2. By
SAJJAD AHMAD (2009-NUST-MS PhD-Str-06)

3. Geometry of frame
DOFs
Load vector
Stiffness matrices of members
Assemblage and Band width
Equilibrium equation for joint “F”

4. FRAME GEOMETRY

5. 4m
2m 2m 2m 2m 2m 2m 2m
4m

6. PART (A)
DEGREE OF FREEDOM (DOFs)

7. 8 17 26 29
7 30
16 25 28
9 18 27
5 14
23
13
4 22
6 15 24
2 11 20
3 1 10 19
12
21

8. DOFs
Total DOFs are 30
Known DOFs at joints “A, D and G” (by support
conditions)
1, 2, 10, 11, 12, 19, 20, 21 = Zero
Unknown DOFs (30 – 8 = 22)

9. PART (B)
LOAD VECTORS

10. LOAD VECTOR
P = F – FEF
P is load vector
F is nodal force
FEF is the force observed at nodes due to
application of loading between nodal points

11. FORMULATION OF LOAD VECTORS
Frame members 1, 2, 3, 4, 7, 8, 9 and 12 have no
loading between nods so
As FEF is null matrix for local axis so for global
FEF will be

12. FORMULATION OF LOAD VECTORS
For member 5 and 6 FEF will be given by
following

13. FORMULATION OF LOAD VECTORS
For member 5 and 6
For transforming it in to global, transformation
matrix will be used

14. FORMULATION OF LOAD VECTORS
For member 5 and 6 θ = 0 degree
FEF will be given by
Which is

15. FORMULATION OF LOAD VECTORS
For member 10 and 11 FEF will be given by

16. FORMULATION OF LOAD VECTORS
As θ = 0 degree so for global FEF will be same

17. STRUCTURE LOAD VECTOR
By combining all nodal
forces and subtracting
all FEFs we get total
structural load vector

18. PART (C)
STIFFNESS MATRIX OF MEMBERS

19. LOCAL STIFFNESS MATRIX
Local stiffness matrix is given by following matrix

20. LOCAL STIFFNESS MATRIX
INPUT CALCULATIONS
Member 1 AE/L 5000
A (m2) 0.01 12EI/L3 5625
E (t/m2) 2E6 6EI/L2 11250
I (m4) 0.015 4EI/L 30000
L (m) 4 2EI/L 15000
FOR MEMBER 01
5000 0 0 -5000 0 0
0 5625 11250 0 -5625 11250
0 11250 30000 0 -11250 15000
-5000 0 0 5000 0 0
0 -5625 -11250 0 5625 -11250
0 11250 15000 0 -11250 30000

21. LOCAL STIFFNESS MATRIX
INPUT CALCULATIONS
Member 2 AE/L 2773.54
A (m2) 0.01 12EI/L3 960.099
E (t/m2) 2000000 6EI/L2 3461.637
I (m4) 0.015 4EI/L 16641.24
L (m) 7.211 2EI/L 8320.621
FOR MEMBER 02
2773.54 0 0 -2773.54 0 0
0 960.099 3461.637 0 -960.099 3461.637
0 3461.637 16641.24 0 -3461.64 8320.621
-2773.54 0 0 2773.54 0 0
0 -960.099 -3461.64 0 960.099 -3461.64
0 3461.637 8320.621 0 -3461.64 16641.24

22. LOCAL STIFFNESS MATRIX
INPUT CALCULATIONS
Member 5 AE/L 3333.333
A (m2) 0.01 12EI/L3 1666.667
E (t/m2) 2000000 6EI/L2 5000
I (m4) 0.015 4EI/L 20000
L (m) 6 2EI/L 10000
FOR MEMBER 05
3333.333 0 0 -3333.33 0 0
0 1666.667 5000 0 -1666.67 5000
0 5000 20000 0 -5000 10000
-3333.33 0 0 3333.333 0 0
0 -1666.67 -5000 0 1666.667 -5000
0 5000 10000 0 -5000 20000

23. GLOBAL STIFFNESS MATRIX

24. GLOBAL STIFFNESS MATRIX
INPUT
Member 1
θ (Degree) 90
cos θ 0
sin θ 1
A (m2) 0.01
E (t/m2) 2000000
I (m4) 0.015
L (m) 4
FOR MEMBER 1
5625 0 -11250 -5625 0 -11250
0 5000 0 0 -5000 0
-11250 0 30000 11250 0 15000
-5625 0 11250 5625 0 11250
0 5000 0 0 5000 0
-11250 0 15000 11250 0 30000

25. GLOBAL STIFFNESS MATRIX
INPUT
Member 2
θ (Degree) 33.69
cos θ 0.83205
sin θ 0.55469
A (m2) 0.01
E (t/m2) 2000000
I (m4) 0.015
L (m) 7.211
FOR MEMBER 2
2215.546 836.9573 -1920.14 -2215.55 -836.957 -1920.14
836.9573 1518.049 2880.255 -836.957 -1518.05 2880.255
-1920.14 2880.255 16641.24 1920.135 -2880.25 8320.621
-2215.55 -836.957 1920.135 2215.546 836.9573 1920.135
-836.957 1518.049 -2880.25 836.9573 1518.049 -2880.25
-1920.14 2880.255 8320.621 1920.135 -2880.25 16641.24

26. GLOBAL STIFFNESS MATRIX
INPUT
Member 5
θ (Degree) 0
cos θ 1
sin θ 0
A (m2) 0.01
E (t/m2) 2000000
I (m4) 0.015
L (m) 6
FOR MEMBER 5
3333.333 0 0 -3333.33 0 0
0 1666.667 5000 0 -1666.67 5000
0 5000 20000 0 -5000 10000
-3333.33 0 0 3333.333 0 0
0 1666.667 -5000 0 1666.667 -5000
0 5000 10000 0 -5000 20000

27. PART (D)
GLOBAL ASSEMBLAGE AND BAND SEMI WIDTH

28. GLOBAL ASSEMBLAGE
DOFs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 7841 837 -13170 -5625 0 -11250 0 0 0 0 0 0 -2216 -837 -1920
2 837 6518 2880 0 -5000 0 0 0 0 0 0 0 -837 -1518 2880
3 -13170 2880 19641 11250 0 15000 0 0 0 0 0 0 1920 -2880 8321
4 -5625 0 11250 14583 0 0 -5625 0 -11250 0 0 0 -3333 0 0
5 0 5000 0 0 11667 5000 0 -5000 0 0 0 0 0 -1667 5000
6 -11250 0 15000 0 5000 80000 11250 0 15000 0 0 0 0 -5000 10000
7 0 0 0 -5625 0 11250 8958 0 11250 0 0 0 0 0 0
8 0 0 0 0 5000 0 0 6667 5000 0 0 0 0 0 0
9 0 0 0 -11250 0 15000 11250 5000 50000 0 0 0 0 0 0
10 0 0 0 0 0 0 0 0 0 5625 0 -11250 -5625 0 -11250
11 0 0 0 0 0 0 0 0 0 0 5000 0 0 -5000 0
12 0 0 0 0 0 0 0 0 0 -11250 0 30000 11250 0 15000
13 -2216 -837 1920 -3333 0 0 0 0 0 -5625 0 11250 20132 837 1920
14 -837 1518 -2880 0 1667 -5000 0 0 0 0 5000 0 837 14851 -2880
15 -1920 2880 8321 0 5000 10000 0 0 0 -11250 0 15000 1920 -2880 116641
16 0 0 0 0 0 0 -3333 0 0 0 0 0 -5625 0 11250
17 0 0 0 0 0 0 0 1667 -5000 0 0 0 0 5000 0
18 0 0 0 0 0 0 0 5000 10000 0 0 0 -11250 0 15000
19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
22 0 0 0 0 0 0 0 0 0 0 0 0 -3333 0 0
23 0 0 0 0 0 0 0 0 0 0 0 0 0 1667 -5000
24 0 0 0 0 0 0 0 0 0 0 0 0 0 5000 10000

29. GLOBAL ASSEMBLAGE
0 -5000 10000 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-5625 0 -11250 0 0 0 -3333 0 0 0 0 0 0 0 0
0 -5000 0 0 0 0 0 -1667 5000 0 0 0 0 0 0
11250 0 15000 0 0 0 0 -5000 10000 0 0 0 0 0 0
12292 0 11250 0 0 0 0 0 0 -3333 0 0 0 0 0
0 8333 0 0 0 0 0 0 0 0 -1667 5000 0 0 0
11250 0 70000 0 0 0 0 0 0 0 -5000 10000 0 0 0
0 0 0 5625 0 -11250 -5625 0 -11250 0 0 0 0 0 0
0 0 0 0 5000 0 0 -5000 0 0 0 0 0 0 0
0 0 0 -11250 0 30000 11250 0 15000 0 0 0 0 0 0
0 0 0 -5625 0 11250 14583 0 0 -5625 0 -11250 0 0 0
0 0 0 0 5000 0 0 11667 -5000 0 -5000 0 0 0 0
0 0 0 -11250 0 15000 0 -5000 80000 11250 0 15000 0 0 0
-3333 0 0 0 0 0 -5625 0 11250 18958 0 11250 -10000 0 0
0 1667 -5000 0 0 0 0 5000 0 0 51667 40000 0 -45000 45000
0 5000 10000 0 0 0 -11250 0 15000 11250 40000 110000 0 -45000 30000
0 0 0 0 0 0 0 0 0 -10000 0 0 10000 0 0
0 0 0 0 0 0 0 0 0 0 45000 -45000 0 45000 -45000
0 0 0 0 0 0 0 0 0 0 45000 30000 0 -45000 60000

30. DEFLECTED FRAME FROM SAP2000

32. COMPARISON BETWEEN DEFORMATION BY HAND CALCULATONS AND SAP2000
HAND CALCULATION SAP2000 ERROR % ERROR
1 0 1 0 1 0 0
2 0 2 0 2 0 0
3 0.000192 3 0.000640 3 0.000448 0.699258328
4 0.001125 4 0.001250 4 0.000125 0.100006278
5 -0.000532 5 -0.000482 5 0.000051 -0.105428095
6 -0.000186 6 -0.002980 6 -0.002794 0.937441803
7 0.001125 7 0.009250 7 0.008125 0.878379833
8 -0.000639 8 -0.000723 8 -0.000084 0.115717957
9 -0.000253 9 -0.003810 9 -0.003557 0.933711891
10 0 10 0 10 0 0
11 0 11 0 11 0 0
12 0 12 0 12 0 0
13 0.001125 13 0.001150 13 0.000025 0.021746083
14 -0.000937 14 -0.001220 14 -0.000283 0.232283224
15 -0.000017 15 0.000874 15 0.000891 1.018878576
16 0.001125 16 0.009090 16 0.007965 0.87623831
17 -0.001161 17 -0.001770 17 -0.000609 0.343998416
18 0.000050 18 0.000010 18 -0.000040 -3.859000675
19 0 19 0 19 0 0
20 0 20 0 20 0 0
21 0 21 0 21 0 0
22 0.001125 22 0.001120 22 -0.000005 -0.004453084
23 -0.000166 23 -0.000815 23 -0.000649 0.796117634
24 0.000412 24 0.001940 24 0.001528 0.787753348
25 0.001125 25 0.009090 25 0.007965 0.876237562
26 -0.000061 26 -0.001360 26 -0.001299 0.954949649
27 0.000412 27 0.004100 27 0.003688 0.899523869
28 0.001125 28 0.009030 28 0.007905 0.87541522
29 0.000808 29 -0.020180 29 -0.020988 1.040056451
30 0.000446 30 0.011570 30 0.011124 0.961432181

33. AXIAL FORCES

35. SHEAR FORCE DIAGRAM

37. BENDING MOMENT DIAGRAM