Use Of Fea In An Introductory Strength Of Materials Course


The new generation of PC-based computer aided engineering software is relatively easy to learn and use. Most have robust graphics user interfaces and adequate CAD modules that greatly enhance the teaching and learning process. These facts coupled with product availability, relatively low cost, and reliability make them a convenient tool to be included within the curriculum of many introductory engineering technology courses. This paper describes how one such computer program in the area of structural finite element modeling and analysis may be included within an introductory course in strength of materials. The software is used to solve one and two-dimensional structural problems. Students enrolled in this course acquire sufficient knowledge within the field of finite element related stress analysis to proceed on to more challenging experiences in other courses or industry. Nomenclature A0 Cross-section area of tension member d Nodal deflection E Modulus of elasticity F Global nodal force f local nodal force L0 Original length of tension member P External tension axial force load u deflection in the local x direction v deflection in the local y direction δ Axial deformation of tension member σ Normal stress θ Orientation angle of truss element measured w/r to the x-axis Introduction The method of finite element analysis may be introduced into the curriculum of an associate degree mechanical engineering technology program. The use of a commercially available, windows-based, FEA computer package could complement other computer software typically used in most engineering technology programs. The NINTENDO factor would thus be reinforced and perhaps additional interest generated. An existing academic program was used as a model to help describe the pedagogical approach taken within this paper. It contains a sequence of courses that are vital to the success of this approach by providing students with the necessary background to gain basic working knowledge of the analysis method. These courses should introduce the following general topics: engineering materials, statics, strength of materials, and machine design. Materials testing such as tension, impact, shear, and fatigue are introduced within the engineering materials course. The P ge 302.1 2 strength of materials course provides instruction in the various mathematical relationships and procedures that may be used to perform stress analysis on structural parts of simple geometry. Instead of the usual test laboratory a computer laboratory module is provided. To help prepare students for use of the computer analysis program the concept of finite element structural analysis is introduced through simple examples of twodimensional truss structures. Two-dimensional matrix truss analysis is introduced early in the course by the definition of a two-dimensional truss element. The necessary computer program commands are introduced through scripted examples that are designed to complement manufacturer supplied documentation. The academic model used allows students to gain additional finite element analysis experience within a first course in machine design. An outline describing the basic sequence of events and the general intent of each is provide. Overview The following outline describes the fundamental pedagogical steps that may be taken to help prepare students to deal with structural analysis problems using the method of finite elements. Clearly the general topics listed below are contained within the curriculum of most associate degree mechanical engineering technology programs. 1. Ample instruction and hands-on experience with materials testing should be provided, especially in the area of tension tests. 2. A firm foundation in the area of equilibrium force analysis of static structures should be provided. The concepts of structural system and structural component equilibrium should be emphasized. 3. A thorough understanding and comprehension of material mechanical properties and Hooke’s law as related to loads and displacements of structural components is necessary. 4. A basic understanding of index notation and matrix algebra operations is necessary. 5. Derivation of the local truss element equations using Hooke’s law and matrix representation of transformation relationships should be presented. 6. Truss element equilibrium equations should be derived in matrix form. 7. The global equilibrium nodal force/displacement equations should be developed in matrix form. 8. The matrix analysis procedure should be illustrated through the use of examples. 9. Hand computed matrix analysis examples should be reinforced with scripted examples of the computer commands necessary to solve the same problems. 10. The finite element analysis concept may be extended by the introduction of the triangular plane stress element. The introduction of this element should be timed to coincide with lecture discussions of plane stress. To help students gain additional appreciation for the increased complexity of the triangular plane stresses element the element stiffness matrix may be introduced. Emphasis should be placed on the necessity of computer program application. 11. Tutorial examples of the computer software may be provided to the students so plane stress structural models and analyses may be performed. 12. Students within the academic model program gain further use of the computer FEA software within a capstone design project. Typically, a simple machine component is designed by hand computation and verified by use of the computer. While each of the items of the above outline are of equal importance only items 3 through 9 are emphasized within this paper. Only enough detail is provided so a general understanding of the proposed pedagogical procedure may be gained. If presented with continuity, the beginnings of a firm foundation in FEA should result. Item 3 Tension Tests The most natural and perhaps the most revealing test of material mechanical properties is the tension test. A strip or cylinder of the material, having length L and cross-section area A is constrained at one end and subjected P ge 302.2 3 to an axial pull-load, P. A gradual increase in the load is accompanied by a corresponding increase in the axial deflection, δ. At some value of P the material begins to behave in a way that would be very detrimental to a structure. The value of P for which this happens is referred to as the ultimate load. This value is dependent not only on the material but also on the size and shape (cross-sectional area) of the test specimen. Thus when reporting the strength of materials loaded in tension it is customary and convenient to account for the area effect by dividing the ultimate load Pu by the original cross-sectional area Ao, i.e. σu = Pu/ Ao The difference between stiffness, which is a measure of the load needed to induce a given deformation in the material, and strength, which usually refers to the material resistance to failure by fracture or excessive deformation, is very important. The stiffness is measured experimentally, by applying known axial loads well short of fracture and measuring the resulting axial deformation. To good approximation, the load P and the resulting deformation δ are related linearly. This relationship is generally known as Hooke’s law and may be written as: P = kδ 1. k is a constant of proportionality called stiffness. The stiffness as defined by k is not a function of the material alone but is also influenced by the specimen shape. A useful way to adjust the stiffness so as to be a purely materials property is to define a parameter that is area independent. This may be accomplished by dividing the load by the stressed area. Further, the deformation δ may be made independent of the original length, Lo, by defining a deformation per unit length ε = δ/Lo . ε is called strain. Using these more general measures of load per unit area and displacement per unit length Hooke’s law becomes: P = (A0E/L0)δ 2. E is a function of material alone and may be determined from tension tests on various materials. Typically E is called the modulus of elasticity. Items 4 Matrix Algebra At the associate degree level, little information regarding index notation and matrix algebraic operations is provided. The rules that apply to multiplication of matrices are very useful in the representation of large numbers of equations in a compact form and in a form that lends itself to index looping within a computer program. The general rule for matrix multiplication may be stated as follows: the multiplication of an M x N matrix, A, containing elements aij by an R x P matrix, B, containing elements bij is defined only if R = N and always results in a new M x P matrix, C, whose elements, cij, are given by the equation cij = Σaikbkj for k = 1 to n and n = P. In general the matrix product AB is not equal to BA. Several examples of matrix multiplication should be provided and explained in detail. Students should be required to hand-compute several practice examples. Items 5-7 Matrix Truss Analysis In matrix truss analysis the preceding rules and conventions may be used to evaluate the forces that accompany known imposed displacements and to evaluate the displacements at the joints or nodes where known external forces are applied.