CRITERIA FOR SELECTION OF FEM MODELS. Prof. P. C.Vasani,Applied Mechanics Department, L. D. College of Engineering,Ahmedabad- 380015 Ph.(079) 7486320 [R] E-mail:pcv-im@eth.net 1. Criteria for Convergence. The finite element method provides a numerical solution to a complex problem. It may, therefore, be expected that the solution must converge to the exact solution under certain circumstances. It can be shown that the displacement formulation of the method leads to an upper bound to the actual stiffness of the structure. Hence the sequence of successively finer meshes is expected to converge to the exact solution if assumed element displacement fields satisfy certain criteria. These are, 1. The displacement field within an element must be continuous. In other words, it does not yield a discontinuous value of function but rather a smooth variation of function, and the variation do not involve openings, overlap, or jumps, which are inherently continuous. This condition can easily be satisfied by choosing polynomials for the displacement model. The function w is indeed continuous if for example it is expressed as, W = C1 + C2 x + C3 x 2 +C4 x 3 +. 2. The approximate function should provide inter-element compatibility up to a degree required by the problem. For instance, for the column problem involving axial deformations, it is necessary to ensure inter-element compatibility at least for displacements of adjacent nodes. That is the approximate function should be such that the nodal displacements between adjacent nodes are the same. Inter-element compatibility must be enforced for displacements and their derivatives up to the order n-1, where n is the highest order derivative in the energy function. The highest order of derivative for a column element, for example, is 1. Hence the inter-element compatibility should
include order of v at least upto 0 (zero), that is, displacement v, In general, the formulation should provide inter-element compatibility upto order n-1, where n is the highest order derivative in the energy function. Approximate functions that satisfy the conditions of compatibility can be called conformable (compatible or conforming elements). In contrast to the plane deformations, for realistic approximation of the physical conditions in the case of bending, it is necessary to satisfy inter-element compatibility with respect to both the displacements and slopes, i.e. first derivative (gradient) of displacement. As a consequence, it becomes necessary to use higher order approximation for the displacement in the case of bending, since it is necessary to provide for inter-element compatibility for slopes also, we can add slope at the node as an additional unknown. This leads to two primary unknowns, displacement w and slope θ = dw/dx, at each node, hence, for an element there are a total of four degrees of freedom: w1, θ1 at node 1 and w2, θ2 at node 2. The approximate function is conformable since it provides for inter-element compatibility upto n-1 = 2-1 = 1 derivative of w, that is, for both w and its first derivative, where n=2 is the highest order of derivative in the potential energy function. The displacement function for a beam element thus can be written as, W = C1 + C2 x +C3 x 2 +C4 x 3 3. The displacement model must include the rigid body displacements of the element. A rigid body displacement is the most elementary deformation an element may undergo. Hence when the nodes are given such displacement corresponding to a rigid motion, the element should not experience any strain and hence leads to zero nodal forces. The constant terms in the polynomials used for the displacement models would usually ensure this condition. For instance the constant term C1 provides for a rigid body
displacement. One such combination should occur for each of the possible rigid body translations and rotations. 4. The displacement models must include the constant strain states within the element. The reason for this requirement can be understood if we imagine the condition when the body or structure is divided into smaller and smaller elements. As these elements approach infinitesimal size, the strains in each element also approach constant values. Hence the assumed displacement function should include terms for representing constant strain states. For one, two and three dimensional elasticity problems, the linear terms present in the polynomial satisfy the requirement. However, in the case of beam, plate and shell elements, this condition will be referred to as constant curvature instead of constant strains. There should exist combinations of values of the generalized coordinates that cause all points on the element to experience the same strain. One such combination should occur for each possible strain. 5. Besides the convergence and compatibility requirements, one of the important considerations in choosing proper terms in the polynomial expansion is that the element should have no preferred direction. That is the displacement shapes will not change with a change in local coordinate system. This property is known as geometric isotropy, or geometric invariance. Geometric invariance is achieved if the polynomial includes all the terms, i.e. the polynomial is a complete one. However, invariance may be achieved if the polynomial is balance in case all the terms cannot be included. This balanced representation can be illustrated with respect to Pascal triangle for twodimensional polynomials:
1 X Y X 2 XY Y 2 X 3 X 2 Y XY 2 Y 3 X 4 X 3 Y X 2 Y 2 XY 3 Y 4 X 5 X 4 Y X 3 Y X 2 Y 3 XY 4 Y 5 Symmetry Axis For example, if we would like to construct a polynomial with four terms, invariance is achieved by selecting U = C1 + C2 X + C3 Y + C4 XY Thus, geometric invariance can be achieved by selecting the corresponding order of terms on either side of the axis of symmetry. 2. FINITE ELEMENT ANALYSIS OF A CONTINUUM. Conventional engineering structures can be visualized as an assemblage of structural elements interconnected at a discrete number of nodal points. If the force-displacement relationships for the individual elements are known it is possible, by using various well-known techniques of structural analysis, to derive the properties and study the behavior of the assembled structure. In elastic continuum the true number of interconnection points is infinite, and here lies the biggest difficulty of its numerical solution. In many phases of engineering the solution of stress and strain distributions in elastic continua is required. Special cases of such problems may range from two-
dimensional plane stress or strain distributions, axisymmetrical solids, plates bending, and shell, to fully three-dimensional solids. In all cases the number of interconnections between any finite element isolated by some imaginary boundaries and the neighboring elements is infinite. It is therefore difficult to see at first glance how such problems may be discretized in the same manner as is done for simpler structures. The difficulty can be overcome (and approximation made) in the following manner. (a) The continuum is separated by imaginary lines or surfaces into a number of finite elements. (b) The elements are assumed to be interconnected at a discrete number of nodal points situated on their boundaries. The displacements of these nodal points will be the basic unknown parameters of the problem, just as in the simple structural analysis. (c) A function (or functions) is chosen to define uniquely the state of displacement within each finite element in terms of its nodal displacements. (d) The displacement functions now define uniquely the state strain within an element in terms of the nodal displacements. These strains, together with any initial strains and the elastic properties of the material will define the state of stress throughout the element and, hence, also on its boundaries. (e) A system of forces concentrated at the nodes and equilibrating the boundary stresses and any distributed loads is determined, resulting in stiffness relationship same as for simpler structure Once this stage has been reached the solution procedure can follow the standard structural routine. So far, the process described is justified only intuitively, but what in fact has been suggested is equivalent to the minimization of the total potential energy of the system in terms of a prescribed displacement field. If this displacement field is defined in a suitable way, then convergence to the correct result must occur. The process is then equivalent to the well-known Ritz procedure. The derivation of characteristics of finite element of a continuum in detail is beyond the scope of this article.
z 3 1 q 2 Axisymmetric element 3. FINITE ELEMENT ANALYSIS OF AXISYMMETRIC SOLIDS. The problem stress distribution in bodies of revolution (axisymmetric solids) under axisymmetric loading is of considerable practical interest. The mathematical problems presented are very similar to those of plane stress and plane strain as, once again, the situation is two-dimensional. By symmetry, the two components of displacements in any plane sectioning the body along its axis of symmetry define completely the state of strain and, therefore, the state of stress. Such a cross-section is shown in Figure above. The volume of material associated with an element is equal to of a body of revolution indicated on fig, and all integration have to be referred to this. In plane stress or strain problems it is being shown that internal work is associated with three strain components in the co-ordinate plane, the stress component normal to this plane not being involved due to zero values of either the stress or the strain. In the axisymmetrical situation any radial displacement automatically induces a strain in the circumferential direction, and as the stresses in this direction are certainly non-zero, this fourth component of strain and of the associated stress has to be considered. Here lies the essential difference in the treatment of the axisymmetric situation. The simplest examples are a circular cylinder loaded by a uniform internal or external pressure, a circular footing resting on a soil mass, pressure vessels, rotating wheels, flywheels etc. (See Fig. below).the deformation being symmetrical with respect to the y-axis, the stress components are independent of the angle θ and all derivatives with respect r
y,v Y X x,u q q AXISYMMETRIC PROBLEMS. To θ vanish. The components w, γ xθ, γ θy, τ xθ and τ θy are zero. The straindisplacement relations are given by x = u ; θ = u ; y = v ; γ xy = u + v x x y y x Thus, the constitutive relation is, σ x σ y σ θ τ x = E (1+ ν)(1-2ν) (1-ν) ν ν 0 x (1-ν) ν 0 y Symmetric (1-ν) 0 θ (1-2ν)/2 γ xy
The stiffness matrix : The stiffness matrix of the any axisymmetric element can now be computed according to the general relationship. Remembering that the volume integral has to be taken over the whole ring of material we have, [K e ] = 2π [B] T [D] [B] r drdz