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Table of Contents
ACKNOWLEDGMENTS _______________________________________________ V
TABLE OF CONTENTS _______________________________________________ VI
LIST OF TABLES ___________________________________________________VIII
LIST OF FIGURES ___________________________________________________ IX
ix
- 1 INTRODUCTION _______________________________________________ ABSTRACT__________________________________________________________ X
- 1.1 THE STEPS INVOLVED I N FINITE ELEMENT M ETHOD _____________________
- 1.1.1 Discretization of The Continuum _______________________________
- 1.1.2 Selection of Interpolation Functions_____________________________
- 1.1.3 Finding The Element Properties________________________________
- 1.1.4 Assembling The Element Properties To Obtain The System Equations __
- 1.1.5 Solution of The System Equations_______________________________
- 1.1.6 Requirement For Additional Computations _______________________
- 1.2 OBJECTIVES ____________________________________________________
- 1.3 THESIS ORGANIZATION ___________________________________________
- 2 MATHEMATICAL FORMULATION ________________________________
- 2.1 M ATHEMATICAL FORMULATION FOR THERMAL A NALYSIS________________
- 2.2 M ATHEMATICAL FORMULATION FOR STRESS ANALYSIS _________________
- 2.3 PLANE STRESS AND PLANE STRAIN ANALYSIS ________________________
- 2.4 THERMAL STRESSES ____________________________________________
- 3 CODE DESCRIPTION ____________________________________________
- 3.1 CODE SPECIFICATION____________________________________________
- 3.2 VARIABLE USED I N THE PROGRAM _________________________________
- 3.3 M AIN PROGRAM DESCRIPTION ____________________________________
- 3.4 DESCRIPTION OF SUBROUTINE PROGRAM ____________________________
- 3.4.1 Subroutine INPUT _________________________________________
- 3.4.2 Subroutine THERMAL ______________________________________
- 3.4.3 Subroutine STRESS_________________________________________
- 3.4.4 Subroutine SOLMIX ________________________________________
- 3.4.5 Subroutine SOLVE _________________________________________
- 3.4.6 Subroutine MATSOLVE _____________________________________
- 4 RESULTS AND DISCUSSION ______________________________________
- 4.1 ONE DIMENSIONAL HEAT FLOW I N A BAR ___________________________
- 4.2 CONVECTION FROM A B LUNT FIN __________________________________
- 4.3 PATCH TEST___________________________________________________
- 4.4 A THIN PLATE WITH A DISTRIBUTED LOAD __________________________
- 4.5 A PLATE WITH A HOLE IN THE CENTER _____________________________ vii
- 4.6 THERMO-M ECHANICAL ANALYSIS _________________________________
- 5 CONCLUSIONS __________________________________________________
- 6 RECOMMENDATIONS FOR FUTURE WORK_______________________
- 7 REFERENCES ___________________________________________________
- 8 APPENDIX 1_____________________________________________________
- 9 APPENDIX 2_____________________________________________________
- Figure 2-1 Coordinate system for the two dimensional problem List of Figures
- Figure 3-1 Flow chart for main program
- Figure 4-1 One dimensional heat flow in a bar.............................................................
- Figure 4-2 A finite element grid for one dimensional bar
- Figure 4-3 A finite element grid for one dimensional bar on ANSYS
- Figure 4-4 Temperature contours in one dimensional bar
- Figure 4-5 Blunt fin problem
- Figure 4-6 A finite element grid for a blunt fin
- Figure 4-7 A finite element grid for a blunt fin on ANSYS
- Figure 4-8 Temperature contours in a blunt fin
- Figure 4-9 A finite element grid for a patch test...........................................................
- Figure 4-10 A finite element grid for a patch test using ANSYS...................................
- Figure 4-11 Distribution of stress in deformed and undeformed shape..........................
- Figure 4-12 Thin plate on a lubricated surface with partial loading...............................
- Figure 4-13 A finite element grid for a partially loaded plate
- Figure 4-14 A finite element grid for a partially loaded plate on ANSYS
- Figure 4-15 Vonmosis stress distribution on a partially loaded plate.............................
- Figure 4-16 A plate with a hole in the center..................................................................
- Figure 4-17 A finite element grid of a quarter plate with a hole in the center................
- Figure 4-18 A finite element grid for a quarter plate using ANSYS
- Figure 4-19 Y-direction stress distribution in a quarter plate
- Figure 4-20 A finite element grid for a thermo-mechanical analysis
- Figure 4-21 X-direction stress distribution in the fin......................................................
- Figure 4-22 Vonmoises stress distribution in the fin
x
Abstract
Thermal and stress analysis is necessary in structures and materials for design purpose. A finite element based computer code ASITM was developed for thermo-mechanical analysis. Mathematics was developed for thermal and stress analysis by using the triangular elements. ASITM code can be used for two dimensional, steady state thermal and stress analysis for regular and irregular bodies and heterogeneous composition. The code can also calculate the thermal stresses and structural stresses in a body. The results are verified with software ANSYS. In case of thermal and stress analysis results match 98 percent with ANSYS and in case of thermal stresses results are 93 percent accurate. These small errors in results are due to the reason that ASITM code uses linear variation between the nodes, while ANSYS uses quadratic variation between the nodes.
is found for a specific problem; there is no closed form expression that permits analytical study of the effects of changing various parameters. Computer and a reliable program are essential. Experiences and good engineering senses are needed to construct a good mesh. Many input data are required, and voluminous out put data must be sorted and under stood. How ever, these drawbacks are not unique to the finite element method. We have been, alluding to the essence of the finite element method, now we will discuss it in greater detail. In a continuum problem of any dimension the field variable whether it is pressure, temperature, displacement, stresses or some other quantities posses infinitely many values because it is a function of each generic point in the body and solution region. Consequently the problem is one with an infinite number of unknowns. The finite element discretization procedure reduces the problem to one of the finite numbers of unknown by dividing the solution region into elements and by expressing the unknowns field variables in term of assumed function with in each element. The approximating function are defined in terms of the field variables at specified points called the nodes or nodal points, Nodes usually lie on the element boundaries where adjacent points are to be considered to be connected. In addition to boundary nodes, an element may have interior nodes. The nodal values of the field variables and the interpolation function for the element completely define the behavior of the field variable with in the elements. For the finite element representation of a problem the nodal values of the field variable become the new unknowns. Once these unknown are found, the interpolation function defines the field variable through the assemblage of elements [2]. Clearly the nature of the solution and the degree of approximation depend not only on the size and the number of element used, but also on the interpolation function selected. As one would expect, we cannot choose function arbitrarily, because certain compatibility conditions should be satisfied. Often function is chosen so that the field variable or its derivatives are continuous across adjoining element boundaries. Another advantage of the finite element method is that the variety of ways in which one can formulate the properties of the individual elements. There are basically four different approaches. The first approach to obtaining element properties is called the direct approach because its origin is traceable to the direct stiffness method of structural analysis. The direct approach also suggests the need for matrix algebra in dealing with
finite element equations. Element properties obtained by the direct approach can also be determined by the more versatile and more advanced variational approach. The variational approach relies on the calculus of variations and involves extremizing a functional. For problems in solid mechanics the functional turns out to be potential energy, or some derivative of these, such as Reissner Variational Principle. Knowledge of the variational approach is necessary to work beyond the introductory level and to extend the finite element method to a wide variety of engineering problems. Where as the direct approach can be used to formulate element properties for only the simplest element shapes. A third and even more versatile approach to deriving element properties has its basis entirely in mathematics and is known as weighted residual approach. The weighted residual approach begins with the governing equations of the problem and proceeds without relying on a functional or a variational statement. This approach is advantageous because it thereby becomes possible to extend the finite element method to problems where no functional is available. For some problems we do not have a functional either because one may not have been discovered or because one does not exist. A fourth approach relies on the balance of thermal and /or mechanical energy of the system. The energy balance approach (like the weighted residual approach) requires no variational statement and hence broadens considerably the range of possible applications of the finite element method [3].
1.1 The Steps Involved In Finite Element Method
1.1.1 Discretization of The Continuum
The first step is to divide the continuum or solution region into elements. In our problem continuum has been divided into M quadrilateral elements to find the solution of conduction equations a variety of element shapes may be used, and with care, different element shapes may be employed in the same solution region although the number and the type of elements to be used in a given problem are matters of engineering judgment, the analyst can rely on the experience of others for guidelines.
1.1.5 Solution of The System Equations
The assembly process of the preceding step gives a set of simultaneous Equations that we can solve to obtain the unknown nodal values of the field variable. If the equations are linear, we can use a number of standard solution techniques, if the equations are nonlinear, their solution is more difficult to obtain.
1.1.6 Requirement For Additional Computations
Some time we may want to use the solution of the system equations to calculate other important parameters. For example, in fluid mechanics problems such as the lubrication problem the solution of the system equations gives the pressure distribution within the system. From the nodal values of the pressure we may then calculate velocity distribution, within the system.
The range of possible applications of the finite element method extends to all engineering problems, but civil and aerospace engineers Concerned with stress analysis are the most frequent users of the method. Major aircraft companies and other organizations involved in the design of structures have developed elaborate finite element programs. From the practitioners point of view, the finite element method, like other numerical technique can always be made more efficient and easier to use. As the method is applied to larger and more complex problems, it becomes increasingly important that the solution process remain economical. This means that studies to find the better ways to solve simultaneous linear and non linear equations will certainly continue. Our efforts are also for the improvement of finite element method, we are trying to reduce the computer time for finding the solution of the problems by finite element method [3].
1.2 Objectives
The main objectives of this thesis work is:
- Development of a computer code for thermo mechanical analysis 2. Bench marking with standard software ANSYS 5.1.
1.3 Thesis Organization
A brief introduction of FEM is given in chapter 1, which is followed by chapter 2 in which mathematical formulation of thermal and stress analysis is given. In this chapter variational approach is used to get the element equation. Chapter 3 contains the code implementation. The detailed description of main program and subroutines are given along with flow chart. Thesis ends with chapter 4 in which results are discussed. Different cases are studied and code results are compared with software ANSYS 5.
C1=boundary on which the value of temperature is specified C2=boundary on which the heat flux q is specified C3=boundary on which the convective heat loss is specified
C1 C
Y-axis
T∞ C
X-axis Figure 2-1 Coordinate system for the two dimensional problem
For the sake of convenience we assume that Kx, Ky=K (x, y) and Q, q and h are known specified function of coordinates. The finite element equations for this problem can be obtained by considering the following functional and minimizing them. For two-dimensional case the respective functional is:
( ) 21 2. 2 12 3 3
2 2
2 2 I kx x ky y f dxdy q dc h dc ∫ A (^) c ∫ c ∫
φ = ∂φ φ φ φ φ (2-5)
In order to obtain the element equations we divide the region into M elements, which are triangular, and they have three nodes each.. The element equations can be obtained after the minimization of the functional. For a triangular element the element equations are as follows.
[ ] { } { } { } [ ] { } { }
(^3 331) () 31 31 33 31 () 3 × 1 ∞
× × × × × × = − − + T
e K (^) T T e KQ Kq Kh T K (2-6)
Where
K (^) Tij = 4 K ∆ ( bibj + ci cj ) Where i,j=1,2,3 ( 2-7)
These are the terms of the thermal stiffness matrix:
[ ] ⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
3
2
1
T
T
T T (^) = Column vector of nodal temperature ( 2-8)
And
{ }
3
2
1
12 Q
Q
Q
KQ = Influence matrix for internal heat generation ( 2-9)
The remaining parts of the above equation will exist only if element boundary is a part of the solution region boundary, and external heat flux or boundary convection is specified on the boundary segment If “i” and “j” designate the nodes that lie on the boundary then
[ ]
K (^) h lij h = Influence matrix for boundary heat convection ( 2-10)
Influence matrix for externally heat flux at boundary nodes
ij
ij
q l
l
K q
[ T ∗^ ] = [ T ∗ x , T ∗ y ] ( 2-17)
Boundary traction component acting on the portion C1of the boundary; these are defined per unit length for a unit thickness.
At equilibrium the potential energy of the system assumes a minimum value.
Taking the first varitation and equating it to zero derive the element equations.
In matrix form the element equations are then as:
[ K ] ap {^ δ } q ={ Fo } q +{ FB } q +{ FT } q ={ F } q (2-18)
Where
[ K ] ap = ∫∫ [ B ] qT [ C ][ B ] ptdA = Stiffness matrix at node q.^ (2-19)
[ F (^) o ] q =∫∫[ B ] Tq [ C ]{ε (^) o^ ∗ } q tdA = Nodal body force vector^ (2-20)
[ FT ] q = (^) ∫ Nq ( x , y )[ T ∗] (^) qdsq = Nodal force vector due to surface loading present only for boundary element (2-21)
{ F }^ q =Resultant external load at node q.
For a system to be analyzed using finite elements, it is first discritized into m nodes. For a system that has m nodes the system of equations become.
[ K ] { } G δ (^) G ={ F } G (2-22)
Where { } δ is a column vector of nodal displacements components for the entire system
and {F} is the column vector of resultant nodal forces. If for a particular geometry displacement field can be calculated, then stress in the element can be calculated from:
{ }σ (^ e )^ =[ C ] ( e )[ B ] ( e ){ }δ ( e )−[ C ] ( e ){ε o ∗ } ( e ) (2-23)
This equation also takes into account initial strains in calculating the stress [2].
2.3 Plane Stress And Plane Strain Analysis
Two-dimensional problems can be classified into two categories, plane stress and plane strain problems. For plane stress problems when the thickness is constant the matrix [K]ij is given by :
[ ] ⎥
= ∆ i ij i b
(e) 2 i i 0 c
b 0 c 4 (1- ) K Et
− (^) j jj
j
c b
c
b v
v
v o 0
(2-24)
For i =1,2,
j =1,2,
Where
a 1 = x 2 y 3 – x 3 y 2 b 1 = y 2 – y 3 c 1 = x 3 – x 2 ( 2-25) a 2 = x 3 y 1 – x 1 y 3 b 2 = y 3 – y 1 c 2 = x 1 – x 3 ( 2-26) a 3 = x 1 y 2 – x 2 y 1 b 3 = y 1 – y 2 c 3 = x 2 – x 1 ( 2-27) Where ‘x 1 ’, ‘x 2 ’, ‘x 3 ’ are the x-coordinate of a triangle and ‘y 1 ’, ‘y 2 ’, ‘y 3 ’ are the
y-coordinate of a triangle.
Where
3 3
2 2
1 1
1
x y
x y
x y ∆ = (2-28)
The element force matrix for a triangle can be obtained as
xyo
o i ci bii yo F c γ
ε
ε (^) xo i 2
(e)
2 0 0 1 - v
v 1 0
1 v 0 0
b 0 2 (1-v )
E t (2-29)
3 Code description
In this chapter complete description of main program and subroutine programs is given. The flow chart of main program is shown in figure 3-1. Detailed explanation of each term used in this code is given one by one
3.1 Code Specification
Name ASITM Language FORTRAN 90 Technique Finite Element Method Geometry 2D Rectangular Length 940 lines Subroutines 6
3.2 Variable Used In The Program
NCASE integer, which specifies the type of problem to be solved NCASE=1 2D plane problem NCASE=2 3D axisymmetric problem NN the number of nodes NE the number of elements XC (I), YC (I) Global coordinate of node I NODE (J,I) J=1,2….NE and I=1,2,3 node number associated with node J NT,NTS (I) node number where temperature is specified TNT specified nodal temperature NNST number of nodes where specified temperature TK (J) thermal conductivity of element J QQ (J) value of internal heat generation of element J NQ, NQS (I) node number where external heat generation is specified QNQ, Q (I) specified external nodal heat flux
NNQS number of node number where external heat flux is specified NC1 (I) , NC2 (I) node number pairs defining boundary segment where convection occur H (I) convective heat transfer coefficient TINF ambient temperature NNHC number of nodes where convection occurs XC$ (I), YC$ (I) local coordinate of node I TM$ element thermal stiffness matrix DEL area of triangular element FQQS$ element influence matrix for internal heat generation FQQ system influence matrix for internal heat generation DNU Poisson ratio TT element thickness ALPH thermal expansion coefficient ROOMT room temperature YM young’s modulus NBFD nodes where boundary force is specified NNBF number of nodes where the boundary force is specified BFX boundary force in X- direction BFY boundary force in Y- direction
