1 Introduction
2 Equivalent single-layer models
3 Plate models considered in the current study
Tag | Transverse shear deformation | Variationalformulation | Number ofunknownsat single node | Transverse shear stress continuity |
---|---|---|---|---|
FOSD_0
| “Pure” FOSD model without transverse shear correction | PVD (22) | 5 | No |
FOSD_1
| PVD (22) | 5 | No | |
FOSD_2
| PVD (22) | 5 | No | |
FOSD_3
| FOSD with global correction of composite transverse shear stiffness following Rolfes and Rohwer [23]; | PVD (22) | 5 | No |
ZZ_1
| Zigzag function of Murakami (18) | RMVT (32) | 7 | Yes |
ZZ_2
| Zigzag function of Tessler (19) | PVD (22) | 7 | No |
ZZ_3
| Zigzag function of Tessler (19) | RMVT (32) | 7 | Yes |
4 Numerical examples
4.1 Simply supported cross-ply (\(0^{\circ }{/}90^{\circ }{/}0^{\circ }\)) square plate
4.2 Simply supported cross-ply (\(0^{\circ }{/}90^{\circ }{/}0^{\circ }\)) rectangular plate
4.3 Simply supported 9-layer composite square plate
4.4 Simply supported square sandwich plate of Pagano [33]
\(E_{x}\)
|
\(E_{y}\)
|
\(G_{xy}\)
|
\(G_{xz}\)
|
\(G_{yz}\)
|
\(\nu _{xy}\)
| |
---|---|---|---|---|---|---|
Faces |
\(25\times E_{b}\)
|
\(E_{b}\)
|
\(0.5\times E_{b}\)
|
\(0.5\times E_{b}\)
|
\(0.2\times E_{b}\)
| 0.25 |
Core |
\(0.04\times E_{b}\)
|
\(0.04\times E_{b}\)
|
\(0.016\times E_{b}\)
|
\(0.06\times E_{b}\)
|
\(0.06\times E_{b}\)
| 0.25 |
4.5 Simply supported 3-layer sandwich square plate with Nomex core
\(E_{x}\) [GPa] | \(E_{y}\)[GPa] | \(G_{xy}\)[GPa] | \(G_{xz}\)[GPa] | \(G_{yz}\)[GPa] |
\(\nu _{xy}\)
| |
---|---|---|---|---|---|---|
Faces | 73 | 73 | 27.239 | 27.239 | 27.239 | 0.34 |
Nomex core |
\(10^{-5}\)
|
\(10^{-5}\)
|
\(22.5\times 10^{-3}\)
|
\(22.5 \times 10^{-3}\)
|
\(22.5\times 10^{-3}\)
| 0.01 |
a/h
| [43] |
FOSD_0
|
FOSD_1
|
FOSD_2
|
FOSD_3
| ||||
---|---|---|---|---|---|---|---|---|---|
\({\bar{w}}\)
| Error (%) |
\({\bar{w}}\)
| Error (%) |
\({\bar{w}}\)
| Error (%) |
\({\bar{w}}\)
| Error (%) | ||
4 |
590.54
| 9.809 | \(-\) 98.34 | 10.654 | \(-\) 98.20 | 1011.942 | 71.36 | 1011.942 | 71.36 |
10 |
149.7
| 6.256 | \(-\) 95.82 | 6.392 | \(-\) 95.73 | 166.598 | 11.29 | 166.598 | 11.29 |
100 |
7.1881
| 5.587 | \(-\) 22.28 | 5.588 | \(-\) 22.26 | 7.190 | 0.03 | 7.190 | 0.03 |
1000 |
5.598
| 5.580 | \(-\) 0.32 | 5.580 | \(-\) 0.32 | 5.596 | \(-\) 0.04 | 5.596 | \(-\) 0.04 |
a/h
| [43] |
ZZ_1
|
ZZ_2
|
ZZ_3
| |||
---|---|---|---|---|---|---|---|
\({\bar{w}}\)
| Error (%) |
\({\bar{w}}\)
| Error (%) |
\({\bar{w}}\)
| Error (%) | ||
4 |
590.54
| 584.208 | \(-\) 1.07 | 584.044 | \(-\) 1.10 | 584.205 | \(-\) 1.07 |
10 |
149.7
| 149.529 | \(-\) 0.11 | 149.524 | \(-\) 0.12 | 149.529 | \(-\) 0.11 |
100 |
7.1881
| 7.188 | 0.00 | 7.188 | 0.00 | 7.188 | 0.00 |
1000 |
5.598
| 5.596 | \(-\) 0.04 | 5.596 | \(-\) 0.04 | 5.596 | \(-\) 0.04 |
4.6 Simply supported 3-layer sandwich square plate with extremely soft core
a/h
| [43] |
FOSD_0
|
FOSD_1
|
FOSD_2
|
FOSD_3
| ||||
---|---|---|---|---|---|---|---|---|---|
\({\bar{w}}\)
| Error (%) |
\({\bar{w}}\)
| Error (%) |
\({\bar{w}}\)
| Error (%) |
\({\bar{w}}\)
| Error (%) | ||
4 |
1370.6
| 9.80860 | \(-\) 99.28 | 9.82401 | \(-\) 99.28 | 100634.6 | 7242.37 | 100634.6 | 7242.37 |
10 |
1260.3
| 6.26023 | \(-\) 99.50 | 6.39599 | \(-\) 99.49 | 16106.22 | 1177.97 | 16106.22 | 1177.97 |
100 |
149.51
| 5.588203 | \(-\) 96.26 | 5.58956 | \(-\) 96.26 | 166.5878 | 11.42 | 166.5878 | 11.42 |
1000 |
7.1901
| 5.58148 | \(-\) 22.37 | 5.58150 | \(-\) 22.37 | 7.19148 | 0.02 | 7.19148 | 0.02 |
a/h
| [43] |
ZZ_1
|
ZZ_2
|
ZZ_3
| |||
---|---|---|---|---|---|---|---|
\({\bar{w}}\)
| Error (%) |
\({\bar{w}}\)
| Error (%) |
\({\bar{w}}\)
| Error (%) | ||
4 |
1370.6
| 1348.789 | \(-\) 1.59 | 1347.957 | \(-\) 1.65 | 1348.779 | \(-\) 1.59 |
10 |
1260.3
| 1257.184 | \(-\) 0.25 | 1257.06 | \(-\) 0.26 | 1257.177 | \(-\) 0.25 |
100 |
149.51
| 149.504 | 0.00 | 149.504 | 0.00 | 149.504 | 0.00 |
1000 |
7.1901
| 7.18959 | \(-\) 0.01 | 7.18959 | \(-\) 0.01 | 7.18959 | \(-\) 0.01 |
4.7 Simply supported 5-layer [0/90/c/90/0] sandwich square plate
\(E_{a}\) [GPa] | \(E_{b}\) [GPa] | \(G_{ab}\) [GPa] | \(G_{az}\) [GPa] | \(G_{bz}\) [GPa] |
\(\nu _{ab}\)
| |
---|---|---|---|---|---|---|
Faces | 50 | 10 | 5 | 5 | 5 | 0.25 |
Nomex core |
\(10^{-5}\)
|
\(10^{-5}\)
|
\(22.5\times 10^{-3}\)
|
\(22.5\times 10^{-3}\)
|
\(22.5\times 10^{-3}\)
| 0.01 |
a/h
| [43] |
FOSD_0
|
FOSD_1
|
FOSD_2
|
FOSD_3
| ||||
---|---|---|---|---|---|---|---|---|---|
\({\bar{w}}\)
| Error (%) |
\({\bar{w}}\)
| Error (%) |
\({\bar{w}}\)
| Error (%) |
\({\bar{w}}\)
| Error (%) | ||
4 |
124.69
| 6.05346 | \(-\) 95.15 | 6.67607 | \(-\) 94.65 | 141.0384 | 13.11 | 141.0360 | 13.11 |
10 |
24.545
| 3.43704 | \(-\) 86.00 | 3.53678 | \(-\) 85.59 | 25.03560 | 2.00 | 25.03528 | 2.00 |
100 |
3.1589
| 2.94316 | \(-\) 6.83 | 2.94416 | \(-\) 6.80 | 3.159224 | 0.01 | 3.159226 | 0.01 |
1000 |
2.936
| 2.93822 | 0.08 | 2.9382 | 0.08 | 2.940382 | 0.15 | 2.940383 | 0.15 |
a/h
| [43] |
ZZ_1
|
ZZ_2
|
ZZ_3
| |||
---|---|---|---|---|---|---|---|
\({\bar{w}}\)
| Error (%) |
\({\bar{w}}\)
| Error (%) |
\({\bar{w}}\)
| Error (%) | ||
4 |
124.69
| 6.43288 | \(-\) 94.84 | 123.6955 | \(-\) 0.80 | 123.7061 | \(-\) 0.79 |
10 |
24.545
| 3.49782 | \(-\) 85.75 | 24.5244 | \(-\) 0.08 | 24.5252 | \(-\) 0.08 |
100 |
3.1589
| 2.94377 | \(-\) 6.81 | 3.15896 | 0.00 | 3.15897 | 0.00 |
1000 |
2.936
| 2.93823 | 0.08 | 2.94038 | 0.15 | 2.94038 | 0.15 |
4.8 Simply supported rectangular composite [90/0] plate
4.9 Simply supported antisymmetric composite [\(-15{{/}}{+}15\)] square plate
a/h
| [41] |
ZZ_2A
|
ZZ_2B
|
ZZ_2A*
|
ZZ_2B*
| ||||
---|---|---|---|---|---|---|---|---|---|
\({\bar{w}}\)
| Error (%) |
\({\bar{w}}\)
| Error (%) |
\({\bar{w}}\)
| Error (%) |
\({\bar{w}}\)
| Error (%) | ||
4 |
4.5548
| 4.460563 | \(-\) 2.07 | 4.435575 | \(-\) 2.62 | 4.391025 | \(-\) 3.60 | 4.391025 | \(-\) 3.60 |
10 |
1.70585
| 1.665781 | \(-\) 2.35 | 1.628672 | \(-\) 4.52 | 1.598906 | \(-\) 6.27 | 1.598906 | \(-\) 6.27 |
100 |
0.802721
| 0.828173 | 3.17 | 0.796743 | \(-\) 0.74 | 0.781512 | \(-\) 2.64 | 0.781514 | \(-\) 2.64 |
1000 |
0.622318
| 0.628638 | 1.02 | 0.623756 | 0.23 | 0.622095 | \(-\) 0.04 | 0.622096 | \(-\) 0.04 |
a/h
| [41] |
ZZ_1=ZZ_3B
|
ZZ_1*= ZZ_3B*
|
ZZ_3A
|
ZZ_3A*
| ||||
---|---|---|---|---|---|---|---|---|---|
\({\bar{w}}\)
| Error (%) |
\({\bar{w}}\)
| Error (%) |
\({\bar{w}}\)
| Error (%) |
\({\bar{w}}\)
| Error (%) | ||
4 |
4.5548
| 4.907563 | 7.74 | 4.864675 | 6.80 | 4.931613 | 8.27 | 4.864675 | 6.80 |
10 |
1.70585
| 1.752464 | 2.73 | 1.722547 | 0.98 | 1.789172 | 4.88 | 1.722547 | 0.98 |
100 |
0.802721
| 0.818604 | 1.98 | 0.802366 | \(-\) 0.04 | 0.851482 | 6.07 | 0.802365 | \(-\) 0.04 |
1000 |
0.622318
| 0.624126 | 0.29 | 0.622309 | 0.00 | 0.629447 | 1.15 | 0.622308 | 0.00 |