behaviour of a double channel beam-to-column connection subjected to in plane bending moment and shear.Simulation of the tests was carried out by means of a nonlinear finite element software ANSYS.Sutherland [4] described the use of a finite element computer code "AXICRP" developed for solving creep problems for plane stress, plane strain, and axisymmetric bodies of revolution.Arya [5] investigated the feasibility of using a viscoplastic model to perform a nonlinear structural analysis.He compared the analytical and FE solutions of three classical problems, viz. a pressurized thick-walled cylinder, a thin rotating disk and a pressurized thick-walled sphere.
Victor and Bikramjit [6] compared numerical results with ANSYS 15.0 results with differences being insignificant.Nirmall and Vimala [7] cantilever beam and tapered cantilever beam, of different sizes were prepared for experimental purpose of free vibration analysis of beams made with different materials such as aluminum, brass and mild steel.The natural frequency of the beams obtained Model for Deflection Analysis in Cantilever Beam Kolawole Adesola Oladejo, Rahaman Abu, and Olufemi Adebisi Bamiro from experimental and theoretical methods were compared with harmonic analysis using ANSYS software.Chaphalkar et al. [8] focuses on the numerical analysis and experimental analysis of transverse vibration of fixed free beam and investigates the mode shape frequency.All the frequency values are analysed with the numerical approach method by using ANSYS package.
Many authors have, on separate basis, carried out study on the finite element analysis of plate bending, using different triangular meshes, and different patterns of presentation of the results [9][10][11].There are many propriety FEA software packages.However, they are very expensive to purchase.The objective of this study is to develop an interactive computer-based package for the analysis of deflection on a cantilever beam, and also to investigate the variation of results due to change in the degree of discretisation of the domain.

A. Exact Solution of Deflection on a Cantilever Beam
The derived displacement function for deflection along the neutral plane (displacement along y-axis) for a beam in Fig. 1 derived by Bamiro [12] is where  =Poisson's ratio, P=Load applied to the beam, E=Modulus of Elasticity, I=Second moment of area, and L= Length of the beam.
The deflection will have a maximum value as derived by Bamiro [12] in (2).

B. FEM Solution of Deflection on a Cantilever Beam
For the development of the model, the following major steps were employed: Step 1: Idealization and discretization The idealization, shown in Fig. 2 and 3, gives the following information (taking the origin of the coordinate system as node 1) For 10-element mesh 0 31 31 33 33 28 28 where u= In-plane displacement in x direction and v=In-plane displacement in y direction.Step 2: Selecting a Displacement Function A quadratic displacement function in each element as This can be expressed in matrix form as Alternatively, ( 7) is expressed as The coefficients a1 through a12 can be obtained by substituting the coordinates into ( 5) and ( 6) and solving for ai's as follows: where ] [X is a 12 x 12 matrix.
Then the ai ' s, in terms of nodal displacements, are substituted into (8) and the general displacement expressions in terms of the shape functions and the nodal degree of freedom as, where, Step 3: Define the Strain and Stress Relationships Element strains and stresses are expressed in terms of the unknown nodal displacements.The strains associated with the two-dimensional element are given by Fenner [13] as Using ( 7) in (13), Substituting ( 9) for ai ' s into (15),    will be obtained in terms of nodal displacements as where [B] is a function of the variables x and y and the coordinates (x1, y1) through (x6, y6) given by In general, the in-plane stresses are given by Daryl [14] as where [D] is the constitutive matrix, and given as Step 4: Generating Element Stiffness Matrix Using the principle of minimum potential energy, the equation for LST element is generated by Daryl [13] as where V=Volume and k=Element stiffness matrix.

Step 5: Assembling the Global Stiffness Matrix
The global structure stiffness matrix and equations is obtained by using the direct stiffness method as derived in Rockey et al. [15] as and where   F = Nodal Force vector Step 6: Solving for the Nodal Displacements The unknown global structure nodal displacements are determined by solving the system of algebraic equations given by ( 22).

Step 7: Calculating Deflection and Stresses
Having solved for the nodal displacements, strains and stresses can be obtained in the global x and y directions in the elements by using ( 16) and ( 18) respectively.

III. DEVELOPMENT OF THE FEA PACKAGE
The common characteristic of all computer applications is their intrinsic ability to carry out complex mathematical computations at high speeds and at a very acceptable degree of accuracy.After critical scrutiny of available software developed on FEA for plate bending, the following features are embedded in the package: (i) a user friendly interface, which presents an easy and FEA flow process data entry; (ii) a visual presentation of the plate model whose solution is being sought; (iii) progressive view of the data input; and (iv) colour-code presentation of the generated output of the Finite Element Method solution The package was developed on the platform of Visual BASIC 6.0.It involved the use of Visual Basic forms, modules and class modules.The forms contain the Graphical User Interface (GUI) objects; the modules contain the functions and subroutines; and the class modules contain user defined objects called classes.The flow of the program involves interaction between the GUI objects, the functions, subroutines and the classes.Fig. 4 shows a simplified framework of the model.Pre-processing stage involves discretization and collection of data needed for the analysis while post-processing stage involves presentation of output of the analysis.In the package, the results are made available in both textual and graphical forms.Codes for each form are attached on individual basis.The important forms in the model are Mainform, frmPlate Dimension, frmDivide, frmLoadBC, frmDisplacementBC, frmMaterialProperties, frmResults and frmPlotView.

Start Input plate dimensions
Specify discritization parameters : no of divisions along length and breadth of the plate Mesh Generation  "set".This is illustrated in Fig. 8.
(vi) Click the "Solve" button to initiate the analysis.
(vii) To view the textual results, click the "Textual Results Menu" and select the appropriate menu item corresponding to the desired view (Fig. 9).The deflections results were obtained for the cantilever problem using both four and ten triangular elements.The exact solutions were computed from (1) using the coordinates of the node along the neutral plane.The deflection curves for points along the neutral plane of the beam are presented in Fig. 10 and 11 using the exact solution and COMSOL Multiphysics software compared with 4-element and 10-element model results respectively.
It was observed from the Fig. 10 that the deflection results for the 4-element model were significantly far from the exact solution and COMSOL Multiphysics results while the 10-element model results almost coincide with that of the exact solution and COMSOL Multiphysics results (Fig. 11).This confirms the fact that the more the number of element, the better the results as reported in literature [11].The present graphical displacement had the same configuration with COMSOL Multiphysics graphical displacement (Fig. 12).This implies that the model can used as alternative to the proprietary software.Visualizing effects of various parameters on the behaviour of the physical system allows analysts to relate the system behaviour to the theoretical concepts.The model may serve as cheap instructional tool for learning in the field of numerical analysis.

Fig. 1 .
Fig. 1.A unit-width cantilever beam with a point load

For
(i) Type of element: linear strain triangular (LST) of uniform thickness; (ii) Number of elements: 4 and 10. (iii) Number of nodes per element: 6. (iv) All elements are isotropic, homogeneous and obey Hooke's law.(v) At each of the six nodes there are two degrees of freedom.For example, at node 3 there is a horizontal deflection component u3 and a vertical deflection component v3.(vi) The Boundary Conditions (BC) is specified as follows: