12.7. Results and Discussion

The following table shows the natural frequencies of the 14 significant modes (below fZPA):

Mode Frequency (Hz)
12.91
24.44
34.86
45.02
56.95
67.58
77.82
810.94
911.65
1011.78
1112.80
1214.32
1315.17
1415.79
 

12.7.1. Results Comparison of the Time-History and Response-Spectrum Analyses

The reactions at the supports obtained in the spectrum analyses A1 through A7 are given in the following topics:

The ratio of the RSA results to the transient results is reported in separate columns. The mean and standard deviation of these ratios are evaluated at the bottom of each table. This form of representation facilitates easy recognition of over-prediction and under-prediction by the RSA method.

A ratio of 1.0 indicates exact agreement, a ratio of > 1.0 indicates RSA over-prediction, and a ratio of < 1.0 indicates RSA under-prediction.

The absolute acceleration solutions are also compared for the x-direction and 3-direction input motions.

 

12.7.1.1. RSAs A1 and A2

The accuracy of the RSA results using SRSS or CQC combination method is assessed.

The modes are closely spaced, as shown:

Mode Number Frequency (Hz) Coupled Modes Coupling Coefficient
Mode No. Freq. (Hz)
34.85745.0170.276
67.58177.8160.299
911.6531011.7750.787
1214.3161315.1730.106

Because of the closely spaced modes, the CQC results are closer to the reference (transient-analysis results). The mean and standard-deviation values of spectrum results using CQC are 1.57 and 1.11, respectively.

Table 12.1: X-Direction Input

S. No. Reaction Forces Reference Full-Transient A1 Spectrum (SRSS) Ratio (A1 / Reference) A2 Spectrum (CQC) Ratio (A2 / Reference)

1

FX1

49.94

31.21

0.62

31.33

0.63

2

FY1

4.40

15.54

3.53

12.86

2.92

3

FZ1

5.95

48.91

8.22

24.88

4.18

4

MX1

180.56

1510.42

8.37

763.03

4.23

5

MY1

846.19

1216.04

1.44

929.90

1.10

6

MZ1

817.39

1120.07

1.37

1116.05

1.37

7

FX4

111.33

45.18

0.41

45.00

0.40

8

FZ4

34.74

82.06

2.36

55.61

1.60

9

FY74

10.94

28.26

2.58

16.75

1.53

10

FY11

11.38

18.21

1.60

16.40

1.44

11

FZ11

68.12

43.51

0.64

42.81

0.63

12

FX15

644.57

368.12

0.57

388.15

0.60

13

FY17

28.09

44.59

1.59

46.16

1.64

14

FZ17

64.67

53.49

0.83

55.37

0.86

15

FY36

60.62

161.97

2.67

168.30

2.78

16

FZ36

55.51

81.73

1.47

82.17

1.48

17

FX38

750.51

116.78

0.16

135.75

0.18

18

FY38

40.04

52.38

1.31

54.17

1.35

19

FZ38

38.43

39.05

1.02

44.45

1.16

20

MX38

787.11

2527.87

3.21

2591.22

3.29

21

MY38

2656.44

2528.86

0.95

2887.73

1.09

22

MZ38

2816.48

3644.84

1.29

3756.73

1.33

23

FX23

259.50

176.84

0.68

193.44

0.75

24

FY23

42.34

144.53

3.41

150.71

3.56

25

FX31

60.09

9.59

0.16

11.23

0.19

26

FY31

13.39

22.28

1.66

24.92

1.86

27

FZ31

15.06

32.85

2.18

33.60

2.23

28

MX31

968.59

2151.45

2.22

2241.86

2.31

29

MY31

546.34

210.58

0.39

236.59

0.43

30

MZ31

2231.74

631.42

0.28

730.46

0.33

Mean of 30 components

1.91

-

1.58

Standard deviation of 30 components

1.95

-

1.12


 

12.7.1.2. RSA A3

To improve accuracy, the missing-mass response is included in the analysis. The standard deviation decreases to 1.05. The mean value of 1.72 still shows over-prediction in the results.

Table 12.2: X-Direction Input (with Missing-Mass Effect)

S. No. Reaction Forces Reference Full-Transient A3 Spectrum Ratio (A3 / Reference)

1

FX1

49.94

42.88

0.86

2

FY1

4.40

15.60

3.55

3

FZ1

5.95

25.01

4.20

4

MX1

180.56

765.49

4.24

5

MY1

846.19

935.02

1.10

6

MZ1

817.39

1126.10

1.38

7

FX4

111.33

75.26

0.68

8

FZ4

34.74

56.13

1.62

9

FY74

10.94

16.75

1.53

10

FY11

11.38

16.40

1.44

11

FZ11

68.12

43.86

0.64

12

FX15

644.57

388.85

0.60

13

FY17

28.09

46.16

1.64

14

FZ17

64.67

55.66

0.86

15

FY36

60.62

168.31

2.78

16

FZ36

55.51

86.07

1.55

17

FX38

750.51

583.19

0.78

18

FY38

40.04

54.18

1.35

19

FZ38

38.43

50.26

1.31

20

MX38

787.11

2591.24

3.29

21

MY38

2656.44

3135.26

1.18

22

MZ38

2816.48

3756.91

1.33

23

FX23

259.50

242.94

0.94

24

FY23

42.34

150.77

3.56

25

FX31

60.09

53.26

0.89

26

FY31

13.39

27.91

2.09

27

FZ31

15.06

34.04

2.26

28

MX31

968.59

2252.48

2.33

29

MY31

546.34

390.59

0.71

30

MZ31

2231.74

1822.15

0.82

Mean of 30 components

1.72

Standard deviation of 30 components

1.06


 

12.7.1.3. RSAs A4 and A5

Both missing-mass and rigid-response effects are taken into account in the analyses. The mean and standard-deviation values using the Gupta method are equal to 1.10 and 0.17, respectively. Using the Lindley-Yow method, the mean and standard-deviation values are 1.19 and 0.24, respectively.

Table 12.3: X-Direction Input (with Missing-Mass and Rigid-Response Effects)

S. No. Reaction Forces Reference Full-Transient A4 Spectrum (Lindley) Ratio (A4 / Reference) A5 Spectrum (Gupta) Ratio (A5 / Reference)

1

FX1

49.94

52.24

1.05

53.42

1.07

2

FY1

4.40

6.08

1.38

3.74

0.85

3

FZ1

5.95

10.23

1.72

5.98

1.01

4

MX1

180.56

317.64

1.76

189.94

1.05

5

MY1

846.19

901.90

1.07

874.82

1.03

6

MZ1

817.39

982.04

1.20

984.43

1.20

7

FX4

111.33

105.26

0.95

102.74

0.92

8

FZ4

34.74

43.09

1.24

42.52

1.22

9

FY74

10.94

11.31

1.03

10.42

0.95

10

FY11

11.38

13.42

1.18

13.32

1.17

11

FZ11

68.12

51.31

0.75

49.51

0.73

12

FX15

644.57

562.49

0.87

546.98

0.85

13

FY17

28.09

35.06

1.25

35.99

1.28

14

FZ17

64.67

52.85

0.82

56.46

0.87

15

FY36

60.62

86.27

1.42

72.88

1.20

16

FZ36

55.51

66.54

1.20

68.13

1.23

17

FX38

750.51

751.13

1.00

768.11

1.02

18

FY38

40.04

46.00

1.15

45.76

1.14

19

FZ38

38.43

43.25

1.13

45.96

1.20

20

MX38

787.11

1188.24

1.51

912.70

1.16

21

MY38

2656.44

3008.28

1.13

3208.35

1.21

22

MZ38

2816.48

3236.99

1.15

3229.79

1.15

23

FX23

259.50

323.39

1.25

344.66

1.33

24

FY23

42.34

67.15

1.59

48.83

1.15

25

FX31

60.09

60.53

1.01

62.19

1.03

26

FY31

13.39

17.09

1.28

16.82

1.26

27

FZ31

15.06

20.49

1.36

20.76

1.38

28

MX31

968.59

1316.65

1.36

1380.33

1.43

29

MY31

546.34

511.67

0.94

551.75

1.01

30

MZ31

2231.74

2193.87

0.98

2313.51

1.04

Mean of 30 components

1.19

-

1.10

Standard deviation of 30 components

0.24

-

0.16


Gupta method, with f1 = 2.80 Hz and f2 = 11.90 Hz.

The absolute acceleration values at support (node 4) and far from support (nodes 16 and 34) are compared in the following table:

Table 12.4: X-Direction Input Motion

Node Reference Full-Transient (absolute) A5 Spectrum ACC (with RIGRESP) Error (%) Reference Full-Transient (absolute) A5 Spectrum ACC (with RIGRESP + MMASS) Error (%)

16

310.21

285.01

8.12

310.21

277.14

10.66

4

207.54

10.62

94.88

207.54

209.36

0.88

34

226.27

233.43

3.16

226.27

227.20

0.41


It is clearly shown that the absolute acceleration value at node 4 (at support) shows close comparison with the full-transient solution after the addition of the missing-mass effect.

 

12.7.1.4. RSA A6

With the Gupta method, there is a limitation lying in the semi-empirical basis of the definition of the rigid-response coefficient αi, as a function of fi.[2][3] The choice of key parameter f2 (RIGRESP,,,,VAL2), which defines the frequency above which modal responses are combined algebraically, has a significant effect on the predicted response.

To show the effect of the f2 value, two different values of f2 are chosen: 6.0 Hz [2 Appendix H] and 11.90 Hz.[4] Both frequencies are within a range where the input acceleration is almost constant and the acceleration value is very close to the ZPA. A value of f1 = 2.80 Hz is maintained for both analyses. The respective results are shown in the following two tables:

For f2 = 6.0 Hz, the mean and standard-deviation values are 0.91 and 0.18, respectively, implying under-prediction of the reaction forces. Conversely, for f2 = 11.90 Hz, the mean and standard-deviation values are 1.10 and 0.17, respectively, implying over-prediction of reaction forces.

Table 12.5: X-Direction Input (with Missing-Mass and Rigid-Response Effects)

S. No. Reaction Forces Reference Full-Transient A6 Spectrum (Gupta) Ratio (A6 / Reference)

1

FX1

49.94

51.17

1.02

2

FY1

4.40

3.13

0.71

3

FZ1

5.95

4.04

0.68

4

MX1

180.56

135.93

0.75

5

MY1

846.19

904.65

1.07

6

MZ1

817.39

885.76

1.08

7

FX4

111.33

111.51

1.00

8

FZ4

34.74

34.90

1.00

9

FY74

10.94

9.20

0.84

10

FY11

11.38

9.38

0.82

11

FZ11

68.12

62.10

0.91

12

FX15

644.57

641.25

0.99

13

FY17

28.09

26.35

0.94

14

FZ17

64.67

46.56

0.72

15

FY36

60.62

44.91

0.74

16

FZ36

55.51

56.27

1.01

17

FX38

750.51

757.30

1.01

18

FY38

40.04

42.35

1.06

19

FZ38

38.43

42.26

1.10

20

MX38

787.11

399.46

0.51

21

MY38

2656.44

2936.13

1.11

22

MZ38

2816.48

2995.73

1.06

23

FX23

259.50

302.56

1.17

24

FY23

42.34

16.35

0.39

25

FX31

60.09

60.12

1.00

26

FY31

13.39

13.52

1.01

27

FZ31

15.06

13.16

0.87

28

MX31

968.59

784.84

0.81

29

MY31

546.34

497.54

0.91

30

MZ31

2231.74

2149.85

0.96

Mean of 30 components

0.91

Standard deviation of 30 components

0.18


Gupta Method, with f1 = 2.80 Hz and f2 = 6.00 Hz.

 

12.7.1.5. RSA A7

This analysis considers the inputs in the X, Y and Z directions. The mean and standard-deviation values obtained are 1.00 and 0.10, respectively, implying that the correlation between the spectrum-analysis and transient-analysis results is better than for single-directional input. The better spectrum-analysis correlation is a result of the reactions having directions orthogonal to the input, which are not significantly improved by including the missing-mass and rigid-response effect; however, these reactions should remain smaller than the primary reactions.

Table 12.6: 3-Direction Input (with Missing-Mass and Rigid-Response Effects)

S. No. Reaction Forces Reference Full-Transient A7 Spectrum (Lindley) Ratio (A7 / Reference)

1

FX1

54.93

56.51

1.03

2

FY1

98.63

98.71

1.00

3

FZ1

37.90

45.23

1.19

4

MX1

722.93

845.20

1.17

5

MY1

1099.20

1207.77

1.10

6

MZ1

1891.99

2013.72

1.06

7

FX4

129.34

117.53

0.91

8

FZ4

318.39

246.76

0.78

9

FY74

120.44

117.45

0.98

10

FY11

226.31

217.24

0.96

11

FZ11

287.89

229.73

0.80

12

FX15

664.97

603.60

0.91

13

FY17

91.60

94.38

1.03

14

FZ17

146.95

153.49

1.04

15

FY36

374.81

348.78

0.93

16

FZ36

830.88

830.99

1.00

17

FX38

751.27

752.08

1.00

18

FY38

181.96

191.64

1.05

19

FZ38

260.43

261.41

1.00

20

MX38

3377.83

3170.57

0.94

21

MY38

11097.98

11184.24

1.01

22

MZ38

7684.11

8443.85

1.10

23

FX23

273.79

336.42

1.23

24

FY23

264.72

233.99

0.88

25

FX31

61.21

61.70

1.01

26

FY31

90.08

92.93

1.03

27

FZ31

138.59

140.44

1.01

28

MX31

9033.85

9045.39

1.00

29

MY31

746.60

714.41

0.96

30

MZ31

2624.24

2584.03

0.98

Mean of 30 components

1.00

Standard deviation of 30 components

0.10


The absolute acceleration values at support (node 4) and far from support (nodes 16 and 34) are compared in the table below:

Table 12.7: 3-Direction Input Motion

Node DOF Reference Full-Transient (absolute) A7 Spectrum ACC (with RIGRESP) Error (%) Reference Full-Transient (absolute) A7 Spectrum ACC (with RIGRESP + MMASS) Error (%)

16

ACCY

518.55

496.17

4.32

518.55

484.7

6.53

4

ACCX

206.26

11.33

94.51

206.26

208.70

1.19

34

ACCZ

203.95

24.772

87.85

203.95

208.77

2.36


It is clearly shown that the absolute acceleration value at node 4 (at support) shows close comparison with the full-transient solution after the addition of the missing-mass effect.