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Matrix Structural Analysis, Steel Frame Analysis in SAP2000
By
SAJJAD AHMAD (2009-NUST-MS PhD-Str-06)
Geometry of frame
DOFs
Load vector
Stiffness matrices of members
Assemblage and Band width
Equilibrium equation for joint “F”
FRAME GEOMETRY
4m




2m   2m   2m   2m   2m   2m        2m



                              4m
PART (A)


DEGREE OF FREEDOM (DOFs)
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
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)
PART (B)


LOAD VECTORS
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
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
FORMULATION OF LOAD VECTORS

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

For member 5 and 6

For transforming it in to global, transformation
matrix will be used
FORMULATION OF LOAD VECTORS

For member 5 and 6 θ = 0 degree
FEF will be given by


Which is
FORMULATION OF LOAD VECTORS

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




As θ = 0 degree so for global FEF will be same
STRUCTURE LOAD VECTOR

 By combining all nodal
 forces and subtracting
 all FEFs we get total
 structural load vector
PART (C)


STIFFNESS MATRIX OF MEMBERS
LOCAL STIFFNESS MATRIX
Local stiffness matrix is given by following matrix
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
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
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
GLOBAL STIFFNESS MATRIX
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
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
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
PART (D)


GLOBAL ASSEMBLAGE AND BAND SEMI WIDTH
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
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
DEFLECTED FRAME FROM SAP2000
Matrix Structural Analysis, Steel Frame Analysis in SAP2000
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
AXIAL FORCES
Matrix Structural Analysis, Steel Frame Analysis in SAP2000
SHEAR FORCE DIAGRAM
Matrix Structural Analysis, Steel Frame Analysis in SAP2000
BENDING MOMENT DIAGRAM
Matrix Structural Analysis, Steel Frame Analysis in SAP2000
Matrix Structural Analysis, Steel Frame Analysis in SAP2000
Matrix Structural Analysis, Steel Frame Analysis in SAP2000

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Matrix Structural Analysis, Steel Frame Analysis in SAP2000

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