HỖ CHÍ MINHTRƯỜNG ĐẠI HỌC BÁCH KHOANGUYÊN MINH NHÂN STATIC AND FREE VIBRATION ANALYSES OFCORRUGATED PANELS USING HOMOGENIZATION MODELS AND A SMOOTHED FINITE ELEMENT METHOD PHAN TICH UNG
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There is a rich literature about homogenization techniques for an extensive range of corrugated structures from single-layer sheets of isotropic [2-12] or composite material [2, 11] to multi-layer structures like corrugated core sandwich plates [l3—
19] and corrugated laminates [20] Some of these homogenization methods are employed in parameter studies and optimization processes of corrugated structures [12, 21, 22] Most of the existing homogenization methods belong to engineering approaches in which different boundary conditions and assumptions of strain/stress distribution are used to derive the formulas of equivalent stiffness terms.
Shimansky and Lele [4] derived an analytical model for initial transverse stiffness of a sinusoidally corrugated panel and then highlighted the impact of plate thickness and degree of corrugation to this stiffness component Also working with a sinusoidal profile, Briassoulis [3] reviewed existing classical equivalent models and modified expressions of some extensional and flexural rigidities Samanta and Mukhopadhyay [5] used a similar approach to determine the equivalent extensional rigidities for trapezoidally corrugated sheets Combining them with the equivalent flexural rigidities derived by McFarland [23], the authors then qualified the resulting model in buckling, linear static, geometric nonlinear, and free vibration analyses Liew et al [6-9] used a meshless Galerkin method to investigate many mechanical behaviors of stiffened corrugated panels of sinusoidal and trapezoidal profiles In their studies, besides modifying formulas of some equivalent flexural rigidities, the authors also employed the derivation of equivalent transverse shear terms [24] in their FSDT plate model Xia et al [2] formulated generalized expressions to estimate equivalent stiffness terms for thin corrugated laminates of any shape This generalized model was then extended to cover the transverse shear stiffness by Park et al [11].
An approach using asymptotic method for homogenization of thin corrugated panels was proposed by Ye et al [10, 25] The method was based on governing equations of a thin shell theory in which the variable fields were represented in asymptotic expansions and then substituted back into the equations The consequently derived systems of governing differential equations were used to find the relationship between the equivalent plate and the original structure The method could handle corrugations of any shape as well as provide a set of formulas to recover the local fields in the corrugated panels Although there are many proposal treatments for corrugated panels, a deeper study of their effectiveness and accuracy is necessary.
After the equivalent models are derived, the shell analyses of the original panel can be replaced by those of a flat orthotropic plate Among various analytical or numerical methods for plate and shell studies, FEM is the most commonly used due to its effectiveness and extensibility for a variety of problems Because of the diversity of plate theories and their broad ranges of applications, FEM still needs improvements in elemental techniques, accuracy, stability, and computational efficiency, etc As a result, researching of new FEM with higher performance still takes a particular concern from scientists around the world during last decades.
On another front of the development of numerical methods, Liu and Nguyen- Thoi [26] have integrated the strain smoothing technique [27] into the FEMs to create a series of smoothed finite element methods (S-FEMs), such as the cell/element-based smoothed FEM (CS-FEM) [28], the node-based smoothed FEM (NS-FEM) [29, 30], the edge-based smoothed FEM (ES-FEM) [31], and the face- based smoothed FEM (FS-FEM) [32] These S-FEMs with different properties have been applied to improve the solution for a wide class of benchmark and practical mechanics problems, especially two-dimensional (2-D) and three-dimensional (3-D) linear elastic mechanics problems Related to analyses of plate and shell structures, the smoothing techniques have improved the performance of traditional plate elements such as DSG3 [33], MITC4 [34] and MIN3 [35] to produce smoothed counterparts such as ES-DSG3 [36], NS-DSG3 [37], CS-DSG3 [38], ES-MIN3
Among these smoothed plate elements, CS-MIN3 [40], which is a combination of the cell-based smoothed technique and the Mindlin plate element,MIN3 [35], possess many significant computational advantages In this method,each triangular element is divided into three sub-triangles on which the MIN3 is employed locally to compute the strains Afterward, the strain smoothing technique on the entire element is applied to smooth the strains on these sub-triangles TheCS-MIN3 is free of shear locking and achieves high accuracy compared to exact solutions and some other existing elements in the literature [40] Compared to otherS-FEMs, the CS-FEM is much simpler in implementation since it is conducted within a target element and does not need any extra information from adjacent elements With the mentioned advantages, the CS-MIN3 has been applied in different analyses of plate structures For examples, isotropic plates [40, 42],laminated composite plates [43, 44], functional graded plates [45], and recently cracked FGM plates [46].
Present Work and Outline - << 0n ng ke 4
The remainder of the present work is organized as follows The next chapter provides a summary of some most cited homogenization models with equivalent stiffness terms given explicitly Chapter 3 represents the Galerkin weak forms for free vibration and static analyses of equivalent orthotropic plates and a brief formulation of the CS-MIN3 Some numerical examples to evaluate the reliability and accuracy of the homogenization methods are described in Chapter 4 andChapter 5 Last but not least, the thesis ends up with some concluding remarks inChapter 6.
Homogenization Methods for Corrugated Plates ôô ô+ 5
Introduction to homogenization of corrugated Structures
In practical applications, depending on the usage and functions, corrugated panel structures possess numerous kinds of shape In this thesis, we only consider shell panels of periodic and symmetric corrugation profiles in one direction Notably, the focus of this work is on trapezoidally and sinusoidally corrugated panels (Fig 1-1b and Fig l-lc) The geometry of such structures can be defined by several parameters such as the half-period c, half-length 7, half-amplitude /f , and the trough angle q@ In several formulas, the period of a corrugation unit £ is used instead of the half one c.
The mid-surface of corrugated panels can be represented analytically using two coordinate systems including a global Cartesian coordinate system xyz anda local coordinate system syn (see Fig 2-la) The corrugation profile is designed in the xz-plane and then extruded in y-direction The local coordinate s represents the arch length at a position on the profile curve The location of a point on the mid- surface in the global coordinate system is defined as r(s,y)=x(s)i+ yj+ z(s)k (2.1) where i, j, and k are unit vectors of x-, y- and z- axes respectively Then we have dx/ds=cos@ and dz/ds=sin@ where @ is the tangential angle at that point.
For instance, the trapezoidal and sinusoidal corrugation profiles in Fig I-[b and Fig 1-lc are in turn represented by following equations
(x, Xp) cosa tx, —x,+(x-x,)/ cosa xe[3;,xg |
` where x, =—c, x,=-c+/f/tana, x,=—f/tana, x, =0, and s(x) = [ i+ (dz(r)/dr) dt = [ xj+( falc) cos? (at/c)dt: | (23) z(x) = f sin(ax/c) element, b) A Reissner-Mindlin plate and its field variables, c) CS-MIN3 element with three sub- triangles.
Although a shell model using FEM can give a precise analysis of corrugated panels, it requires significant computing cost Homogenization is a more practical alternative in which a flat orthotropic plate with equivalent rigidities is derived In many engineering approaches, the rigidities are determined through analyzing a primary unit cell called a Representative Volume Element (RVE) [2] (see Fig.
2-1a) In the local curvilinear coordinates, the strain energy of the RVE is
Q Q where Q) is the mid-surface domain, N=|N, NA i M=|M, M, M,,| and Q=|Q QO ] are the stress consultant vectors given by
In Eq (2.5), [z„[ =|#, E, Vy | lk] =| K, kK, K,, | and bal =| 7, Yr | are respectively the vectors of membrane strains, curvatures, and transverse shear Strains of the shell panel and
(0 0 Ag | 0 0 De “ are the rigidity matrices that consist of extensional, bending, and shearing stiffness components, respectively.
The strain energy of the equivalent plate element is calculated as
Ol Zi Zi Ol Zi A where Q is the mid-plane domain of the equivalent plate element and
INL|A 0 O|fe MI=l0 D, 0lÌkK|=Dz (2.8)
Q| |0 0 Dịy. are the stress consultant vectors in the equivalent plate All components of the matrices and vectors in Eq (2.8) are denoted by adding a bar above their counterparts in Eq (2.6) To obtain a homogenization model, we need to identify the extensional, bending, and transverse shear stiffness terms in A, D, , and D, , respectively.
Equivalent stiffness terms of some typical homogenization models
As mentioned in the introduction section, there are two main approaches for homogenization of corrugated panels In engineering approaches, the RVE is constrained in some specific boundary and loading conditions Equivalent force and energy methods are then applied to derive the equivalent stiffness terms as functions of material and geometric parameters of the original structure (more details in [2]).
This approach is highlighted in many studies about trapezoidally and sinusoidally corrugated panels by Samanta et al [5], Peng et al [6-9] and Xia et al [2] Using thin shell governing equations, the asymptotic approach proposed by Ye et al [10] is also applicable for generalized corrugation profiles Especially, their study finds out that the equivalent orthotropic model with no extension-bending coupling is only suitable for symmetric corrugation profiles For convenience, from this point onwards, we label the above four homogenization models as SamantaHM, PengHM, XiaHM, and YeHM, respectively.
Consider a corrugated panel containing n, corrugation units with half-period c and uniform thickness 4 We assume that the panel is made from an isotropic material with Young’s modulus EF, Poisson’s ratio v , and uniform density po The equivalent models for the panel when its profile is a trapezoidal, sinusoidal, or general one are respectively given in Table 2-1.
Table 2-1 Equivalent extensional, bending, and transverse shear stiffness terms for corrugated panels of trapezoidal, sinusoidal, and general profiles
Sinusoidal Term Trapezoidal profile profile General profile (symmetric) ì _— PengHM [8] PengHM [8] XiaHM [2] YeHM [10]
Ay VA, VA, VA, A, A, (A, VA,
5 Is oy 12(I-v?) 12{1-12) LA, +LD,, GEh+ΠTC
? 2c Eh ,, Ehf* 2c +?D, ca 2 ơ 1 Eh Eh’ Eh’ ly C, Eh ° c6(I+r) 24(+r) 24(I+r) ứ 4 2(I+r)
In Table |, the constant @, and a, are given as follows aL s(t LU Fd = > Bids = B'd° B =| Bi B: Bị | i=1 i=l
We see that the approximated membrane and bending strains of MIN3 are constant, while the approximated transverse shear strain is linear From the Galerkin weak form of free vibration analysis, the stiffness matrix of a MIN3 element can be derived as follows
K""? = A, (B") AB”+A (B°) D,B’+ [ (BY) DBdQ @10) T— where A, is the area of the entire triangular element.
Let the central point of the element ©, be the node 0 Respectively replacing the triangle (1,2,3) in the above formulation by the sub-triangles (1,2,0) , (0,2,3) and (1,0,3) , we can derive the MIN3 approximation of strain fields in each sub- element
K, =B, di i=1,2,3; di =|d;|, di =|d;|, di =|đ¿| GAD Ya =Bi dc do om L1
The cell-based strain smoothing technique on the whole element is then applied to derived the smoothed membrane £”, smoothed curvature K, and smoothed transverse shear strains y Using the constant smoothing function
3 1 ô l ô b r=], Ky OD, (x)dQ=7 DAs Xà `. q | M- ee > ệẦ —Xe ^^~~ [| > | — aa >ằ —Nay ^~~ [| SIH M- I, By (x)dQd,
The displacement vector d, at the center point 0 is simply the average of the three displacement vectors d/, d,, and d;, i.e d, = (d; +d; +d5)/ 3 As a result, the smoothed strains and the smoothed elemental stiffness matrix of the CS-MIN3 are derived as follows zZ"=B”d, £=Bjd, 7 (3.13)
Numerical Examples for Free Vibration AnalÌySIS
Validation example - cọ 0 nọ và 16 4.2 Free vibration analysis of a trapezoidally corrugated panel 18 4.3 Effect of some geometric DafaITẨ€TS - G G0 ng 22
In this example, we will evaluate the accuracy of the CS-MIN3 in the free vibration analysis of a square orthotropic plate whose material and geometric properties are given by
The modal analysis is in turn implemented using CS-DSG3, CS-MIN3, ES-DSG3 and ABAQUS in two cases of boundary conditions: simply supported (SSSS) and clamped (CCCC) at all edges.
Table 4-1 represents the first thirteen non-dimensional frequencies of the plate.
The ABAQUS results are taken as references beside those of the exact 3D analysis[47] Gin SSSS case only) The relative errors between these S-FEMs and ABAQUS are depicted in Fig 4-2 We see that all S-FEMs agree well with ABAQUS and exact 3D analysis The agreement in mode shape is also attained Specially, as
17 shown in Fig 4-3, only CS-MIN3 agrees with ABAQUS at the 10" and 11" mode shapes of the plate in SSSS boundary condition.
Table 41 The first thirteen non-dimensional frequencies of a square orthotropic plate in simply supported (SSSS) and clamped (CCCC) boundary conditions
Non-dimensional frequency Mod SSSS CCCC
€ CS- CS- ES- ABAQU © Exact CS- CS- ES- ABAQU DSG3 MIN3 DSG3 5 [47] DSG3 = MIN3 DSG3 S 1 00467 0.0465 0.0466 0.0465 0.0474 0.0781 0.0778 0.0777 0.0776 2 0.1029 0.1019 0.1024 0.1021 0.1033 0.1409 0.1396 0.1394 0.1396 3 0.1181 0.1171 01175 0.1173 0.1188 0.1524 0.1512 0.1510 0.1513 4 0.1685 0.1662 0.1671 0.1658 0.1694 0.2049 0.2023 0.2022 0.2011 5 0.1906 0.1875 0.1886 0.1892 0.1888 0.2340 0.2304 0.2299 0.2321 6 0.2178 0.2151 0.2155 0.2166 0.2180 0.2483 0.2457 0.2448 0.2467 7 0.2489 0.2442 0.2454 0.2437 0.2475 0.2875 0.2830 0.2819 0.2816 8 0.2629 0.2581 0.2590 0.2575 0.2624 0.2947 0.2894 0.2888 0.2874 9 0.3036 0.2966 0.2979 0.3017 0.2969 0.3489 0.3417 0.3399 0.3469 I0 0.3330 0.3274 0.3268 0.3245 0.3319 0.3566 0.3519 0.3490 0.3547 II 03344 03277 03277 0.3315 0.3320 0.3673 0.3604 0.3581 0.3558 I2 03558 0.3461 0.3471 0.3467 0.3476 0.3949 0.3860 0.3835 0.3857 I3 03735 0.3658 0.3649 0.3658 0.3707 0.3997 0.3915 0.3884 0.3900
5 ® me -2 sp CS-DSG3 Mode
Fig 4-2 Relative errors between non-dimensional frequencies of a square orthotropic plate computed by different methods and those computed by ABAQUS
ABAQUS CS-DSG3 CS-MIN3 ES-DSG3
Fig 4-3 Mode 10 and Mode 11 of a simply supported orthotropic plate which are computed by
ABAQUS and three different smoothed finite element methods 4.2 Free vibration analysis of a trapezoidally corrugated panel
We now conduct a modal analysis of a square trapezoidally corrugated panel using four homogenization methods CS-MIN3-SamantaHM, CS-MIN3-PengHM, CS- MIN3-XiaHM, and CS-MIN3-YeHM The material and geometric parameters of the panel are given as follows
The free vibration of the panel in simply supported (SSSS) and clamped (CCCC) boundary conditions was first studied in Liew et al [9] where only flexural stiffness terms of PengHM were considered and a Galerkin meshfree method was used instead In addition, the authors remained the original density in their homogenization model Now, the accuracy of the full PengHM and other homogenization models is assessed by comparing their frequencies and mode shapes with those of the corresponding ANSYS Workbench shell simulation using a mesh of 5851 nodes.
The equivalent stiffness terms and densities are listed in Table 4-2 We see that the four equivalent models have similar extensional terms The flexural terms
19 of SamantaHM are significantly different from other models Fig 4-4 visualizes the 1st, 34, 4h, 10", and 16" mode shapes of the panel and equivalent plates in SSSS boundary condition Here, the order is defined by increasing frequencies We see that all four equivalent methods agree with ANSYS about the 1*' mode shape while disagree with it about the 16 one Compared to ANSYS result, the first wrong mode of CS-MIN3-SamantaHM is 3 while that of CS-MIN3-XiaHM is 10 Both CS-MIN3-PengHM and CS-MIN3-YeHM begin their failure at the 16'" mode We recognize that the wrong mode shape likely comes from a swap between two close modes For example, the mode 3 and mode 4 of CS-MIN3-SamantaHM (Fig 4-4).
To make the comparison between homogenization models more reasonable, we rearrange the first fifteen modes of CS-MIN3-SamantaHM and CS-MIN3- XiaHM to attain a consistency with the arrangement in ANSYS result In this new arrangement, the first fifteen natural frequencies of the four homogenization models in SSSS and CCCC boundary conditions are listed in Table 4-3 and Table 4-4, respectively With the mean absolute percentage error (MAPE) of 13.68% in SSSS case and of 8.94% in CCCC case, CS-MIN3-SamantaHM cannot produce good prediction for all modes Fig 4-5 shows the relative errors between natural frequencies of CS-MIN3-PengHM, CS-MIN3-XiaHM, and CS-MIN3-YeHM and those of ANSYS The three methods give well-agreed results compared to ANSYS, especially at low frequency modes It should be noticed that the frequencies of CS- MIN3-YeHM are bounded by those of the other two methods.
Table 4-2 Equivalent stiffness terms for a trapezoidally corrugated panel with ten corrugation units
Term SamantaHM PengHM XIaHM YeHM Ay 1.056E+08 1.056E+08 1 OOOE+08 1 OOOE+08
CS-MIN3-PengHM CS-MIN3-SamantaHM ANSYS-Shell
CS-MIN3-YeHM — CS-MIN3-XiaHM
Fig 4-4 The 1°, 3 4°, 10", and 16" mode shapes (ordered by increasing frequencies) of a trapezoidally corrugated panel and its equivalent plates in SSSS boundary condition Table 4-3 The first fifteen natural frequencies of a simply supported (SSSS) trapezoidally corrugated panel with ten corrugation units
CS-MIN3- CS-MIN3- CS-MIN3- Meshfree- ANSYS-PengHM XiaHM YeHM PengHM [9] Shell
| 15.405 12.991 13.610 13.241 13.741 13.171 2 29.016 23.251 23.823 23.612 24.540 23.454 3 50.000 43.090 43.619 43.502 45.353 43.112 4 45.971 44.696 47.130 45 423 47.195 45.192 5 62.034 532.228 34.723 33.245 54.847 53.596 6 86.203 68.190 70.643 69.491 71.190 70.34 7 78.904 71.994 72.541 72.466 75.320 71.526 8 118.227 94.053 96.444 95.578 97.545 96.416 9 96.628 95.200 105.632 99.720 103.310 97.982 10 114.149 104.985 110.556 106.860 109.760 106.94 II 116.165 109.776 110.399 110.349 114.170 108.23 12 141.279 118.526 124.213 120.879 122.830 123.4 13 157.938 129.844 132.236 131.578 133.830 131.67 14 177.119 140.885 146.601 143.727 144.510 148.53 15 162.938 156.556 157.345 157.309 161.340 153.01 MAPE 13.68 % 1.89 % 2.05 % 1.24 % 3.50 % - Table 4-4 The first fifteen natural frequencies of a clamped (CCCC) trapezoidally corrugated panel with ten corrugation units
Mode CS-MIN3- CS-MIN3- CS-MIN3- CS-MIN3- Meshfree- ANSYS-
SamantaHM PengHM XIaHM YeHM PengHM [9] _ Shell Ị 27.997 27.112 28.423 27.516 28.588 27.3072 43.049 38.615 39.755 39.102 40.468 39.1153 67.095 60.831 61.809 61.381 63.276 61.2704 68.823 69.183 72.938 70.264 72.597 68.6415 85.588 77.696 81.445 79.032 80.888 78.6156 100.086 93.079 94.056 93.747 95.97 92.9647 111.899 95.227 98.827 96.825 98.058 97.6318 147 424 123.156 126.642 125.038 125.510 126.5709 130.461 133.537 140.928 135.625 137.850 129.76010 142.268 135.087 136.131 135.936 139.070 133.280II 148.731 141.175 148.666 143.621 159.470 140.28012 177 841 156.116 163.682 159.036 163.370 159.17013 191.511 161.934 165.399 164.118 181.660 165.21014 195.098 186.530 187.657 187.656 187.850 181.79015 217.205 180.416 188.146 183.859 210.490 187.420
+ OS-MIN3-PengHM Mode œ 15 r— —e— CS-MIN3-YeHM | | | | | lạ | | | |
8 10Ƒ x về a an i R ols, - Ke a © 5 ae verge ene ag Sy > ado ee ate
CL agg A ea Ce CBT ee,
Fig 4-5 Relative errors between frequencies of homogenization methods and ANSYS Workbench shell simulation in modal analysis of a trapezoidally corrugated panel in SSSS and CCCC boundary conditions 4.3 Effect of some geometric parameters Using the previous trapezoidally corrugated panel, we evaluate the accuracy of the four homogenization models for various values of the number of corrugation units, corrugation amplitude, trough angle and panel thickness In these parametric studies, we take the results of ANSYS APDL shell simulations as references.
The fundamental frequencies of the panel when the number of corrugation units 7,is modified are listed in Table 4-5 The relative errors (compared to ANSYS APDL results) of four homogenization methods and other benchmark software are represented in Fig 4-6 Except CS-MIN3-SamantaHM, other homogenization methods produce acceptable results at many levels of corrugation degree We also noticed the results of YeHM are softer than XiaHM while stiffer than PengHM.
Table 4-5 Fundamental frequencies of a trapezoidally corrugated panel corresponding to different values of number of corrugation units
CS-MIN3- CS-MIN3- CS-MIN3- ABAQUS- Workbench APDL- M PengHM XIaHM YeHM Shell -Shell Shell
5 SSSS 15.969 13.887 14.414 14.012 14.282 14.341 14.290 CCCC 29.595 29.128 30.309 29.328 29.197 29.399 29.269 10 SSSS 15.405 12.991 13.610 13.241 13.062 13.171 13.074 CCCC 27.977 27.112 28.423 27.516 26.961 27.307 27.138 20 SSSS 14.403 11.322 12.135 11.833 11.300 11.526 11.371 CCCC 25.065 23.314 24.907 24.146 23.028 23.729 23.468 30 SSSS 13.535 9.763 10.796 10.559 9.740 10.196 9.980
CCCC 22.398 19.686 21.622 21.010 19.309 20.520 20.119 MAPE 14.28 % 1.18 % 5.22 % 2.75 % 1.25 % 1.10 % - 10 Simply supported (SSSS) 3 Clamped (CCCC)
——t— CS-MIN3-PengHM g | | 1% CS-MIN3-XiaHM x 6 |
Oo At 5 2L i mH kỹ kp œ© a ©
Fig 4-6 Relative errors between fundamental frequencies of homogenization methods and ANSYS APDL shell simulation of a trapezoidally corrugated panel when the number of corrugation units
The amplitude of the corrugated panel f is replaced by f/2% where the coefficient ry in turn takes value of —1,0,1,2,3 or 4 Table 4-6 represents the ratios of stiffness terms between homogenization models and the flat plate ( ƒ =0) in the
24 case r, =4 Although the plate is nearly flat, many extensional terms and bending terms of SamantaHM are far different from those of the flat plate The failure is obvious due to the convergence to infinity of A, and the convergence to zero of D„ When f tends to zero Inheriting extensional terms from SamantaHM,
PengHM can only approach the bending terms of the flat plate As a result, in this study we only consider bending terms in these two models.
The fundamental frequencies of the panels in SSSS and CCCC boundary condition are reported in Table 4-7 and the relative errors between homogenization models and ANSYS APDL are depicted in Fig 4-7 Once again, the result of CS- MIN3-YeHM is bounded by those of CS-MIN3-PengHM and CS-MIN3-XiaHM and all three methods give acceptable approximations especially when the panel is more shallow.
Table 4-6 Ratios of stiffness terms between homogenization models and the flat plate (f = 0) when the amplitude of corrugation is reduced 16 times
Ratio of stiffness terms (r„ = 4) Term
A,/ A, 34.86 34.86 0.98 0.98A, / Av 34.86 34.86 0.98 0.98A, / Ay 0.92 0.92 1.01 1.01A | Age 0.99 0.99 0.99 0.99D,,/D, 0.90 0.99 0.99 0.99D,, | Di 0.00 0.99 0.99 0.99D,, | Dy 0.02 1.02 1.02 1.02Dye | Deg 4.03 1.00 1.01 1.01Dez | Des 3.98 0.99 0.99 0.99D,,/ Dy 4.03 1.00 1.01 1.01
Table 4-7 Fundamental frequencies of a trapezoidally corrugated panel corresponding to different levels of corrugation amplitude
Bound CS_MIN3- rr Cond SamantaH CS-MIN3- CS-MIN3- CS-MIN3-ABAQUS- APDL-
M PengHM XiaHM YeHM Shell Shell
0 SSSS 15.405 12.991 13.610 13.241 13.062 13.074 CCCC 27.997 27.112 28.423 27.516 26.961 27.138 1 SSSS 12.970 10.401 10.611 10.474 10.313 10.195 CCCC 20.604 20.350 20.746 20.384 20.089 19.982 2 SSSS 12.171 9.538 9.606 9.565 9.371 9.289
MAPE 13.33 % 1.99% 3.00 % 1.64 % 0.39 % - 6 Simply supported (5555) Clamped (CCCC)
AL Xr Xe x b vr "aay
Relative error (%) © Relative error (%) Se al —e— CS-MIN3-YeHM || al
6L— | | | | | -6L— | , ! ! 1 0 1 2 38 4 1 0 1 2 3 4 r/ ry Fig 4-7 Relative errors between fundamental frequencies of homogenization models and ANSYS
APDL shell simulation of a trapezoidally corrugated panel when the corrugation amplitude is changed
43.3 Trough angle The fundamental frequencies of the panel when the trough angle in turn take value œ °,30°,45°,60°,75° or 90° are represented in Table 4-8 The relative errors between homogenization methods and ANSYS APDL shell simulation are depicted in Fig 4-8 We can see that except CS-MIN3-SamantaHM, other three methods show good performance in all cases of trough angles.
Table 4-8 Fundamental frequencies of a trapezoidally corrugated plate corresponding to different values of trough angle
Fundamental frequency (Hz) CS-MIN3- CS-MIN3- CS-MIN3- CS-MIN3- ABAQUS- Trough Bound.
Angle Cond, SamantaHM PengHM XiaHM YeHM shel) PDL-Shell
Simply supported (SSSS) Clamped (CCCC)
1906 CS-MIN3-XiaHM re 2‡ xe 3h —e©— CS-MIN3-YeHM at a —-