Solution Manual
A First Course in the
Finite Element Method
ENHANCED
6th EDITION, SI
DARYL L. LOGAN
SI Edition Prepared by
QABOOS IMRAN
This is the Only Original and Complete Solutions Manual for 6th
Enhanced Edition SI, All other Files in the Market are Fake/Old/
Wrong Edition.
All Chapters Arranged Reverse: 16-1
, Contents
Chapter 1:............................................................................................................................ 1
Chapter 2:............................................................................................................................ 3
Chapter 3:.......................................................................................................................... 25
Chapter 4:........................................................................................................................ 133
Chapter 5:........................................................................................................................ 199
Chapter 6:........................................................................................................................ 291
Chapter 7:........................................................................................................................ 335
Chapter 8:........................................................................................................................ 349
Chapter 9:........................................................................................................................ 363
Chapter 10:...................................................................................................................... 383
Chapter 11:...................................................................................................................... 403
Chapter 12:...................................................................................................................... 425
Chapter 13:...................................................................................................................... 437
Chapter 14:...................................................................................................................... 479
Chapter 15:...................................................................................................................... 501
Chapter 16:...................................................................................................................... 529
Appendix A: ..................................................................................................................... 565
Appendix B: ..................................................................................................................... 571
Appendix D:..................................................................................................................... 577
, Chapter 16
16.1
1 2 2 3
AL 2 1 AL 2 1
[m(1)] = , [m(2)] =
6 1 2 6 1 2
2 1 0
AL
[M] = 1 4 1
6
0 1 2
16.2
(a) Lumped mass matrix
1 2 2 3
AL 1 0 AL 1 0
[m(1)] = , [m(2)] =
2 0 1 2 0 1
3 4
AL 1 0
[m(3)] =
2 0 1
1 0 0 0
0 2 0 0 AL
[M] =
0 0 2 0 2
0 0 0 1
(b) Consistent mass matrix
AL 2 1 AL 2 1
[m(1)] = , [m(2)] =
6 1 2 6 1 2
AL 2 1
[m(3)] =
6 1 2
529
© 2023 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in
whole or in part.
, 2 1 0 0
AL 1 4 1 0
[M] =
6 0 1 4 1
0 0 1 2
16.3
1 1 0
AE
[K] = 1 2 1
L
0 1 1
2 1 0
AL
[M] = 1 4 1
6
0 1 2
([K] – 2 [M]) {X} = 0 with x1 = 0
AE 2 1 2 AL 4 1 x2 0
=
L 1 1 6 1 2 x3 0
Let 2 =
E
divide by AL and let =
L2
2 1 4 1
=0
1 1 6 1 2
2
2 3 6
=0
6 3
2
2 ( – ) – (– – )2 = 0
3 3 6
2
4 2
22 – + 2 – 2 – =0
3 9 3 36
5 7 2
2 – + =0
3 36
60 36 2
or 2 – + =0
7 7
530
© 2023 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in
whole or in part.
A First Course in the
Finite Element Method
ENHANCED
6th EDITION, SI
DARYL L. LOGAN
SI Edition Prepared by
QABOOS IMRAN
This is the Only Original and Complete Solutions Manual for 6th
Enhanced Edition SI, All other Files in the Market are Fake/Old/
Wrong Edition.
All Chapters Arranged Reverse: 16-1
, Contents
Chapter 1:............................................................................................................................ 1
Chapter 2:............................................................................................................................ 3
Chapter 3:.......................................................................................................................... 25
Chapter 4:........................................................................................................................ 133
Chapter 5:........................................................................................................................ 199
Chapter 6:........................................................................................................................ 291
Chapter 7:........................................................................................................................ 335
Chapter 8:........................................................................................................................ 349
Chapter 9:........................................................................................................................ 363
Chapter 10:...................................................................................................................... 383
Chapter 11:...................................................................................................................... 403
Chapter 12:...................................................................................................................... 425
Chapter 13:...................................................................................................................... 437
Chapter 14:...................................................................................................................... 479
Chapter 15:...................................................................................................................... 501
Chapter 16:...................................................................................................................... 529
Appendix A: ..................................................................................................................... 565
Appendix B: ..................................................................................................................... 571
Appendix D:..................................................................................................................... 577
, Chapter 16
16.1
1 2 2 3
AL 2 1 AL 2 1
[m(1)] = , [m(2)] =
6 1 2 6 1 2
2 1 0
AL
[M] = 1 4 1
6
0 1 2
16.2
(a) Lumped mass matrix
1 2 2 3
AL 1 0 AL 1 0
[m(1)] = , [m(2)] =
2 0 1 2 0 1
3 4
AL 1 0
[m(3)] =
2 0 1
1 0 0 0
0 2 0 0 AL
[M] =
0 0 2 0 2
0 0 0 1
(b) Consistent mass matrix
AL 2 1 AL 2 1
[m(1)] = , [m(2)] =
6 1 2 6 1 2
AL 2 1
[m(3)] =
6 1 2
529
© 2023 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in
whole or in part.
, 2 1 0 0
AL 1 4 1 0
[M] =
6 0 1 4 1
0 0 1 2
16.3
1 1 0
AE
[K] = 1 2 1
L
0 1 1
2 1 0
AL
[M] = 1 4 1
6
0 1 2
([K] – 2 [M]) {X} = 0 with x1 = 0
AE 2 1 2 AL 4 1 x2 0
=
L 1 1 6 1 2 x3 0
Let 2 =
E
divide by AL and let =
L2
2 1 4 1
=0
1 1 6 1 2
2
2 3 6
=0
6 3
2
2 ( – ) – (– – )2 = 0
3 3 6
2
4 2
22 – + 2 – 2 – =0
3 9 3 36
5 7 2
2 – + =0
3 36
60 36 2
or 2 – + =0
7 7
530
© 2023 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in
whole or in part.
