Discretized Solution
Balance Equilibrium Stiffness matrix Convection Stiffness Matrix Boundary forces Body forces Initial strain Initial stress Table 12.1 Comparison of Scalar and Vector Field Problems, Revisited
Ia [
12.44-a 12.44-b 12.44-c 12.44-d Convection Matrix The shape functions for the linear element are given by Eq. 12.35-a. 12.45-a 12.45-b 12.45-c 12.45-d With respect to Fig. 12.1 heat transfer by convection occurs across the perimeter of the Figure 12.1 Heat Flow in a Thin Rectangular Fin fin P 2 B 1 however, as noted above, we are considering a unit volume of the fin Figure 12.1 Heat Flow in a Thin Rectangular Fin fin P 2 B 1 however, as noted above, we are considering a unit volume of the fin...
Parasitic ShearIncompatible Elements Q The Problem
37 The shape functions for the bilinear element are given by Eq. 9.19 Ni Z,rj 1 1 - 0 1 - 7 1 1 0 1 - 7 mt,v 1 1 0 1 v 1 1- 0 1 v 38 Imposing the displacement field field of Fig. 13.5 the bilinear element and correct displacements and strains are given by Table 13.2. Table 13.2 Bilinsear and Exact Displacements Strains Table 13.2 Bilinsear and Exact Displacements Strains 39 From this table we note that parasitic shear contributes to the strain energy stored in the element which is equal to PT.d...
Summary 1
1. Two theories thin plates Kirchhoff Euler-Bernouilli , and thick plates Mindlin-Reissner Timoshenko , without and with shear deformations. 2. Strong analogy between plates and beams. 3. If membrane actions are accounted for, lead to formulation of shell elements. 5. If thick plate theory is applied to thin plates, will have shear locking problems. 6. Shear locking is alleviated by reduced integration of the shear components. 7. For thin plate theory, will have non-conforming elements if only...
Applications of the Galerkin Method to D Elasticity Problems
23 We will apply the Galerkin method to two classes of problems. 24 We start with elasticity problems, to show that the Galerkin Method would result in exactly the same formulation as the one obtained from the variational principle. 25 Following this validation of the method, we shall consider in the next chapter a class of problems which does not have a variational principle, and thus the Galerkin method is the only applicable technique. The second application will be the heat equation.
Introduction Uuc
1 With refernce to Fig. 14.1, we distinguish different levels of analysis First Level elastic which excludes anly nonlinearities. This is usually acceptable for service loads. Elastic Critical load is usually determined from an eigenvalue analysis resulting in the buckling load. Secon-order elastic accounts for the effects of finite deformation and displacements, equilibrium equations are written in terms of the geometry of the deformed shape, does not account for material non-linearilties, may...
Strong Form
3 Column buckling theory originated with Leonhard Euler in 1744. 4 An initially straight member is concentrically loaded, and all fibers remain elastic until buckling occur. For buckling to occur, it must be assumed that the column is slightly bent as shown in Fig. 14.2. Note, in reality no column is either perfectly straight, and in all cases a minor 14.1.1.1 Lower Order Differential Equation 5 At any location x along the column, the imperfection in the column compounded by the concentric load...
SecondOrder Elastic Analysis Geometric NonLinearity
49 From Eq. 14.44 it is evident that since kg depends on the magnitude of Px, which itself may be an unknown in a framework, then we do have a geometrically non-linear problem. but since Kg depends on the axial laod P, the preoblem is nonlinear. 51 A simple way to solve this nonlinear equation is to use a step-by-step incremental procedure. The linearized incremental formulation can be obtained by applying an incremental operator A Example 14-3 Effect of Axial Load on Flexural Deformation...
Rotational DOF
53 Drilling d.o.f. are rotations at corner nodes. In the context of plane elements, they are associated with parabolic displaced shapes of the edges, Fig. 13.7. They are of importance in shells defined as an assemblage of flat elements. Hence, the element can model flexure and membrane action. 54 Assuming rotation IJi and ZJj at nodes i and j of an element side of length L, the end moments of an immaginary beam will be Mi 2cJj ZJj 13.18-a Mj uJi 'IuJj 13.18-b Hence, from M Mi Mj Mj and the...
General Formulation Differential Operators
7 Considering a general differential equation where L is a differential operator, assuming a 1D problem 0 0 x is an unknown function which represents the state variable to be determined i.e. displacement, temperature, head, etc , g x is a known forcing function. 8 We assume that the differential operator to be symmetric and positive definite, Symmetric L u vdQ L v udQ Positive Definite L u udQ gt 0 where u and v are any functions which satisfy essential and natural boundary conditions. 11.2.1.1...
ujhk uj skuj
33 For k 0 and k 1, the values of sk are 0.0 and 1.0, respectively. Therefore, uJ 1'0 uJ and uj hl uj l. The orthogonality condition is quantified by a. scalar value gk representing the iterative change in energy, which is defined as n3 hk Txt -rV U' 16.31 are the residual loads at the end of solution iteration j and line search iteration k. 34 gk can be expressed as a function of sk see Figure 16.6 and the object of the line search is to find sk such that gk is zero. An estimate of sk l such...
Info Spd
21 We can also define the gradient of a vector field as that is Vv y gives the rate of change of the ith component of v with respect to the jth coordinate axis. 22 Note the diference between vVx and Vxv. In matrix representation, one is the transpose of the other. 23 The gradient of a vector is a tensor of order 2. 24 We can interpret the gradient of a vector geometrically, Fig. A.7. If we consider two points a and b that are near to each other i.e As is very small , and let the unit vector m...
fmi Afmi
We again neglect the quadratic and higher order terms, leading to is the tangent stiffness matrix of the structure 6 Differentiating Eq. 16.5 which is the well known formula for the stiffness matrix, however De is now replaced by the tangent moduli D 7 Before we discuss solution strategies, it may be helpful to point out the parallelism which exists between numerical solution strategies, and experimental testing methods. Modern testing equipment can be programmed to apply a pre-determined rate...
STRAIGHT SIDED ELEMENTS si GENERATION
1 Having first introduced the method of virtual displacements, than the interpolation or shape functions N , which relate internal to external nodal displacements, and finally having applied the virtual displacement method to finite element in chapter 7, we derive the stiffness matrices of some simple elements. 8.2 Rod Elements 8.2.1 Truss Element 2 The shape functions of the truss element were derived in Eq. 6.11 3 The corresponding strain displacement relation B is given by
FE Discretization 1
13 The temperature field is expressed in terms of the nodal ones where N are the shape functions, and T are the nodal known temperatures. 7 fA 0div qdA fr 0qTndr - fJV0 TqdA 14 The flux is given by q DVT where furthermore, the flux qn at the boundary is given by qn qTn. 15 Substituting 16 Note, this definition of the B matrix for scalar field problems, should not be confused with the one for vector problems, Eq.
Material Nonlinearities
16.1 Introduction 16.1.1 Linearization i We define a constitutive operator as where c denotes the constitutive operator analogous to the L . 2 Given a strain state e, the corresponding stress will be c cr et . 3 The constitutive operator c can be expanded into a Taylor series with respect to e Neglecting quadratic and higher order terms leads to a linearized constitutive law a a e D e fe which approximates Eq. 16.1 for strains in the neighborhood of e, and is the tangent stiffness matrix which...
Strain smoothing
Strain Smoothing Zienkiewicz, Vilotte, Toyoshima and Nakazawa 1985 SS is an indirect procedure within Step 2 that avoids the direct decomposition of the C matrix. Nodal strains are iteratively evaluated until the ratio of the Euclidean norms of strain correction to total strains satisfies a prescribed limit. This technique exploits the diagonal matrix CL previously described and the consistent matrix C defined below. Iteratively the nodal strains are evaluated by where j 0,1,2 is the...
Element Conductivity Matrix
266.67 -200.00 0. -200.00 533.34 -333.34 0. 333.34 333.34 12.37-a 12.37-b 12.37-c 12.37-d Element Convection Matrix Since convection occurs only at one point, 1, we just add the convection term h 25 to K11 -66.67 266.67 -200.00 0. 0. 200.00 533.34 333.34 -66.67 266.67 -200.00 0. 0. -200.00 533.34 -333.34 0. 0. 333.34 333.34 Element heat rate vector Since there is no heat generation Q, fQ 0, and we do not have a distributed convection hence fb - NTqf dr - Ntqndr Tx Nthdr 0 and we only have a...
Plott of Shape Functions
108 Matlab code to generate shape function plots
Csplitting
A new iterative process was recently developed by Cervenka, Keating and Felippa 1993 to solve Step 2. This new technique guarantees faster convergence for linear triangular and tetrahedral elements T3 and T4 . This technique is referred to as C-splitting CS . This method splits the consistent matrix C of Equation 10.13 into two matrices. One matrix is diagonalized and the second is formed such that their algebraic sum is equivalent to the original C matrix a is a splitting coefficient...
Indirect Displacement Control
56 Direct displacement control can be applied only on structures loaded only at one point, or when the load is transmitted by a stiff platen so that all points on the loaded surface exhibit the same displacements. 57 However, this is not always the case. As an example, consider a dam loaded by hydrostatic pressure due to reservoir overflow see Fig. 16.9. Here, the load is applied along a large portion of Figure 16.9 Hydrostatically Loaded Gravity Dam Figure 16.9 Hydrostatically Loaded Gravity...
Computer Implementation Algorithm
107 The computer implementation of a numerically integrated isoparametric element is summarized as follows. But first, it is assumed that this operation is to be performed in a function called stiff and it takes as input arguments elcod, young, poiss, type, ndime, ndofn, ngaus. In turn it will compute the stiffness matrix KELEM of element ielem. 1. Retrieve element geometry and material properties for the current element 3. Call function dmat to set the constitutive matrix De of the element 4....
A Vector Differentiation
2 A field is a function defined over a continuous region. This includes, Scalar Field g x , Vector Field v x , or Tensor Field T x . 3 We first introduce the differential vector operator Nabla denoted by V 4 We also note that there are as many ways to differentiate a vector field as there are ways of multiplying vectors, the analogy being given by Table A.1. V-v divergence V X v curl Vv gradient. Table A.1 Similarities Between Multiplication and Differentiation Operators 5 The derivative of a...
List of Tables
1.1 Summary of Variational Terms Associated with One Dimensional Elements . . . 1-4 3.1 Selected Examples of Diffusion Problems 3-19 3.2 Comparison of Scalar and Vector Field Problems 3-26 3.3 Classification of various Physical Problems, Kardestuncer 1987 3-27 5.1 Functionals in Linear 5.2 Comparison Between Total Potential Energy and Hu-Washizu Formulations . . . 5-12 6.1 Characteristics of Beam Element Shape Functions 6-9 6.2 Interpretation of Shape Functions in Terms of Polynomial Series 1D...
Illustrative Examples
Let us consider a composite wall made up of three materials, k1 20 W moC, k2 30 W moC, k3 50 W moC , The outer temperature is T0 20oC, convection heat transfer takes place on the inner surface of the wall with the fluid temperature T 800oC, the film coefficient or convection term h 25 W m2, C. We seek the temperature distribution across the wall. Solution 1. Selecting linear elements, we have - x a i 1 x .Ti 1 2. The conductivity matrix for the one dimensional element is 3. Substituting into...
Direct Displacement Control
Adapted from Jirasek and Bazant 2001 43 Independently of the choice of iterative algorithm, any solution strategy using load control fails if the prescribed external loads cannot be maintained in equilibrium by the internal forces. This would typically occur if the load is monotonically increased until the load-carrying capacity of the structure is exhausted, Fig. 16.8 Figure 16.8 Divergence of Load-Controled Algorithms Figure 16.8 Divergence of Load-Controled Algorithms 44 In most engineering...
Introduction
1 Whereas the first course focused exclusively on one dimensional rod elements, this course will greatly expand our horizons by considering introducing a methodology to solve partial differential equations, with special emphasis on solid mechanics. 2 The field of mechanics, can itself be subdivided into four major disciplines Theoretical which deals with the fundamental laws and principles of mechanics. A Continuum Mechanics course is a must. Applied mechanics seeks to apply the theoretical...
Uniqueness and Existence of a Solution
The BB condition for uniqueness and existence of a solution of the three-field variational principle is stated in Appendix . This condition was derived by Babuska 1971 , Babuska 1973 and Brezzi 1974 . Xue and Atluri 1985 extended the condition to a general three-field problem, and derived its discrete form. The continuous and discrete forms of the BB conditions are again described in Appendix , and it is shown that they are equivalent to the following three conditions rank C na lt ne nu 10.23...
Elastic Instability Bifurcation Analysis
41 In elastic instability, the intensity of the axial load system to cause buckling is yet unknown, the incremental stiffness matrix must first be numerically evaluated using an arbitrary chosen load intensity since Kg is itself a function of P . 42 For buckling to occur, the intensity of the axial load system must be A times the initially arbitrarily chosen intensity of the force. Note that for a structure, the initial distribution of P must be obtained from a linear elastic analysis. Hence,...
Tangent Stiffness Matrix
23 It should be noted that each iteration involves three computationally expensive steps 1. Evaluation of internal forces fmt or reactions 2. Evaluation of the global tangent stiffness matrix Kt 3. Solution of a system of linear equations 24 This method is essentially the same as the Newton-Raphson however in Eq. 16.23 K is replaced by Kt which is the tangent stiffness matrix of the first iteration of either 1 the first increment Kt Ky0, Fig. 16.4, or 2 current increment, Fig. 16.3 Kt Kyn Fig....
[ n n n n
1 - 1 - n 1 - n n n 1 - n 15.56-a 15.56-b 15.56-c 15.56-d 43 In order to facilitate the integration of Ks, a one point Gauss integration used. Upon substitution where a and 5 a 3. 44 The final element stiffness matrix is and in terms of the global stiffness matrix we will have 45 Through inspection of the previous equation, and noting that we observe that for very thin plates is very large, hence as t gt 0, gt 00. Hence, unrealistic u 0 can be obtained independently of the load. This phenomena...
[uTn UTm
It is these general forms of the constraint equations that are implemented in MERLIN. Figure 16.13 Flow chart for line search with IDC methods 1 1, k l j l,k l Compute D , CT 1 1, k l j l,k l Compute D , CT Figure 16.13 Flow chart for line search with IDC methods Abramowitz, M. and Stegun, I. 1970, Handbook of mathematical functions, Technical report, National Bureau of Standard. Applied Mathematics Series, No. 55. Babuska, I. 1971, Error-bounds for finite element methods, Numer. Math 20 3 ,...
Finite Element Formulations
32 Numerous elements have been proposed for plate bending. An ideal plate element should have the following properties 1. Formulation should be based on continuum mechanics and plate theory, the nodal d.o.f. are the transverse displacement w, and sectional rotations dxx and dyy. 2. The element should be numerically correct, and convergent. The element stiffness matrix must contain the three rigid-body modes and no rank deficiency spurious zero-energy modes . 3. The element should not lock in...
U
Figure 6.3 Rectangular Bilinear Element 6.2.1.5 Rectangular Bilinear Element 25 Next we consider a quadrilateral element, Fig. 6.3 with bi-linear displacement field in both x and y . 26 Using the Lagrangian interpolation function of Eq. 6.25, and starting with the u displacement, we perform two interpolations the first one along the bottom edge 1-2 and along the top one 4-3 . 27 From Eq. 6.25 with m 1 we obtain 29 Next, we interpolate in the y direction along 1-4 and 2-3 between 12 and 43....
Element Evaluation Patch Test
6 The patch test is a check which ascertains whether a patch of infinitesimally small elements subjected to constant strain reproduces exacly the constitutive behavior of the material through correct stresses. 7 It has been argued that an element which passes the patch test satisfies the two essential conditions for convergence. 8 First we assemble a small number of elements into a patch in such a way that there is at least one internal node shared by two or more elements, and that one or more...
Fundamental Relations Equilibrium
2 Considering an arbitrary plate, the stress are given by, Fig. 15.2, resultants per unit width 3 Note that in plate theory, we ignore the effect of the membrane forces, those in turn will be accounted for in shells. 4 The equation of equilibrium is derived by considering an infinitesimal element tdx dy subjected to an applied transverse load pz. We would have to consider three equations of equilibrium, Fig. 15.3 Summation of Forces in the z direction dxdy - z-dxdy pAxdy 0 dx dy Figure 15.3...
Residual Formulation
11 We now alter Eq. 11.1, and for a 1D problem rewrite it as b pb u x v' x dx u x v x a _ I v x u' x dx where w is an arbitrary weight function2. We note that through this equation, the governing differential equation is satisfied in a weak sense. 12 We now seek an approximate numerical solution for the unknown function 0 x , hence, in general we may assume 0 x app ai i x a2 2 x ----- an n x where ai are the unknown parameters, and x are prespecified trial functions3 of x. 13 Substituting the...
Weak Form Strain Energy
24 Considering a uniform section prismatic element, Fig. , subjected to axial and flexural deformation no shear , the Lagrangian finite strain-displacement relation is given by 3.26 25 We note that the first and second terms are the familiar components of axial and flexural strains respectively, and the third one which is nonlinear is obtained from large-deflection strain-displacement. 26 The Strain energy of the element is given by 27 Substituting Eq. 14.23 into Ue we obtain for y measured...
List of Figures
1.1 Summary of Variational 1.2 Duality of Variational Principles 1-3 1.3 Frame 1.4 Example for ID Matrix 1.5 Simple Frame Analyzed with the MATLAB Code 1-11 1.7 Simple Frame Analyzed with the MATLAB Code 1-18 1.8 Stiffness Analysis of one Element Structure 1-21 2.1 Finite Element Process, Bathe 1996 2-2 2.2 Seepage 2.3 One Dimensional Heat 2.4 Rod subjected to Step 2.5 System of Rigid Carts Interconnected by Linear Springs, Bathe 1996 2-9 2.6 Slab Subjected to Temperature Boundary Conditions,...
u Kn
and apply an increment of load Af such that and we seek to determine the corresponding change in displacement We will keep Af reasonably small to capture the full nonlinear response. 16.2.1.1 Newton-Raphson Tangent Stiffness Method 16 This is the most rapidly convergent process albeit computationally expensive of non-linear problems. i- At the beginning of each step n 1, we start from the displacement. u that, were computed in the previous step through equilibrium 7Zn 0 or rn fn . The external...
ISOPARAMETRIC ELEMENTS nd GENERATION
1 We have previously examined simple finite elements, in this lecture we shall distort those simpler elements into others of more arbitrary shape. 2 Correspondingly, the natural coordinates will be distorted into new curvilinear sets when plotted in a cartesian x,y,z space, Fig. 9.1. 3 In the isoparametric formulation, displacements are expressed in terms of natural coordinates, however they must be differentiated with respect to cartesian coordinates x, y, z. This is accomplished through a...
Nodal Equivalent Loads
92 In the finite element formulation, all loads must be replaced by an energy equivalent nodal load. 93 We shall consider the following cases nodal load, gravity, tractions, and thermal. 9.5.1 Gravity Load 94 Gravity forces are equivalent to a body force unit volume acting within the solid in the direction of the gravity axis, Fig. 9.19, which need not be coincident with either of the coordinate axis . where g is the gravitational acceleration and p is the mass density. 95 Recalling from Eq....
Eigenvalue Test
12 The stiffness matrix is, by definition, singular due to the fact that equilibrium relations are embedded in its formulation, or alternatively, the assumed displacement field on which it is based is supposed to provide for rigid body motions translations and or rotations . 13 The augmented stiffness matrix may be expressed as where B is the statics or equilibrium matrix, relating external nodal forces to internal forces d is a flexibility matrix, and d_1 is its inverse or reduced stiffness...
Mesh Generation
Requires further editing 4.1 Introduction 1 Finite element mesh generation is now an integral part of a finite element analysis. With the increased computational capabilities, increasingly more complex structures are being analysed. Those structures must be discretized. 2 The task is one of developing a mathematical model discretization or tessalation of a continuum model. This is not only necessary in finite elment analysis, but in computer graphics rendering also. 3 In computer graphics, we...
Element Formulation
It is necessary to select appropriate interpolation functions for all three elastic fields i.e. u, e and a . The choice of these shape functions must be such that the BB condition is satisfied Appendix . In this work, the same interpolation functions are used for all three fields i.e. displacements, strains and stresses , which implies that there is a full number of unknowns in each node. dim ura N x dim R, dim era dim ara N x dim a 10.1 where N denotes the number of nodes, dim is the problem...
f tndtdn nm P t ndtdn E wmwf f ti Vj
and the total number of integration points will thus be m x n, Fig. 9.16. 9.3.4 Triangular Regions 85 For the numerical integration over a triangle, the Gauss points are shown in Fig. 9.17, and the corresponding triangular coordinates are given by Table 9.5. Figure 9.16 Gauss-Legendre Integration Over a Surface Figure 9.17 Numerical Integration Over a Triangle Table 9.5 Coordinates and Weights for Numerical Integration over a Triangle Table 9.5 Coordinates and Weights for Numerical Integration...
Fundamental Relations
1 Whereas, ideally, a course in Continuum Mechanics should be taken prior to a finite element course, this is seldom the case. Most often, students have had a graduate course in Advanced Strength of Materials, which can only provide limited background to a solid finite element course. 2 Accordingly, this preliminary chapter mostly extracted from the author's lecture notes in Continuum Mechanics will partially remedy for occasional defficiencies and will be often referenced. 3 It should be noted...
Derivation of the Weak Form 1
3 Our starting point will be the field equation, Eq. 3.113 -div DV lt gt -Q pM 12.1 4 Recalling Eq. 3.98 that q -DV0, and that for steady state problems the right hand side reduces to zero, and finally for the mere sake of clarity substituting 0 by T, this equation reduces to 5 The boundary conditions are given by Essential, Temperature rT T g Natural, Flux qn q n qf 12.3 Natural, Convection Flux rc qc h T - T 6 Note that in a fin, the convection heat loss can be considered as a negative heat...
FINITE ELEMENT DISCRETIZATION and REQUIREMENTS
This section is mostly extracted from Reich 1993 7.1.1 Discretization of the Variational Statement for the General TPE Variational Principle 1 The discretization of Equation 5.8 will be performed on an element domain Qe using the procedures described in Chapter 2 of Zienkiewicz and Taylor 1989 2 The surface of the element subjected to surface tractions r comprises one or more surfaces of the element boundary r. For the present time this discussion will be kept on a very general level with no...
L
1. Show that the kinetic energy and the strain energy are given by 2. For Wnc 0 derive an expression for the first variation of the Hamilton's functional 3. If we are interested in determining the periodic motion, which has the form where w is the frequency of natural vibration, and uo x is the amplitude. Show that 4. Show that the Euler equations of the preceding functional are EA PAlo2u0 0 for 0 lt x lt L it - on n - m. - i l k2 , a j , EA , a E 5. Rewrite Eq. 5.24 in terms of x u, 6. Show...