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INSTRUC TO R'S SO LUTIO NS M A NUA LTO A C C O M PA NYM A TRIX A NA LYSISo f STRUC TURESSEC O ND EDITIO NA SLA M KA SSIM A LISo uthe rn Illino is Unive rsity—C a rb o nd a le@Seismicisolation@solutionmanualFOLFNKHUHWRGRZQORDGConte nts####### Chapter 2 1####### Chapter 3 28####### Chapter 5 84####...

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FOLFNKHUHWRGRZQORDG

INSTRUC TO R' S SO LUTIO NS M A NUA L TO

A C C O M PA NY

M A TRIX A NA LYSIS o f STRUC TURES SEC O ND EDITIO N

A SLA M KA SSIM A LI So uthe rn Illino is Unive rsity—C a rb o nd a le @solutionmanual1

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FOLFNKHUHWRGRZQORDG

Contents Cha p te r 2

1

Cha p te r 3

28

Cha p te r 5

84

Cha p te r 6

131

Cha p te r 7

216

Cha p te r 8

278

Cha p te r 9

308

Cha p te r 10

315

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FOLFNKHUHWRGRZQORDG

Chapter 2

2.1

 8 C = A + B = −1  1

17 −1 −7

−3  −1  ; 1 

1 −2 −1  D = A − B = 17 −13 −7  −3 −1 9 

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FOLFNKHUHWRGRZQORDG 2.2

 19 − 14 − 9  2  − − 1 0 C = 2A + B =  ; −10 2 4   20 − 7  − 5

6  − 1 − 12  13 0 − 11  D = A − 3B =  − 12 29 − 19   21  1 − 4

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FOLFNKHUHWRGRZQORDG 2.3

 3   C = AB = [4 −6 2]  1 = 12 − 6 − 10 = −4 −5 6  3  12 −18  1 4  2 D = BA =   [ −6 2 ] =  4 − 6  − 5 − 2 30 − 10

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FOLFNKHUHWRGRZQORDG 2.4

6 22 44  4  −82 −12  7   3 − 5 6 72 0 − 44 − − 5  − 1 −1   C = AB =  =  1 − 9 − 13 − 4 7 2  116 39 − 68 − 52     92 60 − 3 11 − 140 − 53 6  4 − 7 − 5 − 36 − − 1 3 5 2 46    = D = BA =     − 13 − 4 7 6  1 − 9 − 35 − 55   − 3 11

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FOLFNKHUHWRGRZQORDG 2.5

5 0 − 18 − 24 21  4 − 6 1  3  6    5 7 5 7 −2 =  7 C = AB =  − −9 53     1 7 8 0 − 2 9  38 38 58 3  D = BA = 5 0

5 7 −2

0  4 −6 1 −18 7 38      5 7 = −24 −9 38 −2 −6    9  1 7 8  21 53 58

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FOLFNKHUHWRGRZQORDG 2.6

12 −11 10 − 50 287 −10 13 −1 5   0  2 −4  44 −98 28  16 −9  0 =  C = AB =    29 − 7 9 8  86 − 91   −3 20 −7     6 15 −5  333 −241 65

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FOLFNKHUHWRGRZQORDG 2.8

 21 10 16  185  7 − 4   15  11 0  116 −  −1 9  = − AB =     13  44 20 −9    3 − 6   7 −17 14  108

( AB) T

 185 =  −90

− 90 159  182  −265

108 −116 44 159 182 −265

(1)

7 21 −15 13 3  108  7 −1  185 − 116 44 10 11 20 −17 =  B A =    − 90 159 182 − 265 − 4 9 − 6 16 0 −9 14   T

T

T

From Eqs. (1) and (2), we can see that ( AB) = BT A T .

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(2)

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FOLFNKHUHWRGRZQORDG 2.9

 −9  13 [ A][ B][C ] =  8   −11

0 20  15  − 3  6  −5 

 −135  315 =  102   −195

T

( [A ][B ][C ])

36   −7 128    −1 −59     16 −1 

9 307 −56 −69

 1512  − 464 =  −1602   1418  1512  900 − = − 810   270

 −7  10 T T T  [ C] [ B] [ A] =  6   0

−7 −4  −1 9   16

−1 16

− 900 5300 200

−810 − 310 942

−2100

−620 −1602 200

− 310 410

942 −360

2 −8 −2

 −168  100 =  90   −30

86  200 − 9  −74   0  40 

 1512  900 − =  −810   270

−464 5300

−1602 200

−310 410

942 −360

13 20

16    15 12  −1 2    −4 8  

8 −3

6 −8

12

2

10 2

6 −8

12

2

0 −2  8 

270 410  −360  130 

−464 5300

−1

1418  − 2100 − 620  130

6  − 9 16   0 9 

13 20

From Eqs. (1) and (2), we can see that T

T

T

(1)

8 −3

− 11 −5 

− 11 −5 

1418  −2100  −620   130 

( [A ][B ][C ]) = [ C ] [ B] [ A]

0 −2 8

10 2

T

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(2)

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FOLFNKHUHWRGRZQORDG 2.10

3  5 −7  7 T 8 −4 C = B AB =    −3 4 9   195 − 119 − 119  =  300 2 −199    −385 198 393

 40 − 10 − 25  5 7 − 3  10 15 12  7 8 4 −  −   −25 12 30   3 −4 9 889 − 2,132   5 7 −3   1, 451  −7 8 4  =  889 2,912 −2,683      3 −4 9   −2,132 −2,683 5,484 

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FOLFNKHUHWRGRZQORDG 2.11

 0.6 − 0.8  0.8 0.6   C = BT AB =   −0.6 0.8    −0.8 −0.6   260 − 220  180 40   0.6  =   −260 220   −0.8    −180 −40 

 300 − 100     −100 200 

 0.6 0.8 − 0.6 − 0.8     −0.8 0.6 0.8 −0.6 

76 − 332 − 76   332  0.8 − 0.6 − 0.8  76 168 − 76 −168    = 0.6 0.8 −0.6   −332 −76 332 76    76 168   −76 −168

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FOLFNKHUHWRGRZQORDG 2.13

−7 3cos x  −4 x  dA  3cos  2 2cos sin 9 x − x x =  − x  dx  −7 −9 x2 6sin x cos x 

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FOLFNKHUHWRGRZQORDG  2 x − 3x −x + 5   2  4 x − 12 x − x 3 + 8  ; C = A+B =   2 x3 − 7 − 3 x 2 + 5 x    3 − x2 + 6 x   2 x − 1 2

2.14

−1   4x − 3 8  12 x − −3x 2  dC  =  6x 2 −6 x + 5  dx   2  6 x −2 x + 6 

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FOLFNKHUHWRGRZQORDG  12 − 30 x  −30x 3 [ A][B ] =  − 6 x + 14 x 2 + 25 x 5 

− 10 x2 − 20 x3   −2x − 4x 2 + 9 x 5  3  28 x + 8 x 

 − 120 x 3 d [A ][B ]  = −90x 2 dx  − 6+ 28x + 125x 4 

− 20 x − 60 x 2   −2 − 8 x + 45 x4  2 28+ 24x 

4

2.15

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FOLFNKHUHWRGRZQORDG 2.16

2 x 0 0  ∂ [A ]  =  0 3 y 0  ∂x  0 0 4 z  0 − 2 y 0 ∂ [A ]  = −2 y 3 x − z   ∂y 0  0 − z

 0 ∂ [ A]  = 0 ∂z 4z

0 0 −y

4z  − y  4x 

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