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CHAPTER VI

NUMERICAL EXAMPLE AND DISCUSSIONS I. Descriptions problem. The offshore platform is given in Fig. 6.1, it is fabricated form high strength steel, the properties of members and load data are given in the tables below. The analyses are carried out for both frame(Fig. 6.2) and and mixed models(Fig. 6.3).

Fig.6.1 Planar offshore platform structure.

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Table 6.1 Properties of steel

Modulus of Elasticity 2.0×108 kN/m2 (2.0×1011 N/m2

Poisson ratio 0.3

Shear Modulus 0.769×108 kN/m2

Weight density 78 kN/m3

Mass density 7.8 T/m3 Table 6.2 Properties of members

Member Diameter (m) Thickness (m) Columns 1.0 0.025 Beams 0.76 0.015 Braces 0.55 0.012

Fig. 6.2 Topology of the frame model.

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Fig. 6.2 Topology of the mixed model. II. Static analysis II.1. Data of loads For static analysis, the acting loads upon to structure consist of: (*) - Self weight load.

(*) The analyses are carried out on the computer : CPU K6-233K, 32Mb DIM RAM.

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- Vertical Loads, they are placed on the top of level of the structure. The details for position an magnitude of a load are indicated in table 6.3

Table 6.3 Load data

Position (at nodes) Magnitude of load Frame model 27, 68 Fy = -20000 kN Mixed model 1760/2064/19

3760/4064/19 (start node/stop/step)

Fy = -117.647 kN Fy = -117.647 kN

To perform analyses, in the static analysis case, in this study used program of thesis which is developed by author and presented in chapter IV. For the mixed model, joints are replaced by superelements. II.2. Results and comparisons of the two models - Table 6.4 Comparison of size of problem and execution times

Features Frame model Mixed model No. of element 91 27 frame elements

5257 shell elementsNo. of nodal points 82 6163 Total execu-tion times

∗ PRE-Processing ∗ Forming the stiff. matrix. ∗ Solving the sys. of Eq. ∗ POST- Processing

auto 6’’ 10’’ 5’’

10’’ 40’’ 20’’ 30’’

Total 21’’ 1’40’’ Table 6.5a Comparison of the displacements (Vertical load condition)

Disp. Ux (m) Uy (m) Rz (rad) Node Frame Mixed Frame Mixed Frame Mixed

5 -0.002794 -0.002814 -0.009928 -0.010406 0.000761 0.000864 11 0.000524 0.000517 -0.022103 -0.024356 -0.000459 -0.000345 17 -0.000734 -0.000784 -0.035896 -0.039175 0.000165 0.000199 23 -0.000980 -0.001267 -0.048618 -0.049254 0.000441 0.000625 29 0.007524 0.007293 -0.062477 -0.067175 -0.003970 -0.004039 30 -0.000038 -0.000034 -0.018327 -0.019857 -0.000806 -0.000854 33 -0.000004 -0.000004 -0.032568 -0.035084 -0.000648 -0.000696 36 0.000005 -0.000005 -0.044942 -0.048417 -0.000806 -0.000874 39 -0.000060 -0.000061 -0.058269 -0.062668 -0.000998 -0.001029

Table 6.5b Comparison of the internal forces (Vertical load condition)

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Intr. forces Normal force (kN) Shear force (kN) Moment (kNm) Elements Frame Mixed Frame Mixed Frame Mixed

1 (1) -20552.230 -20575.455 259.562 235.688 -401.790 -399.0504 (2) -20450.225 -20465.260 167.107 172.438 -996.352 -1060.9188 (3) -20357.875 -20371.012 -17.805 -12.203 222.104 230.24412 (4) -20213.158 -20219.887 87.902 89.959 -407.247 -413.06016 (5) -20106.525 -20109.289 -242.865 -237.230 298.819 299.90338 (11) 167.127 167.389 71.045 73.356 -295.003 -299.15045 (13) 10.737 9.824 52.553 52.476 -143.949 -148.13852 (15) -16.446 -17.007 81.473 83.677 -271.064 -276.05359 (17) 265.590 258.420 100.804 102.378 -298.471 -293.72066 (19) -1807.503 -1811.438 19.114 19.120 184.596 184.947

Table 6.6a Comparison of the displacements (Horizonal load condition)

Displ. Ux (m) Uy (m) Rz (rad) Node Frame Mixed Frame Mixed Frame Mixed

5 0.058895 0.061257 0.002344 0.002915 -0.006114 -0.005719 11 0.087262 0.088530 0.008270 0.009547 -0.003512 -0.003681 17 0.133879 0.136655 0.012068 0.013678 -0.004879 -0.004966 23 0.177627 0.181817 0.013333 0.015076 -0.003841 -0.003742 29 0.262231 0.268742 0.009335 0.011068 -0.017334 -0.018252 30 0.048609 0.049007 -0.012690 -0.013426 0.005848 0.005692 33 0.086783 0.087551 -0.006640 -0.007436 0.002362 0.002182 36 0.129256 0.131242 -0.004823 -0.005309 0.001809 0.001547 39 0.177150 0.180553 -0.009539 -0.010811 0.002979 0.002694

Table 6.6b Comparison of the internal forces (Horizonal load condition)

Intr. forces Normal force (kN) Shear force (kN) Moment (kNm) Elements Frame Mixed Frame Mixed Frame Mixed

1 (1) 14370.604 14340.535 2730.531 2730.613 -10203.92 -10310.594 (2) 14116.610 14086.533 -663.850 -651.908 3101.859 3103.2298 (3) 11354.311 11346.406 -101.084 -103.524 -61.999 66.12412 (4) 8411.776 8403.356 -176.259 -177.213 672.162 678.48116 (5) 5121.107 5127.464 -802.389 -802.539 995.455 1011.47638 (11) 3345.593 3323.947 -12.853 -13.538 818.139 813.20645 (13) 3119.272 3119.410 71.571 75.706 135.597 136.88352 (15) 3408.468 3408.675 15.499 17.392 415.056 413.98859 (17) 4351.613 4350.984 193.443 194.146 55.451 59.65966 (19) -33.225 -34.302 -266.572 -273.180 1994.148 2012.513

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Fig. 6.3a Vertical displacement of the two models

Fig. 6.3b Horizontal displacement of the two models

II. Eigenfrequency analysis

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II.1. Data of Mass Sssuming two masses are placed on top of level of the structure, their position and values are given in table 6.6. To solving eigenfrequency problem, in this test used SAP90 software [45]. Comparisons of results five first natural frequencies of two models are shown in table 6.7.

Table 6.6 Data of Mass

Position (at nodes) Magnitude of masses Frame model 27, 68 Mx = My = 2000 Ton Mixed model. 1760/2064/19

3760/4064/19 Mx = My = 11.7647 Ton Mx = My = 11.7647 Ton

II.2. Results of eigenfrequency and comparisons of the two models The Comparision of five first natural frequencies and the corresponding modeshapes of the two models are given in Table 6.7 and Fig. 6.4 Table 6.7 Comparison of Eigenfrequencies of the two models

Natural frequencies of the two models (Hz)

Model Mode 1st 2nd 3rd 4th 5th Frame 0.468007 1.613220 2.043541 2.424538 7.608842 Mixed 0.432196 1.583171 1.969066 2.358485 5.935195 Error (%) 8.28 1.89 3.782 2.8 28.19

Circular frequencies of the two models (rad/sec)

Model Mode 1st 2nd 3rd 4th 5th Frame 2.94057 10.1362 12.8399 15.2338 47.8078 Mixed 2.71557 9.94736 12.3780 14.8188 37.2919

Periods of the two models (sec)

Frame 2.1367 0.619878 0.489347 0.412450 0.131426 Mixed 2.313765 0.631644 0.507855 0.424001 0.168486

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mode 1st mode 2nd mode 3rd

mode 4th mode 5th

Fig. 6.4 Five first modes shapes of the frame model

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mode 1st mode 2nd mode 3rd

mode 4th mode 5th

Fig. 6.5 Five first mode shapes of the mixed model

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III. Dynamics Response Analysis III.1. Data of loads • The acting loads upon to the structure consist of: - Static loads are placed on the top of structure. - Wave forces are cosine functions depend on time, in which the circular frequency and the magnitude of the load are: 2.0 Rad/sec and 4000kN respectively. The function of the load is: F = Fo.sin(Ωt) = 4000.sin(2t). The load are placed on the top of structure too. The dynamic response analysis is computed by using SAP90 software. - The solution method for dynamic response are modal superposition, in which + Time step - ∆t = 0.001 sec + Number of modes (or Ritz vector) = 5; + Number of step : 3140 + Damping ratio = 2% In the Fig. 6.6 indicates loading function and its values are given in the table 6.8 Table 6.8 The values of the loading function- Unit : × 4000 (kN)

Time Value Time Value Time Value 0.10 0.19867 1.20 0.67546 2.20 -0.95160 0.20 0.38942 1.20 0.67546 2.30 -0.99369 0.30 0.56464 1.30 0.51550 2.40 -0.99616 0.40 0.71736 1.40 0.33499 2.50 -0.95892 0.50 0.84147 1.50 0.14112 2.60 -0.88345 0.60 0.93204 1.60 -0.05837 2.70 -0.77276 0.70 0.98545 1.70 -0.25554 2.80 -0.63127 0.80 0.99957 1.80 -0.44252 2.90 -0.46460 0.90 0.97385 1.90 -0.61186 3.00 -0.27942 1.00 0.90930 2.00 -0.75680 3.10 -0.08309 1.10 0.80850 2.10 -0.87158 3.20 0.11655

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

0.10

0.40

0.70

1.00

1.30

1.60

1.90

2.20

2.50

2.80

3.10

T ime (sec)

Mag

itude

of l

oad

func

tion

x 40

00 (k

N)

Fig. 6.6 Loading function.

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III.2. Results of response analysis and comparisons of the two models • Horizontal and vertical displacement results of node at the top level of the two model

are compared and introduced in the Fig. 6.7 - 6.8.

-6.00E-1

-4.00E-1

-2.00E-1

0.00E+0

2.00E-1

4.00E-1

6.00E-1

0.00

1

0.20

1

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1

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1.00

1

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2.60

1

2.80

1

3.00

1

Time (sec)

Dis

plac

emen

t (m

)

Frame model

Mixed model

Fig. 6.7 Horizontal displacement at top level of structure.

-4.0E-2

-3.0E-2

-2.0E-2

-1.0E-2

0.0E+0

1.0E-2

2.0E-2

3.0E-2

4.0E-2

0.00

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2.60

1

2.80

1

3.00

1

T ime (sec)

Y-D

ispl

acem

ent (

m)

Frame model

Mixed model

Fig. 6.8 Vertical displacement at top level of structure. • Results of internal force components at the end I of element No. 1 (please see Fig. 6.2)

of the two model are compared and shown in Fig. 6.9-3.11

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-4.0E+4

-3.0E+4

-2.0E+4

-1.0E+4

0.0E+0

1.0E+4

2.0E+4

3.0E+4

4.0E+4

0.00

1

0.20

1

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2.60

1

2.80

1

3.00

1

T ime (sec)N

orm

al fo

rce

(kN

)

Frame model

Mixed model

Fig. 6.10 Normal force of element No.1

-6.0E+3

-4.0E+3

-2.0E+3

0.0E+0

2.0E+3

4.0E+3

6.0E+3

0.00

1

0.20

1

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1

2.80

1

3.00

1

T ime (sec)

She

ar fo

rce

(kN

)

Frame mode

Mixed model

Fig. 6.10 Shear force at the end I of element No.1

-2.5E+4

-2.0E+4

-1.5E+4

-1.0E+4

-5.0E+3

0.0E+0

5.0E+3

1.0E+4

1.5E+4

2.0E+4

2.5E+4

0.00

1

0.20

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1

2.80

1

3.00

1

T ime (sec)

Bend

ing

Mom

ent (

kNm

)

Frame model

Mixed model

Fig. 6.11 Bending moment at the end I of element No.1

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III. Discussions

From the result of this test, may be see that, i). On all of analyses of the tests - There are no chaotic values in results, the calculation process are very stable

and controllable. A lot of data sets were tried and there are no processing cases leading to the global matrices are degraded and to make the problem become dissolvable. - All the changing values follow some specific rules - This seems that the suggested model is correct and stable. ii) The comparison of two models shows us: - The values of the displacement in the mixed model are greater than that of the frame model. This means that the mixed model may more flexible than the frame model - The intenal force values in two cases are different. Their values depend on their positions in the structure. iii) With the eigenfrequency problem, some lowest frequencies of the mixed model are smaller than that of the frame model. A more again, this proves that in this case if structure is modelling by using mixed model are flexible than and accurate than frame modes. This result is a very important point of this thesis. It will affects to the dynamic respond of the structure when the loads is a function of time. iv) The result from the dynamics response analysis shows us external force and displacement components in the mixed model are much greater than that of the frame model. Therefore, if we use the result from the mixed model to design, the structure will be safer.