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Advanced Modeling and Simulation in Engineering Sciences

Table 1 Displacement and stress evaluation on the three-layered cylindrical shells with various \(R_{\beta }/h\) under distributed pressure

From: Evaluation of shear and membrane locking in refined hierarchical shell finite elements for laminated structures

\({{\varvec{R}}}_{\varvec{\beta }}{{\varvec{/h}}}\)

TOS

Integration

\({{\varvec{p}}}\)

\(\bar{{\varvec{w}}}\)

\(\bar{\varvec{\sigma }}_{\varvec{\alpha \alpha }}\)

\(\bar{\varvec{\sigma }}_{\varvec{\beta \beta }}\)

\(\bar{\varvec{\sigma }}_{\varvec{\alpha \beta }}\)

\(\bar{\varvec{\sigma }}_{\varvec{\alpha } {\varvec{z}}}\)

\(\bar{\varvec{\sigma }}_{\varvec{\beta } {\varvec{z}}}\)

\(\bar{\varvec{\sigma }}_{{\varvec{zz}}}\)

\({\varvec{E}}_{{\varvec{memb}}}/{\varvec{E}} {\varvec{(\%)}}\)

\({\varvec{E}}_{{\varvec{bend}}}/{\varvec{E (\%)}}\)

\({\varvec{E}}_{{\varvec{ps}}}/{\varvec{E}} {\varvec{(\%)}}\)

\({\varvec{E}}_{{\varvec{zs}}}/{\varvec{E}} {\varvec{(\%)}}\)

\({\varvec{E}}_{{\varvec{zz}}}/{\varvec{E (\%)}}\)

2

FSDT

FULL

8

8.387

0.1025

2.331

\(-\) 0.2258

0.2140

\(-\) 0.960

–

0.1

4.6

0.7

94.6

–

LE4

FULL

8

10.102

0.1735

7.155

\(-\) 0.2916

0.3009

\(-\) 1.377

\(-\) 0.3368

0.0

9.1

0.4

58.7

31.7

LE4

REDI

8

10.102

0.1735

7.155

\(-\) 0.2916

0.3009

\(-\) 1.377

\(-\) 0.3368

0.0

9.1

0.4

58.7

31.7

LE4

SELI

8

10.102

0.1735

7.155

\(-\) 0.2916

0.3009

\(-\) 1.377

\(-\) 0.3368

0.0

9.1

0.4

58.7

31.7

Varadan and Bhaskar [48]

10.11

0.1761

7.168

\(-\) 0.2922

0.3006

\(-\) 1.379

\(-\) 0.34

–

–

–

–

–

4

FSDT

FULL

8

2.867

0.0722

3.155

\(-\) 0.1131

0.0467

\(-\) 1.150

–

0.1

16.3

0.6

83.0

–

LE4

FULL

8

4.009

0.1267

6.543

\(-\) 0.1609

0.1739

\(-\) 2.349

\(-\) 0.6194

0.1

18.5

0.6

68.8

12.0

LE4

REDI

8

4.009

0.1267

6.543

\(-\) 0.1609

0.1739

\(-\) 2.349

\(-\) 0.6194

0.1

18.5

0.6

68.8

12.0

LE4

SELI

8

4.009

0.1267

6.543

\(-\) 0.1609

0.1739

\(-\) 2.349

\(-\) 0.6194

0.1

18.5

0.6

68.8

12.0

Varadan and Bhaskar [48]

4.009

0.1270

6.545

\(-\) 0.1609

0.1736

\(-\) 2.349

\(-\) 0.62

–

–

–

–

–

10

FSDT

FULL

8

0.938

0.05690

3.718

\(-\) 0.05875

0.06063

\(-\) 1.258

–

0.2

54.9

0.5

44.5

–

LE4

FULL

8

1.223

0.07392

4.682

\(-\) 0.07292

0.08284

\(-\) 3.264

\(-\) 1.270

0.2

43.4

0.5

53.3

2.6

LE4

REDI

8

1.223

0.07392

4.682

\(-\) 0.07292

0.08284

\(-\) 3.264

\(-\) 1.270

0.2

43.4

0.5

53.3

2.6

LE4

SELI

8

1.223

0.07392

4.682

\(-\) 0.07292

0.08284

\(-\) 3.264

\(-\) 1.270

0.2

43.4

0.5

53.3

2.6

Varadan and Bhaskar [48]

1.223

0.0739

4.683

\(-\) 0.0729

0.0826

\(-\) 3.264

\(-\) 1.27

–

–

–

–

–

50

FSDT

FULL

9

0.5372

0.06993

3.887

\(-\) 0.07495

0.04622

\(-\) 1.261

–

2.5

92.4

2.1

3.0

–

LE4

FULL

9

0.5495

0.07127

3.932

\(-\) 0.07600

0.08922

\(-\) 3.492

\(-\) 4.847

2.4

90.4

2.1

5.0

0.1

LE4

REDI

9

0.5495

0.07128

3.932

\(-\) 0.07600

0.08910

\(-\) 3.492

\(-\) 4.847

2.4

90.4

2.1

5.0

0.1

LE4

SELI

9

0.5495

0.07127

3.932

\(-\) 0.07600

0.08910

\(-\) 3.492

\(-\) 4.847

2.4

90.4

2.1

5.0

0.1

Varadan and Bhaskar [48]

0.5495

0.0712

3.930

\(-\) 0.0760

0.0894

\(-\) 3.491

\(-\) 4.85

–

–

–

–

–

100

FSDT

FULL

9

0.4688

0.08344

3.501

\(-\) 0.1035

0.0405

\(-\) 1.127

–

8.8

84.1

6.4

0.7

–

LE4

FULL

9

0.4715

0.08389

3.511

\(-\) 0.1038

0.1219

\(-\) 3.128

\(-\) 8.300

8.2

84.2

6.4

1.2

0.1

LE4

REDI

11

0.4715

0.08384

3.507

\(-\) 0.1038

0.1223

\(-\) 3.127

\(-\) 8.297

8.2

84.2

6.4

1.2

0.1

LE4

SELI

11

0.4715

0.08384

3.507

\(-\) 0.1038

0.1223

\(-\) 3.127

\(-\) 8.297

8.2

84.2

6.4

1.2

0.1

Varadan and Bhaskar [48]

0.4715

0.0838

3.507

\(-\) 0.1038

0.1223

\(-\) 3.127

\(-\) 8.30

–

–

–

–

–

500

FSDT

FULL

12

0.1027

0.05585

0.7898

\(-\) 0.08885

0.0090

\(-\) 0.2486

–

47.9

18.6

33.4

0.0

–

LE4

FULL

12

0.1027

0.05586

0.7895

\(-\) 0.08886

0.1051

\(-\) 0.6907

\(-\) 9.115

44.4

22.2

33.4

0.0

0.0

LE4

REDI

13

0.1027

0.05586

0.7897

\(-\) 0.08886

0.1032

\(-\) 0.6907

\(-\) 9.141

44.4

22.2

33.4

0.0

0.0

LE4

SELI

13

0.1027

0.05586

0.7895

\(-\) 0.08886

0.1078

\(-\) 0.6907

\(-\) 9.120

44.4

22.2

33.4

0.0

0.0

Varadan and Bhaskar [48]

0.1027

0.0559

0.7895

\(-\) 0.0889

0.1051

\(-\) 0.691

\(-\) 9.12

–

–

–

–

–

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