Hutton - Fundamentals of Finite Element Analysis (523155), страница 31

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Solution convergence is depicted in Figure 6.1e in terms of maximum normal stress in thex direction. For this example, the exact solution is taken to be the maximumbending stress computed using elementary beam theory. The true exact solutionis the plane stress solution from the theory of elasticity. However, the maximumnormal stress is not appreciably changed in the elasticity solution.The need for convergence during regular mesh refinement is rather clear.If convergence is not obtained, the engineer using the finite element methodhas absolutely no indication whether the results are indicative of a meaningfulapproximation to the correct solution.

For a general field problem in which thefield variable of interest is expressed on an element basis in the discretizedform␾(e) (x , y, z) =MN i (x , y, z)␾i(6.1)i=1where M is the number of element degrees of freedom, the interpolation functionsmust satisfy two primary conditions to ensure convergence during mesh refinement: the compatibility and completeness requirements, described as follows.6.2.1 CompatibilityAlong element boundaries, the field variable and its partial derivatives up to oneorder less than the highest-order derivative appearing in the integral formulation of the element equations must be continuous. Given the discretized representation of Equation 6.1, it follows that the interpolation functions must meetthis condition, since these functions determine the spatial variation of the fieldvariable.Recalling the application of Galerkin’s method to the formulation of thetruss element equations, the first derivative of the displacement appears in Equation 5.34.

Therefore, the displacement must be continuous across element boundaries, but none of the displacement derivatives is required to be continuousacross such boundaries. Indeed, as observed previously, the truss element is aconstant strain element, so the first derivative is, in general, discontinuous at theboundaries.

Similarly, the beam element formulation, Equation 5.49, includesthe second derivative of displacement, and compatibility requires continuity ofboth the displacement and the slope (first derivative) at the element boundaries.In addition to satisfying the criteria for convergence, the compatibility condition can be given a physical meaning as well. In structural problems, therequirement of displacement continuity along element boundaries ensures thatno gaps or voids develop in the structure as a result of modeling procedure.

Similarly the requirement of slope continuity for the beam element ensures that no“kinks” are developed in the deformed structure. In heat transfer problems, thecompatibility requirement prevents the physically unacceptable possibility ofjump discontinuities in temperature distribution.165Hutton: Fundamentals ofFinite Element Analysis1666. Interpolation Functionsfor General ElementFormulationCHAPTER 6© The McGraw−HillCompanies, 2004TextInterpolation Functions for General Element Formulation6.2.2 CompletenessIn the limit as element size shrinks to zero in mesh refinement, the field variableand its partial derivatives up to, and including, the highest-order derivativeappearing in the integral formulation must be capable of assuming constantvalues.

Again, because of the discretization, the completeness requirement isdirectly applicable to the interpolation functions.The completeness requirement ensures that a displacement field within astructural element can take on a constant value, representing rigid body motion,for example. Similarly, constant slope of a beam element represents rigid body rotation, while a state of constant temperature in a thermal element corresponds tono heat flux through the element. In addition to the rigid body motion consideration, the completeness requirement also ensures the possibility of constant valuesof (at least) first derivatives. This feature assures that a finite element is capable ofconstant strain, constant heat flow, or constant fluid velocity, for example.The foregoing discussion of convergence and requirements for interpolationfunctions is by no means rigorous nor greatly detailed. References [1–5] lead theinterested reader to an in-depth study of the theoretical details.

The purpose hereis to present the requirements and demonstrate application of those requirementsto development of appropriate interpolation functions to a number of commonlyused elements of various shape and complexity.6.3 POLYNOMIAL FORMS:ONE-DIMENSIONAL ELEMENTSAs illustrated by the methods and examples of Chapter 5, formulation of finiteelement characteristics requires differentiation and integration of the interpolation functions in various forms. Owing to the simplicity with which polynomialfunctions can be differentiated and integrated, polynomials are the most commonly used interpolation functions. Recalling the truss element development ofChapter 2, the displacement field is expressed via the first-degree polynomialu(x ) = a0 + a1 x(6.2)In terms of nodal displacement, Equation 6.2 is determined to be equivalent toxxu(x ) = 1 −u1 + u2(6.3)LLThe coefficients a0 and a1 are obtained by applying the nodal conditionsu(x = 0) = u 1 and u(x = L ) = u 2 .

Then, collecting coefficients of the nodaldisplacements, the interpolation functions are obtained asxxN1 = 1 −N2 =(6.4)LLEquation 6.3 shows that, if u 1 = u 2 , the element displacement field corresponds to rigid body motion and no straining of the element occurs. The firstHutton: Fundamentals ofFinite Element Analysis6. Interpolation Functionsfor General ElementFormulation© The McGraw−HillCompanies, 2004Text6.3 Polynomial Forms: One-Dimensional Elementsderivative of Equation 6.3 with respect to x yields a constant value that, as wealready know, represents the element axial strain.

Hence, the truss element satisfies the completeness requirement, since both displacement and strain can takeon constant values regardless of element size. Also note that the truss elementsatisfies the compatibility requirement automatically, since only displacement isinvolved, and displacement compatibility is enforced at the nodal connectionsvia the system assembly procedure.In light of the completeness requirement, we can now see that choice of thelinear polynomial representation of the displacement field, Equation 6.2, was notarbitrary.

Inclusion of the constant term a0 ensures the possibility of rigid bodymotion, while the first-order term provides for a constant first derivative. Further,only two terms can be included in the representation, as only two boundary conditions have to be satisfied, corresponding to the two element degrees of freedom.

Conversely, if the linear term were to be replaced by a quadratic term a2 x 2 ,for example, the coefficients could still be obtained to mathematically satisfy thenodal displacement conditions, but constant first derivative (other than a value ofzero) could not be obtained under any circumstances.Determination of the interpolation functions for the truss element, as justdescribed, is quite simple.

Nevertheless, the procedure is typical of that used todetermine the interpolation functions for any element in which polynomialsare utilized. Prior to examination of more complex elements, we revisit thedevelopment of the beam element interpolation functions with specific referenceto the compatibility and completeness requirements.

Recalling from Chapter 5that the integral formulation (via Galerkin’s method, Equation 5.49 for the twodimensional beam element includes the second derivative of displacement, thecompatibility condition requires that both displacement and the first derivative ofdisplacement (slope) be continuous at the element boundaries. By includingthe slopes at element nodes as nodal variables in addition to nodal displacements,the compatibility condition is satisfied via the system assembly procedure. As wehave seen, the beam element then has 4 degrees of freedom and the displacementfield is represented as the cubic polynomialv(x ) = a0 + a1 x + a2 x 2 + a3 x 3(6.5)which is ultimately to be expressed in terms of interpolation functions and nodalvariables asv 1␪ 1v(x) = N1 v1 + N2 ␪1 + N3 v2 + N4 ␪2 = [N1 N2 N3 N4 ](6.6) v2 ␪2Rewriting Equation 6.5 as the matrix product,v(x) = [1xx2 a 0a3x ] 1 a2 a3(6.7)167Hutton: Fundamentals ofFinite Element Analysis1686.

Interpolation Functionsfor General ElementFormulationCHAPTER 6© The McGraw−HillCompanies, 2004TextInterpolation Functions for General Element Formulationthe nodal conditionsv(x = 0) = v1dv = ␪1dx x=0(6.8)v(x = L ) = v2dv = ␪2dx x=Lare applied to obtainv1 = [1 0 0 0]␪1 = [0 1 0 0]v2 = [1LL2␪2 = [0 1 2La 0a 1(6.9) a2 a3a 0a 1 a2 a3a 0a 1L 3] a2 a33L 2 ]a 0a 1 a2 a3The last four equations are combined into the equivalent matrix form   v1 0 00  a0  1 0  a1␪10 1 0=23 1 L LL v2  a2 ␪20 1 2L 3L 2a3(6.10)(6.11)(6.12)(6.13)The system represented by Equation 6.13 can be solved for the polynomial coefficients by inverting the coefficient matrix to obtain1000   0 100 av1  0  3 ␪1 231a1− = − 2 −(6.14) a2   v2 LL2L L 2a3121  ␪2−L3L2L3L2Hutton: Fundamentals ofFinite Element Analysis6. Interpolation Functionsfor General ElementFormulation© The McGraw−HillCompanies, 2004Text6.3 Polynomial Forms: One-Dimensional ElementsThe interpolation functions can now be obtained by substituting the coefficients given by Equation 6.14 into Equation 6.5 and collecting coefficients of thenodal variables.

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