Finite Element Method (FEM) is an efficient and powerful tool to numerically analyze and solve the problems related to structures and continua.

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Finite Element Method (FEM) is an efficient and powerful tool to numerically analyze and solve the problems related to structures and continua.

Chapter 1

Introduction

Finite Element Method (FEM) is an efficient and powerful tool to numerically analyze and solve the problems related to structures and continua. While there are various numerical analysis methods, FEM attained the largest popularity in many fields of engineering. Usually the problem addressed is too complicated to be solved satisfactorily by classical analytical methods, where FEM provides ways to deal with those problems in a systematic manner (which would be explained later on).

1.1 What is The Finite Element Method (FEM)?

The Finite Element Method originated as a method of stress analysis¹. Today finite elements are also used to analyze problems of heat transfer, fluid flow, lubrication, electric and magnetic fields and many others. This method is used to model a structure as an assemblage of small elements. Each element is of simple geometry and therefore is much easier to analyze than the actual structure. Then analyzing those elements individually and taking into account the interactions between them, the solution of that specified problem can be obtained.

The finite element procedure produces many simultaneous algebraic equations, where the calculations are performed on personal computers, mainframes and all sizes in between. Results are rarely exact. However, errors are decreased by processing more equations and results accurate enough for engineering purposes are obtainable at reasonable cost.

Now-a-days most of the analysis using FEM is done on software packages, which comprises of mainly three components – pre-processor, solver and post-processor. These components usually perform the functions followed by a typical finite element analysis, where the steps that are pursued to do so are given below:

– At first the structure or continuum has to be discredited into finite elements. Mesh generation program, called Pre-processors, help the user in doing so. Then the boundary conditions and known nodal values (for plane stress problem it is zero, for heat transfer problem it is the nodal temperature) are specified on the mesh to assign it the correct nodal degrees of freedom.

– In the next step simultaneous linear algebraic equations, evolved from various elements, are solved by the program called ‘Solver’ to determine the specific nodal values, which depends upon the nature of the problem (for plane stress problem it is nodal stress value, for heat transfer problem it is the nodal heat fluxes, etc.).

– Last step concerns the generation of output (as determined by the solver) in a graphical form with the help of an output interpretation program, called Post-processor.

The power of the finite element method resides principally in its versatility. The method can be applied to various physical problems. The body analyzed can have arbitrary shape, loads and boundary conditions. The mesh can mix elements of different types, shapes and physical properties. This great versatility may be contained within a single computer program. User-prepared input data controls the selection of problem type, geometry, boundary conditions, and element selection and so on.

Another attractive feature of finite element method is the close physical resemblance between the actual structures and its finite element model. The model is not simply an abstraction. This seems especially true in structural mechanics and may account for the finite element method having its origins here.

1.2 How/Why should we study the method?

Whether computer based or not, analytical methods rely on assumptions and on theory that is not universally applicable. That is why its limitations are really a matter to be concerned with. It is far more easy for a user to make silly mistakes like making an error in computer program, input of wrong data or generation of poor mesh, which would lead to the formation of incorrect output with elegant graphic display. The results obtained from the programs cannot be trusted if user has no knowledge of their internal workings and little understanding of the physical theories on which those are based. Moreover, the choice of element for various analyses is crucial. An element that is good in one problem area (such as magnetic fields) may be poor in another (such as stress analysis). Even in a specific problem area an element may behave well or badly, depending on particular geometry, loading and boundary conditions. If an analysis is to be done by numerical methods, finite elements are not the only choice, because there are other methods like finite difference method, boundary element method, finite volume method etc., which would be more effective in some areas of analysis than the finite element method. For example, finite difference methods are effective for shell of revolutions and boundary elements are effective for some problems with boundaries at infinity. But in linear computational solid mechanics problems, finite element methods currently dominate the scene as regards space discretization. Boundary element methods post a strong second choice in specific application areas, where for nonlinear problems the dominance of finite element method is overwhelming. In other cases experiment may be the most appropriate method to obtain data needed for analysis, as well as to compare it with the results obtained from analysis using the discretionary methods, where the analytical process is being pushed beyond previous experience & established practice.

1.3 Present study:

This study concerns mainly with the fundamentals of FEM, where a particular structural problem is analyzed to make the reader acquainted with the steps involved in solving problems using this method. Here the problem concerning stress distribution in an infinitely long plate with a hole subjected to uniform tensile force at the edges of the plate is being reviewed with the help of a FEM software package- ‘LISA’. This is a typical structural discontinuity problem for which the theoretical solutions are obtainable and is very much commonly encountered in the structural construction of ships, aero-planes, cars, etc. The stress concentration factor, which is the ratio of maximum developed stress to the applied uniform stress, is being considered to be the determining factor for the intensity of stress distribution. The value of this factor depends very much on the abruptness of the discontinuity and it follows that it is desirable to design structures in the neighborhood of a discontinuity so as to keep this magnification factor as low as possible². The effect of the high local stress may result in the stress concentration to be so great as to cause direct local failure of the material.

In chapter 2 some primary concepts about the process followed in finite element method are being introduced. Following this, the formulation and examination of the problem, as mentioned above, are being carried out in Chapter 3. And at last some future recommendations on further study in this field of computational mechanics are made in Chapter 4.

Chapter 2

Fundamentals of Finite Element Method

Today the concept of the finite element method is a very broad one. Even when restricting ourselves to the analysis of structural mechanics problems only, the approach towards the formulation of those can differ in nature. Here the potential energy approach is being applied in the derivation of necessary entities, needed in solving those analytical problems.

The formulation of element stiffness matrix and global load vector requires the potential energy or variational approach³, where the potential energy is defined as,

………….. (2.1)

Where, the first term denotes the strain energy equation for a linear elastic body, which is expressed as,

And the rest of the terms together constitute the work potential of the body, where

WP =

In eqn.2.1, f & T denotes the body force components comprising the distributed forces per unit volume and surface traction forces per unit area respectively as shown in Fig-2.1. Here the Fig-2.1 represents a three-dimensional body having a volume and surface area V and S respectively. Traction force per unit area T, distributed body force per unit volume f, are also shown in the figure, where some region of the boundary, S_u, are constrained. The deformation of a point is given by the three components of its displacements,. In the last term of eqn.2.1, P_i represents a force acting at point i.

Fig-2.1: A three dimensional body

As mentioned earlier in section 1.1, that the FEM analysis of the structural problem involves three major steps – the tasks involved in each step require a good understanding of the sequential aggregation of the modeling of the structure, evaluation of Global stiffness matrix and load vector and the handling of specified displacement boundary conditions. Since this study mainly focuses on the two-dimensional problem concerning the analysis of a plate with a hole, the approach towards most of the conceptual review would be to cover only the two-dimensional aspects of finite element analysis.

2.1 Finite Element Modeling

The Finite Element Method is the dominant discretization technique in structural mechanics. The basic concept in FEM is the subdivision of a region into disjoint (non-overlapping) components of simple geometry called finite element or element for short. For 2-D modeling the most commonly employed elements are linear or quadratic triangles and quadrilaterals. In two-dimensional problem, each node is permitted to displace in the two directions. Thus, each node has two degrees of freedom. So, the displacement components of node j are taken as Q_2j-1 in the x direction and Q_2j in the y direction. And the global displacement vector can be represented as

………………….. (2.2)

And global load vector,

………………….. (2.3)

Where, N is the number of degrees of freedom (dof), which is defined as the flexibility of the nodes to displace in permitted numbers of direction. Thus in two dimensional problem, as the nodes are permitted to displace in both x and y direction, hence each node has two degrees of freedom.

Computationally, the information on the discretization is to be represented in the form of nodal coordinates and connectivity. The nodal coordinates are stored in a two-dimensional array represented by the total number of nodes and the two coordinates per node. The element connectivity information is an array of the size and number of elements and the nodes per element, which are the global node numbers of the particular elements that can be derived from the discretized region.

2.2 Evaluation of Global Stiffness Matrix and Load Vector

This section explains the way to assemble the Global Stiffness Matrix and Load Vector.

The total potential energy as in eqn.2.1 can be written in the form,

………………………… (2.4)

by taking element connectivity into account, where K and F are the Global Stiffness Matrix and Load Vector respectively. This step involves assembling K and F from element stiffness and force matrices. Here a one-dimensional approach is being followed in deriving the global stiffness matrix using a spring system with arbitrarily numbered nodes and elements (Fig-2.2), where the basic procedure is the same for two and three dimensional problems in FEM analysis.

Fig-2.2: A Spring system

Here, F₁, F₃ are considered to be the nodal forces and K₁, K₂, K₃ and K₄ are the element stiffness of the four springs. The nodal displacements are defined as u₁, u₂, u₃, u₄ and u₅.

Element connectivity Table can be constructed as follows:

The element stiffness matrices can be expressed as follows:

u₄ u₂ u₂ u₃

u₃ u₅ u₂ u₁

Finally, applying the superposition method, the global stiffness matrix can be obtained,

u₁ u₂ u₃ u₄ u₅.

This matrix is symmetric, banded and singular.

Similarly the global load vector F is assembled from element force vectors and point loads as-

where, & are the element body force vector and element traction force vector respectively and _i represents the point vector

For the spring system, as illustrated in Fig-2.2 the load vector can be represented as,

2.3 Treatment of Boundary conditions:

In dealing with the proper boundary condition and deriving the equilibrium equations the minimum potential energy theorem can be used. This theorem states that: Of all possible displacements that satisfy the boundary conditions of a structural system, those corresponding to equilibrium configurations make the total potential energy assume a minimum value.Consequently, the equations of equilibrium can be obtained by minimizing with respect to Q, the potential energy subject to the boundary conditions. Boundary conditions are usually of the type,

Q_p1 = a₁, Q_p2 = a₂, …….., Q_pr = a_r

Where, P₁,P₂,……, P_r are denoted to be the degrees of freedom and r is judged to be the number of supports in the structure.

For an N-dof structure, let the single boundary condition to be Q₁ = a_1, where the global stiffness matrix is of the form,

…………………………….. (2.5)

Where, K is a symmetric matrix.

Using eqns.2.2, 2.3 and 2.5 in eqn.2.4, the potential energy can be written in the expanded form as,

(2.6)

Substituting the boundary condition Q₁ = a₁ in eqn.2.6, we obtain

(2.7)

In this expression the displacement Q₁ has been eliminated.

Consequently, the requirement that take on a minimum value implies that,

…………. (2.8)

From eqns.2.7 and 2.8 we obtain,

(2.9)

These finite element equations can be expressed in the matrix form as,

…. (2.10)

Which may be denoted as?

……………………. (2.11)

Where, K is a reduced (N-1×N-1) matrix obtained by eliminating the row and column corresponding to the specified degrees of freedom. This process of determining equilibrium equations is referred to as elimination approach.

In the elimination approach, the stiffness matrix K is obtained by deleting rows and columns corresponding to fixed dofs. In the spring system of Fig-2.2, the boundary conditions can be expressed as,

u₄ = u₅ = 0

Thus by deleting 4^th and 5^th rows and columns of original K the modified K is obtained. Also Q and F is obtained by deleting 4^th and 5^th component of the original Q and F respectively, where the equilibrium equations can be expressed as,

u₁ u₂ u₃

Another approach to handle the specified displacement boundary conditions is the penalty approach. In this approach a spring with a large stiffness C is used to model the boundary condition, which may in this case be assumed to be Q₁ = a₁. At the support the spring is supposed to be displaced by an amount of a₁, where the point of support of the structure will have a displacement approximately equal to a₁. Hence, the strain energy in the spring equals,

…………….. (2.12)

This strain energy contributes to the total potential energy. So, from eqn.2.4 we get,

……. (2.13)

The minimization of can be carried out by setting, i = 1,2,…..,N.

The resulting finite element equations are,

……… (2.14)

The only modifications from the elimination approach that take place in this process are the introduction of a large number (say C) which gets added to the first diagonal element of K and that Ca₁ gets added on to F₁. The magnitude of C can be expressed as:

For

For the spring system of Fig-2.2, a large number C gets added to the 4^th and 5^th diagonal elements of original K to determine the modified stiffness matrix, K. And to modify the load vector, C.0 or 0 gets added to the 4^th and 5^th component of F.Here, the value of C is chosen as,

Assuming,

>

Now, the modified stiffness matrix can be expressed as,

And the global load vector transforms into,

Or,

So, the equilibrium equation can be expressed as,

2.4 Element stress calculation

Equation 2.11 and 2.14 can be solved for the displacement vector Q using Gaussian elimination. As the reduced K matrix is a nonsingular one, the boundary condition can be considered to be specified properly. Once Q has been determined, the element stress can be evaluated using the equation derived from Hooke’s law,

…………………. (2.15)

where B is the element strain-displacement matrix, which is defined later in section 2.5(b) and q is the element displacement vector for each element, which is extracted from Q using element connectivity information.

2.5 Formulation of four node quadrilateral element matrices:

The two dimensional Finite Element formulation provides with a family of Isoperimetric

Elements, where the Four-node Quadrilateral Element represents one of the most basic and rudimentary form among those. Since in this study the discretization of the models are made using the quadrilateral elements, the shape functions, element stiffness matrix and element body forces for only this particular element is being figured out here.

Here a general quadrilateral element has been considered as shown in Fig-2.3, having local nodes numbered as 1,2,3 and 4 in a counterclockwise fashion and (x_i, y_i) are the coordinates of node i. The vector q = [q₁, q₂,…….q₈]^T denotes the element displacement vector. The displacement of an interior point P located at (x, y) is represented as u = [u(x, y), v(x, y)]^T.

2.5(a) SHAPE FUNCTIONS

To develop the shape functions³ let us consider a master element (Fig-2.4) having a square shape and being defined in ?-, ?- coordinates (or natural coordinates). The Lagrange shape functions, where i = 1, 2, 3 and 4, are defined such that N_i is equal to unity at node i and is zero at other nodes. In particular:

= 1 at node 1

= 0 at nodes 2, 3 and 4 ………….. (2.16)

Now, the requirement that N₁ = 0 at nodes 2, 3 and 4 is equivalent to requiring that N₁ = 0 along edges ? = +1 and ? = +1 (Fig-2.4).

Thus, N₁ has to be of the form

………………………………… (2.17)

?

(-1,1) (1,1)

q₈ q₆ 4 3

q₇ q₅

v

u

•

q₂ P(x, y) ?

q₁

Y q₄

1 2

X q₃ (-1, -1) (1, -1)

Fig 2.3: Four-node quadrilateral element Fig 2.4: The quadrilateral element

in ?, ? space (master element)

Where, c is some constant. The constant is determined from the condition, N₁ = 1 at node 1. Since, ? = -1, ? = -1 at node 1, we have

……………………… (2.18)

Which yields c = ¼. Thus,

……………… (2.19)

All the four shape functions can be written as

.…………….. (2.20)

Now, to express the displacement field within the element in terms of the nodal values, let u = [u, v]^T represent the displacement components of a point located at (?, ?), and q, dimension (8 × 1), represents the element displacement vector, then

…………….. (2.21)

This can be written in matrix form as,

u = Nq ………………………. (2.22)

Where,

N = …………. (2.23)

In the isoparametric formulation, the same shape functions can also be used to express the coordinates of a point within the element in terms of nodal coordinates.

Thus,

………………. (2.24)

Let a function in view of Eqs.2.24 be considered to be an implicit function of and as Using the chain rule of differentiation, we have

……………… (2.25)

or,

J ……………………… (2.26)

Where, J is the Jacobian matrix.

J = ………………………. (2.27)

In view of Eqs.2.20 & 2.24, we have

J =

= …………………………………………………………………. (2.28)

Equation 2.26 can be written as,

J^-1 ……………………………. (2.29)

or,

……………… (2.30)

An additional relation that is worthy to mention, from calculus we find that

…………………………… (2.31)

2.5(b) ELEMENT STIFFNESS MATRIX

The stiffness matrix for the quadrilateral element can be derived from the strain energy in the body, given by,

………………………………. (2.32)

Or,

………………………….. (2.33)

Where is the thickness of element e.

The strain – displacement relations are

………………………. (2.34)

By considering in Eqn.2.30, we have

……….. (2.35)

Similarly,

…………… (2.36)

Equations 2.34, 2.35 and 2.36 yields,

…………………………… (2.37)

Where A is given by,

A = ………………… (2.38)

Now, from the interpolation equations 2.21, we have

……………………………. (2.39)

Where,

G = (2.40)

From Eqs.2.37 and 2.39 we get,

……………………………………. (2.41)

Where,

……………………………. (2.42)

The relation is the desired result. The strain in the element is expressed in terms of its nodal displacement. The stress is now given by

………………………. (2.43)

Where, D is s (3×3) material matrix. The strain energy in eqn.2.33 becomes,

As, and, using eqn.2.33 we get,

……… (2.44)

…………………………………………. (2.45)

Where,

………………….. (2.46)

is the element stiffness matrix of dimension (8×8).

Here the quantities B and det J in the integral in eqn.2.46 are involved functions of ? and ?.

2.5(c) NUMERICAL INTEGRATION

The one-dimensional integral can be expressed in the form,

…………………………. (2.47)

The Gaussian quadrature approach for evaluating I is adopted here. This method has proved most useful in finite element work.

Let us consider the n-point approximation,

……… (2.48)

Where ?_1, ?₂,……and ?_n are the weights and ?₁, ?₂,……..and ?_n are the sampling point or Gauss points. The idea behind Gaussian quadrature is to select the n Gauss points and n weights such that eqn.2.48 provides an exact answer for polynomials f(?) of as large a degree as possible.

Two Dimensional Integration

The extension of Gaussian quadrature to two-dimensional integrals of the form,

…………………………….. (2.49)

follows readily, since

or,

or,

……………………… (2.50)

2.5(d) STIFFNESS INTEGRATION

From eqn.2.46 we find the element stiffness for a quadrilateral element

Where B and det J are functions of ? and ?. This integral actually consists of the integral of each element in an (8×8) matrix.

Let represent the with element in the integrand, where

……………………. (2.51)

Fig – 2.5: Gaussian quadrature in two dimension using 2×2 rule

Then, if a 2×2 rule is used, then we get

….. (2.52)

where ?₁ = ?₂ = 1.0, ?₁ = ?₁ = –0.57735… and ?₂ = ?₂ = +0.57735…. The Gauss points for the two-point rule used above are shown in Fig-2.5, where the Gauss points are labeled as 1,2,3 and 4.

2.5(e) ELEMENT FORCE VECTORS

Body Force

A body force that is distributed force per unit volume, contributes to the global load vector F. This contribution can be determined by considering the body force term in the potential – energy expressions

………………………………….. (2.53)

Using u = Nq, and treating body force as constant within each element, we get

…………………………… (2.54)

where the (8×1) element body force vector is given by,

…………… (2.55)

Since N^T and det J is the function of ? and ?, the body force vector has to be evaluated numerically.

Traction Force

A traction force is distributed load acting on the surface of the body. Such a force acts on edges connecting boundary nodes. A traction force acting on the edge of an element contributes to the global load vector F. Let us assume a traction force

is applied on edge 2-3 of the quadrilateral element in Fig-2.4. From the potential energy equation we get the traction term,

…………… (2.56)

Along that edge we have ? = 1. If we use the shape functions as in eqn.2.20, this becomes N₁ = N₄ = 0, N₂ = (1-?)/2 and N₃ = (1 + ?)/2, where the shape functions have become linear functions. Consequently, from the potential, the element traction load vector is readily given by,

……………. (2.57)

Where, l_2-3 = length of edge 2-3. , are the traction force components at node 2 and , are the traction force components at node 3.

Finally, point loads are considered in the usual manner by having a structural node at that point and simply adding to the global load vector F.

2.5(f) STRESS CALCULATION

From Eqn.2.43 we get that, the stresses acting in the quadrilateral element are not constant within the element; they are functions of ?and ? and consequently vary within the element. In practice, the stresses are evaluated at the Gauss points, which are also the points used for numerical evaluation of k_e, where they are found to be accurate.

2.6 Convergence

As the results obtained in solving problems by finite element method yield the approximate solution, the convergence towards the exact result necessitates the implementation of number of elements in modeling structures. In this respect, most of the concentration here is being projected on the modification of coarse elements into finer one and the construction of isoparametric elements properly.

2.6 a) REQUIREMENTS FOR MONOTONIC CONVERGENCE

With the idealized structure having been represented as an assemblage of finite elements, the accuracy of the analysis depends mainly on the number of elements used, and on the nature of the assumed displacement functions within the elements⁴. In particular, it is important that the accuracy of analysis can be increased by using more elements in the representation of the structure provided that the elements satisfy certain convergence requirements.

Considering the convergence requirements, the elements should be complete and compatible, which is also called conforming. If the elements used are both complete and compatible, convergence is monotonic; i.e., the accuracy of the analysis results measured in some norm increases continuously as the number of elements is increased. However, if the elements are only complete and not compatible, the analysis results may still converge in the limit to the “exact” results but in general do not converge monotonically.

The requirement of completeness means that the displacement functions of the elements must be able to represent the rigid body displacements and the constant strain states. The rigid body displacements are those displacement modes that the element must be able to undergo without stresses being developed in it. For example, a plane stress element must be able to translate uniformly in either direction of its plane and to rotate without straining.

The necessity for the constant strain states can physically be understood if we imagine that more and more elements are used in the assemblage to represent the structure. Then in the limit as each element approaches a very small size, the strain in each element approaches a constant value, and any complex variation of strain within the structure can be approximated.

The concept of compatibility means that the displacements within the elements and across the elements and across the element boundaries are continuous. Physically, compatibility assures that no gaps occur between elements when the assemblage is loaded. When only translational degrees of freedom are defined at the element nodes (as in the case of two dimensional problems like plane stress, plane strain and ax symmetric analysis), only continuity in the displacements u and v, whichever is applicable, must be preserved.

2.6 b) CONVERGENCE CONSIDERATIONS

Since this study mainly focuses on the two dimensional finite element problems, where the isoparametric four-node quadrilateral element was used in modeling structures, the important issue here would be to investigate whether the isoparametric element formulation satisfies the convergence criteria.

To investigate the compatibility of an element assemblage, each edge or rather face, between adjacent elements are being considered. For compatibility it is necessary that the coordinates and the displacements of the elements at the common face be the same. This is the case if the elements have same face nodes and the coordinates and the displacements along the common face are in each element defined by the same shape functions.

Fig-2.6: Four-node two-dimensional element.

In case of Four-node quadrilateral element as in Fig-2.6, the coordinates and displacements vary linearly along the common element edges and determined only by the coordinates and displacements of the edge nodal points. Thus it can be said that the four-node quadrilateral isoparametric element is compatible.

In order to satisfy the completeness requirements, following condition considering the shape functions (N_i) of any isoparametric element needs to be fulfilled⁴,

……………………. (2.58)

where, i = 1,2,…….q and q = number of nodes of the element.

Therefore, for four-node quadrilateral element from eqn.2.20 and using eqn.2.58 we get that,

= 1

Hence the element is also complete and all requirements for monotonic convergence are satisfied.

Chapter 3

FEM Analysis of a Structural problem

This study is mainly based on the FEM analysis of a plate with a circular hole in it. This chapter investigates some of the criteria related to the analysis of the plate with a hole using Finite Element Method.

3.1 Analytical investigation of a problem concerning the

stress concentration around a circular hole in an infinitely

long plate:

3.1a) Formulation of the problem:

To investigate the convergence of approximate result towards near exact one in Finite Element Analysis it is necessary to confirm that the isoparametric elements, which would be properly constructed for modeling purposes, are complete and compatible. In this respect, here a stress concentration problem has been figured out with the help of a user friendly and compact FEM software LISA.

Let an infinitely long plate be of finite width H as in Fig-3.1, which is being placed under constant tensile stress ? (plane stress), acting in a perpendicular direction to the width at the edges of the plate and having a circular hole of diameter d at the middle of it.

Fig-3.1: Plane stress of a finite width element with a circular hole.

Since the maximum stresses ?_max are produced at the ends m and n of the diameter perpendicular to the direction of the tension⁵ (Fig-3.1), the stress concentration factor, K_tg, can be introduced to correlate the terms ? and ?_max, where for this problem K_tg is defined⁶ as,

……… (3.1)

A number of infinitely long plates with circular holes have been analyzed; where for each d/H ratio three models of plates having same dimensions with different numbers of elements have been examined through the application of uniform stress at the edges (plane stress). For analytical purposes the length of the plates are assumed to have finite length as represented in Fig-3.1 by the term A, which is although large enough compared to the diameter of the hole to be deemed as infinite.

3.1b). Features of the models of plates:

For modeling purposes, one quarter of each plate is being subjected to the analysis due to the symmetrical appearance of the structure about horizontal and vertical axis.

Application of uniform stress at the edges was also being accomplished with the help of symmetrical boundary condition.

The notations that are being used to provide some information on each model are given below:

• NN – Number of Nodes

• NE – Number of Elements

• EN – Element Number

• DF – Distortion Factor*

• ASP – Aspect ratio*

The useful quantities used in this problem are:

• Uniform stress at the edges, ? = 400 psi

• Thickness of the plates, t = 0.4 inch

• Modulus of Elasticity, E = 30 × 10⁶ psi
• Poisson’s ratio, ? = 0.3

Here, the plates are considered to be sufficiently infinite in length compared to the diameter (as would be analyzed in section 3.2), where all the models are configured to adopt a length(A) of 20 inch – large/infinite compared to the diameter (d) of 2 inch. Only the width (H) of the plates is being changed from one configuration to another to analyze the models having different d/H ratio. The dimensions of the plates used in modeling them in section 3.1c) can be illustrated as follows, where each of the figures from Fig-3.2 to 3.6 has three models having same configurations:

*N.B. – Definition of Distortion Factor and Aspect Ratio are given in Appendix A and B.

No.AdHd/H3.2202200.13.3202100.23.42026.670.33.520250.43.620240.5

3.1c). Modeling of plates:

To fulfill the convergence requirements, the models have to show deliberate conformance to the actual phenomenon, where the experimental analysis should nearly approximate the theoretical one. For this purpose, several specific regions of the models are highlighted to show the significant alterations in the model, where same regions are being further refined from model to model to influence the results. Here, the models are being depicted with all their element numbers in model no. (a)s, whereas in the other models only the significant elements, which are used in the discussion of the problem, are numbered. Due to the problem of getting entangled with the number of nodes and the element numbers, the node numbers are being excluded. The models are supplemented by corresponding tables comprising the values of distortion factor and aspect ratio of some significant elements, where the tables have been given the same identification number as their corresponding models. For example, the table comprising distortion factor and aspect ratios of the model of model-1(a) is being dubbed table-1(a). In this case, a program using C++ coding is being used to evaluate those distortion factor and aspect ratio values from nodal coordinates and element connectivity information. The modeling was done on a FEM software named LISA, where the analysis of the models was also been made.

model-1(a) model-1(b)

Table – 1(a) Table – 1(b)

Table -1(c)

Model-1(c)

Fig 3.2: Three different models (no. of elements & nodes changed) having same diameter to width ratio (d/H – 0.1).

model-2(a) model-2(b)

Table – 2(a) Table-2(b)

Table – 2(c)

Model-2(c)

Fig 3.3: Three different models (no. of elements & nodes changed) having same diameter to width ratio (d/H – 0.2).

Model-3(a)

Table – 3(a) Table – 3(b)

model-3(b)

model-3(c)

Table – 3(c)

Fig 3.4: Three different models (no. of elements & nodes changed) having same diameter to width ratio (d/H – 0.3).

model-4(a)

Table – 4(a) Table – 4(b)

model-4(b)

model-4(c)

Table – 4(c)

Fig 3.5: Three different models (no. of elements & nodes changed) having same diameter to width ratio (d/H – 0.4).

model-5(a)

Table – 5(a) Table – 5(b)

model-5(b)

model-5(c)

Table – 5(c)

Fig 3.6: Three different models (no. of elements & nodes changed) having same diameter to width ratio (d/H – 0.5).

3.1 d) Application of boundary conditions:

Since one quarter of each of the models is being used in the analysis, proper boundary conditions are needed to be applied to make the analysis conforming to the real life experiments. In this respect, the treatment of boundary conditions in this problem is illustrated as follows:

Fig-3.7: Boundary conditions of models

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