Finite Element Techniques For Orthotropic Plane Stress And Orthotropic .
U.S. FOREST SERVICERESEARCH PAPERFPL-87JUNE 1968U. S. DEPARTMENT OF AGRICULTUREFOREST SERVICEFOREST PRODUCTS LABORATORYMADISON, WISCONSINFINITE ELEMENT TECHNIQUESFOR ORTHOTROPIC PLANESTRESS ANDORTHOTROPIC PLATE ANALYSIS
ABSTRACTThis paper develops finite element techniques forapplicability to plane stress problems and plateproblems involving orthotropic materials such as woodand plywood. Applications to limited examples showthat the methods have merit especially if means ofhandling very large systems of equations are utilized.
FINITE ELEMENT TECHNIQUES FORORTHOTROPIC PLANE STRESS ANDORTHOTROPIC PLATE ANALYSISA.C. MAKI, EngineerFORESTPRODUCTSLABORATORY1FOREST SERVICEU.S. DEPARTMENT OF AGRICULTUREINTRODUCTIONThe solutions of many plane stress problems are literally impossible when attempted by applyingthe differential equations of the theory of elasticity. For the solution of such problems, stress analystshave sought other methods. One of these methods has been termed the finite element technique, andappears to have merit in yielding approximate solutions to such problems.The basic concept of the method consists of replacing the solid elastic body to be analyzed by anetwork of finite elements. It is believed that as the size of the finite element approaches the differential element stage, the results yielded by the method would compare favorably to those obtainedfrom a rigorous mathematical analysis. By keeping the element finite in size, the network modelwould no longer yield equivalent results but should represent a close approximation.The finite element technique of plane stress analysis has been presented in different papers by2Hrennikoff (2), McCormick (4), Turner (6), and Melosh (5) to name a few. The technique has gainedconsiderable recognition with application to problems associated with the aircraft industry. In all ofthese papers, however, the technique has been applied to problems associated with isotropic materials.In general, the technique can be divided into two separate subcategories: (1) The framework methodand (2) the stiffness element method. The two methods differ principally in the composition of thefinite element. It is the purpose of this paper to examine each of these methods closely and determinetheir applicability in handling problems in orthotropic plane stress.1Maintained at Madison, Wis., in cooperation with the University of Wisconsin.2Underlined numbers in parentheses refer to Literature Cited at the end of this Paper.
METHODSFramework MethodThe framework method is appropriately named since the method consists of replacing the solidelastic body to be analyzed by a mathematical model of an imaginary framework. The framework isofcomposed of a series of pin-connected bars arranged in a definite pattern compatible with theproblem solved, such as the cantilever beam of figure 1.The most common pattern used in the framework method, and the one used by McCormick, is arectangular elementdiagonals connecting the corners, as shown in figure 2.The areas of the bars A in the network system are determined by the requirement of equal deformaibility of the solid elastic body and the framework model under a given stress situation. Computationsfor isotropic materials have been carried out by Hrennikoff and McCormick. In the process of calculation, however, it was discovered that a restriction must be imposed such thatis Poisson's ratio of the solid elastic body.1physical significance of this has not been established, but the proximity of to the actual3value of Poisson's ratio for isotropic materials permits the applicability of the method for suchmaterials.In orthotropic plane stress, however, there are two Poisson’s ratios associated with stress in theplane. This could perhaps be seen more clearly by observing the form of the generalized stressstrain relationships for orthotropic plane stress:whereTheFPL 872
where µeand µ are the two Poisson's ratios associated with the x-y plane and where ε , ε , andyxxyx yare the strains associated with the x, y, and x-y directions, respectively. This might imply an evenxygreater restriction of the framework method when applied to orthotropic materials. Computationswill now be carried out similar to those performed for isotropic material by Hrennikoff and McCormick,to determine the restriction in orthotropic plane stress.Mathematical analysis.--Consider a solid orthotropic elastic element (wood) of thickness t, and thecorresponding framework model subjected to a situation of pure shear stress:3
For such a s t r e s s situation, since 0 (where ε is strain), the diagonal membersmust supply the shear stiffness. The required stiffness (S A E ) of the diagonal members can bed dddetermined by considering equilibrium of a section as:where:(2)From Mohr’s circle (see Appendix I):(3)Also:(4)equating(3) and(4):(5)From consideration of deformability of the solid element:(6)Substituting (6) in (5) results in:(7)FPL 874
Consider now the stress situation existing such a s to create a case of strain in the x-directiononly, i.e.:(8)Therefore for the elastic body and using equations (1):(9)(utilizing the reciprocal relationship for orthotropic materialsThe expression forcan then be written:(10)whereA pictorial representation of this stress situation would look like:5
Considering again the equilibrium of a framework section under this situation:(11)(12)The forces P1and P2can also be determined by realizing that:Therefore:(13)Equating (11) and (13) results in:(14)Also from Mohr’s circle it can be seen:(15)FPL 876
Therefore:(16)since(17)equating (12) and (17) results in:(18)The same expression for S in (18) could be obtained using a stress situation such thatd0. The point is, however, that it is also the function of the diagonal members to providestiffness for the Poisson’s effect when the model i s subjected to extensional strain only. The stiffness(S ) a s given by (18) and (7) must be equal. Hence, equating, we have:dwhich in general i s not true for orthotropic materials. In view of this basic ingenerality, therefore,it. is concluded that the framework method is not applicable to general orthotropic plane stress problems, but should give good results for the special case of orthotropy a s defined by equation (19).Stiffness Element MethodThe stiffness element method differs from that of the framework method in that the elements in thenetwork system are solid or plate elements, and further the elastic properties of the element shouldduplicate the material it replaces. The elements still remain connected to each other only at thecorners or nodes, This perhaps can be visualized more easily if reference is made to a typical orthotropic beam problem shown in figure 3 which will also be the subject of the following discussion onorthotropic plane stress. From figure 3 it is easy to visualize that as the element size decreases, orthe number of elements increases, the behavior of the beam model will tend to approach the truebehavior of the orthotropic beam.7
The stress situation in the beam model will be determined by the manner in which the forces arepropagated from element node to element node. For any one element the forces are related to thedisplacement of the element nodes, since deformability is the physical feature determining the mannerof stress propagation through the stressed medium, Each node of the network may have externalforces applied in the x and y directions (coordinate system as shown in fig. 4). The deformation ofa single element, of thickness t, such as shown in figure 4 is defined by the eight possible nodaldisplacements.The relationships between the forces and displacements of a single element can be convenientlyhandled in matrix form as:(20)where {}, [ ] indicate a column and rectangular matrix, respectively.The [K] matrix in equation (20) is generally termed the stiffness matrix, hence the name given tothis method.In this method the structure is regarded as an assemblage of parts and each component has associated with it a stiffness matrix relating the forces and displacements at its nodes, The stiffnessmatrix for the complete connected structure is then obtained by addition of all the component stiffnessmatrices. For a major portion of plane stress problems and those to be dealt with in this paper, theobject to be analyzed is considered to be homogeneous throughout, which means each componentstiffness matrix is identical.SPECIFIC OBJECTIVESThe remaining objectives of this paper, therefore, are to: A. Develop the stiffness [K]single orthotropic element. B. Determine a technique in which the component stiffnessbe conveniently handled without overflowing available computer capacity, C. Checkby investigating simple problems in which the stress distributions are well established.878matrix for amatrices canthese resultsD. Develop
a stiffness [ K] matrix which might be used in orthotropic plate analysis. And finally, E. Check theresults of the bending stiffness matrix by analyzing a hypothetical orthotropic plate and comparingthe results with a rigorous mathematical analysis.Develop K Matrix for Single ElementConsider for analysis the orthotropic plate element as shown in figure 4 with the forces and displacements shown in their positive directions, The stresses and strains for such an element are relatedby:(21)The coefficients in any one column of thematrix represent physically the forces which must beapplied at the nodes in order to give a displacement of unity for the particular column chosen whilethe remaining displacements remain zero. It is this part of the derivation which determines the numberof strain expressions or alternatively the number of applied stress states which must be used toachieve this. The number is always twice the number of nodes minus three. Hence, for a rectangularelement, five states are required, which will be:9
For ease in derivation, theof superposition will be utilized so that each case may be handledseparately and the results added to determine the combined effect.Consider first:(a) Strain in the 5-direction, i.e.,From equations (21), the stresses for an orthotropic element a r e found to be:It is assumed, however, that these stresses can be replaced by the following equivalent forcesapplied at the nodal points (refer to fig. 4 for sign convention):The expression for E can be written:XSubstituting,therefore:Also:orConsider next:(b) Strain in the y-direction, i.e.,The stresses induced, therefore, are:FPL 8710
The equivalent force system, therefore:where:Then:Also:(c) Shear Strain, i.e.,The expression for shear stress is given by:Considering forces on the plate element:11
from figure 5, therefore, it can be seen that:In equivalent force s y s t e mand also:Similarly:and(d) Pure Bending About x-AxisConsider element with bending forces:SincethereforeIn considering the equivalent force system for this case, care must be taken to insure equilibrium.The typical force relationship becomes:FPL 8712
Therefore:orwhere:Then:(e) Pure Bending About y-AxisThe positive bending action can be represented by:13
SincethereforeThe equivalent force system can be written:whereThen:By superimposing the relationships obtained between forces and displacements for the five stressstates, the final equations can be written in matrix form as:(22)where:FPL 8714
or simply:where the [ K ] matrix represents the matrix of coefficients as given in (22).In order to obtain numerical results in later calculations, it will be assumed that the orthotropicbeam in figure 3 has the following elastic properties;Therefore:It will also be assumed that the finite elements a r e square s o that;15
whereDevelop a Technique for Handling the ComponentStiffness MatricesWith the establishment of the component stiffness matrix it is now desirable to formulate a compositestiffness matrix which would relate, for example, all the nodal forces and deformations occurring inour beam model in figure 3. For many problems, however, it is not necessary to model the wholestructure where in most places the stress distribution is well established, but only adjacent to locations where irregularities occur which might effect the normal stress distribution, For purposes ofverification and discussion, therefore, let us assume it is desirable to model a section of the beamadjacent to thewhere symmetry might be observed, and use can be made of the boundary conditionthat horizontal displacement at themust be zero. The beam model of such a section is shown infigure 6.FPL 8716
The size of a network (9 x 9) of elements was limited to the available computer capacity. Theelements are numbered vertically and the arrows indicate the direction of positive forces and displacements,The composite stiffness matrix, then, will relate all the forces to displacements occurring in themodeled section. Such a matrix, however, would represent the solution of 200 simultaneous equations,which might prove too cumbersome for many small digital computers. It is necessary, therefore, todevelop a technique by which the matrix size might be kept small.It is first necessary to build a typical stiffness matrix for the column of elements being modeled.Since the model is assumed to be homogeneous, all such stiffness matrices are identical, the onlychange being the nodal values. The stiffness matrix, for example, for the first column of elements canbe obtained by combining the elemental component matrices one through nine (see fig. 6). This resultsin a matrix of size 40 x 40, relating the first 40 forces and displacements. The second matrix isidentical to the first with the exception it relates the forces and displacements 21 through 60. Thecomposite matrix for the first two columns of elements can he obtained by combining the individualmatrices, realizing that forces and displacement 21 through 40 are common to both columns of elements, resulting in a matrix of size 60 by 60. This may be represented in the form of a partitionedmatrix as:(23)(where the K matrices a r e of size 20 by 20.)iIf the type of problem to be solved is restricted so that no external loading of the structure willoccur between extreme sections of the model, then use can be made of the fact that internal forceequilibrium must occur at each common node or :(24)17
To make use ofso to read:(24), it is first convenient to interchange rows and columns of equations (23)(25)or(26)where:FPL 8718
Writing equation (24) in matrix form gives:or(27)The forces and displacements 1 through 20 and 41 through 60 can now be written:(28)Substituting equations (27) in equations (28) results in:or finally:(29)19
This process of eliminating the common nodal values can be repeated until the desired network sizeis achieved. For the particular size network chosen here, the final matrix is modified so that thefinal equations become(30)Verifying the Beam ModelWith the establishment of the A modified matrix for the 9 by 9 network, it is now possible to checkthe model’s ability to duplicate the original beam by subjecting it to stress distributions which arewell established and calculate whether the matrix analysis yields equivalent results. For purposes ofcomparison, therefore, the following three loading conditions will be applied to the beam model in 9,000 pounds per squarefigures 3 and 6; (1) a force F acting alone, such that a tensile stress ofinch is induced, (2) the forces P acting alone such that a maximum bending stress of 10,000 poundsper square inch is induced in the extreme fibers of the beam model, and (3) a combination (1) and (2).The given stress loading conditions will be applied as a statically equivalent force system derivedfrom states of loading designated as (a) and (d) in the section “Develop K Matrix for Single Element.”These forces will be applied to the extreme left section of the beam model and then the stress situationcalculated at the centroid of each element in the ninth column. The stress distributions will then becompared with the known distribution.This operation can perhaps be seen more clearly by beginning with the equations:(30)FPL 8720
By observing the boundary conditions at theyielding(31)The { f } matrix now represents the external force matrix which, for the three loading conditions,iis presented on page 22, where (dt) represents the cross-sectional area of the beam.The displacements in equation (31) can now be determined by carrying out the matrix operationdefined, for each of the three cases. The displacement 6through 6can be gotten from node180161elimination equation (27), which for addition of the ninth column would appear as(where the D and C matrices are determined when the ninth column was added.)21
FPL 8722
With the solution of equations (31), all the nodal displacements for column 9 are known. It is nowa relatively easy operation to relate these displacements to corresponding stresses at the centroidsof each element, by making use of the equations already established in the section “Develop K Matrixfor Single Element.”It is found that the stress matrix for any one element of such a 9 element column can be written:where:where thematrix represents the eight nodal displacements about the element maintaining coordinatevalues similar to those in figure 4.The resulting computed stress distributions for column 9 are presented in figure 7. On the basis ofthis figure, therefore, it is concluded that the finite element technique was successful in modelingthese three simple cases of orthotropic plane stress.Figure 7.--Stress distributionsacrossof 9th column of orthotropic beam model.23
Finite Element Technique in OrthotropicPlate AnalysisAnother area in which the finite element technique has gained popularity is in bending problems ofthin plates. Most of these problems again have been associated with the aircraft industry, where itseems most of this work has been pioneered. Melosh (5), derives an elemental stiffness matrixutilizing the bending strain energy expression for a uniform flexurally rigid isotropic plate. Most ofthe derivations performed by Melosh were based on purely geometrical considerations, so that atransformation to derive a plate stiffness matrix for orthotropic thin plates can be accomplishedrather easily. For this section, therefore, an elemental stiffness matrix will be derived for orthotropicthin plates, for which the procedure will parallel that of Melosh with the exception of maintainingorthotropic behavior,Mathematical analysis.--It will be assumed that a given rectangular orthotropic (wood or plywood)plate, as shown in figure 8a, can be modeled by an assemblage of rectangular elements connected attheir nodal points, as in figure 8b.Each element in figure 8 b is assigned a stiffness matrix relating the forces and displacements atits nodal points. Each node will have three degrees of freedom, the angles of rotation and about thex and y axis, respectively, and the lateral displacement of each node, w . Associated with these disiplacements are the forcesat each node, respectively. The stiffness matrix for eachelement will be of size 12 by 12. The stiffness matrix of the complete structure can then be determinedby addition of the individual component matrices.plate in which the axes of orthotropyThe bending strain energy expression for acoincide with the plate boundaries is given by (1):(32)whereFPL 8724
The stiffness matrix will be developed by adding matrices reflecting the stiffness of each term inthe bending energy expression, since each of these terms can be treated separately.In deriving the plate stiffness matrix, it will be assumed that the bending curvature along the edgesof the plate can be expressed in terms of a third-order polynominal, or(33)where w now defines the displacements; for example, along edge 1-3 in Figure 9.The four constants in expression (33) can be determined by applying the boundary conditions whichmust exist along that edge; for example, along edge 1-3, the conditions are:25
where it can be found that:From these expressions, a relationship foralong 1-3 can be written in matrix notation as:’(34)The total strain energy resulting fromcan be found by squaring expression (34) andintegrating the result over half the plate area. (It is also assumed that expression (33) varies uniformlywith y.) The force-displacement relationships can then be found for this portion of the plate by utilizingCastigliano’s complementary relation:in whichrepresent displacement and force components, respectively. Performing theseoperations then yields the following expressions for edge 1-3:(35)3 The s u b s c r i p t o n t h e d i f f e r e n t i a l t e r m s d e n o t e s t h e edge a l o n g w h i c h t h e r e l a t i o n s h i papplies.FPL 8726
A similar set of expressions can be found by treating edge 2-4 and using the remaining half of theplate. By adding the two results, the final bending stiffness matrix due to theequation (32) can be written:(36)27
If it is assumed that displacements along the edge in the Y-direction are of the same general formas given by expression (33), then the bending stiffness matrix for theby observing symmetry as:term can be written(37)To obtain the bending strain energy due to the bending coupling effect, it is necessary to performthe integration:over the area of the plate. This can be accomplished, as outlined by Melosh, by taking the product,i.e.,at each node and integrating over only a quadrant of the plate at a time. The sum of theintegral of the four products then represents the coupling energy. For example, to obtain the energyfor the first quadrant, it is necessary to perform the integration:where the productFPL 87can be written in matrix form as:28
29
Treating the remaining three quadrants in a similar manner and superimposing the results lead tothe following stiffness matrix for the coupling energy t e r m of equation (32):In deriving the stiffness matrix for the torsion t e r m of equation (32), it is necessary to determineits corresponding energy or:(39)An expression forfigure 10, where:can be found by considering the simple torsion of the finite element of(40)FPL 8730
By squaring equation (40), performing the integration of (39), and differentiation as done previously,it can be found that the stiffness matrix for the torsion term can be written:(41)The final stiffness matrix for orthotropic thin plates can now be determined by superposition ofexpressions(36), (37), (38), and (41), and found to be:(42)31
where:and recalling that:Where for plywood plates, these elastic properties represent their effective values in the directiondenoted by the subscripts. These can be determined experimentally or by knowing the properties ofthe individual plies, they can be calculated by existing formula (7).Comparison of Finite Element Model withMathematical AnalysisWith the establishment of the stiffness matrix for orthotropic thin plates, it is now desirable tocompare the finite element model’s ability to duplicate the behavior of an orthotropic plate as describedby a rigorous mathematical solution such as that derived by March (3).For purposes of comparison, therefore, it will be assumed it is desirable to determine themaximum deflection of the simply-supported five-ply plywood plate of Figure 10 under (1) a concentrated center load and (2) a uniformly distributed loadFPL 8732
From March it can be found after considerable calculation that the maximum center deflection W is0given by:for the case of the concentrated load, and by:for the case of the uniformly distributed load, where D is as previously defined.To obtain comparative values by the finite element technique, the plate of Figure 10 will be modeledby 36 finite elements as shown in Figure33
By adding the individual elemental stiffness matrices it can be seen that the final composite matrixThis can be reduced, however, to a 48 by 48 matrix by observing symmetrywill be of size 147 byand analyzing only one quadrant for the particular cases chosen. By applying the appropriate boundaryconditions and inverting the resulting matrix, the maximumdeflections by the finite element model wasfound to be:andfor the cases of the concentrated and uniform loads, respectively.By comparing results of the two analyses it is seen that for the concentrated load problem the difference is in the neighborhood of 1 percent, while for the uniform load case the difference is approximately 4 percent. The greater error found in the uniform load case is attributed to the fact that in thefinite element approach, the uniform load is replaced by a statically equivalent set of concentratedforces acting at the nodal points, and as a result the loads along the boundary of the plate have noeffect on the bending of the plate.FPL 8734
CONCLUSIONSOn the basis of the limited examples examined in this report, it appears the finite element techniquehas merit regarding problems in orthotropic plane stress. It is realized that the networks used in thisreport were extremely coarse, but this should not distract the fact of the model’s ability to duplicateorthotropic behavior. It is felt that the techniques which need to be developed in the future lie in theeffective handling of the large systems of equations that result a s finer networks a r e desired. Theprimary obstacle possibly is the round-off e r r o r occurring in the many digital computations.35
LITERATURE CITED1. Hearmon, R.F.S.1961. Applied anistropic elasticity. Oxford University Press.2. Hrennikoff, A,1941. Solutions of problems of elasticity by the framework method. J. Appl. Mech. 8(4):A169A175 (Dec.)3. March, H.W.1942. Flat plates ofplywoodunder uniformor concentrated loads. Forest Products Lab. Rep. 1312.4. McCormick, C.1963. Plane stress analysis. J. Struc. Div., Proc. Amer. Soc. Civil Eng. (Aug.)5. Melosh, R. J.1961. A stiffness matrix for the analysis ofthinplates in bending. J. Aero. Sci. 28(1):34-42 (Jan.)6. Turner, M. J., Clough, R. W., Martin, H. C., and Topp, L. J.1956. Stiffness and deflection analysis of complex structures. J. Aero. Sci. 23(9): 805-823 (Sept.)7. U.S. Department of Defense1951. Design of wood aircraft structures. Air Force, Navy, Commerce Bulletin ANC-18. 2nd ed.Munitions Board Aircraft Comm.FPL 8736
APPENDIX 1Mohr's c i r c l e s o f s t r a i n for t h e t y p i c a l element;S t r a i n i n t h e x- D i r e c t i o n o n l y .37
S t r a i n i n t h e y- D i r e c t i o n o n l y .Shear S t r a i n i n x - y Plane.FPL 8738
NOTATIONStrains (extension or compression) in the x and y directions, respectively.Strain (shear); the change in angle between lines originally drawn in the x and y directions.Modulus of elasticity of wood in the x and y directions, respectively.Modulus of rigidity associated with shear deformation in the x-y plane resulting from shearstresses in the xz and yz planes.Normal stress components in the x and y directions, respectively,Shear stress associated with the x-y plane.Poisson's ratio of contraction along the direction y to extension along the direction x due toa normal tensile stress in the direction x; similarly,The stiffness values (E A ) of the diagonal and horizontal member in the framework model.i i39
PUBLICATION LISTS ISSUED BY THEFOREST PRODUCTS LABORATORYThe following lists of publications deal with investigative projects of theForest Products Laboratory o r relate to special interest groups and a r e available upon request:Architects, Builders, Engineers,and Retail LumbermenLogging, Milling, and Utilizationof Timber ProductsBox and Crate Construction andPackaging DataMechanical Properties and Structural Uses of Wood and WoodProductsChemistry of WoodModified Woods, Paper - BaseLaminates, and ReinforcedPlastic LaminatesDrying of WoodFire PerformanceSandwich ConstructionFungus and Insect Defects inForest ProductsThermal Properties of WoodFurniture Manufacturers,Woodworkers, and Teachersof Woodshop PracticeWood Fiber ProductsGlue and PlywoodWood PreservationWood Finishing SubjectsGrowth, Structure, andIdentification of WoodNote: Since Forest Products Laboratory publications a r e so varied in subjectmatter, no single catalog of titles is issued. Instead, a listing is made foreach area of Laboratory research. Twice a year, January 1 and July 1,a list is compiled showing new reports for the previous 6 months.This is the only item sent regularly to the Laboratory’s mailing roster,and it serves to keep current the various subject matter listings. Namesmay be added to the mailing roster upon request.FPL 87401.5-40
network of finite elements. It is believed that as the size of the finite element approaches the differ-ential element stage, the results yielded by the method would compare favorably to those obtained from a rigorous mathematical analysis. By keeping the element finite in size, the network model
Finite element analysis DNV GL AS 1.7 Finite element types All calculation methods described in this class guideline are based on linear finite element analysis of three dimensional structural models. The general types of finite elements to be used in the finite element analysis are given in Table 2. Table 2 Types of finite element Type of .
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element type. This paper presents a comprehensive study of finite element modeling techniques for solder joint fatigue life prediction. Several guidelines are recommended to obtain consistent and accurate finite element results. Introduction Finite element method has been used for a long time to study the solder joint fatigue life in thermal .
1 Overview of Finite Element Method 3 1.1 Basic Concept 3 1.2 Historical Background 3 1.3 General Applicability of the Method 7 1.4 Engineering Applications of the Finite Element Method 10 1.5 General Description of the Finite Element Method 10 1.6 Comparison of Finite Element Method with Other Methods of Analysis
Finite Element Method Partial Differential Equations arise in the mathematical modelling of many engineering problems Analytical solution or exact solution is very complicated Alternative: Numerical Solution – Finite element method, finite difference method, finite volume method, boundary element method, discrete element method, etc. 9
3.2 Finite Element Equations 23 3.3 Stiffness Matrix of a Triangular Element 26 3.4 Assembly of the Global Equation System 27 3.5 Example of the Global Matrix Assembly 29 Problems 30 4 Finite Element Program 33 4.1 Object-oriented Approach to Finite Element Programming 33 4.2 Requirements for the Finite Element Application 34 4.2.1 Overall .
2.7 The solution of the finite element equation 35 2.8 Time for solution 37 2.9 The finite element software systems 37 2.9.1 Selection of the finite element softwaresystem 38 2.9.2 Training 38 2.9.3 LUSAS finite element system 39 CHAPTER 3: THEORETICAL PREDICTION OF THE DESIGN ANALYSIS OF THE HYDRAULIC PRESS MACHINE 3.1 Introduction 52
4.3 STAGES OF SOCIAL WORK GROUP FORMATION There are a number of stages or phases in formation of a social work group. Ken Heap (1985) discussed these as group formation and planning; the first meetings; the working phase; use of activities and action; and the termination of the Group. According to Douglas (1979) there are five stages viz., conceptualisation, creation, operation, termination .
