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

Chapter 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 attains 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 analysis 1. 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 â&#x20AC;&#x201C; 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 â&#x20AC;&#x2DC;Solverâ&#x20AC;&#x2122; 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 Postprocessor.

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. Userprepared 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 discretization 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- â&#x20AC;&#x2DC;LISAâ&#x20AC;&#x2122;. 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 possible 2. 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, 1

∏= 2 ∫σ

T

v

ε dV − ∫ u T f dV − ∫u T T dS − ∑u i Pi ………….. (2.1) T

v

s

i

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

1 σTεdV 2 ∫v

And the rest of the terms together constitute the work potential of the body, where T T T u i Pi WP = −∫u f dV −∫u T dS −∑ i v

s

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 x(= [ x, y, z ] T ) is given by the three components of its displacements, u = [ u, v, w ] T . In the last term of eqn.2.1, Pi 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 twodimensional 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 Q2j-1 in the x direction and Q2j in the y direction. And the global displacement vector can be represented as Q = [Q1 , Q2 ,.........., Q N ] ………………….. (2.2) T

F = [ F1 , F2 ,..........., FN ] ………………….. (2.3) T

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, 1

∏= 2 Q

T

KQ − Q T F ………………………… (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 (Fig2.2), 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 K 1, 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:

Element

Node i (1)

Node j (2)

1 2 3 4

4 2 3 2

2 3 5 1

The element stiffness matrices can be expressed as follows: u4

u2

u2

 K1 − K1  u 4 k1 =  − K1 K1  u 2 u3

u3

 K2 − K2  u2 k2 =  − K 2 K 2  u3

u5

u2

 K3 − K3  u3 k3 =  − K3 K3  u 5

u1

 K4 − K4  u2 k4 =  − K 4 K 4  u1

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

u1

u2

u3

u4

u5.

0  K4 − K4 − K  4 K1 + K 2 + K 4 − K 2 K = 0 − K 2 K 2 + K3  − K1 0 0 0 0 − K3 

 u1 u  2 0 − K3  u3 u K1 0  4 0 K 3  u 5

0 0 − K1 0

This matrix is symmetric, banded and singular. Similarly the global load vector F is assembled from element force vectors and point loads asF = ∑ (f e + T e ) + P i e

where, f e & T e are the element body force vector and element traction force vector respectively and P i represents the point vector For the spring system, as illustrated in Fig-2.2 the load vector can be represented as,

[

F = F1 0

F3

0

0]

T

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 Π subject to the boundary conditions. Boundary conditions are usually of the type, Qp1 = a1, Qp2 = a2, …….., Qpr = ar Where, P1,P2,……, Pr 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 Q 1 = a1, where the global stiffness matrix is of the form,

 K11 K K =  21    K N1

K12 K 22  KN2

   

K 1N  K 2N     K NN  ................................... (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, 1 (Q1 K 11Q1 + Q1 K 12 Q 2 + .... + Q1 K 1N Q N 2 +Q 2 K 21Q1 +Q 2 K 22 Q 2 +.... +Q 2 K 2N Q N .................................................................. +Q N K N1Q1 +Q N K N2 Q 2 +.... +Q N K NN Q N ) −(Q 1 F1 +Q 2 F2 +..... +Q N FN )

Π=

(2.6)

Substituting the boundary condition Q1 = a1 in eqn.2.6, we obtain 1 (a 1 K 11a 1 + a 1 K 12 Q 2 + .... + a 1 K 1N Q N 2 +Q 2 K 21 a 1 +Q 2 K 22 Q 2 +.... +Q 2 K 2N Q N .................................................................. +Q N K N1 a 1 +Q N K N2 Q 2 +.... +Q N K NN Q N ) −(a 1 F1 +Q 2 F2 +..... +Q N FN )

Π=

In this expression the displacement Q1 has been eliminated. Consequently, the requirement that Π take on a minimum value implies that, dΠ =0 dQi

From eqns.2.7 and 2.8 we obtain,

i = 2,3,......, N

……….... (2.8)

(2.7)

K 22 Q 2 +K 23 Q 3 +.... +K 2N Q N =F2 −K 21 a 1 K 32 Q 2 +K 33 Q 3 +.... +K 3N Q N =F3 −K 31 a 1 ....................................................................... K N2 Q 2 +K N3 Q 3 +.... +K NN Q N =FN −K N1a 1

(2.9)

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

 K22 K23  K2N   Q2   F2 − K21a1  K K  K  Q  F − K a      32 3 3 31 1 33 3 N    =                   KN2 KN3  KNN   QN   FN − KN1a1

.... (2.10)

Which may be denoted as? KQ = F ……………………. (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 4th and 5th rows and columns of original K the modified K is obtained. Also Q and F is obtained by deleting 4th and 5th component of the original Q and F respectively, where the equilibrium equations can be expressed as, u1

u2

u3

 K4 − K4 0   u1  F1 − K K + K + K − K  u  = 0  4 1 2 4 2   2    0 − K2 K2 + K3  u3  F3 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 Q 1 = 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, Us =

1 C(Q1 − a1 ) 2 …………….. (2.12) 2

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

∏ The minimization of

M

M

=

1 T 1 Q KQ + C(Q1 − a1 ) 2 − Q T F ……. (2.13) 2 2

can be carried out by setting

The resulting finite element equations are,

∂ΠM = 0 , i = 1,2,…..,N. ∂Q i

 (K11 + C) K12  K1N   Q1   F1 + Ca1 K K  K  Q  F      21 22 2 N 2 2    =                   KN1 KN2  KNN   QN   FN 

……… (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: C = max K ij ×10 4

For 1 ≤i ≤ N 1≤ j ≤N

For the spring system of Fig-2.2, a large number C gets added to the 4 th and 5th 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 4th and 5th component of F. Here, the value of C is chosen as, C = ( K 1 + K 2 + K 4 ) × 10 4

Assuming,

K1 + K 2 + K 4 > K 2 + K 3

Now, the modified stiffness matrix can be expressed as,

0  K4 − K4 − K  4 K1 + K 2 + K 4 − K 2 K = 0 − K2 K2 + K3  − K1 0 0 0 0 − K3 

     K1 + C 0  0 K 3 + C  0 0 − K1 0 0 − K3

And the global load vector transforms into,

 F1  0    F =  F3     0 + C.0  0 + C.0 Or,

 F1  0    F =  F3    0   0  So, the equilibrium equation can be expressed as,

 K4 − K4 0 0 0   u1   F1 − K      K 4 1 + K2 + K4 − K2 − K1 0   u2  0   0 − K2 K2 + K3 0 − K3   u3 =  F3      0 − K1 0 K1 + C 0   u4  0   0 0 − K3 0 K3 + C  u5  0    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, σ = EBq ………………….

(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, yi) 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 ξ-, η- 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: N 1 = 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 N 1 = 0 along edges ξ = +1 and η = +1 (Fig-2.4). Thus, N1 has to be of the form N 1 = c(1 − ξ )(1 −η) ……………........................ (2.17)

η q8

(-1,1) 4

q6 q7

(1,1) 3

q5 v

• P (ξ, η)

u • P(x, y)

q2

(0, 0)

ξ

q1 Y

q4 X

q3

1 (-1, -1)

2 (1, -1)

Fig 2.4: The quadrilateral element in ξ, η space (master element)

Where, c is some constant. The constant is determined from the condition, N 1 = 1 at node 1. Since, ξ = -1, η = -1 at node 1, we have 1 = c (2)( 2) …………………...... (2.18)

Which yields c = ¼. Thus, N1 =

1 (1 − ξ )(1 −η) ……………… (2.19) 4

All the four shape functions can be written as

N1 =

1 (1 − ξ )(1 −η) 4

N2 =

1 (1 + ξ )(1 −η) 4

. …………….. (2.20) 1 N 3 = (1 + ξ )(1 +η) 4 N4 =

1 (1 − ξ )(1 +η) 4

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 (ξ, η), and q, dimension (8 × 1), represents the element displacement vector, then u = N 1 q1 + N 2 q3 + N 3 q5 + N 4 q 7

…………..... (2.21)

v = N 1 q 2 + N 2 q 4 + N 3 q 6 + N 4 q8

This can be written in matrix form as, u = Nq

…………………….... (2.22)

Where, N=

N 1 0  0 N1 

N2 0

0 N2

N3 0

0 N3

N4 0  …………. (2.23) 0 N4  

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, x = N 1 x1 + N 2 x 2 + N 3 x3 + N 4 x 4

…………....... (2.24)

y = N 1 y1 + N 2 y 2 + N 3 y 3 + N 4 y 4

Let a function f = f ( x, y ), in view of Eqs.2.24 be considered to be an implicit function of

ξ and η as f = f [ x (ξ,η), y (ξ,η)]. Using the chain rule of differentiation, we have ∂f ∂f ∂x ∂f ∂y = + ∂ξ ∂x ∂ξ ∂y ∂ξ ∂f ∂f ∂x ∂f ∂y = + ∂η ∂x ∂η ∂y ∂η

……………… (2.25)

or,

 ∂f   ∂f   ∂ξ    ∂x       = J   ……………………… (2.26)  ∂f   ∂f     ∂y    ∂η   Where, J is the Jacobian matrix.

J=

In view of Eqs.2.20 & 2.24, we have

x ∂ ∂ ξ  ∂  x ∂  η

∂y  ∂ξ   ………………………. (2.27) ∂y  ∂η 

1  − (1 − η ) x1 + (1 − η ) x 2 + (1 + η ) x3 − (1 + η ) x 4 J=  4  − (1 − ξ ) x1 − (1 + ξ ) x 2 + (1 + ξ ) x3 + (1 − ξ ) x 4 − (1 − η ) y1 + (1 − η ) y 2 + (1 + η ) y 3 − (1 + η ) y 4 

− (1 − ξ ) y1 − (1 + ξ ) y 2 + (1 + ξ ) y 3 + (1 − ξ ) y 4  

J 11

=  J 

21

J 12  ………………………………………………...................... (2.28) J 22  

Equation 2.26 can be written as,

 ∂f   ∂f   ∂ξ     ∂x    -1  ∂f  = J   ………………………….... (2.29)  ∂f       ∂y    ∂η   or,

 ∂f    J 22  ∂x   1  ∂f  = −J   det J  21   ∂y  

 ∂f  − J 12   ∂ξ    ……………… (2.30) J 11   ∂f    ∂η  

An additional relation that is worthy to mention, from calculus we find that dx dy = detJ dξ dη………………………......

(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, U = ∫ 12 σT ε dV …………………………....... (2.32) v

Or,

U = ∑t e e

Where is the thickness of element e. The strain – displacement relations are

∫ e

1 2

σT ε dA ………………………..... (2.33)

 ∂u    ∂ x  εx       ∂v  ε =  εy  =   ∂ y γ     xy   ∂ u ∂ v   +   ∂ y ∂ x

……………….......... (2.34)

By considering f ≡ u in Eqn.2.30, we have

 ∂u   − J 12   ∂x   1  J 22  ∂u  = −J    det J  21 J 11    ∂y  

 ∂u   ∂ξ      ……….. (2.35)  ∂u   ∂η   

Similarly,

 ∂v   − J 12   ∂x   1  J 22  ∂v  = −J    det J  21 J 11    ∂y  

 ∂v   ∂ξ      …………… (2.36)  ∂v    ∂η  

Equations 2.34, 2.35 and 2.36 yields, u  ∂ ∂ ξ   u  ∂ ∂   η ε =A  …………………………... v  ∂ ∂ ξ   v  ∂ ∂  η 

(2.37) Where A is given by,  J 22

1  0 A= det J  

−J 21

− J 12

0

0

− J 21

J 11

J 22

Now, from the interpolation equations 2.21, we have

0  J 11   ……………...... (2.38) − J 12  

 ∂u   ∂ξ     ∂u   ∂η      = Gq ………………………….... (2.39) ∂ v    ∂ξ     ∂v    ∂η  

Where,

G=

0 (1 − η ) 0 (1 + η ) 0 − (1 + η ) 0   − (1 − η )  0 − (1 + ξ ) 0 (1 + ξ ) 0 (1 − ξ ) 0  1  − (1 − ξ ) − (1 − η ) 0 (1 − η ) 0 (1 + η ) 0 − (1 + η ) 4 0   0 − ( 1 − ξ ) 0 − ( 1 + ξ ) 0 ( 1 + ξ ) 0 ( 1 − ξ )  

(2.40) From Eqs.2.37 and 2.39 we get, ε = Bq …...………………………….......

(2.41)

Where, B = AG ……………………………. (2.42)

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

………………………. (2.43)

Where, D is s (3×3) material matrix. The strain energy in eqn.2.33 becomes, U = ∑t e e

1 2

σT ε dx dy

e

As, σT = Dq T B T and ε = Bq , using eqn.2.33 we get,  U = ∑ 12 q T t e e 

1

1

∫ ∫B

−1 −1

T

 DB detJ dξ dη q ……… (2.44) 

U = ∑ 12 q T k e q ....................................……….... (2.45) e

Where, 1

k e = te

1

∫ ∫B

T

DB detJ dξ dη ……………........ (2.46)

−1 −1

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 ξ and η. 2.5(c) NUMERICAL INTEGRATION The one-dimensional integral can be expressed in the form, 1

I = ∫ f (ξ ) dξ ……………………....... (2.47) −1

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, 1

I =

∫ f (ξ ) dξ ≈ ω f (ξ ) +ω 1

1

2

f (ξ 2 ) + ........ +ωn f (ξ n ) ……… (2.48)

−1

Where ω1, ω2,……and ωn are the weights and ξ1, ξ2,……..and ξn 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(ξ) of as large a degree as possible. Two Dimensional Integration The extension of Gaussian quadrature to two-dimensional integrals of the form, 1

I =∫

1

∫ f (ξ,η) dξ ∂η …………………………..... (2.49)

−1 −1

I≈

−1

n  ∑ωi f (ξ i ,η ) dη  i =1 

or, n n  I ≈ ∑ω j ∑ωi f (ξ i ,η j ) j =1  i =1 

or,

n

I ≈∑

n

∑ω ω i

i =1 j =1

j

f (ξ i ,η j ) ……………………… (2.50)

2.5(d) STIFFNESS INTEGRATION From eqn.2.46 we find the element stiffness for a quadrilateral element 1

k e = te

1

∫ ∫B

T

DB detJ dξ dη

−1 −1

Where B and det J are functions of ξ and η. This integral actually consists of the integral of each element in an (8×8) matrix. Let φ represent the with element in the integrand, where φ(ξ,η) = t e (B T DB det J) ij ……………………. (2.51)

η2 =

η1 = −

ξ1 = − ξ2 = 1

1 3

1 3

1 3

1 3 1

∫ ∫ f (ξ ,η) dξ dη ≈ ω

1

2

f (ξ1 ,η1 ) +ω1ω2 f (ξ1 ,η2 ) +ω1ω2 f (ξ 2 ,η1 ) +ω2 f (ξ 2 ,η2 ) 2

−1 −1

Fig – 2.5: Gaussian quadrature in two dimension using 2×2 rule Then, if a 2×2 rule is used, then we get k ij ≈ ω1 φ(ξ1 ,η1 ) + ω1ω2φ(ξ1 ,η 2 ) + ω1ω2φ(ξ 2 ,η1 ) + ω2 φ(ξ 2 ,η 2 ) ..... (2.52) 2

2

where ω1 = ω2 = 1.0, ξ1 = η1 = -0.57735… and ξ2 = η2 = +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

∫u v

T

f dV ………………………………….. (2.53)

T Using u = Nq, and treating body force f = [ f x , f y ] as constant within each element, we get

∫u

T

v

f dV = ∑q T f e …………………………... (2.54) e

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

1 1    fx  e T f = te  ∫ ∫ N det J dξ dη    …………… (2.55)  −1 −1   fy Since NT and det J is the function of ξ and η, 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

[

T = Tx , T y

]

T

is applied on edge 2-3 of the quadrilateral element in Fig-2.4. From the

potential energy equation we get the traction term,

∫u L

T

T t e dl = ∫

l 2 −3

(uTx + vT y )t e dl …………… (2.56)

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

[

]

t e l 2 −3 T 0 0 Tx 2 T y 2 Tx 3 T y 3 0 0 ……………. (2.57) 2

Where, l2-3 = length of edge 2-3. Tx 2 , Tx 3 are the traction force components at node 2 and Ty 2

, T y 3 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 σ = DBq acting in the quadrilateral element are not constant within the element; they are functions of ξ and η 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 ke, 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 elements 4. 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 â&#x20AC;&#x153;exactâ&#x20AC;? 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 (Ni) of any isoparametric element needs to be fulfilled4, q

N ∑ i =1

i

= 1 ……………………. (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, q

N ∑ i =1

i

= N1 + N 2 + N 3 + N 4 =

1 1 1 1 (1 − ξ )(1 −η) + (1 + ξ )(1 −η) + (1 + ξ )(1 +η) + (1 − ξ )(1 +η) 4 4 4 4

=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 Ď&#x192; (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 σmax 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, Ktg, can be introduced to correlate the terms σ and σmax, where for this problem Ktg is defined6 as, 2

K tg

σ 2 d  d    = max = 0.284 + − 0.6001 −  + 1.321 −  ……… (3.1) σ 1−d / H H H  

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, σ = 400 psi

Thickness of the plates,

t = 0.4 inch

Modulus of Elasticity,

E = 30 × 106 psi

Poisson’s ratio,

υ = 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 â&#x20AC;&#x201C; 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. â&#x20AC;&#x201C; Definition of Distortion Factor and Aspect Ratio are given in Appendix A and B. Figure A d H d/H No. 3.2 20 2 20 0.1 3.3 20 2 10 0.2 3.4 20 2 6.67 0.3 3.5 20 2 5 0.4 3.6 20 2 4 0.5

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) Table - 1(a) NN – 54 NE – 40 EN DF ASP 3 50.229056 3.053924 28 72.824843 4.412969 29 72.824871 4.412968 33 16.242640 5.935876 34 24.886230 6.493195 36 39.517247 8.380005

model-1(b) Table - 1(b) NN – 72 NE - 56 EN DF ASP 4 71.132446 2.206484 12 75.576573 1.587011 25 29.078304 4.190003 26 48.310440 2.564504 31 15.631144 2.967938 32 17.121113 1.713116 Table -1(c) NN - 99 NE - 80 EN DF ASP 5 71.132421 2.206482 23 75.576508 1.587012 58 48.310345 2.564508 63 15.631161 2.967940 64 17.121143 1.297745 75 48.358787 1.526960 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) Table – 2(a) NN - 51 NE - 36 EN DF ASP 1 16.850908 5.299267 2 26.278530 5.857696 5 41.623513 7.742361 15 67.975990 5.218801 18 79.699736 2.712324 31 0.000000 1.333333

Table – 2(c)

model-2(b) Table-2(b) NN – 74 EN DF 24 81.009968 28 76.011371 36 65.885228 41 16.159888 42 17.800763 50 50.728857

NE - 56 ASP 1.555281 2.157385 3.479201 2.649633 1.586564 2.458622

NN - 83 NE - 64 EN DF ASP 9 0.000000 1.000000 10 0.000000 1.000000 41 62.87892 1.739600 8 25 17.80076 1.586564 3 57 16.15988 2.649633 8 58 22.30580 2.928848 6

Model-2(c) Fig 3.3: Three different models (no. of elements & nodes changed) having same diameter to width ratio (d/H â&#x20AC;&#x201C; 0.2).

Model-3(a) Table - 3(a) Table - 3(b) NN - 61 NE - 44 EN DF ASP 1 17.131331 3.110818 2 25.291223 3.484188 5 36.569155 4.739484 17 63.555507 4.156144 35 0.000000 1.598801 36 0.000000 1.598801

NN - 71 NE - 52 EN DF ASP 15 0.000000 1.142000 16 0.000000 1.142000 29 17.131331 3.110818 33 36.569155 4.739484 45 63.555507 4.156144 46 78.700952 2.403681

model-3(b)

model-3(c) NN - 89 NE - 68 EN DF ASP 2 60.602540 2.770763 23 73.463230 1.871104 24 80.532040 1.399406 39 0.000000 1.142000 53 16.478337 1.555409 54 17.866588 1.289038 Table â&#x20AC;&#x201C; 3(c)

Fig 3.4: Three different models (no. of elements & nodes changed) having same diameter to width ratio (d/H â&#x20AC;&#x201C; 0.3).

Table - 4(a)

model-4(a) Table - 4(b)

NN - 57 NE - 40 EN DF ASP 13 0.0000000 2.0000000 1 14 0.0000000 2.0000000 1 25 18.601686 4.0270326 7 26 21.459229 1.9247210 2 33 46.652428 6.4670730 6 34 75.089154 3.2240988 1

NN - 66 NE - 48 EN DF ASP 5 77.899807 2.113532 8 21.743288 1.347448 9 20.266902 1.496500 10 17.967072 2.516895 17 37.141791 4.041921 18 62.614051 2.544009

model-4(b)

model-4(c) NN - 85 NE - 64 EN DF ASP 25 0.000000 1.500000 26 0.000000 1.500000 41 56.460674 2.219758 49 17.104592 1.330113 50 18.323224 1.615523 58 38.450791 1.336417

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)

NN - 67 NE - 48 EN DF ASP 1 19.569429 2.713421 3 30.672970 3.164272 9 44.641444 4.663543 15 19.569592 2.713421 16 22.517952 1.529931 33 0.000000 2.000000

Table - 5(b) NN - 86 NE - 64 EN DF ASP 1 18.652263 1.385984 8 18.652367 1.385984 9 20.943507 1.468319 23 20.943665 1.468319 24 22.834750 1.333333 45 0.000000 1.600000

model-5(b)

model-5(c) NN - 114 NE - 88 EN DF ASP 25 0.000000 1.333333 53 20.597685 1.390711 65 22.051494 1.695199 73 18.092690 1.870473 79 19.466331 1.216949 87 18.092790 1.870473

Table â&#x20AC;&#x201C; 5(c)

Fig 3.6: Three different models (no. of elements & nodes changed) having same diameter to width ratio (d/H â&#x20AC;&#x201C; 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 From Fig-3.7 it is evident that the sides of the quarter of plates adjacent to the hole are being constrained to move in such directions which would restrict the rigid body displacement of the models. Taking into consideration the two – dimensional aspect of the problem, the body forces comprising only the weights of the plates are neglected, where the Global Load Vector, F, is deemed to be comprised of only the Traction forces applied at the edges. In Fig-3.2 to 3.6 each and every quarter of a plate with hole is being modeled with Four-node Quadrilateral Element and being imposed at the right hand edges with Traction Forces as derived from eqn.2.57. Since each of the plates are modeled with right hand edges having five nodes equidistant from each other (elements adjacent to the edges have same dimensions), the traction force distribution as Global Load Vectors for each of the Figures can be expressed as: For d/H-0.1:

 200  400   F =  400    400  200

For d/H-0.2:

100   200   F =  200    200 100 

For d/H-0.3:

 66.7   133.4   F =  133.4    133.4  66.7 

For d/H-0.4:

 50  100   F = 100   100  50 

For d/H-0.5:

 40  80    F =  80     80   40

3.1 e) Results obtained from the test: Using the post-processor of the LISA software, the results that are obtained from the analysis are plotted in Table 3.1 to 3.5, where the analytical and theoretical K tg are compared with each other. As well the graphical representation of the post-processor analysis of the closest converging model for each of the d/H ratio is being delineated along with those tables. In the tables the results obtained from the analyses on three of the models (having same d/H ratio) are being plotted along with the information named ‘% difference’, which is defined as,

Percentage of difference(% difference) =

(Theoretical K tg − Analytical K tg ) Theoretical K tg

×100

Fig-3.8: Graphical representation of the stress distribution on the model having a d/H ratio of 0.1

Table-6(a): Results obtained from the analyses of model-1(a), 1(b), 1(c) Model No. of No. of Applied No. Nodes Elements In-plane stress, σ (psi)

Maximum Analytical Theoretical % stress Ktg Ktg difference developed, σmax (psi)

1(a)

54

40

400

1095.097

2.7377

3.035422

9.8%

1(b)

72

56

400

1202.5847

3.006

3.035422

0.97%

1(c)

99

80

400

1207.5418

3.0188

3.035422

0.55%

Fig-3.9: Graphical representation of the stress distribution on the model having a d/H ratio of Table-6(b): Results obtained from the analyses of model-2(a), 2(b), 2(c) Model No. of No. of Applied No. Nodes Elements In-plane stress, σ (psi)

Maximum Analytical Theoretical % stress Ktg Ktg difference developed, σmax (psi)

2(a)

51

36

400

1132.009

2.83

3.1488

10.12%

2(b)

74

56

400

1239.3144

3.0982

3.1488

1.6%

2(c)

83

64

400

1243.3088

3.1082

3.1488

1.3%

Fig-3.10: Graphical representation of the stress distribution on the model having a d/H ratio of 0.3 Table-6(c): Results obtained from the analyses of model-3(a), 3(b), 3(c) Model No. of No. of Applied No. Nodes Elements In-plane stress, Ď&#x192; (psi)

Maximum stress developed, Ď&#x192;max (psi)

Analytical Theoretical % Ktg Ktg difference

3(a)

61

44

400

1289.7447

3.224

3.3675

4.26%

3(b)

71

52

400

1289.8867

3.2247

3.3675

4.24%

3(c)

89

68

400

1350.31

3.3757

3.3675

-0.24%

Fig-3.11: Graphical representation of the stress distribution on the model having a d/H ratio of 0.4 Table-6(d): Results obtained from the analyses of model-4(a), 4(b), 4(c) Model No. of No. of Applied No. Nodes Elements In-plane stress, Ď&#x192; (psi)

Maximum stress developed, Ď&#x192;max (psi)

Analytical Theoretical % Ktg Ktg difference

4(a)

57

40

400

1356.1795

3.3904

3.7325

9.16%

4(b)

66

48

400

1440.0768

3.6001

3.7325

3.54%

4(c)

85

64

400

1479.6616

3.6991

3.7325

0.89%

Fig-3.12: Graphical representation of the stress distribution on the model having a d/H ratio of 0.5 Table-6(e): Results obtained from the analyses of model-5(a), 5(b), 5(c) Model No. of No. of Applied No. Nodes Elements In-plane stress, Ď&#x192; (psi)

Maximum stress developed, Ď&#x192;max (psi)

Analytical Theoretical % Ktg Ktg difference

5(a)

67

48

400

1610.3495

4.025

4.314

6.69%

5(b)

86

64

400

1687.2083

4.218

4.314

2.22%

5(c)

114

88

400

1695.6716

4.2391

4.314

1.73%

3.1 f) Discussions made on the models: In Fig-3.2 to 3.6 the plates are modeled with significant changes in number of elements from model (a) to (c), where the elements near the edge of the hole are increased in number to justify the sensitivity of stress concentration at the top edge of the hole. The values of distortion factor and aspect ratio are used here to judge the change in quality of the elements

from model to model. As the main focus is on the increment of the quantity of elements in the models, the distortion factors may not get improved that much with the refinement of the elements due to the consideration of only the internal angles between the sides of the elements in the definition of it. But the aspect ratio would get better as the elements are further refined, which is being used here as the parameter to influence the results of the analysis. The outcome from the analysis can be illustrated as below: 1) As can be seen from model-1(a) to 1(c) in Fig-3.2, the elements like 33 and 28 in model1(a) are subdivided into 32 & 31 and 4 & 12 respectively in model-1(b) and further being divided in model-1(c) as portrayed in the colored portion of the models. From table 1(a) & 1(b), we can see that Element no. 33 in model-1(a) having an aspect ratio of 5.935876 is being transformed into EN-32(ASP-1.713116) & 31(ASP-2.967938) in model-1(b), which suggest a certain improvement in the quality of the elements with their increase in number from model to model (with same d/H ratio). In Table-6(a) the results are plotted for the models in Fig-3.2, where it can be observed that the analytical K tg gets fairly closer to the theoretical value as the number of elements and their quality are increased from model-1(a) to 1(b). Further improvement of both the no. and quality of elements in model-1(c) suggest slight improvement in the percentage of the difference from model-1(b). Though the distortion factor of the elements do not get changed that much, as can be seen from table-1(a) to 1(c), their values within acceptable limits may contribute to the enhancement of the element quality. 2) Fig-3.3 shows the refinement of elements of the models in model-2(a) to 2(c), where the elements like 1, 2, 5 etc. in model-2(a), having aspect ratios 5.299267, 5.857696 and 7.742361 respectively, are being subdivided into elements like 42(ASP-1.586564), 41(ASP2.649633), 50(ASP-2.45822) etc. in model-2(b) with certain improvements in aspect ratios and increase in number of elements. This in turn provided with significant change in the percentage of difference between the theoretical K tg and the analytical one from 10.12% to 1.6%, as can be seen from Table â&#x20AC;&#x201C; 6(b). The graphical representation of model-2(c) in Fig-3.9 exhibits the maximum stress distribution at the upper edge of the hole, which conforms to the theoretical attribution to the stress concentration around the hole, as being mentioned in section 3.1(a).

3) In Fig-3.4 in the same manner, as being explained so far, the models (having same d/H ratio) are being increased in number of the elements from model to model as can be seen from the model-3(a) to 3(c). Enhancement in quality of elements can be found out from the table-3(a), 3(b) & 3(c), where the aspect ratios are being improved from model to model. The value of distortion factor of the elements are being kept within the specified limit of 105 (explanation are given in Appendix A) to restrict the elements from being distorted too much while the discretization of the models were taking place â&#x20AC;&#x201C; which in turn maintained the quality of the elements to a certain level. Although their occurred a certain distinction in the result for model-3(c), where from table-6(c) it can be seen that, the analytical value of K tg got slightly larger than the theoretical ktg, which is been shown in the table as a negative percentage of difference. 4) As the plates are getting narrower (H decreasing, d/H ratio increasing), the dramatic changes in analytical Ktg values from model to model (with the increase in number of elements) are turning out to show a gradual nature in their alteration. This did happen for the models having a d/H ratio of 0.4, where the % difference for model-4(a) (9.16%) gets decreased to a value of 3.54% in model-4(b) as the numbers of elements are increased from 40 to 48 and again follows that path to a value of 0.89% in model-4(c) â&#x20AC;&#x201C; comprised of 64 elements (Table-6(d)). In this respect it can be said that the gradual refinement of elements around the hole may contribute to this cause, where element 33(ASP-6.4670730) in model4(a) gets divided into 17(ASP-4.041921) and 18(ASP-2.544009) in model-4(b) and further gets divided into elements 41(ASP-2.219758), 57 and 58(ASP-1.336417) in model-4(c) â&#x20AC;&#x201C; a certainly steady enhancements in quality and number of elements from model to model. 5) In Fig-3.6 the models having lowest width are analyzed. Similar to Fig-3.5, the models are gradually increased in number of elements near the edges of the hole, where not that much of significant quality changes of the elements did occur. Although, the coarse elements like 1(ASP-2.713421), 3(ASP-3.164272), 9(ASP-4.663543) etc. in model 5(a) are subdivided into finer elements like EN-53(ASP-1.390711), EN-73(ASP-1.870473), EN-79(ASP-1.216949) etc. in model-5(c), the results as plotted in Table-6(e) puts forward the decision that the analytical Ktg values surely do converging towards the theoretical one. The results plotted in Table-6(a) to 6(e) suggest the complete and compatible refinement of the elements, which served with the converging stress values towards the theoretical ones.

3.2 Verification of the infiniteness of the models: As the formulation in eqn.3.1 requires the plate to be infinite in length, the stress value at the edge of the hole may not conform with the anticipated result if the length is not quite large enough compared to the diameter of the hole. At this stage of the analysis, a test is being carried out using three models having same d/H ratio, where the models are made to differ in length to figure out the acceptable diameter-length ratio for the judgment of infiniteness of the plates. Taking into consideration of the models having same d/H – 0.1, the length of the plates are increased from 10 inch to 30 inch with an increment of 10 inch in each model. The modeling and analysis were done using the same procedures as were discussed in sections 3.1(a) to 3.1(d). In this respect, the approach was to find out the influence on the stress distribution at the right edge of the plate, which is being subjected to the uniform tensile stress σ. Due to the presence of hole at the middle of the plate, the stress distribution in the neighborhood of the hole would get changed, as the stress is applied at both the opposite edges of the plate (Fig-3.1). But according to Saint-Venant’s principle, the change is negligible at distances which are large compared with the radius of the hole 5. It means that for a large plate the length can be judged to be infinite if the stress distribution at the edges under tension is almost identical to the applied stress distribution – i.e., not much being influenced by the stress distribution around the hole. The models are being shown in their graphical form, where the nodal stress values at the right edge of the plates are also been given. Here, the tensile stress is being applied at the right edge in such a fashion (as explained below) that every nodal stress values (in short term it is expressed as NS) should show almost the same values (σ = 400 psi) if the stress distribution isn’t being influenced by the presence of the hole – that is the length can be considered to be infinite in length compared to the radius of the hole. The Global load vector for each of the models can be expressed in the following manner, where the procedures for the evaluation of it are the same as been explained in section 3.1(d): For A-10:

For A-20:

For A-30:

100  200   200   200    F = 200   200   200   200 100    100  200    200    200   F =200    200   200    200  100   

100  200   200   200    F = 200   200   200   200 100   

Fig-3.13: Stress distribution on one quarter of the model-having a length of 10 inch

Fig-3.14: Stress distribution on one quarter of the model-having a length of 20 inch

Fig-3.15: Stress distribution on one quarter of the model-having a length of 30 inch

In Fig-3.13 the stress distribution at the right edge of the model suggest that the presence of hole has certain influence on the stress values, as the values get a bit deviant from the applied stress value (400 psi), where every nodal value should be very much close to 400 psi if the hole is considerably away from the edge. Whereas, in Fig-3.14 the nodal stress values at the edge almost always approximately assume a value of 400 psi. Further increase in length in Fig-3.15 suggests a slight improvement in the approximation in nodal stress values. Hence, it can be agreed upon through the analysis that the adoption of the plate models having a length of 20 inch, as compared to the diameter of the hole of 2 inch, can be considered to be infinite in length for theoretical as well as analytical purposes. This is the reason why, all the models in section-3.1(c) are being constructed with the consideration of having the length of 20 inch.

3.3 Inferences from the FEM analysis: From the overall test it is apparent that, the analysis conforms with the convergence criteria â&#x20AC;&#x201C; the elements used in modeling are quite complete and compatible, which is one of the primary requirements for the mesh generation and analysis in Finite Element Analysis and can be seen from Fig-3.16, where the Stress concentration factor, Ktg, is plotted against

various d/H ratio to compare the analytical results with the theoretical one. It is apparent from the figure that the analytical values very closely approximate the theoretical ones, where the test concerning fulfillment of convergence criteria can be assigned to be a successful one.

5 4.5 4 3.5 Ktg

3 2.5 2 1.5

Analytical Theoretical

1 0.5 0 0.1

0.2

0.3

0.4

0.5

d/H

Fig-3.16: Comparison between the theoretical and analytical Ktg value for various d/H ratio.

3.4 Influence of thickness change on results: In this section, keeping all the dimensions of the plates the same, the thickness of each of the plates was changed to see the effect of the alteration of thickness on the stress values around the edges of the hole. Although the main study is being focused on the two-dimensional treatment of the FEM method, thickness consideration in this problem may contribute to the three-dimensional aspect of the FEM analysis. In this case, three models having same length (A-20 inch), width (H-10 inch) and diameter of the hole (d-2 inch) are used in this test, where the modeling and analysis techniques are quite identical to the procedures those have been followed so far in the analyses of the models in section 3.1 and 3.2. The thicknesses (t) of the models are changed from 0.1 to 0.4 and then to 1 inch, which rendered the evaluation of Global Load Vector for the models using eqn.2.57 in the following manner:

For t = 0.1:

 25  50   F =  50    50  25

For t = 0.4:

 100   200   F =  200    200  100 

For t = 1.0:

 250  500    F =  500     500   250

Since three models show the same type of graphical representation, here in the next page only the stress distribution of one model is being shown:

Fig-3.17: Stress distribution on the models having different thicknesses like t=0.1, t=0.4, t=1.0 The models, though they are varied in their thickness, do not show any effect of this change on the stress distribution around the hole or any other places in the plate. The reason lies mainly on the definition of the traction force (constituting the Global Load Vector for this

analysis) for four-node quadrilateral element, which is being applied at the right edge of the model. From eqn.2.57 we can see that, the traction force is expressed as,

Te =

[

]

t e l 2 â&#x2C6;&#x2019;3 T 0 0 Tx 2 T y 2 Tx 3 T y 3 0 0 2

where, the thickness (te) change contributes to the proportionate change in the distribution of traction force (can be seen from the Global Load Vector values for each of the model) at the nodes on the edges, which in turn provides with the same stress distribution at the edges despite of the alteration in thickness. This is the reason why, it can be considered to be a plane stress problem, which is two-dimensional in nature.

Chapter 4 Conclusion and future recommendation 4.1 Conclusion: In this study the main target was to perform a FEM analysis on a structural problem concerning stress concentration around a hole in the middle of an infinitely long plate to validate the theoretical formulation of the problem as well as to investigate some of the dimensional aspect of the problem. The high stress concentration found at the edge of a hole is of great practical importance. As an example, holes in shipsâ&#x20AC;&#x2122; decks may be mentioned. When the hull of a ship is bent, tension or compression is produced in decks and there is a high stress concentration at the holes. Under the cycles of stress produced by waves, fatigue of the metal at the overstressed portions may result finally in fatigue cracks 1. Also in the aviation industry, the localized stress distribution concept forms a very useful aspect in the building of aircrafts, where the discontinuities in plates are quite commonly encountered in the construction of windows, frames etc of the aero-planes. From the analysis the conclusions that can be made are summarized below: 1) The examination of models with holes for different diameter-width ratio suggested that, as the element number in the models are increased the

analytical stress concentration factor (Ktg) values did show the convergence towards the values found out from the theoretical formulation. 2) Another conclusion, which was arrived concerning the dimension of the plates is that, the stress concentration in the neighborhood of the hole did not effect the stress distribution at the edges where the loads are being applied – confirming the infiniteness in length of the plates. 3) At last, the test regarding the alteration of thickness does show that, the changes of thickness do not really influence the stress distribution in the plates.

4.2 Future plan: Since, in this study only one hole is being considered to accomplish the discontinuity in the middle of the plate. But in real life problems, holes more than one in the plates may be encountered. This is why, it is necessary to have this sorts of analyses been done on the plates having multiple holes using FEM analysis.

References 1.

Robert D. Cook, David S. Malkus, Michael E. Plesha, “Concepts and Applications of Finite Element Analysis”, 3rd edition, Published by John Wiley and Sons, Inc., University of Wisconsin – Madison, 1989.

2.

William Muckle, “Strength of Ships’ Structures”, published by Edward Arnold Ltd., London, 1967.

3.

Tirupathi R. Chandrupatla and Ashok D. Belegundu, “Introduction to Finite Elements in Engineering”, 3rd edition, Published by Prentice Hall, Inc, 2002.

4.

Klaus – Jurgen Bathe and Edward L. Wilson, “Numerical Methods in Finite Element Analysis”, published by Prentice-Hall, Inc., 1976.

5.

S. P. Timoshenko and J. N. Goodier, “Theory of Elasticity”, 3rd edition, published by McGraw-Hill Book Company, International edition, 1970.

6.

Walter D. Pilkey, “Peterson’s Stress Concentration Factors”, 2nd edition, published

by John Wiley and Sons, Inc., 1997. 7.

Yasumi Kawamura and Md. Shahidul Islam, “A method to remove self-intersections from dual cycles of a quadrilateral surface mesh for the generation of a hexahedral mesh”, published on April 2006 by the Japan Society for Computational Engineering and Science.

8.

Yijun Liu, “Lecture Notes: Introduction to Finite Element Method”, 1998, University of Cincinnati.

Appendix A Distortion Factor: The quality of the surface mesh should be good to generate a good quality mesh. To judge if a mesh is of sufficient good quality, it is needed to define a standard7. Zhu et al considered a quadrilateral element satisfactory if all its internal angles θ fall within 90º ± 45º and was considered as unsatisfactory if θ exceeds the limit 90º ± 60º. Lo and Lee found that the first condition appeared to be too strict, so a more flexible range of 90º ± 52.5º was used for quadrilateral interior angles. In the present study Lo and Lee’s range is chosen for acceptable quality of a quadrilateral element. Any element exceeding this range is considered unacceptable. The optimum shape for a quadrilateral is a square with interior angles 90º. The following equations were used to measure the distortion factor of quadrilaterals. The deviation of each interior angle of a quadrilateral, δθi , is defined as,

δθi =

π 2

− θi

i = 1, 2, 3, 4.

The distortion factor for quadrilateral element, Fq is defined as,

Fq =

4

(δθ ) ∑ i =1

2

i

It can be seen that Fq would attain a minimum value of zero for a perfect square and the acceptable range of 90º ± 52.5º defined by Lo and Lee would correspond to Fq

≤105  .

Derivation of Distortion factor for a quadrilateral element can be expressed in the following manner:

The deviations of each interior angle can be expressed as, δθ1 = 90  −99  = 9 

δθ2 = 90  −111 = 21 δθ3 = 90  −57  = 33  δθ4 = 90  −93 = 3

Hence, the distortion factor, Fq = (δθ1 2 + δθ 2 2 + δθ 3 2 + δθ 4 2 ) = (9 2 + 212 + 33 2 + 3 2 )

= 40.25 

where, the value of Fq is below the limit of 105  . So, the element as depicted above can be said to possess acceptable quality in terms of distortion factor according to Lo and Lee.

Appendix B Aspect ratio8: Aspect ratio of a quadrilateral element is defined as the ratio of the largest to the smallest length of the sides of the element. So, Aspect ratio = Lmax / Lmin Where, Lmax and Lmin are the largest and smallest characteristic lengths of an element, respectively. Examples of acceptable and unacceptable elements in terms of aspect ratio are given below: