3D Structure analysis (Kurdish) Prepared by: Eng. Ali Muhammad Shekha Contents: 1. Preface 2. Symbols and Abbreviations ……………………………………………………………………………… 7 3. Matrix (Some Information About Matrix) ……………………………………………………………….… 8 4. Drawing, Supports and Review Some Rules ………………….……12 5. Stiffness Method (Matrix Structural Analysis) ……………………………………………………….….17 6. Example1 …………………………………………………………………………………………………………………………………18 7. Direct Stiffness Method ………………..……………… …………………….……..……19 8. 2D analysis ……………………………………………………………………………………………………………………….………19 9. 2D Beams Analysis by Direct Stiffness Method ………………………………………………………………..…….…18 10. Example2 ……………………………………………………………………………………………………………………………….…22 11. 2D Frames Analysis by Direct Stiffness Method ………………………………………………………………….…….28 12. Example3 …………………………………………………………………………………………………………………………………30 13. 2D Trusses Analysis by Direct Stiffness Method ……………………………………………………………………….38 14. Example4 ………………………………………………………………………………………………………………………………...39 15. 3D analysis ………………………………………………………………………………………………………………………….……43 16. 3D Beams Analysis by Direct Stiffness Method …………………………………………………………………….…..43 17. Example5 …………………………………………………………………………………………………………………………………47 18. 3D Frames Analysis by Direct Stiffness Method ……………………………………………………………………….54 19. Example6 ………………………………………………………………………………………………………………………………...54 20. 3D Trusses Analysis by Direct Stiffness Method ……………………………………………………………….…..….82 21. Example7 ………………………………………………………………………………………………………………………………….84 22. Example8 ………………………………………………………………………………………………………………………………….87 23. Problems ………………………………………………………………………………………………………………………………….98 24. Assembly Stiffness Method ……………………….……….99 25. Symbols and Abbreviations …………………………………………………………………………… 101 …………………………………………………………………………………………………………..…………….111 27. Example1 ……………………………………………………………………………………………………………………..…………112 28. Example2 ……………………………………………………………………………………………………………………..…………123 29. Example3 ……………………………………………………………………………………………………………………..…………139 30. Problem ………………………………………………………………………………………………………………………………….147 31. References …………………………………………………………………………………………….…………..…148

2. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
1














‫إبراهيم‬


























‫الطالق‬
3D Structural Analysis
‫دووريدا‬ َ‫ى‬‫س‬ ‫لة‬ ‫َكهاتةكان‬‫ي‬‫ث‬ ‫شيكارى‬
•
‫َب‬‫ي‬‫كت‬ ‫ناوى‬
:
(
3D Structural Analysis
)
‫دووريدا‬ َ‫ى‬‫س‬ ‫لة‬ ‫َكهاتةكان‬‫ي‬‫ث‬ ‫شيكارى‬
•
: ‫ئامادةكردنى‬
/ ‫ئةندازيار‬
‫َخة‬‫ي‬‫ش‬ ‫حممد‬ ‫على‬
•
: ‫َداضوونةوةى‬‫ي‬‫ث‬
‫سعيد‬ ‫حممد‬ ‫الدين‬ ‫جنم‬.‫د‬
•
‫ابوبكر‬ ‫امساعيل‬ ‫بيالل‬ : ‫بةرط‬ ‫نةخشةسازى‬
•
: ‫َكارى‬‫ل‬َ‫ي‬‫ه‬ ‫و‬ ‫تايث‬
‫ئامادةكار‬
•
‫يةكةم‬ : ‫ضاث‬

3. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
2
‫َزانني‬
‫ي‬‫ث‬ ‫و‬ ‫سوثاس‬
❖
‫لة‬ ‫كردم‬ ‫هاوكاريان‬ ‫كة‬ ‫كةسانةى‬ ‫ئةو‬ ‫هةموو‬ ‫بؤ‬ ‫سوثاس‬
.‫بابةتة‬ ‫ئةم‬ ‫ئامادةكردنى‬
❖
.‫َت‬
‫ي‬‫ب‬ ‫ووشةش‬ ‫يةك‬ ‫بة‬ ‫ئةطةر‬ ‫َبةخشيوم‬
‫ي‬‫ث‬ ‫زانياريان‬ ‫كة‬ ‫ئةكةسانةى‬ ‫هةموو‬ ‫بؤ‬ ‫سوثاس‬
❖
‫بؤ‬ ‫سوثاس‬
.‫خؤى‬ ‫ئةستؤى‬ ‫طرتة‬ ‫َداضونةوةى‬
‫ي‬‫ث‬ ‫ئةركى‬ ‫كة‬ )‫سعيد‬ ‫حممد‬ ‫الدين‬ ‫جنم‬ .‫(د‬

4. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
3
‫َشكةشكردن‬
‫ي‬‫ث‬
❖
‫َة‬
‫ل‬‫مندا‬ ‫و‬ ‫باوكم‬ ‫و‬ ‫دايك‬ ‫بة‬ ‫َشكةشة‬
‫ي‬‫ث‬
‫خؤشةويستةكامن‬
) ‫منصور‬ ‫و‬ ‫ميسور‬ (
.
❖
‫َشكةشة‬
‫ي‬‫ث‬
َ
‫ي‬‫دةخيو‬ ‫كة‬ ‫كةسانةى‬ ‫ئةم‬ ‫هةموو‬ ‫بة‬
‫ن‬
‫ن‬
‫ةوة‬
‫َوةردةطرن‬
‫ي‬‫ل‬ ‫سودى‬ ‫و‬
.
❖
‫الوانى‬ ‫و‬ ‫طةنج‬ ‫بة‬ ‫َشكةشة‬
‫ي‬‫ث‬
‫كوردستان‬
.
❖
.‫َتةكةيانن‬
‫ال‬‫و‬ ‫و‬ ‫ميللـةت‬ ‫َسؤزى‬
‫ل‬‫د‬ ‫كة‬ ‫ئةوكةسانةى‬ ‫هةموو‬ ‫بة‬ ‫َشكةشة‬
‫ي‬‫ث‬
❖
‫بؤ‬ ‫دةدةن‬ َ
‫ل‬‫هةو‬ ‫كة‬ ‫كةسانةى‬ ‫ئةو‬ ‫هةموو‬ ‫بة‬ ‫َشكةشة‬
‫ي‬‫ث‬
.‫زانست‬ ‫َربوونى‬
‫ي‬‫ف‬
❖
.‫برادةران‬ ‫و‬ ‫دؤست‬ ‫و‬ ‫هاوريان‬ ‫سةرجةم‬ ‫بة‬ ‫َشكةشة‬
‫ي‬‫ث‬
❖
.‫حاجياوا‬ ‫شارةوانى‬ ‫سةرؤكايةتى‬ ‫لة‬ ‫َيامن‬
‫ي‬‫هاور‬ ‫سةرجةم‬ ‫بة‬ ‫َشكةشة‬
‫ي‬‫ث‬

5. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
4
Contents: Page
1. Preface )‫َشةكى‬‫ي‬‫(ث‬ …………………………………………………………………………………………………………………........ 5
2. Symbols and Abbreviations (
‫َما‬‫ي‬‫ه‬
‫كو‬
‫ر‬
‫تكراوةكان‬
) ……………………………………………………………………………… 7
3. Matrix (Some Information About Matrix) (
‫ِيزكراوةكان‬‫ر‬
) ……………………………………………………………….… 8
4. Drawing, Supports and Review Some Rules (
َ‫ل‬َ‫ي‬‫ه‬
,‫كارى‬
....‫ياساكاندا‬ ‫بة‬ ‫َداضونةوةيةك‬‫ي‬‫ث‬ ‫و‬ ‫ِاطرةكان‬‫ر‬
) ………………….……12
5. Stiffness Method (Matrix Structural Analysis) (
‫ِةقى‬‫ر‬ ‫َطاى‬‫ي‬ِ‫ر‬
) ……………………………………………………….….17
6. Example1 …………………………………………………………………………………………………………………………………18
7. Direct Stiffness Method (
‫ِةقى‬‫ر‬ ‫ِاستةوخؤى‬‫ر‬ ‫َطاى‬‫ي‬ِ‫ر‬
) ………………..………………‫يةكةم‬ ‫بةشى‬ …………………….……..……19
8. 2D analysis ……………………………………………………………………………………………………………………….………19
9. 2D Beams Analysis by Direct Stiffness Method ………………………………………………………………..…….…18
10. Example2 ……………………………………………………………………………………………………………………………….…22
11. 2D Frames Analysis by Direct Stiffness Method ………………………………………………………………….…….28
12. Example3 …………………………………………………………………………………………………………………………………30
13. 2D Trusses Analysis by Direct Stiffness Method ……………………………………………………………………….38
14. Example4 ………………………………………………………………………………………………………………………………...39
15. 3D analysis ………………………………………………………………………………………………………………………….……43
16. 3D Beams Analysis by Direct Stiffness Method …………………………………………………………………….…..43
17. Example5 …………………………………………………………………………………………………………………………………47
18. 3D Frames Analysis by Direct Stiffness Method ……………………………………………………………………….54
19. Example6 ………………………………………………………………………………………………………………………………...54
20. 3D Trusses Analysis by Direct Stiffness Method ……………………………………………………………….…..….82
21. Example7 ………………………………………………………………………………………………………………………………….84
22. Example8 ………………………………………………………………………………………………………………………………….87
23. Problems ………………………………………………………………………………………………………………………………….98
24. Assembly Stiffness Method (
‫ِةقى‬‫ر‬ ‫طشتى‬ ‫َطاى‬‫ي‬ِ‫ر‬
) …………….……..........…. ‫دوو‬ ‫بةشى‬
‫ة‬
‫م‬ ……………………….……….99
25. Symbols and Abbreviations (
‫َما‬‫ي‬‫ه‬
‫كورتكراوةكان‬
) …………………………………………………………………………… 101
26. ‫شيكاركردن‬ ‫هةنطاوةكانى‬ …………………………………………………………………………………………………………..…………….111
27. Example1 ……………………………………………………………………………………………………………………..…………112
28. Example2 ……………………………………………………………………………………………………………………..…………123
29. Example3 ……………………………………………………………………………………………………………………..…………139
30. Problem ………………………………………………………………………………………………………………………………….147
31. References (
‫سةرضاوةكان‬
) …………………………………………………………………………………………….…………..…148

6. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
5
‫ث‬
‫ـ‬
َ‫ي‬
‫ـ‬
‫ش‬
‫ـ‬
‫ةك‬
‫ـ‬
‫ى‬
( ‫َكهاتةكان‬‫ي‬‫ث‬ ‫شيكارى‬
Structural analysis
)
‫و‬ ‫َز‬‫ي‬‫ه‬ ‫ئةو‬ ‫هةموو‬ ‫دؤزينةوةى‬ ‫بؤ‬ ‫َتةوة‬‫ي‬‫ئةبين‬ ‫خؤى‬ ‫سةرةكى‬ ‫َكى‬‫ي‬‫بابةت‬ ‫وةك‬
‫َزو‬‫ي‬‫ه‬ ‫بةهؤى‬ ‫َت‬‫ي‬‫دةب‬ ‫دروست‬ ‫َكهاتةكاندا‬‫ي‬‫ث‬ ‫ناو‬ ‫لة‬ ‫كة‬ ‫زةبرانةى‬
‫بة‬ ‫كة‬ ‫دةرةكيةوة‬ ‫زةبرى‬
‫رةنطا‬
‫ر‬
‫وايكردووة‬ ‫ئةمةش‬ ,‫َتةوة‬‫ي‬‫دةب‬ ‫َكهاتةكان‬‫ي‬‫ث‬ ‫ى‬
َ‫ي‬‫ئةجنامدةدر‬ ‫َدا‬‫ي‬‫ت‬ ‫َينةوةى‬‫ل‬‫َكؤ‬‫ي‬‫ل‬ ‫سةرةكى‬ ‫َكى‬‫ي‬‫بابةت‬ ‫وةك‬ ‫كة‬
‫بةرفراوانى‬ .‫َكهاتةكان‬‫ي‬‫ث‬ ‫جؤرى‬ ‫َوازو‬‫ي‬‫ش‬ ‫َبذاردنى‬‫ل‬‫هة‬ ‫بؤ‬ ‫َت‬‫ي‬‫َنر‬‫ي‬‫دةه‬ ‫بةكار‬ ‫و‬ ‫ت‬
‫شيكار‬ ‫بؤ‬ ‫َت‬‫ي‬‫هةب‬ ‫جياوازى‬ ‫َطاى‬‫ي‬ِ‫ر‬ ‫ضةندةها‬ ‫كة‬ ‫وايكردووة‬ ‫بابةتة‬ ‫ئةم‬
‫كردن‬
‫ئاسان‬ ‫َكى‬‫ي‬‫كار‬ ‫دةست‬ ‫بة‬ ‫َؤزةكان‬‫ل‬‫ئا‬ ‫َكهاتة‬‫ي‬‫ث‬ ‫شيكارى‬ ‫كة‬ ‫ديارة‬ ,
‫بؤ‬ ‫َرةكان‬‫ي‬‫سؤفتو‬ ‫ِؤدا‬‫ر‬‫ئةم‬ ‫ِؤذطارى‬‫ر‬ ‫لة‬ ‫ئةبينني‬ ,‫نية‬
‫بابةتة‬ ‫ئةم‬
‫كةم‬ ‫بةاليةنى‬ ‫ئةندازيار‬ ‫َم‬‫ال‬‫بة‬ ,‫طرتووةتةوة‬ ‫دةستيان‬ ‫َطاى‬‫ي‬‫ج‬
‫َويستة‬‫ي‬‫ث‬ ‫ةوة‬
‫َكهاتةكان‬‫ي‬‫ث‬ ‫شيكارى‬
‫باشرت‬ ‫و‬ ‫َت‬‫ي‬‫َر‬‫ي‬‫َبذ‬‫ل‬‫هة‬ ‫َكهاتةكان‬‫ي‬‫ث‬ ‫جؤرى‬ ‫َوازو‬‫ي‬‫ش‬ ‫َت‬‫ي‬‫بتوان‬ ‫سةركةووتوانة‬ ‫َكى‬‫ي‬‫َواز‬‫ي‬‫ش‬ ‫بة‬ ‫ئةوةى‬ ‫بؤ‬ ‫َت‬‫ي‬‫بزان‬ ‫دةست‬ ‫بة‬
.‫َت‬‫ي‬‫َن‬‫ي‬‫به‬ ‫بةكار‬ ‫َرةكان‬‫ي‬‫سؤفتو‬
‫هة‬ ‫َمة‬‫ي‬‫ئ‬ ‫ئةمة‬ ‫لةبةر‬
‫بؤ‬ ‫َن‬‫ي‬‫َنر‬‫ي‬‫كاردةه‬ ‫بة‬ ‫كة‬ ‫َطايانةى‬‫ي‬‫ر‬ ‫ئةم‬ ‫هةموو‬ ‫ناو‬ ‫لة‬ ‫َشكةوتوو‬‫ي‬‫ث‬ ‫َطايةكى‬‫ي‬‫ر‬ ‫َبذاردنى‬‫ل‬‫هة‬ ‫بة‬ ‫ستاوين‬
‫شيكار‬
‫كردن‬
,
‫ِوونكردونةتةوة‬‫ر‬ ‫هةنطاوةكامنان‬ ‫كوردى‬ ‫شريينى‬ ‫زمانى‬ ‫بة‬
‫جؤ‬ ‫طشت‬ َ‫ل‬‫لةطة‬ ‫َت‬‫ي‬‫بطوجن‬ ‫كة‬
‫ر‬
( ‫ةكانى‬
Beams, Frames, Trusses
,)
‫بؤ‬
‫لة‬ ‫ئاسانرت‬ ‫َوةيةكى‬‫ي‬‫ش‬ ‫بة‬ ‫َنةر‬‫ي‬‫خو‬ ‫ئةوةى‬
َ‫ي‬‫ت‬ ‫بابةتةكة‬
‫بطات‬
‫ِةقى‬‫ر‬ ‫بةهؤى‬ ‫َكهاتةكان‬‫ي‬‫ث‬ ‫شيكاركردنى‬ ‫َطاى‬‫ي‬ِ‫ر‬ ‫لة‬ ‫بريتية‬ ‫ئةميش‬ ‫كة‬ ,
( ‫َكهاتةكةوة‬‫ي‬‫ث‬
(
Matrix Structural analysis
)
Stiffness Method
)
.
‫ئةندا‬ ‫ِةقى‬‫ر‬ ‫ثةيوةندى‬ ‫َطاية‬‫ي‬ِ‫ر‬ ‫ئةم‬
‫بةكار‬ ‫مةكان‬
‫َت‬‫ي‬‫َن‬‫ي‬‫دةه‬
,‫َكهاتةكة‬‫ي‬‫ث‬ ‫َةى‬‫ل‬‫جو‬ ‫و‬ ‫ئةندامةكان‬ ‫َزى‬‫ي‬‫ه‬ ‫دؤزينةوةى‬ ‫بؤ‬
‫شيكا‬ ‫دةتوانني‬ ‫كة‬
‫ر‬
‫جؤ‬ ‫طشت‬ ‫ى‬
‫ر‬
‫بة‬ ‫بةامبةر‬ ‫بدةين‬ ‫ئةجنام‬ ‫َكاتةكان‬‫ي‬‫ث‬ ‫ى‬
‫َطريةكان‬‫ي‬‫ج‬ ‫زةبرة‬ ‫و‬ ‫َز‬‫ي‬‫ه‬
‫دوو‬ ‫لة‬
‫دوورى‬
(
‫ِووتةخت‬‫ر‬
)
‫دووريدا‬ َ‫ي‬‫س‬ ‫و‬
)‫(بؤشايى‬
,
‫بةكارى‬ ‫َرةكان‬‫ي‬‫سؤفتو‬ ‫زؤربةى‬ ‫كة‬ ‫َطايةشة‬‫ي‬ِ‫ر‬ ‫ئةم‬ ‫هةمان‬ ‫وة‬
‫شي‬ ‫بؤ‬ ‫َنن‬‫ي‬‫دةه‬
‫َؤزةكان‬‫ل‬‫ئا‬ ‫َكهاتة‬‫ي‬‫ث‬ ‫كارى‬
.
‫لةم‬
‫بةشةدا‬
)‫يةكةم‬ ‫(بةشى‬
‫ِاستةوخؤ‬‫ر‬ ‫ِيطاى‬‫ر‬ ‫َمة‬‫ي‬‫ئ‬
‫ى‬
(
Direct Stiffness Method
)
‫َنني‬‫ي‬‫دةه‬ ‫بةكار‬
,
‫ياسا‬ ‫َطاية‬‫ي‬ِ‫ر‬ ‫ئةم‬ ‫كة‬
‫َكهاتةكان‬‫ي‬‫ث‬ ‫شيكارى‬ ‫بؤ‬ ‫َت‬‫ي‬‫َن‬‫ي‬‫بةكاردةه‬ ‫سةرةكيةكة‬
‫تيايدا‬ ‫ِانكاريةك‬‫ر‬‫طؤ‬ ‫هيض‬ َ‫ي‬‫بةب‬
.
‫بة‬ ‫و‬ ‫طشتطريدا‬ ‫منوونةيةكى‬ ‫ضةند‬ ‫لة‬ ‫وة‬
‫ضةند‬
‫دياريك‬‫َكى‬‫ي‬‫هةنطاو‬
‫ر‬
‫َرةكانى‬‫ي‬‫سؤفتو‬ ‫شيكارى‬‫ئةجنامى‬ ‫بة‬‫كردووة‬‫بةراورد‬‫منونةكانيشمان‬ ‫شيكارى‬‫ئةجنامى‬ ‫و‬ ‫شيكردؤتةوة‬‫بابةتةكةمان‬‫او‬
( ‫وةك‬
STAAD. Pro, SAP 2000, ETABS
)
.
‫ئةندازي‬
‫ـ‬
‫عل‬ /‫ار‬
‫ـ‬
‫حم‬ ‫ى‬
‫ـ‬
‫ش‬ ‫مد‬
‫ـ‬
‫َخة‬‫ي‬
07502454161

7. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
6
: ‫زاراوةكان‬ ‫لة‬ َ‫ي‬‫هةند‬ ‫َناسةي‬‫ي‬‫ث‬
❖
‫َكهاتةكان‬‫ي‬‫ث‬
(
Structures
)
‫بريتية‬ :
......‫و‬ ‫تاوةرةكان‬ ,‫ثرد‬ ,‫(بينا‬ ‫وةك‬ ‫وةزنةكان‬ ‫َطرتنى‬‫ل‬‫هة‬ ‫بؤ‬ ‫جياواز‬ ‫َوازى‬‫ي‬‫ش‬ ‫بة‬ ‫لكاو‬ ‫بةيةكةوة‬ ‫َكى‬‫ي‬‫ئةندام‬ ‫َة‬‫ل‬‫كؤمة‬ ‫لة‬
❖
َ‫ل‬‫ِاية‬‫ر‬
(
Beam
)
.‫زةبر‬ ‫يان‬ ‫ستونى‬ ‫وةزنى‬ ‫َطرتنى‬‫ل‬‫هة‬ ‫بؤ‬ ‫َت‬‫ي‬‫َنر‬‫ي‬‫دةه‬ ‫بةكار‬ ‫ئاسؤيية‬ ‫َكى‬‫ي‬‫ئةندام‬ ‫طشتى‬ ‫َوةيةكة‬‫ي‬‫ش‬ ‫بة‬ :
❖
‫ثاية‬
(
Column
)
‫طش‬ ‫َوةيةكة‬‫ي‬‫ش‬ ‫بة‬ :
.‫زةبرةكان‬ ‫يان‬ ‫ستونى‬ ‫وةزنى‬ ‫َطرتنى‬‫ل‬‫هة‬ ‫بؤ‬ ‫َت‬‫ي‬‫َنر‬‫ي‬‫دةه‬ ‫بةكار‬ ‫ستونية‬ ‫َكى‬‫ي‬‫ئةندام‬ ‫تى‬
❖
(‫طورزةكان‬ ‫َى‬‫ل‬‫توو‬
Trusses Bar
: )
‫دةب‬ ‫َك‬‫ي‬‫َز‬‫ي‬‫ه‬ ‫بةرةنطاري‬ ‫َت‬‫ي‬‫الرب‬ ‫يان‬ ‫ستوونى‬ , ‫ئاسؤى‬ ‫َت‬‫ي‬‫دةطوجن‬ ‫كة‬ ‫طورزةية‬ ‫َكى‬‫ي‬‫ئةندام‬ ‫طشتى‬ ‫َوةيةكة‬‫ي‬‫ش‬ ‫بة‬
‫َت‬‫ي‬
‫بة‬ ‫كة‬ ‫ةوة‬
‫َذ‬‫ي‬‫در‬ ‫تةوةرةى‬ ‫ئاراستةى‬
‫يةكةيةتى‬
.
‫َواز‬‫ي‬‫ش‬ ‫ِووى‬‫ر‬‫لة‬ ‫َكهاتةكان‬‫ي‬‫ث‬ ‫جؤرى‬
( ‫َنان‬‫ي‬‫بةكاره‬ ‫و‬
Type of Structures
:)
❖
‫َةكان‬‫ل‬‫ِاية‬‫ر‬
(
Beams
)
‫وةزنةكان‬ ‫َطرتنى‬‫ل‬‫هة‬ ‫بؤ‬ ‫َنراون‬‫ي‬‫لك‬ ‫بةيةكةوة‬ ‫جياواز‬ ‫َوازى‬‫ي‬‫ش‬ ‫و‬ ‫ئاسؤى‬ ‫َوازى‬‫ي‬‫ش‬ ‫بة‬ ‫َك‬‫ي‬َ‫ل‬‫ِاية‬‫ر‬ ‫ضةند‬ ‫يان‬ َ‫ل‬‫ِاية‬‫ر‬ ‫يةك‬ ‫لة‬ ‫بريتية‬ :
.
❖
‫ثةيكةرةكان‬
(
Frames
)
‫لةو‬ ‫بريتية‬ :
.‫َت‬‫ي‬‫بةكارد‬ ‫بينادا‬ ‫درووستكردنى‬ ‫لة‬ ‫زؤرى‬ ‫بة‬ ‫كة‬ ‫َت‬‫ي‬‫دةب‬ ‫دروست‬ ‫ثاية‬ ‫و‬ َ‫ل‬‫ِاية‬‫ر‬ ‫َك‬‫ي‬َ‫ل‬‫كؤمة‬ ‫يةكطرتنى‬ ‫لة‬ ‫كة‬ ‫ثةيكةرةى‬
❖
‫طورزة‬
‫كان‬
(
Trusses
)
( ‫َوازى‬‫ي‬‫ش‬ ‫بة‬ ‫لكاو‬ ‫بةيةكةوة‬ ‫َكى‬‫ي‬‫ئةندام‬ ‫َة‬‫ل‬‫كؤمة‬ ‫لة‬ ‫بريتية‬ :
Pin
)
،
‫َك‬‫ي‬‫َز‬‫ي‬‫ه‬ ‫بةرةنطاري‬ ‫طشتى‬ ‫َوةيةكى‬‫ي‬‫ش‬ ‫بة‬ ‫ئةندامةكانى‬ ‫طشت‬ ‫وة‬
‫دةبنة‬
.‫ئةندامةكةية‬ ‫َذى‬‫ي‬‫در‬ ‫تةوةرةى‬ ‫ئاراستةى‬ ‫بة‬ ‫كة‬ ‫وة‬
❖
‫وةك‬ ‫هةية‬ ‫َكهاتةمان‬‫ي‬‫ث‬ ‫ترى‬ ‫جؤرى‬ ‫ضةند‬
(
…….Cable, Arches, Surface Structures
)
.
( ‫وةزنةكان‬
Loads
)
‫كة‬ ‫وةزنانةى‬ ‫لةم‬ ‫بريتني‬ :
‫دةكةنة‬ ‫كاريطةرى‬
‫دةبنةوة‬ ‫بةرةنطاريان‬ ‫َكهاتةكان‬‫ي‬‫ث‬ ‫وة‬ ‫َكهاتةكان‬‫ي‬‫ث‬ ‫سةر‬
‫بةهؤى‬ ‫يان‬ ‫َكهاتةكة‬‫ي‬‫ث‬ ‫ئةندامةكانى‬ ‫بةهؤى‬
‫وةك‬ ‫وةزنةكانيش‬ ,‫ئةندامةكان‬ ‫بةيةكالكانى‬ ‫َنى‬‫ي‬‫شو‬
(
Deal Load, Live load, Wind load, Snow load, Earthquake load,………
)
,
‫ِى‬‫ر‬‫ب‬ ‫وة‬
‫َت‬‫ي‬‫دادةنر‬ ‫وةزنة‬ ‫ئةو‬ ‫َرةى‬‫ي‬‫بةطو‬ ‫ئةمانةش‬
‫كة‬
‫َت‬‫ي‬‫دةكر‬ ‫َشبينى‬‫ي‬‫ث‬
‫َكهاتةكة‬‫ي‬‫ث‬ ‫سةر‬ ‫كاربكاتة‬
‫موو‬ ‫كةمينةكانى‬ ‫لة‬ ‫و‬
‫ا‬
‫سةرجةميشيان‬ ‫ِةضاوى‬‫ر‬ ‫و‬ ‫َت‬‫ي‬‫نةب‬ ‫كةمرت‬ ‫سةفات‬
.‫َت‬‫ي‬‫دةكر‬
( ‫شيكاريةوة‬ ‫ِووى‬‫ر‬‫لة‬ ‫َكهاتةكان‬‫ي‬‫ث‬ ‫جؤرى‬
Type of Structures
:)
❖
(
Statically Determinate Structures
‫دةتوانني‬ ‫كة‬ ‫َكهاتانةن‬‫ي‬‫ث‬ ‫ئةو‬ : )
.‫هاوسةنطى‬ ‫َزى‬‫ي‬‫ه‬ ‫ياساكانى‬ ‫َنانى‬‫ي‬‫بةكاره‬ ‫بة‬ ‫بكةين‬ ‫شيكاريان‬
❖
(
Statically Indeterminate Structures
)
.‫هاوسةنطى‬ ‫َزى‬‫ي‬‫ه‬ ‫ياساكانى‬ ‫َنانى‬‫ي‬‫بةكاره‬ ‫بة‬ ‫بكةين‬ ‫شيكاريان‬ ‫ناتوانني‬ ‫كة‬ ‫َكهاتانةن‬‫ي‬‫ث‬ ‫ئةو‬ :
‫َكهاتةكان‬‫ي‬‫ث‬ ‫شيكاركردنى‬ ‫َطاكانى‬‫ي‬ِ‫ر‬ ‫لة‬ ‫َك‬‫ي‬‫هةند‬
‫َطريةكان‬‫ي‬‫ج‬ ‫وةزنة‬ ‫بؤ‬
:
1. Deflections Using Energy Method.
2. Approximate Analysis Method.
3. Deflections.
4. Force Method ( Flexibility Method ) .
5. ........
6. …….
7. Stiffness Method (Matrix Structural Analysis)
(
Degree of Freedom
:)
‫َةكان‬‫ل‬‫خا‬ ‫ِانى‬‫ر‬‫سو‬ ‫و‬ ‫َة‬‫ل‬‫جو‬ ‫لة‬ ‫بريتية‬
(
Translation and Rotation
)
‫بؤ‬ ‫ئةندامةكة‬ ‫تةوةرةى‬ ‫يان‬ ‫طشتى‬ ‫تةوةرةى‬ ‫ئاراستةى‬ ‫بة‬
‫هةية‬ ‫ِانيان‬‫ر‬‫سو‬ ‫و‬ ‫َن‬‫ال‬‫جو‬ ‫ئةطةرى‬ ‫كة‬ ‫َكهاتةكة‬‫ي‬‫ث‬ ‫ِاطرةكانى‬‫ر‬ ‫و‬ َ‫ل‬‫خا‬ ‫هةموو‬
.

8. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
7
σ = Stress ‫فشا‬
‫ر‬
‫كورتكراوةكان‬ ‫َما‬‫ي‬‫ه‬
:
A = Area ‫ِووبةر‬‫ر‬
L= Length ‫َذى‬‫ي‬‫در‬
H= Height ‫بةرزى‬
W= Width ‫ثانى‬
D or DOF = Deformation ro Degree Of Freedom(Rotation & Displacement) ‫سةربةست‬ ‫ِانى‬‫ر‬‫سو‬ ‫و‬ ‫َن‬‫ال‬‫جو‬ ‫ِادةى‬‫ر‬
K= Stiffness Matrix ‫ِيزكراوة‬‫ر‬ ‫َوةى‬‫ي‬‫ش‬ ‫بة‬ ‫مةواد‬ ‫ِةقى‬‫ر‬
F= Load ‫بارستايى‬
E= Modulus of Elasticity ‫ِي‬‫ري‬‫ج‬ ‫ِادةى‬‫ر‬
G= Shear Modulus (Modulus of Rigidity) ‫ِين‬‫ر‬‫ب‬ ‫و‬ ‫ِان‬‫ر‬‫سو‬ ‫بؤ‬ ‫ِى‬‫ري‬‫ج‬ ‫ِادةى‬‫ر‬
J= Polar moment of inertia
Ix = moment of inertia about( x)
Iy= moment of inertia about (y)
Iz = moment of inertia about (z)
X, Z, Y= Axis Direction
RX, RZ, RY= Rotation About Axis
B. M. D= Bending Moment Diagram
S. F. D= Shear Force Diagram
A. F. D= Axial Force Diagram
MZ = Moment about (z)
My = Moment about (y)
Mx = Torsion (Moment about (x))
SZ = Shear Force about (z)
Sy = Shear Force about (y)
AX = Axial Force
FER = Fixed End Reaction
PER = Pinned End Reaction
C = Compression
T = Tension
JFV= Joint Force Vector
θx, θy, θz=Rotate Angle About Axis
∆𝑥, ∆𝑦, ∆𝑧 = Displacement Direct Axis
θ= Rotate Angle
∆= Displacement

9. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
8
‫ِيزكراوةكان‬‫ر‬
(
Matrix
)
:
‫ِيزكراو‬‫ر‬ ‫ِى‬‫ر‬‫ب‬ ‫َك‬‫ي‬َ‫ل‬‫كؤمة‬ ‫لة‬ ‫بريتني‬
‫ستوون‬ ‫و‬ ‫ِيز‬‫ر‬ ‫َوةى‬‫ي‬‫ش‬ ‫بة‬
.‫ِيزكراون‬‫ر‬
‫تايبةمتةندى‬
‫ِيزكراوةكان‬‫ر‬
(Matrices Property)
:
( ‫ئةطةر‬
A,B
)
. ‫بن‬ ‫ِيزكراوة‬‫ر‬ ‫دوو‬
A ± B = [
2 3
4 5
] ± [
6 −7
8 9
] = [
2 ± 6 3 ± (−7)
4 ± 8 5 ± 9
]
x ∗ A = 𝑥 ∗ [
2 3
4 5
] = [
𝑥 ∗ 2 𝑥 ∗ 3
𝑥 ∗ 4 𝑥 ∗ 5
] = [
2𝑥 3𝑥
4𝑥 5𝑥
] = A*x
A = [
a11 a12 … … a1𝑛
a21 a22 … … a2𝑛
… … … … …
a𝑚1 a𝑚2 … … a𝑚𝑛
]
|A| = 2 ∗ 5 − 4 ∗ 3 = −2
|B| = 6 ∗ 9 − (8 ∗ −7) = 110
𝟐 −𝐀𝐝𝐝𝐢𝐭𝐢𝐨𝐧 𝐚𝐧𝐝 𝐒𝐮𝐛𝐭𝐫𝐚𝐜𝐭𝐢𝐨𝐧 𝐨𝐟 𝐌𝐚𝐭𝐫𝐢𝐜𝐞𝐬
A2∗2 ∗ B2∗2 = [
2 3
4 5
] ∗ [
6 −7
8 9
] = [
2 ∗ 6 + 3 ∗ 8 2 ∗ (−7) + 3 ∗ 9
4 ∗ 6 + 5 ∗ 8 4 ∗ (−7) + 5 ∗ 9
]
2∗2
𝟑 −𝐌𝐮𝐥𝐭𝐢𝐩𝐥𝐢𝐜𝐚𝐭𝐢𝐨𝐧 𝐨𝐟 𝐌𝐚𝐭𝐫𝐢𝐜𝐞𝐬
[6 8] ∗ A = [6 8] ∗ [
2 3
4 5
] = [6 ∗ 2 + 8 ∗ 4 6 ∗ 3 + 8 ∗ 5]=[44 58]
A ∗ [
6
8
] = [
2 3
4 5
] [
6
8
] = [
2 ∗ 6 + 3 ∗ 8
4 ∗ 6 + 5 ∗ 8
]=[
36
64
]
𝟒 −𝐈𝐧𝐯𝐞𝐫𝐬𝐞 𝐨𝐟 𝐌𝐚𝐭𝐫𝐢𝐱
A−1
=
1
|A|
[
5 −3
−4 2
] =
1
−2
∗ [
5 −3
−4 2
] =[
1
−2
∗ 5
1
−2
∗ −3
1
−2
∗ −4
1
−2
∗ 2
]=[
−2.5 1.5
2 −1
]
𝟓 −𝐓𝐫𝐚𝐧𝐬𝐩𝐨𝐬𝐞 𝐨𝐟 𝐌𝐚𝐭𝐢𝐱
AT
= [
2 4
3 5
]
𝟔 −𝐏𝐫𝐨𝐩𝐞𝐫𝐭𝐲 𝐨𝐟 𝐂𝐚𝐥𝐜𝐮𝐥𝐚𝐭𝐢𝐨𝐧
AB ≠ BA
A(BC)=(AB)C
A(B+C)=AB+AC
(A+B)C=AC+BC
(AB)T
=BT
AT
A*A-1
=1 (aij=0 but for each i=j then aij=1)
A2∗2 = [
2 3
4 5
] , B2∗2 = [
6 −7
8 9
]
𝟏 − 𝐃𝐞𝐭𝐞𝐫𝐦𝐢𝐧𝐚𝐧𝐭 𝐨𝐟 𝐌𝐚𝐭𝐫𝐢𝐱
aij = matrix element
Matrix
‫ستونى‬
1
‫ِيزى‬‫ر‬
1

10. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
9
[
2 6
7 9
]
−1
:‫ِيزكراوةكان‬‫ر‬ ‫سودى‬
.‫َيةكان‬‫ل‬َ‫ي‬‫ه‬ ‫َشة‬‫ي‬‫هاوك‬ ‫سيستةمى‬ ‫كردنى‬ ‫شيكار‬ ‫بؤ‬ ‫َنني‬‫ي‬‫دةه‬ ‫بةكار‬ ‫ِيزكراوةكان‬‫ر‬
.‫ِيزكراوةكان‬‫ر‬ ‫َنانى‬‫ي‬‫كاره‬ ‫بة‬ ‫بة‬ ‫بكة‬ ‫شيكار‬ ‫خووارةوة‬ ‫َشةى‬‫ي‬‫هاوك‬ ‫دوو‬ ‫ئةم‬ :1‫منونة‬
/ ‫شيكار‬
-1
‫ِيزكراوة‬‫ر‬ ‫َوةةى‬‫ي‬‫ش‬ ‫بؤ‬ ‫ِين‬‫ر‬‫دةطؤ‬ ‫َشةكة‬‫ي‬‫هاوك‬ ‫هةردوو‬
:‫خوارةوة‬ ‫َوةى‬‫ي‬‫ش‬ ‫بةم‬
‫هةردوو‬
:‫خوارةوة‬ ‫َوةى‬‫ي‬‫ش‬ ‫بةم‬ ‫دةكةين‬ ‫جارانى‬ ‫َشةكة‬‫ي‬‫هاوك‬ ‫الى‬ -2
.‫َشة‬‫ي‬‫هاوك‬ ‫سيستةمى‬ ‫بؤ‬ ‫ِة‬‫ر‬‫بطؤ‬ ‫ِيزكراوة‬‫ر‬ ‫ئةم‬ :2‫منونة‬
/ ‫شيكار‬
-1
.‫دةدةين‬ ‫ئةجنام‬ ‫ِاست‬‫ر‬ ‫الى‬ ‫بؤ‬ ‫ِيزكراوة‬‫ر‬ ‫جارانكردنى‬ ‫كردارى‬
2𝑥 + 6𝑦 = 62 1
7𝑥 + 9𝑦 = 109 2
[
2 6
7 9
] [
𝑥
𝑦] = [
62
109
]
[
2 6
7 9
] [
2 6
7 9
]
−1
[
𝑥
𝑦] = [
2 6
7 9
]
−1
[
62
109
]
∴ [
𝑥
𝑦] =
1
−24
∗ [
9 −6
−7 2
] [
62
109
]
∴ [
𝑥
𝑦] =
1
−24
∗ [
9 ∗ 62 + (−6) ∗ 109
(−7) ∗ 62 + 2 ∗ 109
] =
1
−24
[
−96
−216
]=[
4
9
] ∴ x = 4 , y = 9
[
F1
F2
] =[
2 6
7 9
] [
𝐷1
𝐷2
]
[
F1
F2
] =[
2 ∗ 𝐷1 + 6 ∗ 𝐷2
7 ∗ 𝐷1 + 9 ∗ 𝐷2
]
∴ F1 =2𝐷1 + 6𝐷2
∴ F2 =7𝐷1 + 9𝐷2

11. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
10
and Inverse Matrix:
nt
General Determina
1 -Minors?
2 -Cofactors?
:
nt
General Determina
or
General Inverse:
A = [
a11 a12 a13
a21 a22 a23
a31 a13 a33
]
𝑚11 = [ 𝐚𝟐𝟐 𝐚𝟐𝟑
𝐚𝟏𝟑 𝐚𝟑𝟑
] = [
a22 a23
a13 a33
]
C𝑖𝑗 = (-1)i+j
*mij
𝑚12 = [𝐚𝟐𝟏 𝐚𝟐𝟑
𝐚𝟑𝟏 𝐚𝟑𝟑
] = [
a21 a23
a31 a33
]
𝐶11 = (−1)2
∗ [ 𝐚𝟐𝟐 𝐚𝟐𝟑
𝐚𝟏𝟑 𝐚𝟑𝟑
] = [
a22 a23
a13 a33
] 𝐶12 = (−1)3
∗ [𝐚𝟐𝟏 𝐚𝟐𝟑
𝐚𝟑𝟏 𝐚𝟑𝟑
] = −1 ∗ [
a21 a23
a31 a33
]
|A| = ∑ 𝑎𝑖𝑗𝑐𝑖𝑗
𝑛
𝑖=1
For j= 1,…………,n-1 or n
|A| = ∑ 𝑎𝑖𝑗𝑐𝑖𝑗
𝑛
𝑗=1
For i= 1,………,n-1 or n
A−1
=
𝐶𝑇
|𝐴|
, 𝐶𝑇
= 𝑇𝑟𝑎𝑛𝑠𝑝𝑜𝑠𝑒 𝐶𝑜𝑓𝑎𝑐𝑡𝑒𝑟 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 .
aij = matrix element

12. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
11
( |A| =? , A−1
=? ) ‫منونة‬
3
:
‫بؤ‬
‫ِيزكراوة‬‫ر‬ ‫ئةم‬
‫بكة‬ ‫ديارى‬ ‫دذةكةى‬ ‫و‬ ‫ئةجنام‬
.
/ ‫شيكار‬
1 -Minors?
2 -Cofactors?
Minor for each element=?
A = [
2 −3 4
0 6 5
8 1 −7
]
𝑚11 = [
2 −3 4
0 6 5
8 1 −7
] = [
6 5
1 −7
] 𝑚12 = [
2 −3 4
0 6 5
8 1 −7
] = [
0 5
8 −7
] 𝑚13 = [
2 −3 4
0 6 5
8 1 −7
] = [
0 6
8 1
]
𝐶11 = [
6 5
1 −7
]= -42-5=-47 𝐶12 = −1 ∗ [
0 5
8 −7
] =-1*(0-40) = 40 𝐶13 = [
0 6
8 1
] =0-48 =-48
∴ |A| =2*-47+(-3) *40+4*-48 = -406
m =
[
[
6 5
1 −7
] [
0 5
8 −7
] [
0 6
8 1
]
[
−3 4
1 −7
] [
2 4
8 −7
] [
2 −3
8 1
]
[
−3 4
6 5
] [
2 4
0 5
] [
2 −3
0 6
]]
= [
−47 −40 −48
17 −46 26
−39 10 12
]
C𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟 = [
−47 40 −48
−17 −46 −26
−39 −10 12
]
C𝑇
𝑇𝑟𝑎𝑛𝑠𝑝𝑜𝑠𝑒 𝑜𝑓 𝐶𝑜𝑓𝑎𝑐𝑡𝑜𝑟 = [
−47 −17 −39
40 −46 −10
−48 −26 12
]
∴ A−1
=
𝐶𝑇
|𝐴|
=
−1
406
[
−47 −17 −39
40 −46 −10
−48 −26 12
]

13. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
12
_ +
+
+
3D Drawing Diagrams of Forces and Moments:
2D Drawing Diagrams of Forces and Moments:
B.M.D(z) B.M.D(y)
B.M.D(x) S.F.D(y)
S.F.D(z) A.F.D(x)
.‫َشني‬‫ي‬‫دةك‬ ‫َنة‬‫ي‬‫و‬ ‫خوارةوة‬ ‫َوةى‬‫ي‬‫ش‬ ‫بةم‬ ‫َكارى‬‫ل‬َ‫ي‬‫ه‬
‫زةبرةكان‬ ‫َزو‬‫ي‬‫ه‬
B.M.D S.F.D

14. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
13
:‫َطرةكان‬‫ل‬‫هة‬ ‫كاردانةوةى‬ ‫ذمارةى‬ ‫و‬ ‫َكارى‬‫ل‬َ‫ي‬‫ه‬ ‫و‬ ‫جؤر‬ ‫بة‬ ‫تايبةت‬ ‫خشتةى‬
‫َداضونةوةيةك‬‫ي‬‫ث‬
‫بؤ‬
‫َزةكان‬‫ي‬‫ه‬ ‫بوونى‬ ‫دابةش‬ ‫و‬ ‫ئةندازةيةكان‬ ‫َوة‬‫ي‬‫ش‬
:
Supports
‫كان‬ ‫َطرة‬‫ل‬‫هة‬ ‫جؤرى‬
‫َنةكةى‬‫ي‬‫و‬
Number of
Reactions
‫كاردانةوةكان‬ ‫ذمارةى‬ ‫َبينى‬‫ي‬‫ت‬
2D 3D
Fix 3 6
( ‫لة‬ ‫كاردانةوةكانى‬ ‫ذمارةى‬
2D
‫يةك‬ ‫َزو‬‫ي‬‫ه‬ ‫دوو‬ )
‫زةبرة‬
( ‫لة‬ ‫َم‬‫ال‬‫بة‬
3D
‫زةبرة‬ َ‫ى‬‫س‬ ‫َزو‬‫ي‬‫ه‬ َ‫ى‬‫س‬ )
.
Pin(Hinge)
(ball &Socket)
2 3 ‫ِاطر‬‫ر‬ ‫وةك‬ ‫سفرة‬ ‫تيايدا‬ ‫زةبر‬
.
Roller 1 1
‫ِاطرة‬‫ر‬ ‫جؤة‬ ‫ئةم‬
‫َرةكاندا‬‫ي‬‫سؤفتو‬ ‫لة‬
‫كة‬ ‫هةية‬ ‫جؤرى‬ ‫زؤرترين‬
‫َت‬‫ي‬ِ‫ر‬‫دةطؤ‬ ‫جؤرةكةى‬ ‫َرةى‬‫ي‬‫طو‬ ‫بة‬ ‫كاردانةوةكانى‬ ‫ذمارةى‬
.
Cable 1 1 ‫َةكةية‬‫ل‬‫َب‬‫ي‬‫ك‬ ‫ئاراستةى‬ ‫بة‬ ‫كاردانةوةكةى‬
.
Internal Hinge 2 3 ‫سفرة‬ ‫تيايدا‬ ‫زةبر‬
.
0,0,9
FAH
FAF
H
G
F
E
D
C
B
A
3,8,0
0,8,0
3,0,9
3,0,0
Z
Y
X
0,0,0
0,8,9
3,8,9
(X,Y,Z)
LAH = √(XH − XA)2 + (YH − YA)2 + (ZH − ZA)2
2
‫دوورى‬ َ‫ى‬‫س‬ ‫بؤ‬
LAF = √(XF − XA)2 + (ZF − ZA)2
2
‫تةوةرةكان‬ ‫َرةى‬‫ي‬‫طو‬ ‫بة‬ ‫دوورى‬ ‫دوو‬ ‫بؤ‬
FX(AH) = FAH ∗
XH − XA
LAH
FY(AH) = FAH ∗
YH − YA
LAH
FZ(AH) = FAH ∗
ZH − ZA
LAH
FX(AF) = FAF ∗
XF − XA
LAF
FZ(AF) = FAF ∗
ZF − ZA
LAF
cos 𝜃𝑋(𝐴𝐻) =
XH − XA
LAH
cos 𝜃𝑌(𝐴𝐻) =
YH − YA
LAH
cos 𝜃𝑍(𝐴𝐻) =
ZH − ZA
LAH
cos 𝜃𝑋(𝐴𝐹) =
XF − XA
LAF
cos 𝜃𝑍(𝐴𝐹) =
ZF − ZA
LAF
θ = 𝐴𝑛𝑔𝑙𝑒 𝐵𝑒𝑡𝑤𝑒𝑒𝑛 𝑙𝑖𝑛𝑒 𝑡𝑜 𝑎𝑥𝑖𝑠

15. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
14
1
2
4
3
5
6
7
‫َداضونة‬‫ي‬‫ث‬
‫ياساكان‬ ‫بؤ‬ ‫وةيةك‬
:
‫لة‬ ‫بريتني‬ ‫سورن‬ ‫ِةنطى‬‫ر‬ ‫بة‬ ‫ياساكاندا‬ ‫لة‬ ‫كة‬ ‫بةشةى‬ ‫ئةم‬
(
K
)
.‫َوازةكةى‬‫ي‬‫ش‬ ‫بؤ‬ ‫ئةندامةكة‬ ‫ى‬
T =
GJ
L
θ
T =
GJ
L
θ
δ𝑆 = ρθ
γ =
𝛿𝑆
L
=
𝜌𝜃
𝐿
τ =
𝑇𝜌
J
T =
GJ
L
θ
G =
𝜏
γ
T = Torque
𝑀𝐴𝐵 =
4𝐸𝐼𝜃
𝐿 𝑀𝐵𝐴 =
2𝐸𝐼𝜃
𝐿
L
A B
θ
𝑆𝐴𝐵 =
6𝐸𝐼𝜃
𝐿2
𝑆𝐵𝐴 =
6𝐸𝐼𝜃
𝐿2
𝑀𝐴𝐵 =
4𝐸𝐼
𝐿
𝜃
𝑀𝐵𝐴 =
2𝐸𝐼
𝐿
𝜃
𝑆𝐴𝐵 =
6𝐸𝐼
𝐿2
𝜃
𝑆𝐵𝐴 = −
6𝐸𝐼
𝐿2
𝜃
𝑀𝐵𝐴 =
6𝐸𝐼
𝐿2
∆
𝑀𝐴𝐵 =
6𝐸𝐼
𝐿2
∆
𝑆𝐴𝐵 =
12𝐸𝐼
𝐿3
∆
𝑆𝐵𝐴 = −
12𝐸𝐼
𝐿3
∆
L
∆
𝑀𝐵𝐴 =
6𝐸𝐼∆
𝐿2
𝑀𝐴𝐵 =
6𝐸𝐼∆
𝐿2
A B
𝑆𝐴𝐵 =
12𝐸𝐼∆
𝐿3
𝑆𝐵𝐴 =
12𝐸𝐼∆
𝐿3
∆
𝐹 =
𝐸𝐴∆
𝐿
A B
𝐹 =
𝐸𝐴∆
𝐿
σ =
F
A
ϵ =
∆
L
E =
𝜎
ϵ
F =
EA
L
∆
ϵ =Unit Tensile or Compressive strain (‫َفشار‬‫ي‬‫)ج‬
A B
L
L
𝐀𝐱𝐢𝐚𝐥 𝐅𝐨𝐫𝐜𝐞
𝐓𝐨𝐫𝐬𝐢𝐨𝐧
𝐑𝐨𝐭𝐚𝐭𝐢𝐨𝐧
𝐃𝐞𝐟𝐨𝐫𝐦𝐚𝐭𝐢𝐨𝐧

16. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
15
W
W
Y
Z
Y
Z
Y
Z
D
H
Square Member
Rectangular member
Circular member
‫بة‬ ‫تايبةت‬ ‫خشتةى‬
‫يةكة‬
‫َماى‬‫ي‬‫ه‬ ‫و‬
‫بةكارهاتوو‬
:
‫بة‬ ‫تايبةت‬ ‫خشتةى‬
‫ِطةى‬‫ر‬‫ب‬ ‫تايبةمتةندى‬
:‫ئةندامةكان‬
(
J
)
x
I
y
I
z
I
Area
Member Type
π
32
D4
π
64
D4
π
64
D4
π
4
D2
HW3
(
1
3
− 0.21
𝑊
𝐻
(1 −
𝑊4
12𝐻4))
HW3
12
WH3
12
H ∗ W
2.25
16
W4
W4
12
W4
12
W2
Quantity Unit Symbol Formula
Length Meter L m
Height Meter H m
Width Meter W m
Area Square Meter A m2
Force Kilogram F kg
Stress Kilogram per Square Meter σ kg/ m2
Modulus of elasticity Kilogram per Square Meter E kg/ m2
Shear modulus Kilogram per Square Meter G kg/ m2
Moment of inertia Meter to fourth power I m4
Polar moment of inertia Meter to fourth power J m4
Rotate angle Rad R R
Displacement Meter ∆ m
Bending moment Kilogram. Meter M kg. m
Torsion Kilogram. Meter T kg. m

17. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
16
3D Structural Analysis
‫دووريدا‬ َ‫ى‬‫س‬ ‫لة‬ ‫َكهاتةكان‬‫ي‬‫ث‬ ‫شيكارى‬
Stiffness Method
‫يةكةم‬ ‫بةشى‬
(
Direct Stiffness Method
)

18. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
17
Stiffness Method ( Matrix Structural Analysis )
‫َكة‬‫ي‬‫يةك‬
( ‫كردنى‬ ‫شيكار‬ ‫َطاكانى‬‫ي‬ِ‫ر‬ ‫لة‬
Structure
‫َؤزةكان‬‫ل‬‫ئا‬ ‫َكهاتة‬‫ي‬‫ث‬ ‫شيكارى‬ ‫بؤ‬ ‫َنن‬‫ي‬‫دةه‬ ‫بةكارى‬ ‫َرةكان‬‫ي‬‫سؤفتو‬ ‫كة‬ ‫تايبةتة‬ ‫َكى‬‫ي‬‫شيكار‬ )
(
complex
structures
)
.
‫َطاي‬‫ي‬‫ر‬ ‫ضةند‬ ‫بة‬ ‫َت‬‫ي‬‫دةب‬ ‫ئةميش‬
‫َكهاتةكان‬‫ي‬‫ث‬ ‫دةتوانني‬ ‫كة‬ ‫ةكةوة‬
(‫دوورى‬ ‫دوو‬ ‫لة‬
2D
(‫دووريدا‬ َ‫ى‬‫س‬ ‫و‬ )
3D
‫شيكار‬ َ‫ى‬‫ث‬ )
‫بكةين‬
.
‫ئةم‬
‫ئةندا‬ ‫ِةقى‬‫ر‬ ‫ثةيوةندى‬ ‫َطاية‬‫ي‬ِ‫ر‬
.‫َكهاتةكة‬‫ي‬‫ث‬ ‫َةى‬‫ل‬‫جو‬ ‫و‬ ‫ئةندامةكان‬ ‫َزى‬‫ي‬‫ه‬ ‫دؤزينةوةى‬ ‫بؤ‬ ‫َت‬‫ي‬‫َن‬‫ي‬‫دةه‬ ‫بةكار‬ ‫مةكان‬
‫بةهؤى‬
‫َينوةى‬‫ل‬‫َكؤ‬‫ي‬‫ل‬
‫لة‬ ‫بابةتة‬ ‫ئةم‬
‫َدةطةين‬‫ي‬‫ت‬ ‫َكهاتةكان‬‫ي‬‫ث‬ ‫ِةفتارى‬‫ر‬ ‫ياساو‬
‫َنني‬‫ي‬‫به‬ ‫بةكار‬ ‫َرةكان‬‫ي‬‫و‬ ‫سؤفت‬ ‫دةتوانني‬ ‫ئاسانرت‬ ‫بة‬ ‫وة‬
.
‫د‬ ‫طؤرانكاريةكاندا‬ ‫يان‬ ‫ئةندامةكاندا‬ ‫بةيةكةوةبةستنى‬ ‫لة‬ ‫كار‬ ‫و‬ ‫َن‬‫ي‬‫دةبر‬ ‫َؤز‬‫ل‬‫ئا‬ ‫َكى‬‫ي‬‫شيكار‬ ‫بةرةو‬ ‫َكهاتةكان‬‫ي‬‫ث‬ ‫َطايةدا‬‫ي‬‫ر‬ ‫ئةم‬ ‫شيكارى‬ ‫لة‬
.‫َت‬‫ي‬‫ةكر‬
‫ئاراستةى‬ ‫بة‬ ‫َةكان‬‫ل‬‫خا‬ ‫لة‬ ‫هةركام‬
‫ث‬ ‫ئةندامةكة‬ ‫يان‬ ‫طشتى‬ ‫تةوةرةى‬
.‫هةية‬ ‫ثلةيةيان‬ ‫ئةم‬ ‫ئةطةرى‬ ‫كة‬ ‫َت‬‫ي‬‫دادةنر‬ ‫بؤ‬ ‫سورانيان‬ ‫َةو‬‫ل‬‫جوو‬ ‫ئازادى‬ ‫لةى‬
‫ِى‬‫ر‬‫ب‬ ‫لة‬ ‫بريتية‬ ‫شيكارةكة‬ ‫نةزانراوى‬
‫َة‬‫ل‬‫جو‬
َ‫ي‬‫ث‬ ‫َكهاتةكةتى‬‫ي‬‫ث‬ ‫ئةندامةكانى‬ ‫ناو‬ ‫زةبرةكانى‬ ‫َزو‬‫ي‬‫ه‬ ‫هةموو‬ ‫دةتوانيت‬ ‫كة‬ ,‫َةكاندا‬‫ل‬‫خا‬ ‫لة‬ ‫ِان‬‫ر‬‫سو‬ ‫و‬
.‫بدؤزيتةوة‬
‫بةرةنطار‬ ‫كة‬ ‫َزانة‬‫ي‬‫ه‬ ‫ئةم‬ ‫هةموو‬ ‫وة‬
َ‫ل‬‫خا‬ ‫هؤى‬ ‫بة‬ ‫كة‬ ‫َتةوة‬‫ي‬‫دةب‬ ‫َكهاتةكة‬‫ي‬‫ث‬
‫َت‬‫ي‬‫ب‬ ‫ئةندامةكانةوة‬ ‫يان‬ ‫ئةندامةكان‬ ‫بةستنى‬ ‫بةيةكةوة‬ ‫ةكانى‬
‫َز‬‫ي‬‫ه‬ ‫َتة‬‫ي‬‫بكر‬ ‫َويستة‬‫ي‬‫ث‬
.‫َةكاندا‬‫ل‬‫خا‬ ‫لة‬ ‫َى‬‫ل‬‫خا‬ ‫َكى‬‫ي‬‫زةبر‬ ‫و‬
ِ‫ر‬ ‫لة‬ ‫بريتية‬ ‫ياساكةش‬ ‫هاوكؤلكةى‬ ‫وة‬
‫بة‬ ‫ِانةكان‬‫ر‬‫سو‬ ‫و‬ ‫َان‬‫ل‬‫جو‬ ‫بارى‬ ‫طشت‬ ‫بؤ‬ ‫كة‬ ‫َكهاتةكة‬‫ي‬‫ث‬ ‫ةقى‬
‫بؤ‬ ‫يةك‬ ‫ذمارة‬ ‫دانانى‬
‫َةكان‬‫ل‬‫خا‬ ‫لة‬ ‫ياساالوةكيةكان‬
‫بؤ‬ ‫ِيزكراوة‬‫ر‬ ‫َوةى‬‫ي‬‫ش‬ ‫بة‬ ‫َكهاتةكةش‬‫ي‬‫ث‬ ‫ِةقى‬‫ر‬ ‫وة‬ .‫َكهاتةكة‬‫ي‬‫ث‬ ‫بؤ‬ ‫خةتى‬ ‫َكى‬‫ي‬‫َز‬‫ي‬‫ه‬ ‫َتة‬‫ي‬‫دةب‬ ‫و‬ ‫َتةوة‬‫ي‬‫دةدؤزر‬ ‫دا‬
‫تر‬ ‫ِةكانى‬‫ر‬‫ب‬ ‫موجةبةو‬ ‫ِيزكراوةكة‬‫ر‬ ‫َذى‬‫ي‬‫ل‬ ‫وة‬ ,‫َت‬‫ي‬‫دةكر‬ ‫ِيز‬‫ر‬ ‫َكهاتةكة‬‫ي‬‫ث‬
‫بةكاره‬ ‫و‬ ‫َيةكان‬‫ل‬َ‫ي‬‫ه‬ ‫َشة‬‫ي‬‫هاوك‬ ‫بوونى‬ ‫بؤية‬ ‫هةر‬ ,‫يةكسانن‬ ‫دةوريدا‬ ‫بة‬
‫َنانى‬‫ي‬
.‫ِيطاية‬‫ر‬ ‫ئةم‬ ‫بؤ‬ ‫َك‬‫ي‬‫ناو‬ ‫بؤتة‬ ‫ِيزكراوةكان‬‫ر‬
‫سنوورى‬ ‫لة‬ ‫كة‬ , ‫بكةيت‬ ‫َكهاتةكان‬‫ي‬‫ث‬ ‫جؤرةكانى‬ ‫هةموو‬ ‫شيكارى‬ ‫دةتوانيت‬ ‫َطاية‬‫ي‬ِ‫ر‬ ‫ئةم‬ ‫َنانى‬‫ي‬‫بةكاره‬ ‫بة‬
.‫َكهاتةكانن‬‫ي‬‫ث‬ ‫َطريى‬‫ي‬‫ج‬ ‫َساكانى‬‫ي‬‫ر‬

19. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
18
/ ‫َبينى‬‫ي‬‫ت‬
1
-
( ‫لة‬ ‫هةركام‬
DOF
( ‫بة‬ ‫بكةين‬ ‫كان‬ ‫ة‬ )
1
.‫َت‬‫ي‬‫دةب‬ ‫دروست‬ ‫ماتريكسةكةى‬ ‫ستيفنس‬ ‫ِيزى‬‫ر‬ ‫و‬ ‫َم‬‫ل‬‫كؤ‬ ‫ذمارةى‬ ‫ئةوا‬ ‫بن‬ ‫سفر‬ ‫تر‬ ‫ئةوانى‬ )
2
-
( ‫َوان‬‫ي‬‫ن‬ ‫ثةيوةندى‬
F
( ‫و‬ )
D
( ‫ِانى‬‫ر‬‫نةطؤ‬ ‫بة‬ ‫راستةوانةية‬ )
K
.)
‫يةكةم‬ ‫منوونةى‬
:
‫ِيزكراوةى‬‫ر‬
( ‫َى‬‫ل‬‫خا‬ ‫ئاسؤى‬ ‫َةى‬‫ل‬‫جوو‬
1
‫و‬
2
( ‫َزى‬‫ي‬‫ه‬ ‫هؤى‬ ‫بة‬ ‫بكة‬ ‫ديارى‬ )
2
,F
1
F
( ‫َك‬‫ي‬‫كات‬ )
2
,K
1
K
)
.‫َت‬‫ي‬‫زانراب‬
:‫شيكار‬
‫سثرينطةكان‬ ‫َنى‬‫ال‬‫جو‬ ‫لةبةرئةوةى‬
‫هةية‬ ‫َمان‬‫ل‬‫خا‬ ‫دوو‬ ‫ئاسؤييةو‬ ‫تةنها‬
.
‫َتة‬‫ي‬‫ئةب‬ ‫و‬ ‫َت‬‫ي‬‫دةب‬ ‫دروست‬ ‫بؤ‬ ‫َشةمان‬‫ي‬‫هاوك‬ ‫دوو‬ ‫كةواتا‬
‫ِيزكراوةيةكى‬‫ر‬
.‫دوو‬ ‫بة‬ ‫دوو‬
‫يةكةم‬ ‫هةنطاوى‬
‫(طرميان‬
D1=1, D2=0
)
1
K
1
+ D
2
K
1
+ D
- D1K2
‫هةنطاوى‬
‫دووةم‬
‫(طرميان‬
D1=0, D2=1
)
2
F2
F1
1
K1 K2
D2
D1
K1
F1 F2
K2
D1=1
F1
K1
K1
K2
K2
D2
D1
K1
F1 F2
K2
D1=0
F1
D2=1
K2
K2
Point(1) D1(K1+K2)
Point(2) - D1K2
Point(1)
- D2K2 - D2K2
Point(2) +D2K2 +D2K2
+F1 = D1 (K1+K2) - D2K2
Point(1)
Point(2) +F2 = - D1K2+D2K2
+F1
+F2
+F2
=
K1+K2 - K2
- K2 K2
D1
D2
F=KD D=K-1
F

20. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
19
Set
Typical Beam
θB ΘC
θB θB
Θ1 Θ2
D3
θB
A
A D1
D2
D3
1- Direct Stiffness Method :
‫َك‬‫ي‬‫يةك‬
‫لة‬
ِ‫ر‬
( ‫َطاكانى‬‫ي‬
Stiffness Method
‫لة‬ ‫بريتية‬ )
(
Direct Stiffness Method
)
( ‫َت‬‫ي‬‫دةوتر‬ ‫َشى‬‫ي‬‫ث‬ ‫وة‬
Matrix Stiffness Method
)
( ‫سةرةكيةكةى‬ ‫ياسا‬ ‫َطايةدا‬‫ي‬ِ‫ر‬ ‫لةم‬ ‫كة‬
Stiffness Method
‫َكهاتةك‬‫ي‬‫ث‬ ‫ئةندامى‬ ‫و‬ ‫ِانكاريةك‬‫ر‬‫طؤ‬ ‫هيض‬ َ‫ي‬‫بةب‬ ‫َت‬‫ي‬‫َنر‬‫ي‬‫دةه‬ ‫بةكار‬ )
َ‫ل‬‫لةطة‬ ‫َن‬‫ي‬‫َكدةخر‬‫ي‬ِ‫ر‬ ‫ان‬
‫وة‬ .‫ياساكة‬
‫َؤزةكان‬‫ل‬‫ئا‬ ‫َكهاتة‬‫ي‬‫ث‬ ‫شيكارى‬ ‫بؤ‬ ‫َنن‬‫ي‬‫دةه‬ ‫بةكارى‬ ‫َرةكان‬‫ي‬‫سؤفتو‬ ‫كة‬ ‫تايبةتة‬ ‫َكى‬‫ي‬‫شيكار‬
(
Complex Structures
)
.
‫باوترين‬ ‫َطاية‬‫ي‬ِ‫ر‬ ‫ئةم‬
( ‫َكارى‬‫ي‬‫َبةج‬‫ي‬‫ج‬
finite element method
)
.‫ة‬
: 2D Structure
‫كة‬ ‫َكهاتانةن‬‫ي‬‫ث‬ ‫ئةو‬
‫ِووتةخدان‬‫ر‬ ‫يةك‬ ‫لة‬
‫َت‬‫ي‬‫دةطوجن‬
‫َز‬‫ي‬‫ه‬
‫زةبر‬ ‫و‬
(
Load, Moment
)
‫لة‬
‫ِوتةختدا‬‫ر‬ ‫هةمان‬
‫كاري‬
‫خبةنة‬ ‫طةرى‬
‫سةر‬
.
1- 2D Beams Analysis by Direct Stiffness Method
‫هةنطاوةكانى‬
‫شيكاركردن‬
:
1
-
( Degree of Freedom )
‫ى‬
(
Structure
)
‫دةكةين‬ ‫ديارى‬ ‫كة‬ ‫ة‬
‫َة‬‫ل‬‫جو‬ ‫لة‬ ‫بريتية‬ ‫كة‬
‫ِانى‬‫ر‬‫سو‬ ‫و‬
‫َةكان‬‫ل‬‫خا‬
(
Translation and Rotation
)
‫بةم‬
‫قةبارةى‬ ‫َوة‬‫ي‬‫ش‬
‫ستيفنس‬
‫َت‬‫ي‬‫دةردةكةو‬ ‫بؤ‬ ‫ماتريكسةكةمان‬
,
( ‫بؤ‬
Beam
‫َوةى‬‫ي‬‫ش‬ ‫بةم‬ )
‫خوارةوةية‬
.
2
-
(
Structure
‫ةكة‬ )
‫و‬ ‫دةكةين‬ ‫بةش‬ ‫بةش‬ ‫َةكاندا‬‫ل‬‫خا‬ ‫لة‬
(
Fixed End Reaction
)
‫بؤ‬ ‫ئةدؤزينةوة‬
‫يةكةلة‬ ‫يةك‬
Deformation
(
Rotation, Shear , Axial
)
( ‫َى‬‫ي‬‫ث‬ ‫بة‬
DOF
‫دةدؤزينةوة‬ ‫ماتريكسةكة‬ ‫ستيفنس‬ ‫بةمةش‬ ‫كان‬ ‫ة‬ )
‫خوارةوة‬ ‫َوةى‬‫ي‬‫ش‬ ‫بةم‬
.
Rotation Deformation for Fixed-Fixed (θ=1)
L L
(0,0,0) (8,0,0) (14,0,0) (20,0,0)
(22,0,0)
θ
A D
B C A B A B
DOF=2 or 3 DOF=1 or 4 DOF=3 or DOF=1
DOF=1 or DOF=0
2𝐸𝐼𝜃
𝐿
A B
θ
4𝐸𝐼𝜃
𝐿
6𝐸𝐼𝜃
𝐿2
6𝐸𝐼𝜃
𝐿2
2𝐸𝐼𝜃
𝐿
A B
θ
4𝐸𝐼𝜃
𝐿
6𝐸𝐼𝜃
𝐿2
6𝐸𝐼𝜃
𝐿2
Y X
Z
Global Axis
‫طشتى‬ ‫تةوةرةى‬
X
y
Local Axis
‫تةوةرةى‬
‫ئةندامةكة‬
DOF=3

21. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
20
Rotation Deformation for Fixed-Hinge (θ=1)
Shear Deformation for Fixed-Fixed(∆=1)
Shear Deformation for Fixed-Hinge (∆=1)
Axial Deformation for Fixed-Fixed (∆=1)
Axial Deformation for Fixed-Hinge (∆=1)
( ‫َوازى‬‫ي‬‫ش‬ ‫تةنها‬ ‫ئةطةر‬
Fixed-Fixed
)
‫َرين‬‫ي‬‫َبذ‬‫ل‬‫هة‬
‫ئةندامةكان‬ ‫طشت‬ ‫بؤ‬ ‫ِدةكةينةوة‬‫ر‬‫ث‬ ‫خوارةوة‬ ‫خشتةى‬ ‫ئةم‬ ‫ئةوا‬
.
EA/L
3
12EI/L
2
6EI/L
4EI/L
2EI/L
I
E
A
Height(H)
Width(W)
Length(L)
Member
A-B
A B
θ
3𝐸𝐼𝜃
𝐿
3𝐸𝐼𝜃
𝐿2
3𝐸𝐼𝜃
𝐿2
A B
θ
3𝐸𝐼𝜃
𝐿
3𝐸𝐼𝜃
𝐿2
3𝐸𝐼𝜃
𝐿2
∆
6𝐸𝐼∆
𝐿2
A B 6𝐸𝐼∆
𝐿2
12𝐸𝐼∆
𝐿3
12𝐸𝐼∆
𝐿3
∆
6𝐸𝐼∆
𝐿2
6𝐸𝐼∆
𝐿2
A B
12𝐸𝐼∆
𝐿3
12𝐸𝐼∆
𝐿3
∆
3𝐸𝐼∆
𝐿2
A B
3𝐸𝐼∆
𝐿3
3𝐸𝐼∆
𝐿3
3𝐸𝐼∆
𝐿2
A B
3𝐸𝐼∆
𝐿3
3𝐸𝐼∆
𝐿3
∆
∆
𝐸𝐴∆
𝐿
A B 𝐸𝐴∆
𝐿
∆
𝐸𝐴∆
𝐿
A B 𝐸𝐴∆
𝐿
∆
𝐸𝐴∆
𝐿
A B 𝐸𝐴∆
𝐿
∆
𝐸𝐴∆
𝐿
A B 𝐸𝐴∆
𝐿
L L
L
L L
L
L
L
L
L

22. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
21
1
-
.‫َت‬‫ي‬‫بكر‬ ‫يةكة‬ ‫ِةضاوى‬‫ر‬ ‫َويستة‬‫ي‬‫ث‬ ‫َك‬‫ي‬‫ثرسيار‬ ‫هةر‬ ‫شيكاركردنى‬ ‫بؤ‬
2
-
‫لة‬ ‫بريتية‬ ‫ئةندامةكان‬ ‫ناو‬ ‫َكى‬‫ي‬َ‫ل‬‫خا‬ ‫هةر‬ ‫َزى‬‫ي‬‫ه‬ ‫طشتى‬ ‫َوةيةكى‬‫ي‬‫ش‬ ‫بة‬, ‫َوازةن‬‫ي‬‫ش‬ ‫ئةم‬ ‫دروستكراوى‬ ‫و‬ ‫نني‬ ‫َطري‬‫ي‬‫ج‬ ‫سةرةوة‬ ‫ياسايانةى‬ ‫ئةم‬
‫سةرجةمى‬
‫َزةكان‬‫ي‬‫ه‬
( ‫طشت‬ ‫بؤ‬ ‫َةكةدا‬‫ل‬‫خا‬ ‫لة‬
DOF
( ‫و‬ ‫ةكان‬ )
FER
)
.‫ةكة‬
∆
𝐸𝐴∆
𝐿
A B 𝐸𝐴∆
𝐿
- 3
(
Fixed End Reaction
)
‫بؤ‬ ‫ئةدؤزينةوة‬
(
Member Loads
)
:‫خوارةوة‬ ‫َوةى‬‫ي‬‫ش‬ ‫بةم‬
-4
‫سةرئةجنامى‬
(
FER
)
‫كان‬ ‫ة‬
‫ئةدؤزينةوة‬
‫بؤ‬ ‫ِين‬‫ر‬‫ئةيطؤ‬ ‫و‬
(
Joint Force Vector
)
‫ماتريكس‬ ‫ناو‬ ‫دةخيةينة‬ ‫نيشانةيةو‬ ‫َضةوانةى‬‫ي‬‫ث‬ ‫كة‬
‫َزةكةوة‬‫ي‬‫ه‬ ‫ى‬
َ‫ل‬‫خا‬ ‫ناو‬ ‫ترى‬ ‫َكى‬‫ي‬‫زةبر‬ ‫َزو‬‫ي‬‫ه‬ ‫هةر‬ َ‫ل‬‫لةطة‬
‫ةكان‬
‫بة‬ ‫بةرامبةر‬
(
Degree of Freedom
)
.‫كة‬ ‫ة‬
.‫دةكةين‬ ‫شيكار‬ ‫ماتريكسةكة‬ ‫َشة‬‫ي‬‫هاوك‬ ‫دواتر‬
5
-
‫زةبرى‬ ‫َزو‬‫ي‬‫ه‬
‫ئة‬ ‫ئةندامةكان‬ ‫ناو‬
(‫دؤزينةوة‬
Internal Force of Members
)
:
Fixed
-
Fixed
For
Bending Moment
:
Fixed
-
Fixed
For
Shear Force
Axial Force:
‫َبينى‬‫ي‬‫ت‬
//
SBA =S(FER)BA -
6𝐸𝐼
𝐿2 (𝜃𝐴 + 𝜃𝐵 −
2∆
𝐿
)
A B
R1 R2
q
𝑞𝑑
𝐿2
ቈ𝑎𝑏2
+
(𝑎 − 2𝑏)𝑑2
12
቉
𝑎 𝑏
𝑞𝑑
𝐿2
ቈ𝑎2
𝑏 +
(𝑏 − 2𝑎)𝑑2
12
቉
𝑑
q
q
F
MBA =M(FER)BA +
2𝐸𝐼
𝐿
(𝜃𝐴 + 2𝜃𝐵 −
3∆
𝐿
)
A B
θ
4𝐸𝐼𝜃
𝐿
2𝐸𝐼𝜃
𝐿
6𝐸𝐼𝜃
𝐿2
6𝐸𝐼𝜃
𝐿2
L
6𝐸𝐼∆
𝐿2
6𝐸𝐼∆
𝐿2
12𝐸𝐼∆
𝐿3
12𝐸𝐼∆
𝐿3
∆
MAB =M(FER)AB+
2𝐸𝐼
𝐿
(2𝜃𝐴 + 𝜃𝐵 −
3∆
𝐿
) ,
FA =
EA∆
L
𝐹𝐿
8
𝐿
2
𝐿
2 𝐹
2
𝐹
2
𝐹𝐿
8
𝑎 𝑏
𝐹𝑎2(𝑎 + 3𝑏)
𝐿3
𝐹𝑎𝑏2
𝐿2
𝐹𝑎2
𝑏
𝐿2
𝐹𝑏2(3𝑎 + 𝑏)
𝐿3
F
𝑎 𝑏
𝑀𝑏(2𝑎 − 𝑏)
𝐿2
6𝑀𝑎𝑏
𝐿3
6𝑀𝑎𝑏
𝐿3
𝑀𝑎(2𝑏 − 𝑎)
𝐿2
M
∆
6𝐸𝐼∆
𝐿2
6𝐸𝐼∆
𝐿2
12𝐸𝐼∆
𝐿3
12𝐸𝐼∆
𝐿3
𝑞𝐿2
12
𝐿
𝑞𝐿
2
𝑞𝐿
2
𝑞𝐿2
12
q
5𝑞𝐿2
96
𝐿
𝑞𝐿
4
𝑞𝐿
4
5𝑞𝐿2
96
𝑞𝐿2
20
𝐿
7𝑞𝐿
20
3𝑞𝐿
20
𝑞𝐿2
30
𝑅1 =
𝑞𝑑
𝐿3
[(2𝑎 + 𝐿)𝑏2
+ (
𝑎 − 𝑏
4
)𝑑2
] 𝑅2 =
𝑞𝑑
𝐿3
[(2𝑏 + 𝐿)𝑎2
+ (
𝑎 − 𝑏
4
)𝑑2
]
SAB =S(FER)AB+
6𝐸𝐼
𝐿2 (𝜃𝐴 + 𝜃𝐵 −
2∆
𝐿
) ,
L
L

23. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
22
[
K11 K12 K13 K14 K15 K16
K21 K22 K23 K24 K25 K26
K31 K32 K33 K34 K35 K36
K41 K42 K43 K44 K45 K46
K51 K52 K53 K54 K55 K56
K61 K62 K63 K64 K65 K66 ]
[
1.6EI
0
−0.96EI
0
0
0 ]
[
K11
K21
K31
K41
K51
K61]
=
Example (2) Draw B.M.D & calculate S.F for the Beam shown due to the Loads shown and
vertical settlement at Support (D =0.01m), where cross section of members are (0.6*0.3)m and
E=2.2e+9kg/m2.
olution:
S
DOF?
-
1
Rad)
M,
kg,
s
(Unit
=? When DOF=1
K
,
=?
(FER)
-
2
I
E
A
W
H(m)
L(m)
Member
0.0054
2.2e+9
0.18
0.3
0.6
2.5
AB
0.0054
2.2e+9
0.18
0.3
0.6
2.5
BC
0.0054
2.2e+9
0.18
0.3
0.6
6
CD
0.0054
2.2e+9
0.18
0.3
0.6
4
DE
//‫َبينى‬‫ي‬‫ت‬
( ‫ئةوةى‬ ‫لةبةر‬
EA,EI
‫طشت‬ )
‫ئةندامةكان‬
‫ن‬ ‫دةتوانني‬ ‫يةكرتى‬ ‫بة‬ ‫يةكسانن‬
‫ر‬
‫خ‬
‫ةكةى‬
‫نةدؤزينةوة‬
.‫كؤتاى‬ ‫تا‬
EA/L
3
12EI/L
2
6EI/L
4EI/L
2EI/L
Member
0.4EA
0.768EI
0.96EI
1.6EI
0.8EI
AB
0.4EA
0.768EI
0.96EI
1.6EI
0.8EI
BC
EA /6
EI /18
EI /6
2 EI /3
EI /3
CD
0.25EA
3 EI /16
3 EI /8
EI
0.5EI
DE
for stiffness matrix
to obtain Column (1)
=0)
2,3,4,5,6
=1, (D
1
D
D2
D1
D3
D4 D5 D6
E
A D
C
5m
(0,0,0) (5,0,0) (11,0,0) (15,0,0)
(17,0,0)
Internal hinge
1000kg/m
1000kg/m 500kg
300kg
B F
DOF=6 Stiffness matrix= (6*6)
[
F1
F2
F3
F4
F5
F6 ]
=
[
D1
D2
D3
D4
D5
D6 ]
(F=KD)
2𝐸𝐼𝜃
𝐿
A B
θ
4𝐸𝐼𝜃
𝐿
6𝐸𝐼𝜃
𝐿2
6𝐸𝐼𝜃
𝐿2
1.6EI
0.96EI
0.8EI
A B
θ
0.96EI

24. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
23
[
K14
K24
K34
K44
K54
K64]
=
[
0
0.8EI
0.96EI
6.8EI/3
EI/3
0 ]
[
K12
K22
K32
K42
K52
K62]
=
[
0
1.6EI
0.96EI
0.80EI
0
0 ]
[
K13
K23
K33
K43
K53
K63]
=
[
−0.96EI
0.96EI
1.536EI
0.96EI
0
0 ]
=0) to obtain Column (2)
1,3,4,5,6
=1, (D
2
D
obtain Column (3)
=0) to
1,2,4,5,6
=1, (D
3
D
=0) to obtain Column (4)
1,2,3,5,6
=1, (D
4
D
∆
6𝐸𝐼∆
𝐿2
6𝐸𝐼∆
𝐿2
A B
12𝐸𝐼∆
𝐿3
12𝐸𝐼∆
𝐿3
∆
6𝐸𝐼∆
𝐿2
6𝐸𝐼∆
𝐿2
B C
12𝐸𝐼∆
𝐿3
12𝐸𝐼∆
𝐿3
∆
0.96EI
0.96EI
A B
12𝐸𝐼∆
𝐿3 0.768EI
∆
0.96EI
B C 0.96EI
0.768EI
12𝐸𝐼∆
𝐿3
2/3EI
C D
θ
1/3EI
1/6EI 1/6EI
4𝐸𝐼𝜃
𝐿
2𝐸𝐼𝜃
𝐿
B C
θ
6𝐸𝐼𝜃
𝐿2
6𝐸𝐼𝜃
𝐿2
1.6EI
0.8EI
B C
θ
0.96EI
6𝐸𝐼𝜃
𝐿2
B C
θ
2𝐸𝐼𝜃
𝐿
4𝐸𝐼𝜃
𝐿
6𝐸𝐼𝜃
𝐿2
6𝐸𝐼𝜃
𝐿2
C D
θ
4𝐸𝐼𝜃
𝐿
2𝐸𝐼𝜃
𝐿
6𝐸𝐼𝜃
𝐿2
6𝐸𝐼𝜃
𝐿2
0.96EI
B C
θ
1.6EI
0.96EI

25. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
24
K =
[
0
0
0
0
0.5EI
EI ]
[
K16
K26
K36
K46
K56
K66]
=
[
K11 K12 K13 K14 K15 K16
K21 K22 K23 K24 K25 K26
K31 K32 K33 K34 K35 K36
K41 K42 K43 K44 K45 K46
K51 K52 K53 K54 K55 K56
K61 K62 K63 K64 K65 K66 ]
=
[
1.6EI 0 −0.96EI 0 0 0
0 1.6EI 0.96EI 0.80EI 0 0
−0.96EI 0.96EI 1.536EI 0.96EI 0 0
0 0.8EI 0.96EI 6.8EI/3 EI/3 0
0 0 0 1EI/3 5EI/3 0.5EI
0 0 0 0 0.5EI EI ]
[
0
0
0
EI/3
5EI/3
0.5EI]
[
K15
K25
K35
K45
K55
K65]
=
*
‫ستونةكان‬ ‫طشت‬ ‫ِى‬‫ر‬‫ب‬ ‫كة‬ ‫ئةوةى‬ ‫دواى‬
‫دؤزيةوة‬ ‫ِيزكراوةكةمان‬‫ر‬ ‫ى‬
,
‫ِةكان‬‫ر‬‫ب‬
‫ِيزكراوةكة‬‫ر‬ ‫ناو‬ ‫دةخةينةوة‬
‫خواةوة‬ ‫َوةى‬‫ي‬‫ش‬ ‫بةم‬
.
=0) to obtain Column (5)
1,2,3,4,6
=1, (D
5
D
obtain Column (6)
=0) to
1,2,3,4,5
=1, (D
6
D
D E
θ
4𝐸𝐼𝜃
𝐿
2𝐸𝐼𝜃
𝐿
6𝐸𝐼𝜃
𝐿2
6𝐸𝐼𝜃
𝐿2
D E
θ
2𝐸𝐼𝜃
𝐿
4𝐸𝐼𝜃
𝐿
6𝐸𝐼𝜃
𝐿2
6𝐸𝐼𝜃
𝐿2
D E
θ
EI
3/8EI 3/8EI
C D
θ
2𝐸𝐼𝜃
𝐿
4𝐸𝐼𝜃
𝐿
6𝐸𝐼𝜃
𝐿2
6𝐸𝐼𝜃
𝐿2
C D
θ
2/3EI
1/6EI 1/6EI
EI
D E
θ
0.5EI
3/8EI 3/8EI

26. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
25
[
1.6EI 0 −0.96EI 0 0 0
0 1.6EI 0.96EI 0.80EI 0 0
−0.96EI 0.96EI 1.536EI 0.96EI 0 0
0 0.8EI 0.96EI 6.8EI/3 EI/3 0
0 0 0 EI/3 5EI/3 0.5EI
0 0 0 0 0.5EI EI ]
[
1.6 0 −0.96 0 0 0
0 1.6 0.96 0.80 0 0
−0.96 0.96 1.536 0.96 0 0
0 0.8 0.96 6.8/3 1/3 0
0 0 0 1/3 5/3 0.5
0 0 0 0 0.5 1 ]
−1
17925
253850/12
6250/12 6250/12
A B C D E
21675 600
A B
E
C D E
D
C
17925 600
B
ixed End Reaction ? for member Loads:
F
-
3
FER=
4- Joint Force Vector?
Invert of FER=
Joint Force Vector =
1
𝐸𝐼
[
6250/12
−6250/12
−2500
−253850/12
26375
44200 ]
=
[
D1
D2
D3
D4
D5
D6 ]
[
D1
D2
D3
D4
D5
D6 ]
=
[
6250/12
−6250/12
−2500
−253850/12
26375
44200 ]
1
𝐸𝐼
6250/12
44800
44300 44800
[
D1
D2
D3
D4
D5
D6 ]
=
22825
27125
9350
2500
1250
6250/12 6250/12 6250/12
[
9739.2739
−1510.3135
15689.5885
−16457.9208
6890.0990
40754.9505 ]
44300
6250/12
F
B
A
1250
1250
1000kg/m
6250/12
6250/12
E
300
300kg
600
E
D
250
250
500kg
250
250
C
B
1250
1250
1000kg/m
6250/12
6250/12
D
C
1500
1500
1000kg/m
1875
1875
D
C
6600
6600
For Settlement
19800
19800
E
D
22275
22275
For Settlement
44550
44550
44200
26375
2500
22825
27125
9350
2500
1250
6250/12 6250/12 21675
6250/12

27. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
26
=?
t
Momen
-
5
∴ 𝐌𝐀𝐁 = −𝟔𝟕𝟒𝟗. 𝟕𝟓 𝐤𝐠. 𝐦
MAB = MFER(AB) +
2EI
L
(2θA + θB −
3∆
L
) → MAB =
6250
12
+
2
2.5
(2 ∗ 0 + 9739.2739 −
3 ∗ 15689.5885
2.5
)
∴ 𝐌𝐁𝐀 = 𝟎
MBC = MFER(BC) +
2EI
L
(2θB + θC −
3∆
L
) → MBC =
6250
12
+
2
2.5
(2 ∗ (−1510.3135) − 16457.9208 −
3 ∗ − 15689.5885
2.5
)
∴ 𝐌𝐁𝐂 = 𝟎
MBA = MFER(BA) +
2EI
L
(2θB + θA −
3∆
L
) → MBA = −
6250
12
+
2
2.5
(2 ∗ 9739.2739 + 0 −
3 ∗ 15689.5885
2.5
)
MCB = MFER(CB) +
2EI
L
(2θC + θB −
3∆
L
) → MCB = −
6250
12
+
2
2.5
(2 ∗ (−16457.9208) − 1510.3135 −
3 ∗ −15689.5885
2.5
)
∴ 𝐌𝐂𝐁 = −𝟏𝟐𝟗𝟗𝟗. 𝟕𝟓 𝐤𝐠. 𝐦
∴ 𝐌𝐂𝐃 = +𝟏𝟐𝟗𝟗𝟗. 𝟕𝟓 𝐤𝐠. 𝐦
MCD = MFER(CD) +
2EI
L
(2θC + θD −
3∆
L
) → MCD = 21675 +
2
6
(2 ∗ (−16457.9208) + 6890.0990 − 0)
MDC = MFER(DC) +
2EI
L
(2θD + θC −
3∆
L
) → MDC = 17925 +
2
6
(2 ∗ (6890.0990) − 16457.9208 − 0)
∴ 𝐌𝐃𝐄 = −𝟏𝟕𝟎𝟑𝟐. 𝟒𝟑 𝐤𝐠. 𝐦
∴ 𝐌𝐃𝐂 = 𝟏𝟕𝟎𝟑𝟐. 𝟒𝟑 𝐤𝐠. 𝐦
MDE = MFER(DE) +
2EI
L
(2θD + θE −
3∆
L
) → MDE = −44300 +
2
4
(2 ∗ (6890.0990) + 40754.9505 − 0)
MED = MFER(ED) +
2EI
L
(2θE + θD −
3∆
L
) → MED = −44800 +
2
4
(2 ∗ (40754.9505) + 6890.0990 − 0)
∴ 𝐌𝐄𝐃 = −𝟔𝟎𝟎 𝐤𝐠. 𝐦
∴ 𝐌𝐄𝐅 = +𝟔𝟎𝟎 𝐤𝐠. 𝐦
B.M.D (kg. m)

28. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
27
= ?
Shear
‫بةراور‬
‫د‬
‫ى‬
‫منوونةكة‬ ‫شيكارى‬ ‫ئةجنامى‬
‫بة‬
‫ئةجنامى‬
‫سؤفت‬
:‫َرةكان‬‫ي‬‫و‬
Note
F
E
D
C
B
A
m)
(kg.
Moment
0
600
-17032.43
12999.75
0
-6749.75
‫منوونةكة‬ ‫ئةجنامى‬
OK
0
600
-17032.42
12999.75
0
-6749.75
STAAD V8i
OK
0
600
-17032.43
12999.75
0
-6749.75
SAP2000
‫بة‬ ‫ئةجنامةكان‬ ‫ئايا‬
(
ETABS
‫ضةند‬ )
‫ن‬
‫؟‬
?
?
?
?
?
?
ETABS
SAB =S (FER)AB+
6EI
L2 (θA + θB −
2∆
L
) → SAB = 1250+
6
6.25
(0 + 9739.2739 −
2∗15689.5885
2.5
)
∴ 𝐒𝐀𝐁 = −𝟏𝟒𝟒𝟗. 𝟗𝟎 𝐤𝐠
SBA=S (FER)BA-
6EI
L2 (θA + θB −
2∆
L
) → SAB = 1250-
6
6.25
(0 + 9739.2739 −
2∗15689.5885
2.5
)
∴ 𝐒𝐁𝐀 = 𝟑𝟗𝟒𝟗. 𝟗𝟎 𝐤𝐠
SBC =S (FER)BC+
6EI
L2 (θB + θC −
2∆
L
) → SBC = 1250+
6
6.25
((−1510.3135) − 16457.9208 −
2∗−15689.5885
2.5
)
∴ 𝐒𝐁𝐂 = −𝟑𝟗𝟒𝟗. 𝟗𝟎 𝐤𝐠
SCB =S (FER)CB-
6EI
L2 (θB + θC −
2∆
L
) → SCB = 1250-
6
6.25
((−1510.3135) − 16457.9208 −
2∗−15689.5885
2.5
)
∴ 𝐒𝐂𝐁 = 𝟔𝟒𝟒𝟗. 𝟗𝟎 𝐤𝐠
SCD=S (FER) CD+
6EI
L2 (θC + θD −
2∆
L
) → SCD = 8100+
6
36
(−16457.9208 + 6890.0990 − 0)
∴ 𝐒𝐂𝐃 = 𝟔𝟓𝟎𝟓. 𝟑𝟔 𝐤𝐠
SDC=S (FER)DC-
6EI
L2 (θC + θD −
2∆
L
) → SCD = -5100-
6
36
(−16457.9208 + 6890.0990 − 0)
∴ 𝐒𝐃𝐂 = 𝟑𝟓𝟎𝟓. 𝟑𝟔 𝐤𝐠
SED=S(FER)ED-
6EI
L2 (θD + θE −
2∆
L
) → SDE = +22525-
6
16
(6890.0990 + 40754.9505 − 0)
∴ 𝐒𝐄𝐃 = 𝟒𝟔𝟓𝟖. 𝟏𝟏𝐤𝐠
SDE=S(FER)DE+
6EI
L2 (θD + θE −
2∆
L
) → SDE = -22025+
6
16
(6890.0990 + 40754.9505 − 0)
∴ 𝐒𝐃𝐄 = −𝟒𝟏𝟓𝟖. 𝟏𝟏 𝐤𝐠
𝐀𝐧𝐝 𝐒𝐄𝐅 = 𝟑𝟎𝟎𝐤𝐠

29. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
28
∆
D1
D2
D3 D1
D1
D2
ϴ
D4
D5
D6
D1
D2
D3
θ
[
∑ −
6𝐸𝐼
𝐿2
sin 𝜃
∑
6𝐸𝐼
𝐿2
cos 𝜃
∑
4𝐸𝐼
𝐿
+
6𝐸𝐼
𝐿2
sin 𝜃
−
6𝐸𝐼
𝐿2
cos 𝜃
2𝐸𝐼
𝐿 ]
2- 2D Frames Analysis by Direct Stiffness Method
‫هةنطاوةكانى‬
‫شيكاركردن‬
:
1
-
( Degree of Freedom )
‫ى‬
(
Structure
)
‫دةكةين‬ ‫ديارى‬ ‫كة‬ ‫ة‬
‫َةكان‬‫ل‬‫خا‬ ‫َةى‬‫ل‬‫جو‬ ‫لة‬ ‫بريتية‬ ‫كة‬
(
Translations and Rotations
)
‫َوة‬‫ي‬‫ش‬ ‫بةم‬
‫َت‬‫ي‬‫دةردةكةو‬ ‫بؤ‬ ‫ماتريكسةكةمان‬ ‫ستيفنس‬ ‫قةبارةى‬
,
( ‫بؤ‬
Frame
.‫خوارةوةية‬ ‫َوةى‬‫ي‬‫ش‬ ‫بةم‬ )
‫كةلة‬ ‫تر‬ ‫بةشانةى‬ ‫ئةو‬ ‫ِةضاوكردنى‬‫ر‬ ‫ِايى‬‫ر‬‫سةرة‬
(
Beam
)
.‫كردووة‬ ‫بامسان‬
2
-
( ‫َى‬‫ل‬‫خا‬ ‫ِةضاوكردنى‬‫ر‬
,3,4,5
2
( ‫بةشى‬ ‫لة‬ ‫كة‬ )
Beam
.‫بامسانكرد‬ )
//‫َبينى‬‫ي‬‫ت‬
( ‫دؤزينةوةى‬ ‫بؤ‬
FER
‫ئةم‬ ‫بؤ‬ ‫ةكان‬ )
‫ئةندامانةى‬
‫َزةكانيان‬‫ي‬‫ه‬ ‫بونى‬ ‫دابةش‬ ‫بؤ‬ ‫َت‬‫ي‬‫بكر‬ ‫طؤشةكةيان‬ ‫ِةضاوى‬‫ر‬ ‫َويستة‬‫ي‬‫ث‬ ‫نني‬ ‫ئاسؤيي‬ ‫كة‬
‫لة‬ ‫كة‬
‫ئاسؤييةوة‬
‫َر‬‫ي‬‫ذم‬ ‫كات‬ ‫ميلى‬ ‫َضةوانة‬‫ي‬‫ث‬ ‫بة‬
‫بؤ‬
‫ئةندامة‬
‫كة‬
‫َت‬‫ي‬‫َور‬‫ي‬‫دةث‬
.‫خوارةوة‬ ‫َوةى‬‫ي‬‫ش‬ ‫بةم‬
)
=1
D3
(
ixed
F
-
for Fixed
Deformation
Rotation
B
6𝐸𝐼
𝐿2
cos θ
6𝐸𝐼
𝐿2
cos 𝜃
6𝐸𝐼
𝐿2
sin θ
6𝐸𝐼
𝐿2
sin θ
B
θ
DOF= 3 DOF= 1 or 0 DOF= 2 or 1
A
Typical Frame

30. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
29
D5
D6
D4
D2 ө
Ø
D4
D5
D6
D2
1
D1
D3
D1
D3
[
∑
EA
L
sin 𝜃 cos θ −
12EI
L3 sin𝜃 cosθ
∑
𝐸𝐴
𝐿
sin2
𝜃 +
12EI
L3 cos2
𝜃
∑
6𝐸𝐼
𝐿2 cosθ
−
EA
L
sin 𝜃 cos θ +
12EI
L3 sin𝜃 cos θ
−
𝐸𝐴
𝐿
sin2
𝜃 −
12EI
L3 cos2
𝜃
6𝐸𝐼
𝐿2 cosθ ]
1
[
∑
𝐸𝐴
𝐿
cos
2
𝜃 +
12EI
L3
sin
2
𝜃
∑
EA
L
sin 𝜃 cos θ −
12EI
L3
sin 𝜃 cos θ
∑ −
6𝐸𝐼
𝐿2
sin 𝜃
−
𝐸𝐴
𝐿
cos
2
𝜃 −
12EI
L3
sin
2
𝜃
−
EA
L
sin 𝜃 cos θ +
12EI
L3
sin 𝜃 cos θ
6𝐸𝐼
𝐿2
sin 𝜃 ]
=1
1
Fixed) D
-
Horizontal Deformation for (Fixed
Fixed) D2=1
-
Vertical Deformation for (Fixed
EA
L
sin
2
θ
EA
L
sin
2
θ
12EI
L
3
sin
𝜃
cos
𝜃
12EI
L
3
sin
𝜃
cos
𝜃
EA
L
sin
𝜃
cos
θ
EA
L
sin
𝜃
cos
θ
𝐸𝐴
𝐿
cos2
𝜃
12EI
L3
sin2
𝜃
12EI
L
3
sin
𝜃
cos
𝜃
B
A
EA
L
sin
𝜃
cos
θ
A
B
𝐸𝐴
𝐿
cos2
𝜃
12EI
L3
sin 𝜃 cos 𝜃
12EI
L3
sin2
𝜃
EA
L
sin
𝜃
cos
θ
A
B
B
A
12EI
L3
sin2
θ
12EI
L3
sin2
θ

31. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
30
1000 kg
A
B C
D
L=5m
L=4m
1m
D1
D2
D3
D4
D5
D6
D7
[
K11 K12 K13 K14 K15 K16 K17
K21 K22 K23 K24 K25 K26 K27
K31 K32 K33 K34 K35 K36 K37
K41 K42 K43 K44 K45 K46 K47
K51 K52 K53 K54 K55 K56 K57
K61 K62 K63 K64 K65 K66 K67
K71 K72 K73 K74 K75 K76 K77]
A
B C
D
Example (3) Draw B.M.D and S.F.D for the Frame shown due to the Loads Shown and
where cross section of members (600*200)mm and E=2200kg/mm2
Solution:
DOF?
-
1
Rad)
Cm,
kg,
s
(Unit
K =? When DOF=1
,
=?
(FER)
-
2
)
4
cm
(
I
)
2
cm
kg/
(
E
)
2
(cm
A
(cm)
W
(cm)
H
(cm)
L
Member
4
36*10
4
*10
22
1200
20
60
500
AB
4
36*10
4
*10
22
1200
20
60
500
BC
4
36*10
4
*10
22
1200
20
60
400
CD
EA/L
3
12EI/L
2
6EI/L
4EI/L
2EI/L
Member
528000
7603.2
1900800
6336*105
3168*105
AB
528000
7603.2
1900800
6336*105
3168*105
BC
660000
14850
2970000
7920*105
3960*105
CD
DOF=7 Stiffness matrix=(7*7)
(F=KD)
[
F1
F2
F3
F4
F5
F6
F7 ]
=
[
D1
D2
D3
D4
D5
D6
D7 ]

32. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
31
1
[
K11
K21
K31
K41
K51
K61
K71]
=
[
722946.048
249790.464
1520640
−528000
0
0
0 ]
1
[
K12
K22
K32
K42
K52
K62
K72]
=
[
249790.464
348260.352
760320
0
−7603.2
1900800
0 ]
6𝐸𝐼
𝐿2
6𝐸𝐼
𝐿2
Column (1)
obtain
=0) to
7
,
2,3,4,5,6
=1, (D
1
D
=0) to obtain Column (2)
1,3,4,5,6,7
=1, (D
2
D
Cos(θ) Sin(θ)
)
θ
(
2
Cos
)
θ
(
2
Sin
Cos(θ)
Sin(θ)
Angle(θ)
Member
0.48
0.36
0.64
-0.6
-0.8
233.130
BA
0
1
0
1
0
0
BC
0
1
0
-1
0
180
CB
0
0
1
0
-1
270
CD
EA
L
sinθ cos θ
12EI
L3
sin θ cos θ
12EI
L3
sin2
θ
12EI
L3
cos2
θ
EA
L
cos2
θ
EA
L
sin2
θ
6EI
L2
cos θ
6EI
L2
sin θ
Member
253440
3649.536
4866.048
2737.152
190080
337920
-1140480
-1520640
BA
0
0
0
7603.2
528000
0
1900800
0
BC
253440
3649.536
4866.048
10340.352
718080
337920
760320
-1520640
Sum
0
0
0
7603.2
528000
0
-1900800
0
CB
0
0
14850
0
0
660000
0
-2970000
CD
0
0
14850
7603.2
528000
660000
-1900800
-2970000
Sum
12EI
L
3
EA
L
EA
L
C
B
A
B C
D
A
B C
D
B
12EI
L
3

33. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
32
[
K43
K24
K34
K44
K54
K64
K74]
=
1
[
−528000
0
0
542850
0
2970000
2970000]
6𝐸𝐼
𝐿2
6𝐸𝐼
𝐿2
6𝐸𝐼
𝐿2
6𝐸𝐼
𝐿2
6𝐸𝐼
𝐿2
[
K13
K23
K33
K43
K53
K63
K73]
=
[
1520640
760320
1267200000
0
−1900800
316800000
0 ]
[
K15
K25
K35
K45
K55
K65
K75]
=
1
[
0
−7603.2
−1900800
0
667603.2
−1900800
0 ]
6𝐸𝐼
𝐿2
6𝐸𝐼
𝐿2
=0) to obtain Column (3)
1,2,4,5,6,7
=1, (D
3
D
D4=1, (D1,2,3,5,6,7=0) to obtain Column (4)
D5=1, (D1,2,3,4,6,7=0) to obtain Column (5)
EA
L
12EI
L3
EA
L
EA
L
4𝐸𝐼
𝐿
2𝐸𝐼
𝐿
4𝐸𝐼
𝐿
B
A
D
A
B C
D
C
C
B
B C
θ
A
B C
D
D
12EI
L3
B
12EI
L3

34. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
33
[
K16
K26
K36
K46
K56
K66
K76]
=
[
0
1900800
316800000
2970000
−1900800
1425600000
396000000 ]
θ
[
K17
K27
K37
K47
K57
K67
K77]
=
[
0
0
0
2970000
0
396000000
792000000]
θ
2𝐸𝐼
𝐿
2𝐸𝐼
𝐿
[
722946.048 249790.464 1520640 −528000 0 0 0
249790.464 348260.352 760320 0 −7603.2 1900800 0
1520640 760320 1267200000 0 −1900800 316800000 0
−528000 0 0 542850 0 2970000 2970000
0 −7603.2 −1900800 0 667603.2 −1900800 0
0 1900800 316800000 2970000 −1900800 1425600000 396000000
0 0 0 2970000 0 396000000 792000000]
[
K11 K12 K13 K14 K15 K16 K17
K21 K22 K23 K24 K25 K26 K27
K31 K32 K33 K34 K35 K36 K37
K41 K42 K43 K44 K45 K46 K47
K51 K52 K53 K54 K55 K56 K57
K61 K62 K63 K64 K65 K66 K67
K71 K72 K73 K74 K75 K76 K77]
=
4𝐸𝐼
𝐿
6𝐸𝐼
𝐿2
6𝐸𝐼
𝐿2
6𝐸𝐼
𝐿2
2𝐸𝐼
𝐿
4𝐸𝐼
𝐿
4𝐸𝐼
𝐿
6𝐸𝐼
𝐿2
D6=1, (D1,2,3,4,5,7=0) to obtain Column (6)
D7=1, (D1,2,3,4,5,6=0) to obtain Column (7)
//‫َبينى‬‫ي‬‫ت‬
( ‫خوارةوة‬ ‫َوةى‬‫ي‬‫ش‬ ‫بةم‬
∆
( ‫هؤى‬ ‫َتة‬‫ي‬‫دةب‬ ‫كة‬ ‫دةدؤزينةوة‬ )
Shear
)
‫ئةندامةكاندا‬ ‫لة‬
.
( ‫ئةم‬ ‫ئةوةى‬ ‫بةر‬ ‫لة‬
∆
‫شيكا‬ ‫لة‬ ‫كة‬ )
‫ر‬
‫ى‬
‫ِيزكر‬‫ر‬
‫ئةي‬ ‫اوةكةدا‬
‫تةوة‬ ‫ئاراستةى‬ ‫بة‬ ‫دؤزينةوة‬
‫ر‬
.‫ئةندامةكة‬ ‫تةوةرةى‬ ‫ئاراستةى‬ ‫بؤ‬ ‫ِين‬‫ر‬‫بيانطؤ‬ ‫َويستة‬‫ي‬‫ث‬ ‫طشتني‬ ‫ةى‬
𝜃
∆Ax
∆Ay
∆Bx
∆By
A
B
C
A
B C
D
B
D
D
A
B
C
D
C
∆= (∆By − ∆Ay) cos 𝜃 − (∆Bx − ∆Ax) sin 𝜃

35. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
34
[
722946.048 249790.464 1520640 −528000 0 0 0
249790.464 348260.352 760320 0 −7603.2 1900800 0
1520640 760320 1267200000 0 −1900800 316800000 0
−528000 0 0 542850 0 2970000 2970000
0 −7603.2 −1900800 0 667603.2 −1900800 0
0 1900800 316800000 2970000 −1900800 1425600000 396000000
0 0 0 2970000 0 396000000 792000000]
Fixed End Reaction ? for member Loads:
-
3
4- Joint Force Vector?
Invert of FER=
Joint Force Vector =
C
B
104kg
896kg
1000kg
16000kg.cm
64000kg.cm
D
C
0
0
0
0
150kg
200kg
C
B
104kg
896kg
16000kg.cm
64000kg.cm
D
C
0
0
0
0
200kg
150kg
B
200kg
1046kg
C
0
104kg
0
D
[
F1
F2
F3
F4
F5
F6
F7 ]
=
[
200
−1046
−32750
0
−104
16000
0 ]
[
200
−1046
−32750
0
−104
16000
0 ]
=
[
D1
D2
D3
D4
D5
D6
D7 ]
F=KD

36. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
35
-1
[
722946.048 249790.464 1520640 −528000 0 0 0
249790.464 348260.352 760320 0 −7603.2 1900800 0
1520640 760320 1267200000 0 −1900800 316800000 0
−528000 0 0 542850 0 2970000 2970000
0 −7603.2 −1900800 0 667603.2 −1900800 0
0 1900800 316800000 2970000 −1900800 1425600000 396000000
0 0 0 2970000 0 396000000 792000000]
Moment = ?
-
5
[
D1
D2
D3
D4
D5
D6
D7 ]
=
[
200
−1046
−32750
0
−104
16000
0 ]
[
D1
D2
D3
D4
D5
D6
D7 ]
=
[
0.0525065441654327𝑐𝑚
−0.0406883975484445 𝑐𝑚
−7.32555333685725 ∗ 10−5
𝑟𝑎𝑑
0.0520538439417292 𝑐𝑚
−0.000739980109877633 𝑐𝑚
3.08254696846886 ∗ 10−5
𝑟𝑎𝑑
−2.10614649623829 ∗ 10−4
𝑟𝑎𝑑]
MAB = MFER(AB) +
2EI
L
(2θA + θB −
3∆
L
) , MBA = MFER(BA) +
2EI
L
(2θB + θA −
3∆
L
)
∴ 𝐌𝐀𝐁 = 𝟏𝟑𝟒𝟐𝟗𝟎. 𝟓𝟎𝟐 𝐤𝐠. 𝐜𝐦
MAB = 31250 +
2 ∗ 220000 ∗ 36 ∗ 104
500
(2 ∗ 0 − 7.32555333685725 ∗ 10−5
−
3 ∗ −0.0664182738614128
500
)
MBA = −31250 +
2 ∗ 220000 ∗ 36 ∗ 104
500
(2 ∗ −7.32555333685725 ∗ 10−5
+ 0 −
3 ∗ −0.0664182738614128
500
)
∴ 𝐌𝐁𝐂 = −𝟒𝟖𝟓𝟖𝟑. 𝟏𝟓 𝐤𝐠. 𝐜𝐦
MBC = MFER(BC) +
2EI
L
(2θB + θC −
3∆
L
) , MCB = MFER(CB) +
2EI
L
(2θC + θB −
3∆
L
)
MBC = 64000 +
2 ∗ 220000 ∗ 36 ∗ 104
500
(2 ∗ −7.32555333685725 ∗ 10−5
+ 3.08254696846886 ∗ 10−5
−
3 ∗ 0.0399484174385668
500
)
∴ 𝐌𝐁𝐀 = 𝟒𝟖𝟓𝟖𝟑. 𝟏𝟓 𝐤𝐠. 𝐜𝐦
MCB = −16000 +
2 ∗ 220000 ∗ 36 ∗ 104
500
(2 ∗ 3.08254696846886 ∗ 10−5
− 7.32555333685725 ∗ 10−5
−
3 ∗ −0.0399484174385668
500
)
∴ 𝐌𝐂𝐁 = −𝟗𝟓𝟔𝟏𝟎. 𝟐𝟗 𝐤𝐠. 𝐜𝐦
MCD = MFER(CD) +
2EI
L
(2θC + θD −
3∆
L
) , MDC = MFER(DC) +
2EI
L
(2θD + θC −
3∆
L
)
MCD = 0 +
2 ∗ 220000 ∗ 36 ∗ 104
400
(2 ∗ 3.08254696846886 ∗ 10−5
− 2.10614649623829 ∗ 10−4
−
3 ∗ −0.0520538439417292
400
)
∴ 𝐌𝐂𝐃 = 𝟗𝟓𝟔𝟏𝟎. 𝟐𝟗 𝐤𝐠. 𝐜𝐦 𝐌𝐃𝐂 = 𝟎 (It must be zero, since it is hinge )

37. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
36
Shear = ?
‫يان‬:
𝟎
𝐁
−𝟒𝟖𝟓𝟖𝟑. 𝟏𝟓
𝐀
−𝟏𝟗𝟔𝟒𝟔. 𝟑𝟐
𝟒𝟐𝟖𝟓𝟑. 𝟔𝟖
𝟗𝟓𝟔𝟏𝟎. 𝟐𝟗
−𝟏𝟗𝟕𝟒𝟒. 𝟒𝟔 𝐂
𝐃
−𝟗𝟗𝟕𝟒𝟒. 𝟒𝟔
B.M.D( kg.cm)
𝐁
𝐀
(134290.29+48583.15)/500=365.75kg
365.75 250+365.75 =615.75
365.75 250-365.75 =-115.75
-
S.F.D for A -B
( ‫دةتوانني‬
Shear
)
‫هؤى‬ ‫بة‬ ‫بدؤزينةوة‬
‫زةبرةوة‬
(
Moment
)
‫خوارةوة‬ ‫َوةى‬‫ي‬‫ش‬ ‫بةم‬
:
SAB =S(FER)AB+
6EI
L2 (θA + θB −
2∆
L
) → SAB = 250+
6∗220000∗36∗104
250000
(0 + −7.32555333685725 ∗ 10−5
−
2∗−0.0664182738614128
500
)
∴ 𝐒𝐀𝐁 = 𝟏𝟔𝟏𝟓. 𝟕𝟓 𝐤𝐠
SBA =S(FER)BA-
6EI
L2 (θA + θB −
2∆
L
) → SBA = 250-
6∗220000∗36∗104
250000
(0 + −7.32555333685725 ∗ 10−5
−
2∗−0.0664182738614128
500
)
∴ 𝐒𝐁𝐀 = −𝟏𝟏𝟓. . 𝟕𝟓 𝐤𝐠
SBC =S(FER)BC+
6EI
L2 (θC + θB −
2∆
L
)
→ SBC = 896+
6∗220000∗36∗104
250000
(3.08254696846886 ∗ 10−5
− 7.32555333685725 ∗ 10−5
−
2∗0.03994841743857
500
)
∴ 𝐒𝐁𝐂 = 𝟓𝟏𝟏. 𝟔𝟏 𝐤𝐠
SCB =S(FER)BC-
6EI
L2 (θC + θB −
2∆
L
)
→ SCB = 104-
6∗220000∗36∗104
250000
(3.08254696846886 ∗ 10−5
− 7.32555333685725 ∗ 10−5
−
2∗0.03994841743857
500
)
∴ 𝐒𝐂𝐁 = 𝟒𝟖𝟖. 𝟑𝟗 𝐤𝐠

38. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
37
‫بةراوردى‬
‫بة‬ ‫منوونةكة‬ ‫شيكارى‬ ‫ئةجنامى‬
‫ئةجنامى‬
:‫َرةكان‬‫ي‬‫و‬ ‫سؤفت‬
Note
D
C
B
A
(kg.cm)
Moment
0
95610.29
48583.15
-
134290.50
‫ئةجنامى‬
‫منوونةكة‬
OK
0
95610.29
48583.15
-
134290.51
STAAD V8i
OK
0
95610.29
48583.15
-
134290.50
SAP2000
( ‫بة‬ ‫ئةجنامةكان‬ ‫ئايا‬
ETABS
‫ضةند‬ )
‫ن‬
‫؟‬
?
?
?
?
ETABS
Note
D
C
B
A
)
(kg
Shear
CD
CB
BC
BA
239.03
239.03
488.39
511.61
115.75
615.75
‫ئةجنامى‬
‫منوونةكة‬
OK
239.03
239.03
488.39
511.61
115.75
615.75
STAAD V8i
OK
239.03
239.03
488.39
511.61
115.75
615.75
SAP2000
( ‫بة‬ ‫ئةجنامةكان‬ ‫ئايا‬
ETABS
‫ضةند‬ )
‫ن‬
‫؟‬
?
?
?
?
?
?
ETABS
239.03
-488.39
511.61
𝐃
𝐁
𝐀
𝐂
S.F.D(kg)
SDC =S(FER)DC-
6EI
L2 (θC + θD −
2∆
L
)
→ SDC = 0-
6∗220000∗36∗104
160000
( 3.08254696846886 ∗ 10−5
+ −2.10614649623829 ∗ 10−4
−
2∗−0.0520538439417292
400
)
∴ 𝐒𝐃𝐂 = −𝟐𝟑𝟗. 𝟎𝟑 𝐤𝐠
SCD =S(FER)CD+
6EI
L2 (θC + θD −
2∆
L
)
→ SCD = 0+
6∗220000∗36∗104
160000
( 3.08254696846886 ∗ 10−5
+ −2.10614649623829 ∗ 10−4
−
2∗−0.0520538439417292
400
)
∴ 𝐒𝐂𝐃 = 𝟐𝟑𝟗. 𝟎𝟑 𝐤𝐠

39. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
38
D1
D2
D1
1
EA
L
sin
𝜃
cos
θ
EA
L
sin
𝜃
cos
θ
𝐸𝐴
𝐿
cos2
𝜃
𝐸𝐴
𝐿
cos2
𝜃
[
∑
𝐸𝐴
𝐿
cos2
𝜃
∑
𝐸𝐴
𝐿
sin 𝜃 cos 𝜃
−
𝐸𝐴
𝐿
cos2
𝜃
−
𝐸𝐴
𝐿
sin 𝜃 cos 𝜃]
3- 2D Trusses Analysis by Direct Stiffness Method
‫هةنطاوةكانى‬
‫شيكاركردن‬
:
1
-
( Degree of Freedom )
‫ى‬
(
Structure
)
‫دةكةين‬ ‫ديارى‬ ‫كة‬ ‫ة‬
(‫َةكان‬‫ل‬‫خا‬ ‫َةى‬‫ل‬‫جو‬ ‫لة‬ ‫بريتية‬ ‫كة‬
Translations
)
‫ستيفنس‬ ‫قةبارةى‬ ‫َوة‬‫ي‬‫ش‬ ‫بةم‬
‫َت‬‫ي‬‫دةردةكةو‬ ‫بؤ‬ ‫ماتريكسةكةمان‬
,
( ‫بؤ‬
Truss
.‫خوارةوةية‬ ‫َوةى‬‫ي‬‫ش‬ ‫بةم‬ )
2
-
(
Structure
‫و‬ ‫دةكةين‬ ‫بةش‬ ‫بةش‬ ‫َةكاندا‬‫ل‬‫خا‬ ‫لة‬ ‫ةكة‬ )
(
Pinned End Reaction
)
‫بؤ‬ ‫ئةدؤزينةوة‬
‫يةكةلة‬ ‫يةك‬
(
Axial
Deformation
)
( ‫َى‬‫ي‬‫ث‬ ‫بة‬
DOF
‫دةدؤزينةوة‬ ‫ماتريكسةكة‬ ‫ستيفنس‬ ‫بةمةش‬ ‫كان‬ ‫ة‬ )
‫خوارةوة‬ ‫َوةى‬‫ي‬‫ش‬ ‫بةم‬
.
- 3
(
Joint Force Vector
)
‫بؤ‬ ‫ئةدؤزينةوة‬
(
Joint Loads
)
‫ئةو‬ ‫بؤ‬ ‫تايبةت‬ ‫بة‬
( ‫بة‬ ‫بةرامبةر‬ ‫كة‬ ‫َزانةى‬‫ي‬‫ه‬
DOF
.‫نني‬ ‫كان‬ ‫ة‬ )
4
-
‫سةرئةجنامى‬
(
Joint Force Vector
)
‫كان‬ ‫ة‬
‫ئةدؤزينةوة‬
‫و‬
‫بة‬ ‫بةرامبةر‬ ‫َزةكةوة‬‫ي‬‫ه‬ ‫ماتريكسى‬ ‫ناو‬ ‫دةخيةينة‬
(
Degree of Freedom
)
.‫كة‬ ‫ة‬
.‫دةكةين‬ ‫شيكار‬ ‫ماتريكسةكة‬ ‫َشة‬‫ي‬‫هاوك‬ ‫دواتر‬
5
-
‫ئةندامةكان‬ ‫ناو‬ ‫َزى‬‫ي‬‫ه‬
‫ئةدؤزينةوة‬
(
s
Internal Force of Member
)
A
F
//‫َبينى‬‫ي‬‫ت‬
( ‫دؤزينةوةى‬ ‫بؤ‬
PER
‫ئةم‬ ‫بؤ‬ ‫ةكان‬ )
‫ئةنداما‬
‫كة‬ ‫نةى‬
‫لة‬ ‫كة‬ ‫َزةكانيان‬‫ي‬‫ه‬ ‫بونى‬ ‫دابةش‬ ‫بؤ‬ ‫َت‬‫ي‬‫بكر‬ ‫طؤشةكةيان‬ ‫ِةضاوى‬‫ر‬ ‫َويستة‬‫ي‬‫ث‬ ‫نني‬ ‫ئاسؤيي‬
‫َضةوانة‬‫ي‬‫ث‬ ‫بة‬ ‫ئاسؤييةوة‬
‫ى‬
‫بؤ‬ ‫َر‬‫ي‬‫ذم‬ ‫كات‬ ‫ميلى‬
‫َت‬‫ي‬‫َور‬‫ي‬‫دةث‬ ‫ئةندامةكة‬
.‫خوارةوة‬ ‫َوةى‬‫ي‬‫ش‬ ‫بةم‬
=
𝐸𝐴∆
𝐿
DOF= 2 DOF= 0 DOF= 1
Some Typical Truss
∆
𝐸𝐴∆
𝐿
A B 𝐸𝐴∆
𝐿
A
∆
B
𝐸𝐴∆
𝐿
A B 𝐸𝐴∆
𝐿
Horizontal Deformation for (Pinned- Pinned) D1=1
D4
D3
𝜃
D2
D1=1

40. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
39
EA
L
sin
2
𝜃
𝐸𝐴
𝐿
sin 𝜃 cos θ
EA
L
sin
2
𝜃
𝐸𝐴
𝐿
sin 𝜃 cos θ
[
∑
𝐸𝐴
𝐿
sin 𝜃 cos 𝜃
∑
𝐸𝐴
𝐿
sin2
𝜃
−
𝐸𝐴
𝐿
sin 𝜃 cos 𝜃
−
𝐸𝐴
𝐿
sin2
𝜃 ]
[
K11 K12 K13 K14 K15
K21 K22 K23 K24 K25
K31 K32 K33 K34 K35
K41 K42 K43 K44 K45
K51 K52 K53 K54 K55]
hown due to the
s
russ
T
at (D) and internal forces for the
Calculate the Joint Displacement
(4)
Example
all members are pin connected.
ll members,
a
for
2
A=0.05m
,
2
ton/m
7
2*10
where E=
,
shown
Load
Solution:
?
DOF
-
1
C
C
A
B
Verticall Deformation for (Pinned - Pinned) D2=1
D3
𝜃
D4
D2=1
D1
//‫َبينى‬‫ي‬‫ت‬
( ‫خوارةوة‬ ‫َوةى‬‫ي‬‫ش‬ ‫بةم‬
∆
( ‫هؤى‬ ‫َتة‬‫ي‬‫كةدةب‬ ‫دةدؤزينةوة‬ )
Axial Force
‫لة‬ )
‫ئةندامةكاندا‬
( ‫ئةم‬ ‫ئةوةى‬ ‫بةر‬ ‫لة‬.
∆
‫شيكارى‬ ‫لة‬ ‫كة‬ )
‫ئةندامةكة‬ ‫تةوةرةى‬ ‫ئاراستةى‬ ‫بؤ‬ ‫ِين‬‫ر‬‫بيانطؤ‬ ‫َويستة‬‫ي‬‫ث‬ ‫طشتني‬ ‫تةوةرةى‬ ‫ئاراستةى‬ ‫بة‬ ‫ئةيدؤزينةوة‬ ‫ِيزكراوةكةدا‬‫ر‬
.
𝜃
∆Ax
∆Ay
∆Bx
∆By
A
B
∆= (∆By − ∆Ay) sin 𝜃 + (∆Bx − ∆Ax) cos 𝜃
1
4m
4m
3m
50 ton
A
B
D
D3
D4
D5
D2
D1
DOF=5 Stiffness matrix=(5*5)
[
F1
F2
F3
F4
F5 ]
=
[
D1
D2
D3
D4
D5 ]
(F= KD)
A
B
D

41. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
40
[
∑
𝐸𝐴
𝐿
cos2
𝜃{𝐴𝐵,𝐴𝐷}
−
𝐸𝐴
𝐿
cos2
𝜃
−
𝐸𝐴
𝐿
𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃
−
𝐸𝐴
𝐿
cos2
𝜃
−
𝐸𝐴
𝐿
𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃 ]
[
K11
K21
K31
K41
K51]
= =
[
0.378EA
−0.25EA
0
−0.128EA
−0.096EA]
[
−
𝐸𝐴
𝐿
cos2
𝜃
∑
𝐸𝐴
𝐿
cos2
𝜃{𝐴𝐵,𝐵𝐶,𝐵𝐷}
∑
𝐸𝐴
𝐿
𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃{𝐵𝐴,𝐵𝐶,𝐵𝐷}
−
𝐸𝐴
𝐿
cos2
𝜃
−
𝐸𝐴
𝐿
𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃 ]
[
K12
K22
K32
K42
K52]
= =
[
−0.25EA
0.5EA
0
0
0 ]
[
−
𝐸𝐴
𝐿
𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃
∑
𝐸𝐴
𝐿
𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃{𝐵𝐴,𝐵𝐶,𝐵𝐷}
∑
𝐸𝐴
𝐿
sin2
𝜃{𝐵𝐴,𝐵𝐶,𝐵𝐷}
−
𝐸𝐴
𝐿
𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃
−
𝐸𝐴
𝐿
sin2
𝜃 ]
[
K13
K23
K33
K43
K53]
= =
[
0
0
1/3EA
0
−1/3EA]
ton, m)
s
(Unit
K =? When DOF=1
,
=?
ER)
P
(
-
2
EA
L
sinθ cos θ
EA
L
cos2
θ
EA
L
sin2
θ
Angle(θ)
EA/L
)
(ton/m
)
2
ton/m
(
E
)
2
(m
A
(m)
L
Member
Point
0
0.25 EA
0
0
0.25 EA
2*107
0.05
4
AB
A
0.096EA
0.128EA
0.072EA
36.87
0.2 EA
2*107
0.05
5
AD
0.096EA
0.378EA
0.072EA
∑ 𝐴𝐵, 𝐴𝐷
0
0.25 EA
0
180
0.25 EA
2*107
0.05
4
BA
B 0
0.25 EA
0
0
0.25 EA
2*107
0.05
4
BC
0
0
1/3EA
90
1/3 EA
2*107
0.05
3
BD
0
0.5EA
1/3EA
∑ 𝐵𝐴, 𝐵𝐶, 𝐵𝐷
0
0
1/3EA
270
1/3EA
2*107
0.05
3
DB
D
0.096EA
0.128EA
0.072EA
216.87
0.2 EA
2*107
0.05
5
DA
-0.096EA
0.128EA
0.072EA
323.13
0.2 EA
2*107
0.05
5
DC
0
0.256EA
1.432/3EA
∑ 𝐷𝐴, 𝐷𝐵
,DC
=0) to obtain Column (1)
2,3,4,5
=1, (D
1
D
=0) to obtain Column (2)
1,3,4,5
=1, (D
2
D
)
3
=0) to obtain Column (
1,2,4,5
=1, (D
3
D

42. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
41
[
−
𝐸𝐴
𝐿
cos2
𝜃
−
𝐸𝐴
𝐿
cos2
𝜃
−
𝐸𝐴
𝐿
𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃
∑
𝐸𝐴
𝐿
cos2
𝜃{𝐷𝐴,𝐷𝐶,𝐷𝐵}
∑
𝐸𝐴
𝐿
𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃{𝐷𝐴,𝐷𝐶,𝐷𝐵}]
[
K14
K24
K34
K44
K54]
= =
[
−0.128EA
0
0
0.256EA
0 ]
[
−
𝐸𝐴
𝐿
𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃
−
𝐸𝐴
𝐿
𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃
−
𝐸𝐴
𝐿
sin2
𝜃
∑
𝐸𝐴
𝐿
𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃{𝐷𝐴,𝐷𝐶,𝐷𝐵}
∑
𝐸𝐴
𝐿
sin2
𝜃{𝐷𝐴,𝐷𝐶,𝐷𝐵} ]
[
K15
K25
K35
K45
K55]
= =
[
−0.096EA
0
−1/3EA
0
1.432/3EA]
[
K11 K12 K13 K14 K15
K21 K22 K23 K24 K25
K31 K32 K33 K34 K35
K41 K42 K43 K44 K45
K51 K52 K53 K54 K55]
=
[
0.378𝐸𝐴 −0.25𝐸𝐴 0 −0.128𝐸𝐴 −0.096𝐸𝐴
−0.25𝐸𝐴 0.5𝐸𝐴 0 0 0
0 0 1/3𝐸𝐴 0 −1/3𝐸𝐴
−0.128𝐸𝐴 0 0 0.256𝐸𝐴 0
−0.096𝐸𝐴 0 −1/3𝐸𝐴 0. 1.432/3𝐸𝐴]
=0) to obtain Column (4)
1,2,3,5
=1, (D
4
D
=0) to obtain Column (5)
4
1,2,4,
=1, (D
5
D
Loads:
Joint
At
?
Joint Force Vector
-
3
4-
50 ton
[
F1
F2
F3
F4
F5 ]
=
[
0
0
−50
0
0 ]

43. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
42
[
𝐷1
𝐷2
𝐷3
𝐷4
𝐷5]
[
𝐷1
𝐷2
𝐷3
𝐷4
𝐷5]
=
-1
[
𝐷1
𝐷2
𝐷3
𝐷4
𝐷5]
=
[
0.378𝐸𝐴 −0.25𝐸𝐴 0 −0.128𝐸𝐴 −0.096𝐸𝐴
−0.25𝐸𝐴 0.5𝐸𝐴 0 0 0
0 0 1/3𝐸𝐴 0 −1/3𝐸𝐴
−0.128𝐸𝐴 0 0 0.256𝐸𝐴 0
−0.096𝐸𝐴 0 −1/3𝐸𝐴 0 1.486/3𝐸𝐴]
[
0
0
−50
0
0 ]
=
= ?
nternal Forces
I
-
5
F (Ton) =
EA
L
∆
(m)
)
∆
(
1
𝐸𝐴
)
2
(ton/m
EA/L
)
2
ton/m
(
E
)
2
(m
A
(m)
L
Member
sion
ten
4
33.3
133.4
0.25 EA
2*107
0.05
4
AB
tension
33.33
133.3
0.25 EA
2*107
0.05
4
BC
presion
com
41.67
-
-208.33
0.2 EA
2*107
0.05
5
CD
presion
com
7
41.6
-
-208.34
0.2 EA
2*107
0.05
5
DA
tension
50
150
EA /3
2*107
0.05
3
BD
Joint Displacement at (D)
=
(D)
Horizontal Displacement at
=
(D)
Vertical Displacement at
‫بةراوردى‬
‫منوونةكة‬ ‫شيكارى‬ ‫ئةجنامى‬
‫بة‬
‫ئةجنامى‬
‫سؤفت‬
:‫َرةكان‬‫ي‬‫و‬
Note
A
D
BD
D
C
C
B
B
A
Force (ton)
-41.67
50
-41.67
33.33
33.34
‫منوونةكة‬ ‫ئةجنامى‬
OK
-41.67
50
-41.67
33.33
33.33
STAAD V8i
OK
-41.67
50
-41.67
33.33
33.33
SAP2000
‫بة‬ ‫ئةجنامةكان‬ ‫ئايا‬
(
ETABS
‫ضةند‬ )
‫ن‬
‫؟‬
?
?
?
?
?
ETABS
[
0
0
−50
0
0 ]
1
𝐸𝐴
[
0.378 −0.25 0 −0.128 −0.096
−0.25 0.5 0 0 0
0 0 1/3 0 −1/3
−0.128 0 0 0.256 0
−0.096 0 −1/3 0 1.486/3]
1
𝐸𝐴
[
−266.67
−133.33
−675
−133.33
−525 ]
m
1
𝐸𝐴
∗ (−133.33) = -1.3333*10-4
= 1.3333*10-4
m
1
𝐸𝐴
∗ (−525) = -5.25*10-4
= 5.25*10-4
m

44. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
43
A
Y
Local Axis
A
DOF=6
RY
RX
RZ
RZ
RX
RY
‫َماى‬‫ي‬‫ه‬
‫َوة‬‫ي‬‫ش‬ ‫بةم‬ ‫زةبرةكان‬ ‫ئاراستةى‬
‫َنة‬‫ي‬‫و‬
‫دةكةين‬
:
x
: ( 3D Structure )
‫دةطوجن‬ ‫كة‬ ‫َكهاتانةن‬‫ي‬‫ث‬ ‫ئةو‬
‫َت‬‫ي‬
‫َز‬‫ي‬‫ه‬
‫زةبر‬ ‫و‬
(
Load, Moment
)
‫َكدا‬‫ي‬‫ِوتةخت‬‫ر‬ ‫هةر‬ ‫لة‬
‫كار‬
‫ي‬
‫خبةنة‬ ‫طةريان‬
‫سةر‬
.
//‫َبينى‬‫ي‬‫ت‬
‫شيكارى‬ ‫لة‬
(
Beam, Frame
)
‫لة‬
(
3D Structure
( ‫بة‬ ‫راورد‬ ‫بة‬ )
2D Structure
( َ‫ى‬‫س‬ )
DOF
)
‫وةك‬ ‫َت‬‫ي‬‫دةب‬ ‫زياد‬
‫سةرةوةدا‬ ‫َنةكةى‬‫ي‬‫و‬ ‫لة‬
‫دةيبينني‬
.
‫ئةندا‬ ‫َذاى‬‫ي‬‫در‬ ‫بة‬ ‫كة‬ ‫ِانةى‬‫ر‬‫سو‬ ‫ئةو‬ ‫َت‬‫ي‬‫ِةضاوبكر‬‫ر‬ ‫َويستة‬‫ي‬‫ث‬ ‫كة‬ ‫ئةوةى‬ ‫وة‬
( ‫َتة‬‫ي‬‫دةب‬ ‫ِوودةدات‬‫ر‬ ‫مةكة‬
Torsion
)
( ‫ِةضاوكردنى‬‫ر‬ َ‫ل‬‫لةطة‬
moment of Inertia
)
.‫َةتةكان‬‫ل‬‫حا‬ ‫طشت‬ ‫بؤ‬
4- 3D Beams Analysis by Direct Stiffness Method
‫هةنطاوةكانى‬
‫شيكاركردن‬
:
1
-
( Degree of Freedom )
‫ى‬
(
Structure
)
‫دةكةين‬ ‫ديارى‬ ‫كة‬ ‫ة‬
‫َةى‬‫ل‬‫جو‬ ‫لة‬ ‫بريتية‬ ‫كة‬
‫َةكان‬‫ل‬‫خا‬
(
Translations and Rotations
)
‫َوة‬‫ي‬‫ش‬ ‫بةم‬
‫َت‬‫ي‬‫دةردةكةو‬ ‫بؤ‬ ‫ماتريكسةكةمان‬ ‫ستيفنس‬ ‫قةبارةى‬
.
2
-
(
Structure
‫و‬ ‫دةكةين‬ ‫بةش‬ ‫بةش‬ ‫َةكاندا‬‫ل‬‫خا‬ ‫لة‬ ‫ةكة‬ )
(
Fixed End Reaction
)
‫بؤ‬ ‫ئةدؤزينةوة‬
‫يةكةلة‬ ‫يةك‬
Deformation
(
Rotation, Shear , Axial
)
( ‫َى‬‫ي‬‫ث‬ ‫بة‬
DOF
‫دةدؤزينةوة‬ ‫ماتريكسةكة‬ ‫ستيفنس‬ ‫بةمةش‬ ‫كان‬ ‫ة‬ )
‫خوارةوة‬ ‫َوةى‬‫ي‬‫ش‬ ‫بةم‬
.
D6 D5
D4
D3
D2
D1
Z
Y
X
Z
Global Axis

45. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
44
)
1
=
z
θ
(
ixed
F
-
for Fixed
Deformation
Rotation
`
)
=1
y
θ
(
fixed
-
for Fixed
Deformation
Rotation
Z
Z
Y
6𝐸𝐼𝑦𝜃𝑦
𝐿2
6𝐸𝐼𝑦𝜃𝑦
𝐿2
4𝐸𝐼𝑦𝜃𝑦
𝐿
2𝐸𝐼𝑦𝜃𝑦
𝐿
Z
Y
6𝐸𝐼𝑦
𝐿2
6𝐸𝐼𝑦
𝐿2
4𝐸𝐼𝑦
𝐿
2𝐸𝐼𝑦
𝐿
Z
Y
Z
Y
6𝐸𝐼𝑦𝜃𝑦
𝐿2
6𝐸𝐼𝑦𝜃𝑦
𝐿2
2𝐸𝐼𝑦𝜃𝑦
𝐿
4𝐸𝐼𝑦𝜃𝑦
𝐿
Z
Y
6𝐸𝐼𝑦
𝐿2
6𝐸𝐼𝑦
𝐿2
2𝐸𝐼𝑦
𝐿
4𝐸𝐼𝑦
𝐿
Z
Y
Z
Y
Z
Y
6𝐸𝐼𝑧𝜃𝑧
𝐿2
6𝐸𝐼𝑧𝜃𝑧
𝐿2
4𝐸𝐼𝑧𝜃𝑧
𝐿
2𝐸𝐼𝑧𝜃𝑧
𝐿
Z
Y
6𝐸𝐼𝑧
𝐿2
6𝐸𝐼𝑧
𝐿2
2𝐸𝐼𝑧
𝐿
4𝐸𝐼𝑧
𝐿
Z
Y
6𝐸𝐼𝑧𝜃𝑧
𝐿2
6𝐸𝐼𝑧𝜃𝑧
𝐿2
2𝐸𝐼𝑧𝜃𝑧
𝐿
4𝐸𝐼𝑧𝜃𝑧
𝐿
Y
6𝐸𝐼𝑧
𝐿2
6𝐸𝐼𝑧
𝐿2
4𝐸𝐼𝑧
𝐿
2𝐸𝐼𝑧
𝐿
Z
Y

46. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
45
)
1
=
x
θ
(
ixed
F
-
for Fixed
Deformation
Rotation
=1)
y
(∆
Fixed
-
for Fixed
Deformation
Shear
Z
𝐺𝐼𝑥𝜃𝑥
𝐿
𝐺𝐼𝑥𝜃𝑥
𝐿
∆y
∆y
6𝐸𝐼𝑧∆𝑦
𝐿2
6𝐸𝐼𝑧∆𝑦
𝐿2
12EIz∆y
L3
12EIz∆y
L3
Z
Y
Z
𝐺𝐼𝑥𝜃𝑥
𝐿
𝐺𝐼𝑥𝜃𝑥
𝐿
𝐺𝐼𝑥
𝐿
𝐺𝐼𝑥
𝐿
Y
Z
6𝐸𝐼𝑧
𝐿2
6𝐸𝐼𝑧
𝐿2
12EIz
L3
12EIz
L3
Y
Y Y
6𝐸𝐼𝑧∆𝑦
𝐿2
6𝐸𝐼𝑧∆𝑦
𝐿2
12EIz∆y
L3
12EIz∆y
L3
Z
Y
12EIz
L3
6𝐸𝐼𝑧
𝐿2
6𝐸𝐼𝑧
𝐿2
12EIz
L3
Z
𝐺𝐼𝑥
𝐿
𝐺𝐼𝑥
𝐿
Y
Z
Y
Z
Y
Y
Z

47. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
46
Shear Deformation for Fixed-Fixed(∆z=1)
=1)
x
(∆
Fixed
-
for Fixed
Deformation
Axial
( ‫َوازى‬‫ي‬‫ش‬ ‫تةنها‬ ‫ئةطةر‬
Fixed-Fixed
)
‫َرين‬‫ي‬‫َبذ‬‫ل‬‫هة‬
‫ئةندامةكان‬ ‫طشت‬ ‫بؤ‬ ‫ِدةكةينةوة‬‫ر‬‫ث‬ ‫خوارةوة‬ ‫خشتةى‬ ‫ئةم‬ ‫ئةوا‬
.
3
-
( ‫َى‬‫ل‬‫خا‬ ‫ِةضاوكردنى‬‫ر‬
3,4,5
( ‫بةشى‬ ‫لة‬ ‫كة‬ )
Beam
.‫َبينيةكان‬‫ي‬‫ت‬ ‫طشت‬ َ‫ل‬‫لةطة‬.‫بامسانكرد‬ )
x
I
y
I
z
I
E
A
H
W
L
Member
A-B
EA/L
/L
x
I
G
3
/L
y
12EI
3
/L
z
12EI
2
/L
y
6EI
2
/L
z
6EI
/L
y
EI
4
/L
y
EI
2
/L
z
EI
4
/L
z
2EI
Member
A-B
𝐸𝐴
𝐿
𝐸𝐴∆𝑥
𝐿
Y
Z
Z
Z
Y
𝐸𝐴∆𝑥
𝐿 𝐸𝐴
𝐿
Z
6𝐸𝐼𝑦∆𝑧
𝐿2
12𝐸𝐼𝑦∆𝑧
𝐿3
12𝐸𝐼𝑦∆𝑧
𝐿3
6𝐸𝐼𝑦∆𝑧
𝐿2
∆z
∆x
∆x
𝐸𝐴∆𝑥
𝐿
𝐸𝐴∆𝑥
𝐿
𝐸𝐴
𝐿
𝐸𝐴
𝐿
Y
Z
Y
Y
Z
∆z
Y
6𝐸𝐼𝑦
𝐿2
12𝐸𝐼𝑦
𝐿3
12𝐸𝐼𝑦
𝐿3
6𝐸𝐼𝑦
𝐿2
6𝐸𝐼𝑦
𝐿2
12𝐸𝐼𝑦
𝐿3
12𝐸𝐼𝑦
𝐿3
6𝐸𝐼𝑦
𝐿2
Y
Z
Y
Z
Y
Z
Y
Z
6𝐸𝐼𝑦∆𝑧
𝐿2
12𝐸𝐼𝑦∆𝑧
𝐿3
12𝐸𝐼𝑦∆𝑧
𝐿3
6𝐸𝐼𝑦∆𝑧
𝐿2
Y
Z
Y
Z

48. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
47
(0,6,0)
D2
(4,0,0)
(4,0,0)
A
A
B
C
B
C
6m
2000kg
1000kg
A
B
C
(0,0,0)
(0,6,0)
D1
D3
D4
D5
D6
[
K11 K12 K13 K14 K15 K16
K21 K22 K23 K24 K25 K26
K31 K32 K33 K34 K35 K36
K41 K42 K43 K44 K45 K46
K51 K52 K53 K54 K55 K56
K61 K62 K63 K64 K65 K66 ]
(0,0,0)
(X,Y,Z)
Example (5) Draw B.M.D(z)&(y) for the Beams shown Space due to the Loads shown, where
cross section of members are (0.9*0.4)m and E=2.2e+9kg/m2, G=9.167e+8 kg/m2 (member
Loads are at the center of members).
4m
3D View Top View
Solution:
1- DOF?
``
Rad)
kg, m,
s
(Unit
When DOF=1
K =?
,
=?
(FER)
-
2
`
)
4
(m
x
I
)
4
(m
y
I
)
4
(m
z
I
)
2
( kg/m
G
)
2
kg/m
(
E
)
2
(m
A
(m)
W
(m)
H
(m)
L
Member
0.01384148026
0.0048
0.0243
9.167e+8
2.2e+9
0.36
0.4
0.9
6
AB
0.01384148026
0.0048
0.0243
9.167e+8
2.2e+9
0.36
0.4
0.9
4
BC
Z
X
Y
Global Axis
DOF=6 Stiffness matrix=(6*6)
(F=KD) [
F1
F2
F3
F4
F5
F6 ]
=
[
D1
D2
D3
D4
D5
D6 ]
Y
W
H Z
Iz =
WH3
12
Iy =
HW3
12
Ix =J = HW3
(
1
3
− 0.21
𝑊
𝐻
(1 −
𝑊4
12𝐻4))

49. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
48
[K11 … . … K16] =[198586666.66667 0 0 0 0 −1760000]
[K21 … . … K26] = [0 133980000 0 0 0 3960000]
[K31 … . … K36] = [0 0 12993750 8910000 −20047500 0]
D1=1, (D2,3,4,5,6=0) to obtain Row(1)
D2=1, (D1,3,4,5,6=0) to obtain Row (2)
D3=1, (D1,2,4,5,6=0) to obtain Row (3)
EA/L
/L
x
I
G
3
/L
y
12EI
3
/L
z
12EI
2
/L
y
6EI
2
/L
z
6EI
/L
y
EI
4
/L
y
EI
2
/L
z
EI
4
/L
z
2EI
Member
132 e+6
2114747.49272
586666.67
2970000
1760000
8910000
7040000
3520000
35640000
17820000
AB
198 e+6
3172121.23908
1980000
10023750
3960000
20047500
10560000
5280000
53460000
26730000
BC
B
A
A
A
A
C
A
C
B
1
c
B
A
B
1
1
B
C
B Z
Y
Z
Y
6𝐸𝐼𝑦
𝐿2
12𝐸𝐼𝑦
𝐿3
Z
Y
𝐸𝐴
𝐿
Z
Y
B
C
B Z
Y
Z
Y
B
Global Axis
Z
X
Y
1
12𝐸𝐼𝑦
𝐿3
6𝐸𝐼𝑦
𝐿2
Z
Y
𝐸𝐴
𝐿
Z
Y
B
C
B Z
Y
Z
Y
1
6𝐸𝐼𝑧
𝐿2
12EIz
L3
Z
Y
B
6𝐸𝐼𝑧
𝐿2
12EIz
L3
Z
Y

50. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
49
B
[K41 … . … K46] =[0 0 8910000 38812121.23908 0 0]
[K51 … . … K56] =[0 0 −20047500 0 55574747.49272 0]
1
C
[K61 … . … K66] =[−1760000 3960000 0 0 0 17600000]
C
D4=1, (D1,2,3, 5,6=0) to obtain Row (4)
D5=1, (D1,2,3, 4,6=0) to obtain Row(5)
1
D6=1, (D1,2,3, 4,5=0) to obtain Row(6)
B
A
B
B
A C
B
A
B
C
B Z
Y
1 1
Z
Y
Global Axis
Z
X
Y
Z
Y
𝐺𝐼𝑥
𝐿
6𝐸𝐼𝑧
𝐿2
4𝐸𝐼𝑧
𝐿
Z
Y
B
A
B
C
B Z
Y
1
Z
Y
Z
Y
𝐺𝐼𝑥
𝐿
𝐺𝐼𝑥
𝐿
6𝐸𝐼𝑧
𝐿2
4𝐸𝐼𝑧
𝐿
Z
Y
A
B
C
B
Z
Y
1
Z
Y
6𝐸𝐼𝑦
𝐿2
4𝐸𝐼𝑦
𝐿
Z
Y
6𝐸𝐼𝑦
𝐿2
4𝐸𝐼𝑦
𝐿
Z
Y
X

51. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
50
[
K11 K12 K13 K14 K15 K16
K21 K22 K23 K24 K25 K26
K31 K32 K33 K34 K35 K36
K41 K42 K43 K44 K45 K46
K51 K52 K53 K54 K55 K56
K61 K62 K63 K64 K65 K66 ]
=
[
198586666.66667 0 0 0 0 −1760000
0 133980000 0 0 0 3960000
0 0 12993750 8910000 −20047500 0
0 0 8910000 38812121.23908 0 0
0 0 −20047500 0 55574747.49272 0
−1760000 3960000 0 0 0 17600000 ]
1000 kg.m
[
1000
1000
−2500
−3750
0
1000 ]
=
[
198586666.66667 0 0 0 0 −1760000
0 133980000 0 0 0 3960000
0 0 12993750 8910000 −20047500 0
0 0 8910000 38812121.23908 0 0
0 0 −20047500 0 55574747.49272 0
−1760000 3960000 0 0 0 17600000 ][
D1
D2
D3
D4
D5
D6 ]
for member Loads:
Reaction?
Fixed End
-
3
4- Joint Force Vector?
Invert of FER=
Joint Force Vector =
3750 kg.m
A
5000 kg
3750 kg. m
2500 kg
B
2500 kg
2500 kg
3750 kg.m
1000 kg.m
C
B
1000 kg.m
1000 kg.m
C
B
3750 kg.m
A
B
2500 kg
2500 kg
3750 kg.m
1000 kg. m
B
[
F1
F2
F3
F4
F5
F6 ]
=
[
1000
1000
−2500
−3750
0
1000 ]

52. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
51
-1
[
D1
D2
D3
D4
D5
D6 ]
=
[
198586666.66667 0 0 0 0 −1760000
0 133980000 0 0 0 3960000
0 0 12993750 8910000 −20047500 0
0 0 8910000 38812121.23908 0 0
0 0 −20047500 0 55574747.49272 0
−1760000 3960000 0 0 0 17600000 ] [
1000
1000
−2500
−3750
0
1000 ]
[
D1
D2
D3
D4
D5
D6 ]
=
[
5.5324674E − 06
5.8067094𝐸 − 06
−4.4103185𝐸 − 04
4.6272605𝐸 − 06
−1.5909359𝐸 − 04
5.6064919𝐸 − 05 ]
5- Moment =?
MZ=? For Member AB
FER=
MZ=? For Member BC
FER=
D3=1
Z
Y
6𝐸𝐼𝑧∆𝑦
𝐿2
4𝐸𝐼𝑧𝜃𝑧
𝐿
D4=1
A
C
6𝐸𝐼𝑧∆𝑦
𝐿2
3750 kg.m
2𝐸𝐼𝑧𝜃𝑧
𝐿
Z
Y
B
D5=1
C
B
4𝐸𝐼𝑧𝜃𝑧
𝐿
2𝐸𝐼𝑧𝜃𝑧
𝐿
6𝐸𝐼𝑧∆𝑦
𝐿2
D3=1
C
B 6𝐸𝐼𝑧∆𝑦
𝐿2
5000 kg
B
2500 kg
2500 kg
3750 kg.m
B
A
M(Z)AB = MFER(AB) +
2EIzθz
L
+
6EIz∆y
L2 = -3750 +
2∗2.2∗109∗0.0243∗4.6272605E−06
6
+
6∗2.2∗109∗0.0243∗(−4.4103185E−04)
62
∴ 𝐌(𝐙)𝐀𝐁 = −𝟕𝟓𝟗𝟕. 𝟏𝟒 𝐤𝐠. 𝐦
M(Z)CB = MFER(CB) +
2EIzθz
L
−
6EIz∆y
L2 = 0 +
2∗2.2∗109∗0.0243∗−1.5909359𝐸−04
4
−
6∗2.2∗109∗0.0243∗(−4.4103185𝐸−04)
42
∴ 𝐌(𝐙)𝐂𝐁 = −𝟒𝟓𝟖𝟗. 𝟎𝟏 𝐤𝐠. 𝐦
M(Z)BA = MFER(BA) +
4EIzθz
L
+
6EIz∆y
L2 =+3750 +
4∗2.2∗109∗0.0243∗4.6272605E−06
6
+
6∗2.2∗109∗0.0243∗(−4.4103185E−04)
62
∴ 𝐌(𝐙)𝐁𝐀 = −𝟏𝟒. 𝟔𝟖 𝐤𝐠. 𝐦

53. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
52
MY=? For Member AB
FER =
MY=? For Member CB
FER=
1000 kg. m
C
B
1000 kg. m
B
6𝐸𝐼𝑦∆𝑧
𝐿2
2𝐸𝐼𝑦𝜃𝑦
𝐿
6𝐸𝐼𝑦∆𝑧
𝐿2
A
4𝐸𝐼𝑦𝜃𝑦
𝐿
2𝐸𝐼𝑦𝜃𝑦
𝐿
B
MBC = MFER(BC) +
4EIzθz
L
−
6EIz∆y
L2 = 0 +
4∗2.2∗109∗0.0243∗−1.5909359𝐸−04
4
−
6∗2.2∗109∗0.0243∗(−4.4103185𝐸−04)
42
∴ 𝐌(𝐙)𝐁𝐂 = 𝟑𝟑𝟔. 𝟒𝟒 𝐤𝐠. 𝐦
Z
Y
A
D6=1
A
6𝐸𝐼𝑦∆𝑧
𝐿2
M(Y)AB = MFER(AB) +
2EIyθy
L
+
6EIy∆z
L2 = 0+
2∗2.2∗109∗0.0048∗ 5.6064919E−05
6
−
6∗2.2∗109∗0.0048∗ 5.5324674E−06
62
B
∴ 𝐌(𝐘)𝐀𝐁 = 𝟏𝟖𝟕. 𝟔𝟏 𝐤𝐠. 𝐦
D1=1
M(Y)BA = MFER(BA) +
4EIyθy
L
+
6EIy∆z
L2 = 0+
4∗2.2∗109∗0.0048∗ 5.6064919E−05
6
−
6∗2.2∗109∗0.0048∗ 5.5324674E−06
62
∴ 𝐌(𝐘)𝐁𝐀 = 𝟑𝟖𝟒. 𝟗𝟔 𝐤𝐠. 𝐦
B
4𝐸𝐼𝑦𝜃𝑦
𝐿
Z
Y
C
C
B
D6=1
6𝐸𝐼𝑦∆𝑧
𝐿2
D2=1
M(Y)CB = MFER(CB) +
2EIyθy
L
+
6EIy∆z
L2 = +1000+
2∗2.2∗109∗0.0048∗ 5.6064919E−05
4
+
6∗2.2∗109∗0.0048∗ 5.8067094E−06
42
∴ 𝐌(𝐘)𝐂𝐁 = 𝟏𝟑𝟏𝟗. 𝟎𝟐 𝐤𝐠. 𝐦
M(Y)BC = MFER(BC) +
4EIyθy
L
+
6EIy∆z
L2 = -1000+
4∗2.2∗109∗0.0048∗ 5.6064919E−05
4
+
6∗2.2∗109∗0.0048∗ 5.8067094E−06
42
∴ 𝐌(𝐘)𝐁𝐂 = −𝟑𝟖𝟒. 𝟗𝟔 𝐤𝐠. 𝐦

54. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
53
‫بةراوردى‬
‫ئ‬
‫بة‬ ‫منوونةكة‬ ‫شيكارى‬ ‫ةجنامى‬
‫ئةجنامى‬
‫سؤفت‬
:‫َرةكان‬‫ي‬‫و‬
Note
C
C
B
A
B
A
m)
Moment(kg.
4589.01
-336.44
-14.68
7597.14
z
M
‫ئ‬
‫منوونةكة‬ ‫ةجنامى‬
-1319.02
-384.96
-384.96
187.61
y
M
OK
4589.02
-336.44
-14.68
7597.14
z
M
STAAD V8i
OK
-1319.02
-384.96
-384.96
187.61
y
M
OK
4589.02
-336.43
-14.68
7597.14
z
M
SAP2000
OK
-1319.02
-384.96
-384.96
187.61
y
M
?
?
?
?
ETABS
187.61
851.99
384.96 1319.02
3708.76
-7597.14
14.68
336.44
B.M.D(z)( kg. m)
-4589.01
-3791.24
B.M.D(y)( kg. m)
1148.01

55. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
54
5- 3D Frames Analysis by Direct Stiffness Method
//‫َبينى‬‫ي‬‫ت‬
‫هةنطاوةكانى‬
( ‫شيكاركردنى‬
3D Frame
‫تةنها‬ )
(
5
‫َشوو‬‫ي‬‫ث‬ ‫هةنطاوةكانى‬ )
.‫َبينيةكان‬‫ي‬‫ت‬ ‫طشت‬ َ‫ل‬‫طة‬ ‫لة‬ ‫َنني‬‫ي‬‫دةه‬ ‫بةكار‬
Example (6) Draw all diagrams for the Space Frame shown due to the Loads Shown, where Section of
beams are (0.6*0.4)m and Columns (0.4*0.4)m, E=3*109
kg/m2
, G=1.25*109
kg/m2
.
Solution:
1- DOF?
F
D22
D23
D19
D20
D21
D24
D16
D17
D13
D14
D15
D18
D10
D11
D7
D8
D9
D12
D4
D5
D1
D2
D3
5 ton
2 ton
1 ton
1 ton
F
E
G
H
D
C
B
A
6m
8m
4m
Y
X
Z
Global Axis
E
G
H
D
C
B
A
D6
Y
X
Z
Global Axis
DOF=24 Stiffness matrix= (24*24)

56. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
55
G =
E
2.4
//‫َبينى‬‫ي‬‫ت‬
( ‫َوان‬‫ي‬‫ن‬ ‫ثةيوةندى‬
E,G
( ‫تةنها‬ ‫ئةوةى‬ ‫بؤ‬ ‫ثرسيارةكة‬ ‫َرةى‬‫ي‬‫طو‬ ‫بة‬ ‫دةدؤزينةوة‬ )
E
‫َنني‬‫ي‬‫به‬ ‫بةكار‬ )
.
2- (FER) =? , K =? When DOF=1 (Units Ton, m, Rad)
4
x m
I
4
y m
I
4
m
z
I
kg/m2
G
kg/m2
E
(m2)
A
(m)
W
(m)
H
(m)
L
Member
0.00360533333
0.0256/12
0.0256/12
E/2.4
E
0.16
0.4
0.4
4
AB
0.00751249383
0.0032
0.0072
E/2.4
E
0.24
0.4
0.6
6
BC
0.00360533333
0.0256/12
0.0256/12
E/2.4
E
0.16
0.4
0.4
4
CD
0.00751249383
0.0032
0.0072
E/2.4
E
0.24
0.4
0.6
8
CE
0.00360533333
0.0256/12
0.0256/12
E/2.4
E
0.16
0.4
0.4
4
EF
0.00751249383
0.0032
0.0072
E/2.4
E
0.24
0.4
0.6
6
EG
0.00360533333
0.0256/12
0.0256/12
E/2.4
E
0.16
0.4
0.4
4
GH
0.00751249383
0.0032
0.0072
E/2.4
E
0.24
0.4
0.6
8
GB
2
/L
y
6EI
2
/L
z
6EI
/L
y
EI
4
/L
y
EI
2
/L
z
EI
4
/L
z
2EI
Member
0.0008E
0.0008E
E*0.0064/3
E*0.0032/3
E*0.0064/3
E*0.0032/3
AB
E*0.0016/3
0.0012E
E*0.0064/3
E*0.0032/3
0.0048E
0.0024E
BC
0.0008E
0.0008E
E*0.0064/3
E*0.0032/3
E*0.0064/3
E*0.0032/3
CD
0.0003E
0.000675E
0.0016E
0.0008E
0.0036E
0.0018E
CE
0.0008E
0.0008E
E*0.0064/3
E*0.0032/3
E*0.0064/3
E*0.0032/3
EF
E*0.0016/3
0.0012E
E*0.0064/3
E*0.0032/3
0.0048E
0.0024E
EG
0.0008E
0.0008E
E*0.0064/3
E*0.0032/3
E*0.0064/3
E*0.0032/3
GH
0.0003E
0.000675E
0.0016E
0.0008E
0.0036E
0.0018E
GB
EA/L
/L
x
I
G
3
/L
y
12EI
3
/L
z
12EI
Member
0.04E
E*0.00360533333/9.6
0.0004E
0.0004E
AB
0.04E
E*0.00751249383/14.4
E*0.0016/9
0.0004E
BC
0.04E
E*0.00360533333/9.6
0.0004E
0.0004E
CD
0.03E
E*0.00751249383/19.2
0.000075E
E*0.0216/128
CE
0.04E
E*0.00360533333/9.6
0.0004E
0.0004E
EF
0.04E
E*0.00751249383/14.4
E*0.0016/9
0.0004E
EG
0.04E
E*0.00360533333/9.6
0.0004E
0.0004E
GH
0.03E
E*0.00751249383/19.2
0.000075E
E*0.0216/128
GB
[
F1
F2
F3
F4
F5
.
.
.
F24]
=
[
K11 K12 K13 K14 K15 K16 . . . K124
K21 K22 K23 K24 K25 K26 . . . K224
K31 K32 K33 K34 K35 K36 . . . K324
K41 K42 K43 K44 K45 K46 . . . K424
K51 K52 K53 K54 K55 K56 . . . K524
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
K241 K242 K243 K244 K245 K246 . . . K2424] [
D1
D2
D3
D4
D5
.
.
.
D24]
(F=KD)

57. Stiffness Method (Matrix Structural Analysis) Ali Muhammad Shekha
56
D1=1 to obtain Row(1)
D2=1 to obtain Row (2)
D3=1 to obtain Row (3)
12𝐸𝐼𝑦
𝐿3
C
6𝐸𝐼𝑧
𝐿2
6𝐸𝐼𝑧
𝐿2
6𝐸𝐼𝑧
𝐿2
6𝐸𝐼𝑧
𝐿2
12𝐸𝐼𝑧
𝐿3
12𝐸𝐼𝑧
𝐿3
12𝐸𝐼𝑧
𝐿3
12𝐸𝐼𝑧
𝐿3
12𝐸𝐼𝑦
𝐿3
12𝐸𝐼𝑦
𝐿3
6𝐸𝐼𝑦
𝐿2
6𝐸𝐼𝑦
𝐿2
𝐸𝐴
𝐿
𝐸𝐴
𝐿
C
𝐸𝐴
𝐿
12𝐸𝐼𝑦
𝐿3
12𝐸𝐼𝑦
𝐿3
Row3=
𝐸𝐴
𝐿
1 Y
A
B
X
Z
12𝐸𝐼𝑧
𝐿3
12𝐸𝐼𝑧
𝐿3
6𝐸𝐼𝑦
𝐿2
𝐸𝐴
𝐿
C
6𝐸𝐼𝑦
𝐿2
6𝐸𝐼𝑦
𝐿2
Row1=
[0.040475E 0 0 0 −0.0008E −0.0003E −0.04𝐸 0 0 0 0 0 0 0 0 0 0 0 −0.000075E 0 0 0 0 −0.0003E]
[0 E ∗ 0.2752/9 0 0.0008E 0 E ∗ 0.0016/3 0 −E ∗ 0.0016/9 0 0 0 E ∗ 0.0016/3 0 0 0 0 0 0 0 −0.03E 0 0 0 0]
B
B Z
A
B
Z
X
Y
6𝐸𝐼𝑧
𝐿2
6𝐸𝐼𝑧
𝐿2
Y
X
Z
Global Axis
1
Y
A
B
X
Z
Y
A
B
X
Z
B
G
1
1
B Z
Y
Z
Y
[0 0 E ∗ 5.1928/128 0.000675𝐸 −0.0012E 0 0 0 −0.0004E 0 −0.0012E 0 0 0 0 0 0 0 0 0 −E ∗ 0.0216/128 0.000675E 0 0]
G
Y
Row2=
Z
Y
C
1
Y
A
B
X
Z
B
G
1
1 B Z
Y
Z
Y
Y
A
B
X
Z
B
B
G
Z
Y
Z
Y
C
12𝐸𝐼𝑦
𝐿3
1
B
G
1
B Z
Y
Z
Y
B
B
G
Z
Y
Z
Y
C