chapter 2 of Finite element coure in structural Eng

This presentation is about the finite element course in structural Engineering

1. یلیبدرا ققحم هاگشناد
University of Mohaghegh Ardabili
رازگرکش نمحر دماح رتکد
دودحم ءازجا شور
Finite Element Method
مود لصف : همدقم و اهسیرتام ربج رب یا لح یاهشور تلاداعم نامزمهیطخ

2. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Matrix Algebra
Definition of a Matrix
A matrix is an m×narray of numbers arranged in m rows and n columns. The matrix
is then described as being of order m×n. Equation (A.1.1) illustrates a matrix withm
rows and n columns.
•If m ≠ n in matrix Eq. (A.1.1), the matrix is called rectangular.
•If m=1 and n > 1, the elements of Eq. (A.1.1) form a single row called a row matrix.
•If m > 1 and n=1 , the elements form a single column called a column matrix.
•If m = n, the array is called a square matrix.

3. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Matrix Algebra
Definition of a Matrix
To identify an element of matrix [a], we represent the element by aij, where the
subscripts iand j indicate the row number and the column number, respectively, of
[a]. Hence, alternative notations for a matrix are given by:
Matrix Operations
Multiplication of a Matrix by a Scalar
If we have a scalar k and a matrix [c]then the product [a]= k[c] is given by

4. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Matrix Algebra
Addition of Matrices
Multiplication of Matrices
For two matrices [a] and [b]to be multiplied in the order shown in Eq. (A.2.4), the number of
columns in [a] must equal the number of rows in [b]For example, consider
If [a] s an m×nmatrix, then [b] must have n rows. Using subscript notation, we can write the product
of matrices [a] and [b] as
For matrix [a]of order 2×2
and matrix [b]of order 2×2,

5. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Matrix Algebra
Multiplication of Matrices
In general, matrix multiplication is not commutative; that is
The validity of the product of two matrices [a] and [b] is commonly illustrated
by
Transpose of a Matrix
The transpose of a matrix is obtained by interchanging rows and columns; that is, the first row
becomes the first column, the second row becomes the second column, and so on

6. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Matrix Algebra
Transpose of a Matrix
Another important relationship that involves the transpose is
Symmetric Matrices
If a square matrix is equal to its transpose, it is called a symmetric matrix; that is, if

7. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Matrix Algebra
Unit Matrix
The unit (or identity) matrix [I ] is such that:
Inverse of a Matrix
The inverse of a matrix is a matrix such that
Orthogonal Matrix
A matrix [T]is an orthogonal matrix if
Hence, for an orthogonal matrix, we have
An orthogonal matrix frequently used is the transformation or rotation matrix [T ]

8. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Matrix Algebra
Orthogonal Matrix
Anorthogonalmatrixfrequentlyusedisthetransformationorrotationmatrix[T].
Intwo-dimensionalspace,thetransformationmatrixrelatescomponentsofavectorinone
coordinatesystemtocomponentsinanothersystem.Forinstance,thedisplacement(andforce
aswell)vectorcomponentsofdexpressedinthex-ysystemarerelatedtothoseinthex’-y’
systemby
where [T] is the square matrix on the right side of Eq.

9. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Matrix Algebra
Orthogonal Matrix
Anotheruseofanorthogonalmatrixistochangefromthelocalstiffnessmatrixtoa
globalstiffnessmatrixforanelement.Thatis,givenalocalstiffnessmatrix[k’]foran
element,iftheelementisarbitrarilyorientedinthex-yplane,then
Above Equation is used throughout this text to express the stiffness matrix [k]in the x-y
plane.

10. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Matrix Algebra
Differentiating a Matrix
Amatrixisdifferentiatedbydifferentiatingeveryelementinthematrixintheconventionalmanner.
Forexample,if

11. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Matrix Algebra
Differentiating a Matrix
Instructuralanalysistheory,wesometimesdifferentiateanexpressionoftheform:
whereUmightrepresentthestrainenergyinabar.Expressionaboveisknownasa
quadraticform.BymatrixmultiplicationofEq.above,weobtain:
Differentiating U now yields
Equation above in matrix form becomes
EQ. 1
EQ. 2
EQ. 3
EQ. 4

12. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
اهسیرتام ربج رب یا همدقم
Differentiating a Matrix
A general form of equation 1 is:
Then, by comparing Eq. (1) and (5), we obtain
EQ. 5
EQ. 6
where x
idenotes x and y.
Here Eq. (6) depends on matrix [a] in Eq. (5) being symmetric.

13. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Matrix Algebra
Integrating a Matrix
Just as in matrix differentiation, to integrate a matrix, we must integrate every element
in the matrix in the conventional manner. For example, if
In our finite element formulation of equations, we often integrate an expression of the form
The triple product in above Equation will be symmetric if [A] is symmetric. The form
[X]
T
[A][X] is also called a quadratic form. For example, letting

14. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Matrix Algebra
Integrating a Matrix
which is in quadratic form.

15. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Matrix Algebra
Cofactor or AdjointMethod to Determine the Inverse of a Matrix
Wewillnowintroduceamethodforfindingtheinverseofamatrix.Thismethodisuseful
forlonghanddeterminationoftheinverseofsmaller-ordersquarematrices(preferablyof
order4×4orless).Amatrix[a]mustbesquareforustodetermineitsinverse.
We must first define the determinant of a matrix. This concept is necessary in
determining the inverse of a matrix by the cofactor method. A determinant is a square
array of elements expressed by
where the straight vertical bars,| |, on each side of the array denote the determinant.
The resulting determinant of an array will be a single numerical value when the array
is evaluated

16. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Matrix Algebra
Cofactor or AdjointMethod to Determine the Inverse of a Matrix
To evaluate the determinant of [a], we must first determine the cofactors of [a
ij]. The
cofactors of [a
ij] are given by
where the matrix [d] called the first minor of [a
ij] is matrix [a] with row iand column j deleted.
The inverse of matrix [a] is then given by:
where [C] is the cofactor matrix and |[a]| is the determinant of [a]

17. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Matrix Algebra
Cofactor or AdjointMethod to Determine the Inverse of a Matrix

18. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Matrix Algebra
Cofactor or AdjointMethod to Determine the Inverse of a Matrix

19. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Matrix Algebra
Inverse of a Matrix by Row Reduction (Gauss–Jordan method )
The inverse of a nonsingular square matrix [a] can be found performing identical
simultaneous operations on the matrix [a] and the identity matrix [a] (of the same order
as [a]) such that the matrix [a] becomes an identity matrix and the original identity
matrix becomes the inverse of [a].
We will invert the following matrix by row reduction.
To find [a]
-1
, we need to find [x] such that [a][x]=[I] where

20. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Matrix Algebra
Inverse of a Matrix by Row Reduction (Gauss–Jordan method )

21. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Matrix Algebra
Inverse of a Matrix by Row Reduction (Gauss–Jordan method )

22. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Matrix Algebra
Inverse of a Matrix by Row Reduction (Gauss–Jordan method )

23. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Matrix Algebra
Inverse of a Matrix by Row Reduction (Gauss–Jordan method )

24. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Matrix Algebra
Inverse of a Matrix by Row Reduction (Gauss–Jordan method )

25. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Matrix Algebra
Inverse of a Matrix by Row Reduction (Gauss–Jordan method )

26. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solution of Simultaneous Linear Equations
General Form of the Equations
In general, the set of equations will have the form

27. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solution of Simultaneous Linear Equations
General Form of the Equations
•Ifthec’sarenotallzero,thesetofequationsisnonhomogeneous,andall
equationsmustbeindependenttoyieldauniquesolution.Stressanalysisproblems
typicallyinvolvesolvingsetsofnonhomogeneousequations.
•Ifthec’sareallzero,thesetofequationsishomogeneous,andnontrivialsolutions
existonlyifallequationsarenotindependent.Bucklingandvibrationproblems
typicallyinvolvehomogeneoussetsofequations.

28. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solution of Simultaneous Linear Equations
Uniqueness, Nonuniqueness,and Nonexistence of Solution
•Uniqueness of Solution: A unique solution exists if and only if the determinant of
the square coefficient matrix is not equal to zero
Uniqueness: درف هب رصحنم

29. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solution of Simultaneous Linear Equations
Uniqueness, Nonuniqueness,and Nonexistence of Solution
•Nonuniquenessof Solution :
Nonuniqueness: درف هب رصحنم ریغ
The determinant of the coefficient matrix iszero:

30. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solution of Simultaneous Linear Equations
Uniqueness, Nonuniqueness,and Nonexistence of Solution
•Nonuniquenessof Solution :
Again,thedeterminantofthecoefficient
matrixiszero.Inthiscase,nosolution
existsbecausewehaveparallellines(no
commonpointofintersection),asshown
inFigure.

31. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solution of Simultaneous Linear Equations
Methods for Solving Linear Algebraic Equations
•Cramer’s Rule
WebeginbyintroducingamethodknownasCramer’srule,whichisusefulforthelonghand
solutionofsmallnumbersofsimultaneousequations.Considerthesetofequations
or,inindexnotation,
We first let [d
(i)
] be the matrix [a] with column ireplaced by the column matrix [c]. Then the
unknown x
i’s are determined by:

32. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solution of Simultaneous Linear Equations
Methods for Solving Linear Algebraic Equations
•Cramer’s Rule
AsanexampleofCramer’srule,considerthefollowingequations:
Inmatrixform,Eqs.(B.3.4)become

33. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Cramer’s Rule
ByEq.(B.3.3),wecansolvefortheunknownx
i’sas
Methods for Solution of Simultaneous Linear Equations

34. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Cramer’s Rule
Ingeneral,tofindthedeterminantofann×nmatrix,wemustevaluatethedeterminantsofn
matricesoforder(n-1)×(n-1).Ithasbeenshownthatthesolutionofnsimultaneousequations
byCramer’srule,evaluatingdeterminantsbyexpansionbyminors,requires(n-1)×(n+1)!
multiplications.Hence,thismethodtakeslargeamountsofcomputertimeandthereforeisnot
usedinsolvinglargesystemsofsimultaneousequationseitherlonghandorbycomputer.
Methods for Solution of Simultaneous Linear Equations

35. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Inversion of the Coefficient Matrix
Thesetofequations[a]{x}={c}canbesolvedfor{x}byinvertingthecoefficient
matrix[a]andpremultiplyingbothsidesoftheoriginalsetofequationsby[a]
-1
,
suchthat:
Methods for Solution of Simultaneous Linear Equations

36. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Gaussian Elimination
Itisbasedontriangularizationofthecoefficientmatrixandevaluationoftheunknownsbyback-
substitutionstartingfromthelastequation.
hegeneralsystemofnequationswithnunknownsgivenby
1.Eliminatethecoefficientofx
1ineveryequationexceptthefirstone.
Todothis,selecta
11asthepivot,and
a.Addthemultiple-a
21/a
11ofthefirstrowtothesecondrow.
b.Addthemultiple-a
31/a
11ofthefirstrowtothethirdrow.
c.Continuethisprocedurethroughthenthrow.
Methods for Solution of Simultaneous Linear Equations

37. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Gaussian Elimination
Thesystemofequationswillthenbereducedtothefollowingform:
2.Eliminatethecoefficientofx
2ineveryequationbelowthesecondequation.Todothis,selecta’
22as
thepivot,and
a.Addthemultiple-a’
32/a’
220ofthesecondrowtothethirdrow.
b.Addthemultiple-a’
42/a’
220ofthesecondrowtothefourthrow.
c.Continuethisprocedurethroughthenthrow.
Methods for Solution of Simultaneous Linear Equations

38. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Gaussian Elimination
Thesystemofequationswillthenbereducedtothefollowingform:
Werepeatthisprocessfortheremainingrowsuntilwehavethesystemofequations(called
triangularized)as
Methods for Solution of Simultaneous Linear Equations

39. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Gaussian Elimination
3.Determinex
nfromthelastequationas
anddeterminetheotherunknownsbyback-substitution.
Methods for Solution of Simultaneous Linear Equations

40. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Gaussian Elimination
.
Methods for Solution of Simultaneous Linear Equations

41. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Gaussian Elimination
.
Methods for Solution of Simultaneous Linear Equations

42. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Gaussian Elimination
These steps are summarized in general form by
where a
i,n+1represent the latest right side c’s given by Eq. (B.3.13) a
i,n+1=C
i
Methods for Solution of Simultaneous Linear Equations

43. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Gaussian Elimination
These steps are summarized in general form by
where a
i,n+1represent the latest right side c’s given by Eq. (B.3.13) a
i,n+1=C
i
Methods for Solution of Simultaneous Linear Equations

44. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Gaussian Elimination
To illustrate the use of the index Eqs. (B.3.15), we re-solve the same example as follows. The
ranges of the indexes in Eqs. (B.3.15) are (k = 1; 2); (i= 2; 3) and (j = 1; 2; 3; 4).
a
34=c
3=6
Methods for Solution of Simultaneous Linear Equations

45. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Gaussian Elimination
Methods for Solution of Simultaneous Linear Equations

46. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Gaussian Elimination
Note: that the pivot element was the diagonal element in each step.
1.The diagonal element (pivot element) must be nonzero.
2.The pivots element should be selected as the largest (in absolute value) of the elements in
any column
Methods for Solution of Simultaneous Linear Equations

47. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Gaussian Elimination
Note 1: The diagonal element (pivot element) must be nonzero.
Methods for Solution of Simultaneous Linear Equations

48. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Gaussian Elimination
Note 1: The diagonal element (pivot element) must be nonzero.
Methods for Solution of Simultaneous Linear Equations

49. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Gaussian Elimination
Note 1: The diagonal element (pivot element) must be nonzero.
Methods for Solution of Simultaneous Linear Equations

50. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Gaussian Elimination
Note 2: The pivots element should be selected as the largest of the elements in any column.
A second problem when selecting the pivots in sequential manner without testing for the best possible
pivot is that loss of accuracy due to rounding in the results can occur. for example, consider the set of
equations given by
The solution by Gaussian elimination without testing for the
largest absolute value of the element in any column is
Methods for Solution of Simultaneous Linear Equations

51. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Gaussian Elimination
Note 2: The pivots element should be selected as the largest of the elements in any column.
This solution does not agree with the actual solution. The solution by interchanging equations is
This solution agree with the actual.
Methods for Solution of Simultaneous Linear Equations

52. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Gauss–Seidel Iteration
Anothergeneralclassofmethods(otherthantheeliminationmethods)usedtosolvesystemsoflinear
algebraicequationsistheiterativemethods.TheGauss–Seidelmethodstartswiththeoriginalsetof
equations[a]{x}={c}writtenintheform
Methods for Solution of Simultaneous Linear Equations

53. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Gauss–Seidel Iteration
Another general class of methods (other than the elimination methods) used to solve systems of linear
The following steps are then applied.
1. Assume a set of initial values for the unknowns x
1; x
2; . . . ; x
n, and substitute them into the right side of
the first of Eqs. (B.3.29) to solve for the new x
1.
2. Use the latest value for x
1obtained from Step 1 and the initial values for x
3; x
4; . . . ; x
nin the right side
of the second of Eqs. (B.3.29) to solve for the new x
2.
3. Continue using the latest values of the x’s obtained in the left side of Eqs. (B.3.29) as the next trial
values in the right side for each succeeding step.
4. Iterate until convergence is satisfactory.
A good initial set of values (guesses) is often x
i= c
i/a
ii
Methods for Solution of Simultaneous Linear Equations

54. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Gauss–Seidel Iteration
EXAMPLE:
Methods for Solution of Simultaneous Linear Equations

55. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Gauss–Seidel Iteration
EXAMPLE:
The first iteration has now been completed.
Methods for Solution of Simultaneous Linear Equations

56. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Gauss–Seidel Iteration
EXAMPLE:
The second iteration yields
Methods for Solution of Simultaneous Linear Equations

57. Finite Element Method
BY: Dr.Hamed R.Shokrgozar
Methods for Solving Linear Algebraic Equations
•Gauss–Seidel Iteration
EXAMPLE:
Methods for Solution of Simultaneous Linear Equations