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Frontiers of Structural and Civil Engineering

Front. Struct. Civ. Eng.    2019, Vol. 13 Issue (2) : 251-272     https://doi.org/10.1007/s11709-017-0456-0
Research Article |
Static analysis of corrugated panels using homogenization models and a cell-based smoothed mindlin plate element (CS-MIN3)
Nhan NGUYEN-MINH1, Nha TRAN-VAN2,3, Thang BUI-XUAN2, Trung NGUYEN-THOI4,5()
1. Faculty of Applied Science, Bach Khoa University (BKU), Ho Chi Minh City, Vietnam
2. Faculty of Mathematics and Computer Science, Ho Chi Minh City University of Science (HCMUS), Ho Chi Minh City, Vietnam
3. Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA
4. Division of Computational Mathematics and Engineering, Institute of Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam
5. Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam
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Abstract

Homogenization is a promising approach to capture the behavior of complex structures like corrugated panels. It enables us to replace high-cost shell models with stiffness-equivalent orthotropic plate alternatives. Many homogenization models for corrugated panels of different shapes have been proposed. However, there is a lack of investigations for verifying their accuracy and reliability. In addition, in the recent trend of development of smoothed finite element methods, the cell-based smoothed three-node Mindlin plate element (CS-MIN3) based on the first-order shear deformation theory (FSDT) has been proposed and successfully applied to many analyses of plate and shell structures. Thus, this paper further extends the CS-MIN3 by integrating itself with homogenization models to give homogenization methods. In these methods, the equivalent extensional, bending, and transverse shear stiffness components which constitute the equivalent orthotropic plate models are represented in explicit analytical expressions. Using the results of ANSYS and ABAQUS shell simulations as references, some numerical examples are conducted to verify the accuracy and reliability of the homogenization methods for static analyses of trapezoidally and sinusoidally corrugated panels.

Keywords homogenization      corrugated panel      asymptotic analysis      smoothed finite element method (S-FEM)      cell-based smoothed three-node Mindlin plate element (CS-MIN3)     
Corresponding Authors: Trung NGUYEN-THOI   
Online First Date: 15 January 2018    Issue Date: 12 March 2019
 Cite this article:   
Nhan NGUYEN-MINH,Nha TRAN-VAN,Thang BUI-XUAN, et al. Static analysis of corrugated panels using homogenization models and a cell-based smoothed mindlin plate element (CS-MIN3)[J]. Front. Struct. Civ. Eng., 2019, 13(2): 251-272.
 URL:  
http://journal.hep.com.cn/fsce/EN/10.1007/s11709-017-0456-0
http://journal.hep.com.cn/fsce/EN/Y2019/V13/I2/251
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Nhan NGUYEN-MINH
Nha TRAN-VAN
Thang BUI-XUAN
Trung NGUYEN-THOI
Fig.1  (a) A corrugated panel and its representative volume element (RVE); (b) a unit of trapezoidal corrugation; (c) a unit of sinusoidal corrugation
Fig.2  (a) A representative volume element (RVE) of a corrugated panel and its equivalent plate element; (b) a Reissner-Mindlin plate and its field variables; (c) the CS-MIN3 element with three sub-triangles
terms trapezoidal profile sinusoidal profile general profile (symmetric)
SamantaHM [5] PengHM [8] PengHM [8] XiaHM [2] YeHM [10]
A11 2 cI2 Eh312 2 cI2 Eh312 E h/αS 2c/( I1 A11+I2D 11) 1C6Eh1 ν2
A12 ν A11 ν A11 ν A11 A11A12/A 11S ν A11
A22 l cEh l cEh l cEh A12A12+l(A11A 22A122) /cA 11 C2E h+ν2A11
A66 c l Eh2( 1+ν) c l Eh2( 1+ν) Eh2(1 +ν) c lA 66 C7Eh2(1 +ν)
D11 c l Eh312 c l Eh312( 1 ν2 ) c l Eh312( 1 ν2 ) c lD 11 1C2D11
D12 0 ν D11 ν D11 D11D12/D 11 νD 11
D22 I22cE h Eh 312( 1 ν2 ) +EhcαT Eh 312( 1 ν2 ) +Ehf22 12c( I2A22+I1D 22) C8ε2Eh+C5E h312+ ν2 D11
D66 l c Eh36( 1+ν) Eh 324( 1+ν) Eh 324( 1+ν) l cD 66 C94 Eh2( 1+ν)
D55 - κ D66/ k2 κ D66/ k2 - -
D44 - κ k1 D66 κ k1 D66 - -
Tab.1  Equivalent extensional, bending and transverse shear stiffness terms for corrugated panels of trapezoidal, sinusoidal and general profiles
Fig.3  Constructive meshes for (a) a trapezoidally corrugated panel, (b) a sinusoidally corrugated panel, and (c) a flat orthotropic plate
Case L/W h/L
1 0.5 0.01
2 0.5 0.05
3 0.5 0.1
4 0.5 0.14
5 0.5 0.2
6 1 0.01
7 1 0.05
8 1 0.1
9 1 0.14
10 1 0.2
11 2 0.01
12 2 0.05
13 2 0.1
14 2 0.14
15 2 0.2
Tab.2  Fifteen cases for static analysis of an orthotropic plate
case DSG3 MIN3 CSDSG3 CSMIN3 ESDSG3 ABAQUS exact [64]
1 13142943.16 13398382.38 13287116.70 13445381.13 13431467.95 13477156.51 -
2 21758.69 21873.69 21777.65 21876.63 21863.65 21924.14 21542.00
3 1430.93 1437.51 1431.67 1437.56 1437.07 1440.65 1408.50
4 395.92 397.58 396.07 397.59 397.56 398.43 387.23
5 106.98 107.36 107.00 107.36 107.40 107.58 -
6 6423898.88 6584293.66 6513920.54 6620442.62 6601674.75 6652258.20 -
7 10735.70 10805.14 10748.57 10807.54 10786.71 10854.89 10443.00
8 712.41 716.34 712.92 716.38 715.32 719.53 688.57
9 199.02 200.01 199.13 200.02 199.82 200.91 191.07
10 54.62 54.86 54.65 54.86 54.85 55.11 -
11 1252006.74 1271158.13 1261410.74 1274681.05 1272364.84 1278477.10 -
12 2093.66 2105.06 2095.23 2105.31 2103.30 2111.25 2048.70
13 143.68 144.26 143.70 144.27 144.33 144.69 139.08
14 41.61 41.73 41.60 41.73 41.81 41.86 39.79
15 12.10 12.12 12.09 12.12 12.16 12.16 -
MAPE 1.14 % 0.40 % 0.90 % 0.32 % 0.37 % - -
Tab.3  Non-dimensional central deflection of a simply supported (SSSS) orthotropic plate under uniform pressure
case DSG3 MIN3 CSDSG3 CSMIN3 ESDSG3 ABAQUS exact
1 3041127.02 3091123.66 3076137.81 3098273.45 3123995.13 3111853.38 -
2 5358.96 5367.60 5362.28 5368.09 5416.90 5388.53 -
3 414.71 415.10 414.82 415.15 418.46 416.59 -
4 134.32 134.40 134.34 134.42 135.37 134.85 -
5 45.49 45.49 45.48 45.50 45.78 45.64 -
6 1900392.22 1975228.75 1949238.42 1995320.60 2005825.20 2028460.70 -
7 3436.30 3451.91 3443.41 3453.55 3483.44 3501.42 -
8 261.58 262.29 261.88 262.37 264.54 265.51 -
9 82.48 82.64 82.55 82.67 83.33 83.55 -
10 26.62 26.65 26.63 26.66 26.87 26.90 -
11 331688.92 337477.23 335382.13 338867.14 341501.64 341849.96 -
12 604.42 605.68 604.90 605.87 611.79 610.20 -
13 49.64 49.61 49.61 49.62 50.13 49.88 -
14 16.77 16.73 16.75 16.74 16.91 16.81 -
15 5.97 5.95 5.96 5.95 6.01 5.97 -
MAPE 1.37 % 0.85 % 1.05 % 0.72 % 0.44 % - -
Tab.4  Non-dimensional central deflection of a clamped (CCCC) orthotropic plate under uniform pressure
Fig.4  Convergences of the non-dimensional central deflections by CS-MIN3 and other FEMs compared to ABAQUS solution
stiffness terms SamantaHM PengHM XiaHM YeHM VAPAS [65]
A11 3.776E+06 3.776E+06 4.052E+06 4.052E+06 4.118E+06
A12 1.133E+06 1.133E+06 1.216E+06 1.216E+06 1.235E+06
A22 1.610E+08 1.610E+08 1.613E+08 1.613E+08 1.613E+08
A66 4.249E+07 4.249E+07 4.249E+07 4.249E+07 4.330E+07
D11 3.712E+02 4.079E+02 4.079E+02 4.079E+02 4.149E+02
D12 0.000E+00 1.224E+02 1.224E+02 1.224E+02 1.248E+02
D22 1.582E+04 1.483E+04 1.781E+04 1.624E+04 1.659E+04
D66 8.321E+02 1.723E+02 2.080E+02 2.080E+02 2.103E+02
D55 9.305E+06 1.927E+06 2.326E+06 2.326E+06 2.352E+06
D44 2.823E+08 5.846E+07 7.057E+07 7.057E+07 7.134E+07
Tab.5  Equivalent stiffness terms for a trapezoidally corrugated panel with nine corrugation units
Fig.5  Deflection along central lines of simply supported trapezoidally corrugated panels under a uniform pressure when the number of corrugation units is changed (ABAQUS-VAPAS is the analysis of the homogenization model VAPAS using ABAQUS plate simulation)
Fig.6  Deflection along central lines of clamped trapezoidally corrugated panels under a uniform pressure when the number of corrugation units is changed (ABAQUS-VAPAS is the analysis of the homogenization model VAPAS using ABAQUS plate simulation)
homogenization method relative error (%)
SSSS CCCC
nc=3 nc=9 nc=15 nc=3 nc=9 nc=15
CS-MIN3-SamantaHM 3.56 ?6.63 ?12.49 ?14.68 ?3.27 ?2.96
CS-MIN3-PengHM 12.11 11.68 23.56 ?11.97 6.87 21.83
CS-MIN3-XiaHM ?0.29 ?7.27 ?8.65 ?21.82 ?11.56 ?10.72
CS-MIN3-YeHM 9.29 1.47 ?0.34 ?14.20 ?2.90 ?2.02
Tab.6  Relative errors of central deflections between homogenization methods and ANSYS shell simulation when the number of corrugation units is changed
Fig.7  Contour plots of the deflections of (a) a simply supported trapezoidally corrugated panel (by ANSYS shell simulation) and (b) its equivalent plate (by CS-MIN3-YeHM)
r f central deflection (m)
CS-MIN3-SamantaHM CS-MIN3-PengHM CS-MIN3-XiaHM CS-MIN3-YeHM ABAQUS-shell ANSYS-shell
?1 2.258E?05 3.287E?05 2.103E?05 2.315E?05 2.358E?05 2.320E?05
0 5.464E?05 6.536E?05 5.427E?05 5.938E?05 5.953E?05 5.844E?05
1 1.366E?04 1.739E?04 1.566E?04 1.680E?04 1.697E?04 1.660E?04
2 2.445E?04 3.568E?04 3.388E?04 3.522E?04 3.551E?04 3.578E?04
3 3.102E?04 4.981E?04 4.886E?04 4.956E?04 5.052E?04 5.092E?04
4 3.329E?04 5.540E?04 5.505E?04 5.527E?04 5.659E?04 5.676E?04
5 3.396E?04 5.695E?04 5.683E?04 5.689E?04 5.827E?04 5.735E?04
6 3.416E?04 5.732E?04 5.727E?04 5.729E?04 5.867E?04 5.760E?04
7 3.424E?04 5.740E?04 5.738E?04 5.738E?04 5.874E?04 5.764E?04
inf (flat) - - - - 5.768E?04 5.739E?04
Tab.7  Central deflection of a simply supported (SSSS) trapezoidally corrugated panel under a uniformly distributed load when the corrugation amplitude tends to zero (rf+)
r f central deflection (m)
CS-MIN3-SamantaHM CS-MIN3-PengHM CS-MIN3-XiaHM CS-MIN3-YeHM ABAQUS-shell ANSYS-shell
?1 4.496E?06 6.404E?06 4.143E?06 4.543E?06 4.880E?06 4.779E?06
0 1.162E?05 1.284E?05 1.062E?05 1.166E?05 1.223E?05 1.196E?05
1 3.596E?05 3.767E?05 3.369E?05 3.651E?05 3.788E?05 3.683E?05
2 8.726E?05 9.016E?05 8.519E?05 8.957E?05 9.125E?05 9.121E?05
3 1.386E?04 1.423E?04 1.395E?04 1.423E?04 1.455E?04 1.459E?04
4 1.635E?04 1.667E?04 1.658E?04 1.668E?04 1.713E?04 1.750E?04
5 1.721E?04 1.740E?04 1.738E?04 1.741E?04 1.795E?04 1.782E?04
6 1.746E?04 1.758E?04 1.758E?04 1.759E?04 1.809E?04 1.793E?04
7 1.753E?04 1.762E?04 1.762E?04 1.763E?04 1.812E?04 1.796E?04
inf (flat) - - - - 1.798E?04 1.793E?04
Tab.8  Central deflection of a clamped (CCCC) trapezoidally corrugated panel under a uniformly distributed load when the corrugation amplitude tends to zero (rf+)
Fig.8  Relative errors of central deflections between homogenization methods and ANSYS shell simulation when the corrugation amplitude tends to zero (rf+)
trough angle Boun. Cond. central deflection (m)
CS-MIN3-SamantaHM CS-MIN3-PengHM CS-MIN3-XiaHM CS-MIN3-YeHM ABAQUS-shell ANSYS-shell
30° SSSS 8.010E?05 9.877E?05 8.296E?05 9.023E?05 9.024E?05 8.903E?05
CCCC 1.811E?05 1.996E?05 1.662E?05 1.820E?05 1.892E?05 1.870E?05
45° SSSS 5.464E?05 6.536E?05 5.427E?05 5.938E?05 5.954E?05 5.853E?05
CCCC 1.162E?05 1.284E?05 1.062E?05 1.166E?05 1.223E?05 1.201E?05
60° SSSS 4.523E?05 5.438E?05 4.429E?05 4.856E?05 5.042E?05 4.796E?05
CCCC 9.446E?06 1.061E?05 8.633E?06 9.481E?06 1.034E?05 9.824E?06
75° SSSS 3.958E?05 4.829E?05 3.842E?05 4.218E-05 4.345E?05 4.176E?05
CCCC 8.200E?06 9.403E?06 7.487E?06 8.223E?06 8.938E?06 8.580E?06
90° SSSS 3.528E?05 4.393E?05 3.401E?05 3.737E?05 3.842E?05 3.712E?05
CCCC 7.281E?06 8.555E?06 6.637E?06 7.288E?06 7.963E?06 7.680E?06
Tab.9  Central deflection of trapezoidally corrugated panels under uniformly distributed load when the trough angle is varied
Fig.9  Relative errors of central deflections between homogenization methods and ANSYS shell simulations when the trough angle is changed
stiffness terms PengHM XiaHM YeHM VAPAS [47]
A11 3.504E+04 4.761E+04 4.761E+04 4.815E+04
A12 7.008E+03 9.523E+03 9.523E+03 9.630E+03
A22 1.871E+08 1.871E+08 1.871E+08 1.869E+08
A66 6.250E+07 5.011E+07 5.011E+07 5.010E+07
D11 2.610E+02 2.610E+02 2.610E+02 2.640E+02
D12 5.220E+01 5.220E+01 5.220E+01 5.295E+01
D22 9.078E+05 1.068E+06 1.026E+06 1.023E+06
D66 1.302E+02 1.624E+02 1.624E+02 1.634E+02
D55 1.905E+04 2.375E+04 2.375E+04 2.390E+04
D44 7.795E+07 9.722E+07 9.722E+07 9.781E+07
Tab.10  Equivalent stiffness terms for a sinusoidally corrugated panel with eleven corrugation units
Fig.10  Deflection along central lines of simply supported sinusoidally corrugated panels under a uniform pressure when the number of corrugation units is changed (ABAQUS-VAPAS is the analysis of the homogenization model VAPAS using ABAQUS plate simulation)
Fig.11  Deflection along central lines of a clamped sinusoidally corrugated panel under a uniform pressure when the number of corrugation units is changed (ABAQUS-VAPAS is the analysis of the homogenization model VAPAS using ABAQUS plate simulation)
homogenization method relative error (%)
SSSS CCCC
nc=4 nc=11 nc=18 nc=4 nc=11 nc=18
CS-MIN3-PengHM ?7.25 9.38 26.99 ?30.34 5.59 24.52
CS-MIN3-XiaHM ?12.60 ?7.06 ?6.62 ?34.34 ?10.33 ?8.54
CS-MIN3-YeHM ?8.97 ?3.20 ?2.74 ?31.65 ?6.65 ?4.77
Tab.11  Relative errors of central deflections between homogenization methods and ANSYS shell simulation when number of corrugation units is changed
Fig.12  Contour plots of the deflections of (a) a clamped sinusoidally corrugated panel (by ANSYS shell simulation) and (b) its equivalent plate (by CS-MIN3-YeHM)
r h central deflection (m)
CS-MIN3-PengHM CS-MIN3-XiaHM CS-MIN3-YeHM ABAQUS-shell ANSYS-shell
?1 3.527E?03 2.996E?03 3.121E?03 3.298E?03 3.245E?03
0 1.763E?03 1.498E?03 1.560E?03 1.643E?03 1.612E?03
1 8.781E?04 7.466E?04 7.775E?04 8.267E?04 8.045E?04
2 4.455E?04 3.774E?04 3.935E?04 4.159E?04 4.075E?04
3 2.293E?04 1.948E?04 2.029E?04 2.098E?04 2.070E?04
4 9.933E?05 8.629E?05 8.917E?05 9.176E?05 9.109E?05
5 2.890E?05 2.604E?05 2.665E?05 2.878E?05 2.786E?05
6 5.346E?06 4.958E?06 5.127E?06 6.467E?06 5.499E?06
7 7.722E?07 7.224E?07 8.226E?07 1.190E?06 8.586E?07
Tab.12  Central deflection of a simply supported (SSSS) sinusoidally corrugated panel under uniformly distributed load when the thickness is changed
r h central deflection (m)
CS-MIN3-PengHM CS-MIN3-XiaHM CS-MIN3-YeHM ABAQUS-shell ANSYS-shell
?1 7.073E?04 6.006E?04 6.253E?04 7.024E?04 6.915E?04
0 3.536E?04 3.002E?04 3.126E?04 3.396E?04 3.349E?04
1 1.766E?04 1.500E?04 1.562E?04 1.685E?04 1.662E?04
2 8.781E?05 7.467E?05 7.773E?05 8.432E?05 8.327E?05
3 4.423E?05 3.763E?05 3.920E?05 4.204E?05 4.154E?05
4 2.112E?05 1.836E?05 1.905E?05 2.016E?05 1.995E?05
5 7.326E?06 6.738E?06 6.917E?06 7.460E?06 7.269E?06
6 1.586E?06 1.547E?06 1.585E?06 1.888E?06 1.708E?06
7 2.568E?07 2.554E?07 2.765E?07 3.683E?07 2.990E?07
Tab.13  Central deflection of a clamped (CCCC) sinusoidally corrugated panel under uniformly distributed load when the thickness is changed
Fig.13  Relative errors of central deflections between homogenization methods and ANSYS shell simulation when the thickness is changed
Trapezoidally corrugated panel Sinusoidally corrugated panel
XiaHM:
I1=2c 4f( 1cos ?α)Tan ?αI2= 4f33sin?α+2f2(c 2ftan?α)
YeHM:
C1=0,?C2= 1+4 f(1cos?α)εsin?α, ?C3=0, ?C4=f2(4f12fcos?α+3εsin?α)3 ε3sin?α,C5= 14 f(1cos?α)εTan?α,? C6=12 C4 (ε /h)2 C5,C7=1/C2,? C8=C4,?C9=h23C2
XiaHM:
z(x)=fsin? (πx/c)I 1=cc dx/1+ ( dzdx)2?I2=cc z2 1+ (dzdx) 2dx
YeHM:
C1=0,?C2= 2π1+m2E(π2, m1+m2) ,?C3= 0,? C4= ψ3(fεcos?(2π X) )φ(X),?C5=2π 11+m2 F(π 2,m1 +m2 )=I1ε, ?C6=12C4( εh) 2 C5,C7=1/ aψ1 ,?C8= I2ε3, ?C9= h23 a 1 a ψ2ψ1
where
m= 2πfε?ψ3(T) =12TT 2+ (12π)2+ 12 (12π)2ln ?(t+T2+(12π)2)
  
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