Sample introduction to finite element analysis and design Nam Ho Kim solution manual pdf PDF

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Authors: Nam-Ho Kim | Bhavani V. Sankar
Published: Wiley 2008
Edition: 1st
Pages: 397
Type: pdf
Size: 3.69MB
Content: Chapter 0 to 8 problem answers...

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tps://gioumeh.com/product/introduction-to-finite-element-analysis-and-design-soluti

FOLFNKHUHWRGRZQORDG

SOL SOLUTION UTION MAN MANU UAL

Nam H. Kim and Bhavani V. Sankar Department of Mechanical & Aerospace Engineering College of Engineering University of Florida P.O. Box 116250 Gainesville, FL 32611

Introduction to Finite Element Analysis and Design

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FOLFNKHUHWRGRZQORDG

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CHAP 0. FOLFNKHUHWRGRZQORDG MA MATHEMA THEMA THEMATICAL TICAL PRE PRELIMINARIES LIMINARIES 1. Consider the following 3×3 matrix [T]: 1 7 2    [T ]   3 4 3  6 5 7   

(a) Write the transpose TT. (b) Show that the matrix [S] = [T] + [T]T is a symmetric matrix. (c) Show that the matrix [A] = [T] − [T]T is a skew-symmetric matrix. What are the diagonal components of the matrix [A]? Solution: (a) The transpose of a matrix [A] is defined as AT ij  Aji where i and j denote the row and column of the matrix [A], therefore:

T

   T

 1 3 6     7 4 5     2 3 7 

(b) A symmetric matrix [A] is defined as Aij = Aji where i and j denote the row and column of the matrix [A], using the definition of the transpose above gives: [S]  [T ]  [T ]T

1 7 2   [ S]  [ T]  [ T]  3 4 3   6 5 7    T

 1 3 6  2 10 8      7 4 5   10 8 8      2 3 7  8 8 14     

(c) A skew-symmetric matrix [A] is defined as one, which obeys the relation Aij = –Aji, using the definition of the transpose above gives: [A ]  [T ]  [T ]T

1 7 2   [ A]  [ T]  [ T]T  3 4 3   6 5 7   

1 3 6   7 4 5     2 3 7  

Note that the diagonal components of [A] are all zero.

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 0 4 4    4 0 2     4 2 0   

2 Finite Element Analysis and Design tps://gioumeh.com/product/introduction-to-finite-element-analysis-and-design-soluti 2. Consider the following two 3×3 matrices [A] and [B]:

FOLFNKHUHWRGRZQORDG

1 7 2    [A ]   3 4 3  , 6 5 7   

3 7 2   [B]   2 1 8  7 4 5  

(a) Calculate [C] = [A] + [B]. (b) Calculate [D] = [A] – [B]. (c) Calculate the scalar multiple [D] = 3[A]. Solution:  1 7 2   3 7 2  4 14 4         C   A   B   3 4 3    2 1 8    5 5 11              6 5 7   7 4 5  13 9 12        1 7 2    D   A    B    3 4 3            6 5 7  

 3 7 2    2 1 8     7 4 5  

 2 0 0     1 3  5   1 1 2   

1 7 2   3 21 6       D   3  A   3  3 4 3    9 12 0      6 5 7  19 15 21     

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CHAP 0 Mathematical Preliminaries 3 tps://gioumeh.com/product/introduction-to-finite-element-analysis-and-design-soluti 3. Consider the following two three–dimensional vectors a and b:

FOLFNKHUHWRGRZQORDG a  {1, 4, 6}T and b  {4, 7, 2}T

(a) Calculate the scalar product c = a ∙ b. (b) Calculate the norm of vector a. (c) Calculate the vector product of a and b Solution: (a) c  a  b  1  4  4  7  6  2  44 ; (b)

a  12  42  62 

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i j k (c) a  b  1 4 6  (8  42)i  (2  24)j  (7  16)k   34i  22j  9k 4 7 2

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4 Finite Element Analysis and Design tps://gioumeh.com/product/introduction-to-finite-element-analysis-and-design-soluti 4. For the matrix [T] in Problem 1 and the two vectors a and b in Problem 3, answer the following questions. FOLFNKHUHWRGRZQORDG (a) Calculate the product of the matrix–vector multiplication [T] ∙a. (b) Calculate b∙[T] ∙a. Solution: The matrix-vector multiplication  T   a is    1  1  28  12   41      1 7 2       3  16  18    37    T   a   3 4 3   4                6 5 7  6    6  20  42     68          

b   T   a can be obtained by scalar product of two vectors b and  T   a       4 41               b   T   a   7    37    4  41  7  37  2  68  559      2     68      

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CHAP 0 Mathematical Preliminaries 5 tps://gioumeh.com/product/introduction-to-finite-element-analysis-and-design-soluti 5. For the two matrices [A] and [B] in Problem 2, answer the following questions.

FOLFNKHUHWRGRZQORDG (a) Evaluate the matrix–matrix multiplication [C] = [A][B]. (b) Evaluate the matrix–matrix multiplication [D] = [B][A]. Solution:

(a)

(b)

1 7        A   B   3 4 6 5   3  14  14    9  8  21 18  10  49 

2   3 7 2  3    2 1 8  7  7 4 5   7 7 8 2  56  10   31 22 68     21  4  12 6  32  15    38 37 53    42  5  28 12  40  35   77 75 87  

 3 7 2  1 7 2   36 59 41         B   A    2 1 8    3 4 3    53 58 63      7 4 5  6 5 7   49 90 61       

Note that in general, [A ][B ]  [B][A ] .

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6 Finite Element Analysis and Design tps://gioumeh.com/product/introduction-to-finite-element-analysis-and-design-soluti 6. Calculate the determinant of the following matrices:

FOLFNKHUHWRGRZQORDG  4 2 , [A ]     3 7 

1 3 2    [B]   1 4 5     2 6 7 

Solution: Determinant of  A  is 4 2 3 7

 (4  7)  (2  3)  28  6  22

Determinant of  B  is 1 3 2 1 4 5  1  (4  7  5  6)  3  (1  7  5  2)  2  (1  6  4  2)  3 2 6 7

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CHAP 0 Mathematical Preliminaries 7 tps://gioumeh.com/product/introduction-to-finite-element-analysis-and-design-soluti 7. Calculate the inverse of the matrix [A] in Problem 6.

FOLFNKHUHWRGRZQORDG

Solution: 1

   A

1

4 2      3 7  

 7 1    7 2   7 2   1 1 11       22  2  4  7  2  3   3 4   22   3 4    3    22 11 

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8 Finite Element Analysis and Design tps://gioumeh.com/product/introduction-to-finite-element-analysis-and-design-soluti 8. Matrices [A] and [B] are defined below. If B = A−1, determine the values of p, q, r and s. FOLFNKHUHWRGRZQORDG  2 4 p   [ A]   1 2 3  ,    q 5 6

 3 1 2   [ B]   3  3  1  .  1  r s  

Solution: If B  A1 , then AB  I .  AB  2  ( 3)  4  3  p  ( 1)  1  p  5   11  AB  q  ( 3)  5  3  6  ( 1)  0  q  3   31  AB   1  1  2  (3)  3  r  1  r  2   22  AB  1 2  2  ( 1)  3  s  0  s  0   23

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CHAP 0 Mathematical Preliminaries 9 tps://gioumeh.com/product/introduction-to-finite-element-analysis-and-design-soluti 9. Solve the following simultaneous system of equations using the matrix method:

FOLFNKHUHWRGRZQORDG 4 x 1  3x 2  3 x 1  3x 2  3 Solution: The given system can be represented by matrix form as follows,  4 3  x1       1 3  x 2      

x 1   4 3             x 1 3  2    

1

3     3   

  3 3   3    0     3 1                  3  1 3   4  3  3  1  1 4           

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10 Finite Element Analysis and Design tps://gioumeh.com/product/introduction-to-finite-element-analysis-and-design-soluti 10. Consider the following row vectors and matrices

FOLFNKHUHWRGRZQORDG a  [5 12 3] 1 6 3   A    2 8 4 

Using

MATLAB,

calculate

b  [1 0 4] 2 5 4   B     7 2 0  AT , aT , A  B, A  B, abT , aT b, AT B

C  BAT , AB, C 1, det(C) . Test the following commands: a. * b, A. * B and

explain the difference between them and a * b, A * B , respectively. Solution: Note that the command A*B gives error since the multiplication size doesn‟t match. MATLAB has two different types of arithmetic operations. Matrix arithmetic operations are defined by the rules of linear algebra. Array arithmetic operations are carried out element-by-element, and can be used with multidimensional arrays. The period character (.) distinguishes the array operations from the matrix operations. However, since the matrix and array operations are the same for addition and subtraction, the character pairs .+ and .- are not used. For more information and examples, use the MATLAB help (go to the index and search for “array”).

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