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The transition from thin plates to moderately thick plates by using finite element analysis and the shear locking problem
Thin-Walled Structures 42 (2004) 1405–1430 www.elsevier.com/locate/tws
The transition from thin plates to moderately thick plates by using finite element analysis and the shear locking problem Tulay Aksu Ozkul , Umit Ture Department of Civil Engineering, Istanbul Technical University, 34469 Maslak-Istanbul, Turkey Received 1 August 2003; received in revised form 27 April 2004; accepted 13 May 2004
Abstract Two simple plate bending elements, based on Mindlin theory for analysis of both moderately thick and thin plates, are presented in this paper. These elements have either four nodes or eight nodes with 12 and 24 DOF, respectively. To illustrate the accuracy of these finite elements named as TURE12 and TURE24, several numerical examples of displacements and stresses for both thin and moderately thick plate bending problems are presented and discussed with a range of finite element meshes and thickness-to-plate length ratios. In addition, the bending and shearing behaviours of a Mindlin plate are analyzed with respect to shear locking. In order to test the shear locking, the results obtained from the Mindlin plate analysis using 4- or 8-noded elements with full, reduced, and selective reduced integration are compared with the exact classical thin plate solution. # 2004 Elsevier Ltd. All rights reserved. Keywords: Mindlin plate theory; Finite element; Shear locking; Thin plates; Thick plates
1. Introduction Great effort has been spent on the development of finite elements which are suitable for the bending of plates. Most of these efforts have been oriented to the thin plate theory, known as Kirchoff plate theory. However, when a plate is moderately thick (thickness=plate length > 0:05 or 0.1), this theory is not accurate enough to solve the problem due to the contribution of transverse shear to the deformation and stress distributions, and for this reason, a new approach is needed. Accordingly,
Corresponding author. Tel.: +90-212-285-3773; fax: +90-212-285-6587. E-mail address: [email protected] (T. Aksu Ozkul).
0263-8231/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2004.05.003
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Nomenclature U,V,W displacement components in the x, y and z directions w transverse displacement hx, hy normal rotations of the mid-plane ex, ey, cxy normal and shearing strains of the middle surface cxz, cyz thickness shearing strains of the middle surface E, m, G Young’s modulus, Poisson’s ratio, shearing modulus of elasticity h thickness of the plate p total potential energy [K]e element stiffness matrix K shear modification factor ½Keb element bending stiffness matrix ½Kes element shear stiffness matrix shape functions of the finite element [Ni]
the Mindlin plate theory has been improved for this purpose, and a number of papers about this theory have appeared [1–7]. v Mindlin plate element theory needs only to be C continuous, i.e. the displacements and the rotations are independent. For this reason, the behaviours of many Mindlin plate elements for moderately thick plates are usually very satisfactory. However, some of the Mindlin plate elements may cause a problem, known as shear locking, when they are used to analyse thin plate structures. To alleviate the shear locking problem, some methods have been suggested, such as, reduced and selective reduced integration techniques [8–10]. In this paper, two displacement-based Mindlin plate elements, which have indev pendent displacement w and rotations hx and hy based on C continuous shape functions, are briefly reviewed for analysis of both moderately thick and thin plates. These elements, named as TURE12 and TURE24, are 4-noded and 8-noded rectangular finite elements, with the basic 3 DOF per node and 12 and 24 DOF for each element, respectively. Numerical results are presented to show the behaviour of these elements involving some plates of different aspect ratios and support conditions with either uniformly distributed load or concentrated load. The results are compared with those obtained using other solution techniques such as analytical solution and different finite elements. The shear locking problem, appeared in application to thin plates, is alleviated by separating the shear and bending energies and by using selective reduced integration technique on the shear term. Several examples are then presented, in which the behaviours of shear locking of the two elements are given, by using full, reduced, and selective reduced integration techniques, and the results are compared with those of the classical thin plate solution. In each example, the transition from thin to moderately thick plates are calculated and compared with each other.
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2. Mindlin plate element formulation 2.1. Fundamental relations At a typical point in a Mindlin plate, the displacement components may be represented as U ¼ zhx ðx; yÞ V ¼ zhy ðx; yÞ W ¼ wðx; yÞ
ð1Þ
where U, V, and W are the displacement components in the x, y, and z directions, respectively, w is the transverse displacement, and hx and hy are the normal rotations of the mid-plane, respectively. The rotations hx and hy can be expressed in the form @w cxz @x @w cyz hy ¼ @y hx ¼
ð2Þ
where @w=@x and @w=@y are the slopes of the deformed median surface in the x and y directions and cxz and cyz are the shear strains. For a displacement-based, n-noded Mindlin plate element, the displacement and normal rotations may be represented by the expressions w¼
n X
hx ¼
Ni w i ;
i¼1
n X Ni hxi ;
hy ¼
i¼1
n X Ni hyi
ð3Þ
i¼1
where wi, hxi and hyi are the displacement and rotation values at node i and Ni is the shape function with node i. 2.2. Strain–displacement relations For the Mindlin plate theory, the strain components may be written in terms of the displacements of the middle surface as follows: @hx ; @x @hy @hx þ cxy ¼ z @x @y @x hx ; cxz ¼ @x ex ¼ z
ey ¼ z
@hy @y ð4Þ
cyz ¼
@x hy @y
Curvature–displacement relations and shear strain–displacement relations are then written in the matrix form X n Bbi eb ð5Þ ½e ¼ ¼ ½di Bsi es i¼1
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in which the curvatures [eb], the shear strains [es], and the unknowns at node i [di] are expressed as ½eb ½es ½di
T ex ey cxy ¼ T cxz cyz ¼ T ¼ ½ xi hxi hyi
ð6Þ
where T shows the transpose, as usual. The curvature–displacement matrix ½Bbi and the shear strain–displacement matrix ½Bsi associated with node i may be written as follows: 2 3 0 Ni;x 0 Ni;y 5 ½Bbi ¼ 4 0 0 0 N Ni;x ð7Þ i;y Ni;x Ni 0 ½Bbi ¼ Ni;y 0 Ni
2.3. Stress–strain relations The moment–curvature relationships and the bending moments are ½rb ¼ ½Db ½eb ;
½rb ¼ ½ Mx
My
Mxy T
ð8Þ
and the matrix of flexural rigidities may be expressed for an isotropic material as follows: 2 3 1 t 0 Eh3 4t 1 0 5 ½Db ¼ ð9Þ 12ð1 m2 Þ 0 0 1 t=2 where E is Young’s modulus, v is Poisson’s ratio and h is the plate thickness. The shear force–shear strain relationships and the shear forces are given as ½rs ¼ ½Ds ½es ;
½rs ¼ ½ Qx
Qy T
ð10Þ
where the matrix of shear rigidities for an isotropic material may be expressed as kEh 1 0 ð11Þ ½Ds ¼ 2ð1 þ mÞ 0 1 Here, k is the shear modification factor and is normally set equal to 5/6 for homogeneous isotropic plates. 2.4. The total potential energy The total potential energy, p, can be written as ð 1 p¼ ðrx ex þ ry ey þ sxy cxy þ sxz cxz þ syz cyz Þdv W 2 v
ð12Þ
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where W is the work done by the applied loads. Considering the strains and inserting in the above expression, ð ð Eh3 kEh 2 2 2 2 e þ 2me e þ e c þ c p¼ dA þ x y y yz dA W 2ð1 þ mÞ A xz 24ð1 m2 Þ A x and p¼
1 2
ð
½eb T ½Db ½eb dA þ A
1 2
ð
½es T ½Ds ½es dA W
ð13Þ
A
the total potential energy for moderately thick plates is derived in the matrix form. By substituting the displacement functions in Eq. (13) and integrating over the element fields, the following equations are obtained for an element ½Ke fdge ¼ ff ge e
e
ð14Þ e
where [K] , {d} and {f} are the element stiffness matrix, displacement vector and the element force vector, respectively.
3. Finite elements Two elements are examined with the formulation of the Mindlin plate element given in Section 2. The first one is the 4-noded rectangular plate element (TURE12) which has linear shape functions, and the second one is the 8-noded rectangular plate element (TURE24) which has second-degree shape functions, respectively, as shown in Fig. 1. The shape functions of these elements are given as follows. Degrees of freedom of these finite elements at each node are the transverse displacement w and the two rotations hx and hy about the x and y axes, respectively. Thus, TURE12 and TURE24 elements have 3 DOF at each node, and 12 DOF and 24 DOF for each element, respectively.
4. Shear locking v
Plate finite elements based on Mindlin theory require only C continuity of displacement and independent rotations, and soon appeared as an alternative according to the plate elements with C0 continuity. Therefore, the behaviour of the Mindlin plate elements is usually very good for a moderately thick plate situation. However, when a thin plate is considered, these displacement-based elements cause a problem known as ‘‘shear locking’’. When the full integration of the stiffness matrices is used with standard Mindlin finite elements, very stiff results may be obtained in applications to thin plates. This means that the bending energy, which should dominate the shear terms, will be incorrectly estimated to be zero in thin plate problems.
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Fig. 1. The finite elements and the shape functions: (a) linear, (b) quadratic.
To avoid the shear locking problem in thin plates, the reduced and selective reduced integration techniques were proposed in the early 1970s [8,9]. The reduced integration procedure is the reduction in the order of integration in computing the stiffness matrix of the finite element. Similarly, the selective integration procedure is also a kind of reduced integration rule which is used to evaluate the stiffness matrix associated with the troublesome shear strain energy. That is to say, this has been adopted to the shear stiffness matrix only and full integration is used on the remaining terms [8–10]. Therefore, the [K]e element stiffness matrix can be obtained by separating into bending terms and shear terms. With these definitions, [K]e element stiffness matrix is given by simplifying the expression ½Ke ¼ ½Kb e þ ½Ks e where ½Kb e ¼ ½Ks e ¼
ð ð
½Bb T ½Db ½Bb dA A
½Bs T ½Ds ½Bs dA
ð15Þ
A
½Kb e and ½Ks e are the bending and shear stiffness matrices, respectively. Table 1 shows the full, reduced, and selective reduced integration rules used to test the shear locking response of the TURE12 and TURE24 elements in this paper.
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Table 1 Integration rules Thick plate
Thin plate
Full integration ½Kb TURE12 TURE24
e
2 2 3 3
½Ks
Reduced integration e
2 2 3 3
½Kb
e
1 1 2 2
½Ks
e
1 1 2 2
Selective reduced integration ½Kb e
½Ks e
2 2 3 3
1 1 2 2
5. Numerical examples A number of examples are taken to examine the performance of the elements of TURE12 and TURE24, which include tests on convergence of both displacements and stresses, and shear locking with a derived computer programme. The results are compared with those obtained by other analytical and numerical methods. 5.1. Example 1: square plate A square plate which is subjected to a uniformly distributed load or a concentrated load is modelled with two different boundary conditions, i.e. either a simply supported or a clamped boundary on all four edges, to evaluate the convergence of the solutions obtained with TURE12 and TURE24. The geometric and material properties of this problem, which has been most frequently examined in the literature, are E ¼ 106 kN=m2 , v ¼ 0:30, a ¼ 8 m, P ¼ 1 kN, q ¼ 1 kN=m2 , and k ¼ 5=6, where P is the concentrated load, q is the uniformly distributed load, and a is the length of the plate. Due to symmetry, only a quarter of the plate is taken into consideration for the analysis by a series of meshes of N N element (N denotes the number of elements in one direction) with different ratios for h=a. The displacements at the center are computed with TURE12 and TURE24 elements for thickness/length ratios (h=a) varying from 0.001 to 0.20 and three mesh densities. The results obtained by TURE12 and TURE24 are presented in dimensionless form in Tables 2–5 with the exact solution and with those given by other authors for the cases of simply supported and clamped edges subjected to uniform load or concentrated load. In addition, the convergence of the central displacement and the central moment obtained by these elements are presented in Figs. 2–5 for h=a ¼ 0:10 with four mesh densities, for comparison. It is obvious that the element TURE24 is a high precision and fast convergence element if it is compared with the element TURE12, which yielded the best result using 8 8 mesh. On the contrary, the error ratio of TURE24 is less than that of TURE12 in some results, even with 2 2 mesh, as shown in Tables 2–5. It should also be noted that as the plate becomes thicker, the convergence of the element TURE24 is faster than that of TURE12. Tables 6 and 7 show the results of the central displacements and the central moments, obtained by the elements TURE12 and TURE24, for uniformly loaded
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Table 2 Central displacements wc ð100D=qa4 Þ for clamped square plate subjected to uniform load q h=a
0.001 0.002 0.005 0.01 0.02 0.05 0.10 0.15 0.20
TURE12
TURE24
Mesh number
Mesh number
Exact solution [11]
2 2
4 4
8 8
2 2
4 4
8 8
0:285 104 0:114 103 7:07 104 2:759 103 0.0102 0.0444 0.1005 0.1049 0.1628
0.1250
0.1261
0.0389
0.0831
0.1256
0.1250
0.1261
0.0411
0.1088
0.1262
0.1250
0.1261
0.0535
0.1228
0.1265
0.1250
0.1263
0.0786
0.1258
0.1267
0.1265
0.1252 0.1251 0.1255 0.1257 0.1987
0.1274 0.1323 0.1504 0.1733 0.2139
0.1076 0.1291 0.1501 0.1745 0.2146
0.1276 0.1326 0.1506 0.1785 0.2170
0.1278 0.1327 0.1506 0.1787 0.2172
0.1499
0.1265
0.2167
Thin plate solution (Timoshenko and Woinowsky-Krieger [15]): 0.1265.
clamped square plates and uniformly loaded simply supported square plates from very thin to moderately thick thicknesses with 8 8 mesh density. The results obtained by the presented formulation are in good agreement with those from the other investigations. 5.2. Example 2: circular plate A circular plate, subjected to a uniformly distributed load, is modelled with two different boundary conditions, i.e. simply supported and clamped conditions by Table 3 Central displacements wc ð100D=qa4 Þ for simply supported square plate subjected to uniform load q h=a
0.001 0.002 0.005 0.01 0.02 0.05 0.10 0.15 0.20
TURE12
TURE24
Mesh number
Mesh number
Exact solution [12]
2x2
4x4
8x8
2x2
4x4
8x8
0.2306 0.2306 0.2310 0.2323 0.2375 0.2624 0.3203 0.2664 0.3563
0.4039 0.4039 0.4039 0.4041 0.4057 0.4093 0.4262 0.4153 0.4671
0.4055 0.4055 0.4055 0.4057 0.4073 0.4107 0.4273 0.4434 0.4845
0.1866 0.3083 0.3876 0.4017 0.4065 0.4107 0.4273 0.4517 0.4891
0.4043 0.4054 0.4059 0.4061 0.4078 0.4112 0.4277 0.4535 0.4903
0.4060 0.4060 0.4061 0.4062 0.4078 0.4112 0.4278 0.4536 0.4904
Thin plate solution (Timoshenko and Woinowsky-Krieger [15]): 0.4057.
0.4062
0.4062 0.4107 0.4273 0.4536 0.4906
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Table 4 Central displacements wc ðEh3 =Pa2 Þ for clamped square plate for concentrated load h=a
TURE12
TURE24
Yuan and Miller [13]
Mesh number
Mesh number
Mesh number
2 2 0.001 0.002 0.005 0.01 0.02 0.05 0.10 0.15 0.20
6
12:44 10 49:77 106 3:096 106 0.01216 0.01311 0.02158 0.05423 0.05679 0.09087
4 4
8 8
2 2
4 4
8 8
6 6
0.05898 0.05898 0.05905 0.05927 0.06027 0.06607 0.08716 0.09269 0.12792
0.06065 0.06065 0.06067 0.06093 0.06213 0.06934 0.09562 0.10799 0.14707
0.02354 0.02439 0.02957 0.04006 0.05239 0.06326 0.07878 0.09905 0.13141
0.04263 0.05375 0.05946 0.06075 0.06192 0.06661 0.08332 0.10995 0.14756
0.06059 0.06096 0.06116 0.06141 0.06220 0.06749 0.08714 0.11806 0.16161
0.05855 0.05856 0.05862 0.05882 0.05963 0.06522 0.08493 0.1173 0.1623
Thin plate solution (Flu¨gge [14]): 0.0611.
using a series of meshes of TURE12 and TURE24 elements. The meshes for one quadrant, and the geometric and material properties are defined in Fig. 6. The central displacement obtained from the analyses are presented in Tables 8 and 9 for different h=R values varying from 0.001 to 0.30, and for three different mesh densities together with those of analytical solutions and a finite element solution namely ARS-Q12 given by Soh et al. [11], for comparison. As shown in Tables 8 and 9, error percentage for TURE24 element is much less than error percentage for TURE12 element, and for former element, convergence of the central displacement is very good even with three elements when it is compared with the latter, which gives the best result with as many as 27 elements. Table 5 Central displacements wc ðEh3 =Pa4 Þ for simply supported square plate for concentrated load h=a
0.001 0.002 0.005 0.01 0.02 0.05 0.10 0.15 0.20
TURE12
TURE24
Yuan and Miller [13]
Mesh number
Mesh number
Mesh number
2 2
4 4
8 8
2 2
4 4
8 8
6 6
0.06692 0.06693 0.06707 0.06765 0.06984 0.08159 0.11227 0.12348 0.14113
0.12595 0.12592 0.12601 0.12618 0.12736 0.13277 0.15346 0.15472 0.19134
0.12638 0.12638 0.12595 0.12634 0.12806 0.13484 0.16071 0.17147 0.21056
0.06281 0.09551 0.11805 0.12314 0.12569 0.12891 0.14338 0.16363 0.19529
0.12350 0.12503 0.12582 0.12634 0.12697 0.13164 0.14823 0.17395 0.21104
0.12622 0.12639 0.12654 0.12682 0.12762 0.13194 0.15145 0.18204 0.22505
0.12441 0.12442 0.12453 0.12491 0.12632 0.13432 0.15754 0.19283 0.24026
Thin plate solution (Timoshenko and Woinowsky-Krieger [15]): 0.1267.
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Fig. 2. Central displacement convergence test for uniformly loaded clamped square plate with h=a ¼ 0:10.
Fig. 3. Central moment convergence test for uniformly loaded clamped square plate with h=a ¼ 0:10.
Fig. 4. Central displacement convergence test for uniformly loaded simply supported square plate with h=a ¼ 0:10.
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Fig. 5. Central moment convergence test for uniformly loaded simply supported square plate with h=a ¼ 0:10.
Furthermore, in Table 10, the results of the central bending moment for the two elements, TURE12 and TURE24, are shown together with those of the exact solutions and some other finite element solutions given in [11] and the results obtained with SAP2000. All results were obtained for the thin (h=R ¼ 0:02) and moderately thick (h=R ¼ 0:20) plates for the cases of simply supported and clamped edges subjected to uniform loading. Table 10 shows that the results obtained using the presented elements are in excellent agreement with those of the other solutions. The results of the central bending moment found with TURE24 element have converged very fast to the exact solutions, like the previous example. As the plate becomes thicker (h=R ¼ 0:20), the convergence of the element TURE24 is much faster than that of TURE12. Comparing with the result of exact solution given by Batoz and Dahn [11], TURE12 element yielded the best central bending moment with 20.69% error for clamped circular plate and with 18.85% error for simply supported circular plate, respectively. While the error ratio of TURE24 element is only 1.35% for the clamped circular plate and 0.44% for the simply supported circular plate, respectively. For a thin circular plate (h=R ¼ 0:02), with a clamped boundary, the convergence for central displacements, which are normalized by the thin plate theory solution, is plotted in Fig. 7 by using different mesh patterns. Furthermore, the convergence result for central displacements for a moderately thick clamped circular plate with h=R ¼ 0:20 is shown in Fig. 8. The accuracy and reliability of these two elements with different mesh patterns are again clearly illustrated in these figures. The displacement distributions along the radius of the uniformly loaded clamped circular plates with two h=R ratios (h=R ¼ 0:02 and h=R ¼ 0:20) are also plotted. The results obtained using the two elements are compared, with the corresponding analytical solutions and with those given in [11,17], in Figs. 9 and 10, respectively. Close agreement between the present solutions and the reference solutions is obtained.
0.1261 0.1261 0.1261 0.1263 0.1274 0.1323 0.1504 0.1733 0.2139
0.1256 0.1262 0.1265 0.1267 0.1278 0.1327 0.1506 0.1787 0.2172
0.1279 – – 0.1281 – – 0.1515 – 0.2183
0.1293 – – 0.1293 – – 0.1521 0.1801 0.2181
0.1234 0.1234 0.1234 0.1236 0.1245 0.1298 0.1482 0.1776 0.2171
[13]
0.122 – – 0.123 0.123 – 0.146 – 0.211
[16]
0.1265 – – 0.1265 – – 0.1499 0.1798 0.2167
Analytic solution [11] 2.3004 2.3003 2.3001 2.2999 2.3013 2.3089 2.3055 2.3058 2.3065
2.2939 2.2993 2.3008 2.3014 2.3028 2.3108 2.3314 2.3524 2.3702
TURE24
Central moments [12]
TURE12
[11]
TURE12
TURE24
Central displacements
Thin plate solution (Timoshenko and Woinowsky-Krieger [15]): 0.1265.
0.001 0.002 0.005 0.01 0.02 0.05 0.10 0.15 0.20
h=a
2.0685 – – 2.0685 – – 2.0701 – 2.0707
[11]
Table 6 Central displacements wc ð100D=qa4 Þ and central moments (100Mx =qa2 ) for clamped square plate subjected to uniform load q
2.27 – – 2.27 2.27 – 2.36 – 2.50
[16]
2.31 2.31 2.31 2.31 2.31 2.31 2.31 2.31 2.31
Thin plate solution [16]
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0.4055 0.4055 0.4055 0.4057 0.4073 0.4107 0.4273 0.4434 0.4845
0.4060 0.4060 0.4061 0.4062 0.4078 0.4112 0.4278 0.4536 0.4904
0.4062 – – 0.4064 – – 0.4271 – 0.4903
0.407 – – 0.407 0.408 – 0.423 – 0.48
0.4043 – – 0.4045 – – 0.4242 0.4502 0.4869
0.4054 0.4055 0.4057 0.4067 0.4101 0.4258 0.4596 0.5018 0.5511
[13] 0.4062 – – 0.4062 0.4062 0.4108 0.4273 0.4536 0.4906
Exact [12] 4.7889 4.7889 4.7889 4.7889 4.7889 4.7889 4.7889 4.7686 4.7263
4.7951 4.7952 4.7953 4.7953 4.7953 4.7953 4.7953 4.7951 4.7952
4.789 – – 4.802 – – 5.096 – –
TURE12 TURE24 [11]
[12]
TURE12 TURE24 [11]
[16]
Central moments
Central displacements
Thin plate solution (Timoshenko and Woinowsky-Krieger [15]): 0.4057.
0.001 0.002 0.005 0.01 0.02 0.05 0.10 0.15 0.20
h=a
4.82 – – 4.82 4.82 – 4.84 – 4.87
[16]
4.779 4.779 4.781 4.788 4.813 4.915 5.079 5.223 5.35
[13]
Table 7 Central displacements wc ð100D=qa4 Þ and central moments (100Mx =qa2 ) for simply supported square plate subjected to uniform load q
4:792=4:789 – – 4:820=4:789 – – 5:096=4:789 – –
Exact [11] Thick=thin
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Fig. 6. Circular plate with mesh patterns.
5.3. Example 3: skew plate v
A 60 skew plate, simply supported on two opposite edges and free on the other two edges, is analyzed. A symmetric quadrant of this skew plate, under uniform loading, which was originally studied by Razzaque [11], is idealized with meshes involving 2 2, 4 4, and 8 8 elements, respectively (Fig. 11). Convergence of the central displacement and the central bending moment obtained with the two elements, TURE12 and TURE24, is shown in Tables 11 and 12, and Fig. 12, respectively. Results obtained by these elements are compared with those given by other researchers. In all cases, reasonably good convergence is obtained. As it is expected, TURE24 element has converged much faster than TURE12 element like the previous two examples. TURE12 element yielded the best result for the central displacement by using (8 8) mesh with 1.08% error with respect to
0.001 0.01 0.02 0.10 0.20 0.30
h=R
0.01298 0.01298 0.01300 0.01369 0.00657 0.01181
3 elements
TURE12
0.01433 0.01437 0.01447 0.01527 0.01225 0.01786
0.01385 0.01408 0.01447 0.01593 0.01487 0.01990
11.39 9.92 7.42 2.51 19.53 18.34
0.00783 0.01300 0.01485 0.01637 0.01784 0.02182
12 elements 27 elements Error % 3 elements (27 elements)
TURE24
0.01519 0.01568 0.01568 0.01634 0.01848 0.02207
0.01563 0.01563 0.01565 0.01634 0.01848 0.02207
0.00 0.00 0.13 0.00 0.00 9.44
12 elements 27 elements Error % (27 elements)
Table 8 Central displacements wc ðD=qR4 Þ for uniformly loaded clamped circular plate
0.01610 0.01611 0.01612 0.01675 0.01881 0.02233
0.01575 0.01576 0.01577 0.01643 0.01856 0.02212
0.01563 0.01563 0.01563 0.01634 0.01848 0.02437
Analytical solution 12 elements 48 elements [11]
[11] ARS-Q12
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0.001 0.01 0.02 0.10 0.20 0.30
h=R
0.04821 0.04821 0.04824 0.04889 0.30092 0.67191
3 elements
TURE12
0.05917 0.05917 0.05914 0.05990 0.04374 0.05543
0.06176 0.06176 0.06178 0.06247 0.05467 0.06317
3.05 3.06 3.06 3.03 17.86 9.92
0.06446 0.06441 0.06435 0.06464 0.06606 0.07003
12 elements 27 elements Error % 3 elements (27 elements)
TURE24
0.06151 0.06354 0.06372 0.06443 0.06654 0.07016
0.06242 0.06363 0.06370 0.06441 0.06656 0.07010
2.00 0.12 0.04 0.01 0.00 0.04
12 elements 27 elements Error % (27 elements)
Table 9 Central displacements wc ðD=qR4 Þ for uniformly loaded simply supported circular plate
0.06305 0.06306 0.06308 0.06370 0.06577 0.06928
0.06354 0.06355 0.06357 0.06422 0.06635 0.06991
0.06370 0.06371 0.06373 0.06442 0.06656 0.07013
Analytical solution 12 elements 48 elements [11]
[11] ARS-Q12
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Clamped cir- 0.0894 cular plate Simply sup- 0.2026 ported circular plate
0.0828
0.2073
0.0821
0.2065
0.2069
0.0826 0.2056
0.0816
TURE12 TURE24 SAP2000 ARS-Q12 MITC4 (27 ele(27 ele- (27 ele- (48 ele- (48 elements) ments) ments) ments) ments) [11] [11]
h=R ¼ 0:02
0.2068
0.0824
DKMQ (48 elements) [11]
0.2064
0.0812
Exact solution (Batoz and Dahn [11])
0.1675
0.0644
0.2073
0.0823
0.2071
0.0828
TURE12 TURE24 ARS-Q12 (27 ele(27 ele(48 elements) ments) ments) [11]
h=R ¼ 0:20
Table 10 Central moment ðMc =qR2 Þ for clamped circular plate and for simply supported circular plate subjected to uniform load q
0.2056
0.0816
MITC4 (48 elements) [11]
0.2072
0.0828
DKMQ (48 elements) [11]
0.2064
0.0812
Exact solution (Batoz and Dahn [11])
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Fig. 7. Central displacement convergence test for uniformly loaded clamped circular plate with h=R ¼ 0:02.
Fig. 8. Central displacement convergence test for uniformly loaded clamped circular plate with h=R ¼ 0:20.
Fig. 9. The displacement distributions along the radius of the uniformly loaded clamped circular plates for h=R ¼ 0:02.
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Fig. 10. The displacement distributions along the radius of the uniformly loaded clamped circular plate for h=R ¼ 0:20.
Fig. 11. Razzaque’s skew plate (2 2 mesh).
the theoretical solution given by Razzaque [11] as shown in Table 11. On the contrary, error ratio of TURE24 element, even with (2 2) mesh 0.0755%, is less than error ratio of TURE12 element. 5.4. Example 4: locking test Two typical examples are taken to test the shear locking response of the present elements. The first one is a square plate, which is subjected to a uniform load or a concentrated load at the center and idealized with two different boundary conditions, i.e. a simply supported and a clamped boundary. The second one is a circular plate, simply supported or clamped boundary, and subjected to uniform
TURE12
0.6787 0.7665 – 0.7859 –
Mesh
2 2 4 4 6 6 8 8 12 12
0.7939 0.7974 – 0.7981 –
TURE24
0.7727 0.7907 – 0.7949 –
SAP2000
0.6666 0.7691 0.7829 0.7876 0.7909
0.3976 0.6737 0.7371 0.7610 0.7785
0.6666 0.7695 0.7829 0.7876 0.7908
0.812 0.792 – 0.794 –
[17] C8
DKMQ
ARS-Q12
MITC4
[11]
Table 11 Central displacement wc ð100D=qL4 Þ for Razzaque’s skew plate
0.806 0.769 – 0.788 –
S8
0.385 0.753 – 0.759 –
H4 (lumped)
0.551 0.710 – 0.769 –
H4 (consistent)
0.7945
Razzaque [11]
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TURE12
72.203 89.453 – 94.358 –
Mesh
2 2 4 4 6 6 8 8 12 12
96.625 97.375 – 96.297 –
TURE24
95.781 95.937 – 96.015 –
SAP2000
92.46 95.95 96.09 96.05 96.02
37.90 77.60 87.10 90.90 93.70
92.20 96.00 96.10 96.10 96.00
92.82 94.83 – 95.79 –
C8
DKMQ
ARS-Q12
MITC4
[17]
[11]
Table 12 Central moment ð103 My =qL2 Þ for Razzaque’s skew plate
91.57 91.29 – 95.99 –
S8
41.62 79.49 – 91.48 –
H4 (lumped)
59.16 83.62 – 92.53 –
H4 (consistent)
95.89
Razzaque [11]
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Fig. 12. Central displacement convergence test for Razzaque skew plate with h=a ¼ 0:01.
Table 13 Central displacement wc ð100D=qa4 Þ locking test for uniformly loaded clamped square plate h=a
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.20
TURE12
TURE24
Full integration
Reduced Selective Full inteintegration reduced gration integration
Reduced Selective integration reduced integration
0.0132 0.0406 0.0661 0.0853 0.0993 0.1100 0.1186 0.1260 0.1327 0.1396 0.2196
0.1269 0.1277 0.1289 0.1307 0.1329 0.1355 0.1386 0.1424 0.1460 0.1511 0.2106
0.1268 0.1276 0.1288 0.1303 0.1328 0.1355 0.1386 0.1418 0.1460 0.1499 0.2198
0.1264 0.1272 0.1284 0.1302 0.1323 0.1350 0.1382 0.1418 0.1460 0.1499 0.2106
0.1255 0.1278 0.1289 0.1305 0.1326 0.1352 0.1386 0.1418 0.1460 0.1499 0.2198
0.1267 0.1275 0.1288 0.1303 0.1328 0.1355 0.1386 0.1418 0.1460 0.1499 0.2198
Analytical solution
Thin plate solution (Timoshenko and WoinowskyKrieger [15]): 0.1265
0.1499 [11] 0.2167 [11]
loading. The geometric and material properties used are the same as Examples 1 and 2. In the locking test, a symmetric quadrant of the square plate and the circular plate are modelled with various thickness ratios by using the 8 8 finite element meshes for the former plate and 27 element meshes for the latter, respectively. The results from the Mindlin plate bending analysis using 4-noded (TURE12) and 8-noded (TURE24) elements with full integration (FI), reduced integration (RI), and selective reduced integration (SRI) are compared with those of the analytical solution. The integral rules used in the examples are given in Table 1. The solutions were obtained for the boundary conditions and loading types mentioned above for the square or circular plates; however, the results are given in Tables 13 and 14 and Figs. 13–16 for only one type, i.e. a uniform loading under
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Table 14 Central displacement wc ðD=qR4 Þ locking test for uniformly loaded clamped circular plate h=R
0.001 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.20
TURE12
TURE24
Full integration
Reduced integration
Selective reduced integration
Full Reduced Selective integration integration reduced integration
0:1678 105 0:16607 105 0:64359 105 0:13765 104 0:22916 104 0:33138 104 0:43813 104 0:54418 104 0:64703 104 0:74435 104 0:83563 104 0.01832
0.01384 0.01407 0.01447 0.01478 0.01499 0.01518 0.01533 0.01548 0.01573 0.01577 0.01593 0.01813
0.01489 0.01493 0.01503 0.01515 0.01527 0.01539 0.01552 0.01564 0.01578 0.01592 0.01607 0.01816
0.01463 0.01539 0.01546 0.01553 0.01562 0.01571 0.01581 0.01592 0.01603 0.01617 0.01630 0.01849
0.01563 0.01563 0.01565 0.01569 0.01574 0.01580 0.01588 0.01597 0.01608 0.01621 0.01635 0.01849
0.01563 0.01563 0.01565 0.01569 0.01574 0.01580 0.01588 0.01597 0.01608 0.01621 0.01635 0.01849
Analytical solution [11]
Thin plate solution 0.01563
0.01634 0.01848
clamped boundary, because the results found for the other conditions are similar with the first one. Shear locking problem, in the TURE12 element, which started from assumed thin plate limit h=a or h=R 0:10, were observed very clearly and prevented by using selective reduced integration as shown in Tables 13 and 14. On the other hand, it was found that the shear locking problem in the TURE24 element is not as evident as that in the TURE12 element and the thin plate transition limit can be reduced to 0.05 for this element. Besides, the full integration rule has provided good results even in thin plate application for TURE24 element. Consequently, prevention of shear locking in the TURE24 element is much easier than that in the TURE12 element.
Fig. 13. Occurrence of shear locking for uniformly loaded clamped square plate.
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Fig. 14. Prevention of shear locking for uniformly loaded clamped square plate.
Fig. 15. Occurrence of shear locking for uniformly loaded clamped circular plate.
Fig. 16. Prevention of shear locking for uniformly loaded clamped circular plate.
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It can be concluded that the selective reduced integration rule should be used for h=a 0:10 for TURE12 element and for h=a ¼ 0:05 0:06 for TURE24 element, respectively. But for h=a > 0:10, the full integration rule must be employed definitely for both TURE12 and TURE24 elements. In addition, occurrence of shear locking with full integration and prevention of it with selective reduced integration are plotted with the corresponding analytical solutions in Figs. 13–16 for both square and circular plates subjected to uniform loading with the clamped boundary condition. It can be said that the tested elements with the selective reduced integration rule give an optimal performance.
6. Conclusion The approach of proposed two finite element solutions, which are 4-noded and 8-noded Mindlin plate bending elements, presented in the analysis of both moderately thick and thin plates. Convergence of these elements is discussed for different mesh densities and different h=a ratios. As can be seen from the examples presented, both the displacements and the moments obtained with the present elements are in close agreement with the results of previous investigations for the moderately thick plates as well as for thin plates. It has been noted that the element (TURE24) with relatively few elements has better and more rapid convergence characteristics than the other element (TURE12). But these elements have the problem of shear locking under the thin plate limit. However, the selective reduced integration rule can be successfully used to overcome this problem for thin plates, and 8-noded finite element gives better results than 4-noded one.
References [1] Reissner E. The effect of transverse shear deformation on the bending of elastic plates. Journal of Applied Mechanics 1945;12:69–77. [2] Mindlin RD. Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. Journal of Applied Mechanics 1951;18:31–8. [3] Hughes TJR, Taylor RL, Kanoknukulchai W. A simple and efficient finite element for plate bending. International Journal for Numerical Methods in Engineering 1977;11:1529–43. [4] Hrabok MM, Hrudey TM. A review and catalogue of plate bending finite elements. Computers and Structures 1984;19:479–95. [5] Bergan PG, Wang X. Quadrilateral plate bending elements with shear deformations. Computers and Structures 1984;19:25–34. [6] Hinton E, Huang HC. A family of quadrilateral Mindlin plate elements with substitute shear strain fields. Computers and Structures 1985;23:409–31. [7] Liu I, Keih T, Lin C. A conforming quadrilateral plate bending element with shear deformations. Computers and Structures 1995;56:93–100. [8] Zienkiewicz OC, Taylor RL, Too JM. Reduced integration technique in general analysis of plates and shells. International Journal for Numerical Methods in Engineering 1971;3:275–90. [9] Pawsey SF, Clough RW. Improved numerical integration of thick shell finite elements. International Journal for Numerical Methods in Engineering 1971;3:575–86.
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[10] Briassoulis D. On the basics of the shear locking problem of C isoparametric plate elements. Computers and Structures 1989;33:169–85. [11] Soh AK, Cen S, Long Y, Long Z. A new 12 dof quadrilateral element for analysis of thick and thin plates. European Journal of Mechanics 2001;20:299–326. [12] Yuqiu L, Fei X. A universal method for including shear deformation in thin plate elements. International Journal for Numerical Methods in Engineering 1992;34:171–7. [13] Yuan F, Miller RE. A rectangular finite element for moderately thick flat plates. Computers and Structures 1988;30:1375–87. [14] Flu¨gge W. Handbook of engineering mechanics. New York: McGraw-Hill; 1962. [15] Timoshenko SP, Woinowsky-Krieger S. Theory of plates and shells. New York: McGraw-Hill; 1959. v [16] Owen DRJ, Zienkiewicz OC. A refined higher-order C plate bending element. Computers and Structures 1982;15:177–83. [17] Crisfield MA. A quadratic Mindlin element using shear constraints. Computers and Structures 1984;18:833–52.
The transition from thin plates to moderately thick plates by using finite element analysis and the shear locking problem Tulay Aksu Ozkul , Umit Ture Department of Civil Engineering, Istanbul Technical University, 34469 Maslak-Istanbul, Turkey Received 1 August 2003; received in revised form 27 April 2004; accepted 13 May 2004
Abstract Two simple plate bending elements, based on Mindlin theory for analysis of both moderately thick and thin plates, are presented in this paper. These elements have either four nodes or eight nodes with 12 and 24 DOF, respectively. To illustrate the accuracy of these finite elements named as TURE12 and TURE24, several numerical examples of displacements and stresses for both thin and moderately thick plate bending problems are presented and discussed with a range of finite element meshes and thickness-to-plate length ratios. In addition, the bending and shearing behaviours of a Mindlin plate are analyzed with respect to shear locking. In order to test the shear locking, the results obtained from the Mindlin plate analysis using 4- or 8-noded elements with full, reduced, and selective reduced integration are compared with the exact classical thin plate solution. # 2004 Elsevier Ltd. All rights reserved. Keywords: Mindlin plate theory; Finite element; Shear locking; Thin plates; Thick plates
1. Introduction Great effort has been spent on the development of finite elements which are suitable for the bending of plates. Most of these efforts have been oriented to the thin plate theory, known as Kirchoff plate theory. However, when a plate is moderately thick (thickness=plate length > 0:05 or 0.1), this theory is not accurate enough to solve the problem due to the contribution of transverse shear to the deformation and stress distributions, and for this reason, a new approach is needed. Accordingly,
Corresponding author. Tel.: +90-212-285-3773; fax: +90-212-285-6587. E-mail address: [email protected] (T. Aksu Ozkul).
0263-8231/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2004.05.003
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Nomenclature U,V,W displacement components in the x, y and z directions w transverse displacement hx, hy normal rotations of the mid-plane ex, ey, cxy normal and shearing strains of the middle surface cxz, cyz thickness shearing strains of the middle surface E, m, G Young’s modulus, Poisson’s ratio, shearing modulus of elasticity h thickness of the plate p total potential energy [K]e element stiffness matrix K shear modification factor ½Keb element bending stiffness matrix ½Kes element shear stiffness matrix shape functions of the finite element [Ni]
the Mindlin plate theory has been improved for this purpose, and a number of papers about this theory have appeared [1–7]. v Mindlin plate element theory needs only to be C continuous, i.e. the displacements and the rotations are independent. For this reason, the behaviours of many Mindlin plate elements for moderately thick plates are usually very satisfactory. However, some of the Mindlin plate elements may cause a problem, known as shear locking, when they are used to analyse thin plate structures. To alleviate the shear locking problem, some methods have been suggested, such as, reduced and selective reduced integration techniques [8–10]. In this paper, two displacement-based Mindlin plate elements, which have indev pendent displacement w and rotations hx and hy based on C continuous shape functions, are briefly reviewed for analysis of both moderately thick and thin plates. These elements, named as TURE12 and TURE24, are 4-noded and 8-noded rectangular finite elements, with the basic 3 DOF per node and 12 and 24 DOF for each element, respectively. Numerical results are presented to show the behaviour of these elements involving some plates of different aspect ratios and support conditions with either uniformly distributed load or concentrated load. The results are compared with those obtained using other solution techniques such as analytical solution and different finite elements. The shear locking problem, appeared in application to thin plates, is alleviated by separating the shear and bending energies and by using selective reduced integration technique on the shear term. Several examples are then presented, in which the behaviours of shear locking of the two elements are given, by using full, reduced, and selective reduced integration techniques, and the results are compared with those of the classical thin plate solution. In each example, the transition from thin to moderately thick plates are calculated and compared with each other.
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2. Mindlin plate element formulation 2.1. Fundamental relations At a typical point in a Mindlin plate, the displacement components may be represented as U ¼ zhx ðx; yÞ V ¼ zhy ðx; yÞ W ¼ wðx; yÞ
ð1Þ
where U, V, and W are the displacement components in the x, y, and z directions, respectively, w is the transverse displacement, and hx and hy are the normal rotations of the mid-plane, respectively. The rotations hx and hy can be expressed in the form @w cxz @x @w cyz hy ¼ @y hx ¼
ð2Þ
where @w=@x and @w=@y are the slopes of the deformed median surface in the x and y directions and cxz and cyz are the shear strains. For a displacement-based, n-noded Mindlin plate element, the displacement and normal rotations may be represented by the expressions w¼
n X
hx ¼
Ni w i ;
i¼1
n X Ni hxi ;
hy ¼
i¼1
n X Ni hyi
ð3Þ
i¼1
where wi, hxi and hyi are the displacement and rotation values at node i and Ni is the shape function with node i. 2.2. Strain–displacement relations For the Mindlin plate theory, the strain components may be written in terms of the displacements of the middle surface as follows: @hx ; @x @hy @hx þ cxy ¼ z @x @y @x hx ; cxz ¼ @x ex ¼ z
ey ¼ z
@hy @y ð4Þ
cyz ¼
@x hy @y
Curvature–displacement relations and shear strain–displacement relations are then written in the matrix form X n Bbi eb ð5Þ ½e ¼ ¼ ½di Bsi es i¼1
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in which the curvatures [eb], the shear strains [es], and the unknowns at node i [di] are expressed as ½eb ½es ½di
T ex ey cxy ¼ T cxz cyz ¼ T ¼ ½ xi hxi hyi
ð6Þ
where T shows the transpose, as usual. The curvature–displacement matrix ½Bbi and the shear strain–displacement matrix ½Bsi associated with node i may be written as follows: 2 3 0 Ni;x 0 Ni;y 5 ½Bbi ¼ 4 0 0 0 N Ni;x ð7Þ i;y Ni;x Ni 0 ½Bbi ¼ Ni;y 0 Ni
2.3. Stress–strain relations The moment–curvature relationships and the bending moments are ½rb ¼ ½Db ½eb ;
½rb ¼ ½ Mx
My
Mxy T
ð8Þ
and the matrix of flexural rigidities may be expressed for an isotropic material as follows: 2 3 1 t 0 Eh3 4t 1 0 5 ½Db ¼ ð9Þ 12ð1 m2 Þ 0 0 1 t=2 where E is Young’s modulus, v is Poisson’s ratio and h is the plate thickness. The shear force–shear strain relationships and the shear forces are given as ½rs ¼ ½Ds ½es ;
½rs ¼ ½ Qx
Qy T
ð10Þ
where the matrix of shear rigidities for an isotropic material may be expressed as kEh 1 0 ð11Þ ½Ds ¼ 2ð1 þ mÞ 0 1 Here, k is the shear modification factor and is normally set equal to 5/6 for homogeneous isotropic plates. 2.4. The total potential energy The total potential energy, p, can be written as ð 1 p¼ ðrx ex þ ry ey þ sxy cxy þ sxz cxz þ syz cyz Þdv W 2 v
ð12Þ
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where W is the work done by the applied loads. Considering the strains and inserting in the above expression, ð ð Eh3 kEh 2 2 2 2 e þ 2me e þ e c þ c p¼ dA þ x y y yz dA W 2ð1 þ mÞ A xz 24ð1 m2 Þ A x and p¼
1 2
ð
½eb T ½Db ½eb dA þ A
1 2
ð
½es T ½Ds ½es dA W
ð13Þ
A
the total potential energy for moderately thick plates is derived in the matrix form. By substituting the displacement functions in Eq. (13) and integrating over the element fields, the following equations are obtained for an element ½Ke fdge ¼ ff ge e
e
ð14Þ e
where [K] , {d} and {f} are the element stiffness matrix, displacement vector and the element force vector, respectively.
3. Finite elements Two elements are examined with the formulation of the Mindlin plate element given in Section 2. The first one is the 4-noded rectangular plate element (TURE12) which has linear shape functions, and the second one is the 8-noded rectangular plate element (TURE24) which has second-degree shape functions, respectively, as shown in Fig. 1. The shape functions of these elements are given as follows. Degrees of freedom of these finite elements at each node are the transverse displacement w and the two rotations hx and hy about the x and y axes, respectively. Thus, TURE12 and TURE24 elements have 3 DOF at each node, and 12 DOF and 24 DOF for each element, respectively.
4. Shear locking v
Plate finite elements based on Mindlin theory require only C continuity of displacement and independent rotations, and soon appeared as an alternative according to the plate elements with C0 continuity. Therefore, the behaviour of the Mindlin plate elements is usually very good for a moderately thick plate situation. However, when a thin plate is considered, these displacement-based elements cause a problem known as ‘‘shear locking’’. When the full integration of the stiffness matrices is used with standard Mindlin finite elements, very stiff results may be obtained in applications to thin plates. This means that the bending energy, which should dominate the shear terms, will be incorrectly estimated to be zero in thin plate problems.
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Fig. 1. The finite elements and the shape functions: (a) linear, (b) quadratic.
To avoid the shear locking problem in thin plates, the reduced and selective reduced integration techniques were proposed in the early 1970s [8,9]. The reduced integration procedure is the reduction in the order of integration in computing the stiffness matrix of the finite element. Similarly, the selective integration procedure is also a kind of reduced integration rule which is used to evaluate the stiffness matrix associated with the troublesome shear strain energy. That is to say, this has been adopted to the shear stiffness matrix only and full integration is used on the remaining terms [8–10]. Therefore, the [K]e element stiffness matrix can be obtained by separating into bending terms and shear terms. With these definitions, [K]e element stiffness matrix is given by simplifying the expression ½Ke ¼ ½Kb e þ ½Ks e where ½Kb e ¼ ½Ks e ¼
ð ð
½Bb T ½Db ½Bb dA A
½Bs T ½Ds ½Bs dA
ð15Þ
A
½Kb e and ½Ks e are the bending and shear stiffness matrices, respectively. Table 1 shows the full, reduced, and selective reduced integration rules used to test the shear locking response of the TURE12 and TURE24 elements in this paper.
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Table 1 Integration rules Thick plate
Thin plate
Full integration ½Kb TURE12 TURE24
e
2 2 3 3
½Ks
Reduced integration e
2 2 3 3
½Kb
e
1 1 2 2
½Ks
e
1 1 2 2
Selective reduced integration ½Kb e
½Ks e
2 2 3 3
1 1 2 2
5. Numerical examples A number of examples are taken to examine the performance of the elements of TURE12 and TURE24, which include tests on convergence of both displacements and stresses, and shear locking with a derived computer programme. The results are compared with those obtained by other analytical and numerical methods. 5.1. Example 1: square plate A square plate which is subjected to a uniformly distributed load or a concentrated load is modelled with two different boundary conditions, i.e. either a simply supported or a clamped boundary on all four edges, to evaluate the convergence of the solutions obtained with TURE12 and TURE24. The geometric and material properties of this problem, which has been most frequently examined in the literature, are E ¼ 106 kN=m2 , v ¼ 0:30, a ¼ 8 m, P ¼ 1 kN, q ¼ 1 kN=m2 , and k ¼ 5=6, where P is the concentrated load, q is the uniformly distributed load, and a is the length of the plate. Due to symmetry, only a quarter of the plate is taken into consideration for the analysis by a series of meshes of N N element (N denotes the number of elements in one direction) with different ratios for h=a. The displacements at the center are computed with TURE12 and TURE24 elements for thickness/length ratios (h=a) varying from 0.001 to 0.20 and three mesh densities. The results obtained by TURE12 and TURE24 are presented in dimensionless form in Tables 2–5 with the exact solution and with those given by other authors for the cases of simply supported and clamped edges subjected to uniform load or concentrated load. In addition, the convergence of the central displacement and the central moment obtained by these elements are presented in Figs. 2–5 for h=a ¼ 0:10 with four mesh densities, for comparison. It is obvious that the element TURE24 is a high precision and fast convergence element if it is compared with the element TURE12, which yielded the best result using 8 8 mesh. On the contrary, the error ratio of TURE24 is less than that of TURE12 in some results, even with 2 2 mesh, as shown in Tables 2–5. It should also be noted that as the plate becomes thicker, the convergence of the element TURE24 is faster than that of TURE12. Tables 6 and 7 show the results of the central displacements and the central moments, obtained by the elements TURE12 and TURE24, for uniformly loaded
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Table 2 Central displacements wc ð100D=qa4 Þ for clamped square plate subjected to uniform load q h=a
0.001 0.002 0.005 0.01 0.02 0.05 0.10 0.15 0.20
TURE12
TURE24
Mesh number
Mesh number
Exact solution [11]
2 2
4 4
8 8
2 2
4 4
8 8
0:285 104 0:114 103 7:07 104 2:759 103 0.0102 0.0444 0.1005 0.1049 0.1628
0.1250
0.1261
0.0389
0.0831
0.1256
0.1250
0.1261
0.0411
0.1088
0.1262
0.1250
0.1261
0.0535
0.1228
0.1265
0.1250
0.1263
0.0786
0.1258
0.1267
0.1265
0.1252 0.1251 0.1255 0.1257 0.1987
0.1274 0.1323 0.1504 0.1733 0.2139
0.1076 0.1291 0.1501 0.1745 0.2146
0.1276 0.1326 0.1506 0.1785 0.2170
0.1278 0.1327 0.1506 0.1787 0.2172
0.1499
0.1265
0.2167
Thin plate solution (Timoshenko and Woinowsky-Krieger [15]): 0.1265.
clamped square plates and uniformly loaded simply supported square plates from very thin to moderately thick thicknesses with 8 8 mesh density. The results obtained by the presented formulation are in good agreement with those from the other investigations. 5.2. Example 2: circular plate A circular plate, subjected to a uniformly distributed load, is modelled with two different boundary conditions, i.e. simply supported and clamped conditions by Table 3 Central displacements wc ð100D=qa4 Þ for simply supported square plate subjected to uniform load q h=a
0.001 0.002 0.005 0.01 0.02 0.05 0.10 0.15 0.20
TURE12
TURE24
Mesh number
Mesh number
Exact solution [12]
2x2
4x4
8x8
2x2
4x4
8x8
0.2306 0.2306 0.2310 0.2323 0.2375 0.2624 0.3203 0.2664 0.3563
0.4039 0.4039 0.4039 0.4041 0.4057 0.4093 0.4262 0.4153 0.4671
0.4055 0.4055 0.4055 0.4057 0.4073 0.4107 0.4273 0.4434 0.4845
0.1866 0.3083 0.3876 0.4017 0.4065 0.4107 0.4273 0.4517 0.4891
0.4043 0.4054 0.4059 0.4061 0.4078 0.4112 0.4277 0.4535 0.4903
0.4060 0.4060 0.4061 0.4062 0.4078 0.4112 0.4278 0.4536 0.4904
Thin plate solution (Timoshenko and Woinowsky-Krieger [15]): 0.4057.
0.4062
0.4062 0.4107 0.4273 0.4536 0.4906
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Table 4 Central displacements wc ðEh3 =Pa2 Þ for clamped square plate for concentrated load h=a
TURE12
TURE24
Yuan and Miller [13]
Mesh number
Mesh number
Mesh number
2 2 0.001 0.002 0.005 0.01 0.02 0.05 0.10 0.15 0.20
6
12:44 10 49:77 106 3:096 106 0.01216 0.01311 0.02158 0.05423 0.05679 0.09087
4 4
8 8
2 2
4 4
8 8
6 6
0.05898 0.05898 0.05905 0.05927 0.06027 0.06607 0.08716 0.09269 0.12792
0.06065 0.06065 0.06067 0.06093 0.06213 0.06934 0.09562 0.10799 0.14707
0.02354 0.02439 0.02957 0.04006 0.05239 0.06326 0.07878 0.09905 0.13141
0.04263 0.05375 0.05946 0.06075 0.06192 0.06661 0.08332 0.10995 0.14756
0.06059 0.06096 0.06116 0.06141 0.06220 0.06749 0.08714 0.11806 0.16161
0.05855 0.05856 0.05862 0.05882 0.05963 0.06522 0.08493 0.1173 0.1623
Thin plate solution (Flu¨gge [14]): 0.0611.
using a series of meshes of TURE12 and TURE24 elements. The meshes for one quadrant, and the geometric and material properties are defined in Fig. 6. The central displacement obtained from the analyses are presented in Tables 8 and 9 for different h=R values varying from 0.001 to 0.30, and for three different mesh densities together with those of analytical solutions and a finite element solution namely ARS-Q12 given by Soh et al. [11], for comparison. As shown in Tables 8 and 9, error percentage for TURE24 element is much less than error percentage for TURE12 element, and for former element, convergence of the central displacement is very good even with three elements when it is compared with the latter, which gives the best result with as many as 27 elements. Table 5 Central displacements wc ðEh3 =Pa4 Þ for simply supported square plate for concentrated load h=a
0.001 0.002 0.005 0.01 0.02 0.05 0.10 0.15 0.20
TURE12
TURE24
Yuan and Miller [13]
Mesh number
Mesh number
Mesh number
2 2
4 4
8 8
2 2
4 4
8 8
6 6
0.06692 0.06693 0.06707 0.06765 0.06984 0.08159 0.11227 0.12348 0.14113
0.12595 0.12592 0.12601 0.12618 0.12736 0.13277 0.15346 0.15472 0.19134
0.12638 0.12638 0.12595 0.12634 0.12806 0.13484 0.16071 0.17147 0.21056
0.06281 0.09551 0.11805 0.12314 0.12569 0.12891 0.14338 0.16363 0.19529
0.12350 0.12503 0.12582 0.12634 0.12697 0.13164 0.14823 0.17395 0.21104
0.12622 0.12639 0.12654 0.12682 0.12762 0.13194 0.15145 0.18204 0.22505
0.12441 0.12442 0.12453 0.12491 0.12632 0.13432 0.15754 0.19283 0.24026
Thin plate solution (Timoshenko and Woinowsky-Krieger [15]): 0.1267.
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Fig. 2. Central displacement convergence test for uniformly loaded clamped square plate with h=a ¼ 0:10.
Fig. 3. Central moment convergence test for uniformly loaded clamped square plate with h=a ¼ 0:10.
Fig. 4. Central displacement convergence test for uniformly loaded simply supported square plate with h=a ¼ 0:10.
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Fig. 5. Central moment convergence test for uniformly loaded simply supported square plate with h=a ¼ 0:10.
Furthermore, in Table 10, the results of the central bending moment for the two elements, TURE12 and TURE24, are shown together with those of the exact solutions and some other finite element solutions given in [11] and the results obtained with SAP2000. All results were obtained for the thin (h=R ¼ 0:02) and moderately thick (h=R ¼ 0:20) plates for the cases of simply supported and clamped edges subjected to uniform loading. Table 10 shows that the results obtained using the presented elements are in excellent agreement with those of the other solutions. The results of the central bending moment found with TURE24 element have converged very fast to the exact solutions, like the previous example. As the plate becomes thicker (h=R ¼ 0:20), the convergence of the element TURE24 is much faster than that of TURE12. Comparing with the result of exact solution given by Batoz and Dahn [11], TURE12 element yielded the best central bending moment with 20.69% error for clamped circular plate and with 18.85% error for simply supported circular plate, respectively. While the error ratio of TURE24 element is only 1.35% for the clamped circular plate and 0.44% for the simply supported circular plate, respectively. For a thin circular plate (h=R ¼ 0:02), with a clamped boundary, the convergence for central displacements, which are normalized by the thin plate theory solution, is plotted in Fig. 7 by using different mesh patterns. Furthermore, the convergence result for central displacements for a moderately thick clamped circular plate with h=R ¼ 0:20 is shown in Fig. 8. The accuracy and reliability of these two elements with different mesh patterns are again clearly illustrated in these figures. The displacement distributions along the radius of the uniformly loaded clamped circular plates with two h=R ratios (h=R ¼ 0:02 and h=R ¼ 0:20) are also plotted. The results obtained using the two elements are compared, with the corresponding analytical solutions and with those given in [11,17], in Figs. 9 and 10, respectively. Close agreement between the present solutions and the reference solutions is obtained.
0.1261 0.1261 0.1261 0.1263 0.1274 0.1323 0.1504 0.1733 0.2139
0.1256 0.1262 0.1265 0.1267 0.1278 0.1327 0.1506 0.1787 0.2172
0.1279 – – 0.1281 – – 0.1515 – 0.2183
0.1293 – – 0.1293 – – 0.1521 0.1801 0.2181
0.1234 0.1234 0.1234 0.1236 0.1245 0.1298 0.1482 0.1776 0.2171
[13]
0.122 – – 0.123 0.123 – 0.146 – 0.211
[16]
0.1265 – – 0.1265 – – 0.1499 0.1798 0.2167
Analytic solution [11] 2.3004 2.3003 2.3001 2.2999 2.3013 2.3089 2.3055 2.3058 2.3065
2.2939 2.2993 2.3008 2.3014 2.3028 2.3108 2.3314 2.3524 2.3702
TURE24
Central moments [12]
TURE12
[11]
TURE12
TURE24
Central displacements
Thin plate solution (Timoshenko and Woinowsky-Krieger [15]): 0.1265.
0.001 0.002 0.005 0.01 0.02 0.05 0.10 0.15 0.20
h=a
2.0685 – – 2.0685 – – 2.0701 – 2.0707
[11]
Table 6 Central displacements wc ð100D=qa4 Þ and central moments (100Mx =qa2 ) for clamped square plate subjected to uniform load q
2.27 – – 2.27 2.27 – 2.36 – 2.50
[16]
2.31 2.31 2.31 2.31 2.31 2.31 2.31 2.31 2.31
Thin plate solution [16]
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0.4055 0.4055 0.4055 0.4057 0.4073 0.4107 0.4273 0.4434 0.4845
0.4060 0.4060 0.4061 0.4062 0.4078 0.4112 0.4278 0.4536 0.4904
0.4062 – – 0.4064 – – 0.4271 – 0.4903
0.407 – – 0.407 0.408 – 0.423 – 0.48
0.4043 – – 0.4045 – – 0.4242 0.4502 0.4869
0.4054 0.4055 0.4057 0.4067 0.4101 0.4258 0.4596 0.5018 0.5511
[13] 0.4062 – – 0.4062 0.4062 0.4108 0.4273 0.4536 0.4906
Exact [12] 4.7889 4.7889 4.7889 4.7889 4.7889 4.7889 4.7889 4.7686 4.7263
4.7951 4.7952 4.7953 4.7953 4.7953 4.7953 4.7953 4.7951 4.7952
4.789 – – 4.802 – – 5.096 – –
TURE12 TURE24 [11]
[12]
TURE12 TURE24 [11]
[16]
Central moments
Central displacements
Thin plate solution (Timoshenko and Woinowsky-Krieger [15]): 0.4057.
0.001 0.002 0.005 0.01 0.02 0.05 0.10 0.15 0.20
h=a
4.82 – – 4.82 4.82 – 4.84 – 4.87
[16]
4.779 4.779 4.781 4.788 4.813 4.915 5.079 5.223 5.35
[13]
Table 7 Central displacements wc ð100D=qa4 Þ and central moments (100Mx =qa2 ) for simply supported square plate subjected to uniform load q
4:792=4:789 – – 4:820=4:789 – – 5:096=4:789 – –
Exact [11] Thick=thin
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Fig. 6. Circular plate with mesh patterns.
5.3. Example 3: skew plate v
A 60 skew plate, simply supported on two opposite edges and free on the other two edges, is analyzed. A symmetric quadrant of this skew plate, under uniform loading, which was originally studied by Razzaque [11], is idealized with meshes involving 2 2, 4 4, and 8 8 elements, respectively (Fig. 11). Convergence of the central displacement and the central bending moment obtained with the two elements, TURE12 and TURE24, is shown in Tables 11 and 12, and Fig. 12, respectively. Results obtained by these elements are compared with those given by other researchers. In all cases, reasonably good convergence is obtained. As it is expected, TURE24 element has converged much faster than TURE12 element like the previous two examples. TURE12 element yielded the best result for the central displacement by using (8 8) mesh with 1.08% error with respect to
0.001 0.01 0.02 0.10 0.20 0.30
h=R
0.01298 0.01298 0.01300 0.01369 0.00657 0.01181
3 elements
TURE12
0.01433 0.01437 0.01447 0.01527 0.01225 0.01786
0.01385 0.01408 0.01447 0.01593 0.01487 0.01990
11.39 9.92 7.42 2.51 19.53 18.34
0.00783 0.01300 0.01485 0.01637 0.01784 0.02182
12 elements 27 elements Error % 3 elements (27 elements)
TURE24
0.01519 0.01568 0.01568 0.01634 0.01848 0.02207
0.01563 0.01563 0.01565 0.01634 0.01848 0.02207
0.00 0.00 0.13 0.00 0.00 9.44
12 elements 27 elements Error % (27 elements)
Table 8 Central displacements wc ðD=qR4 Þ for uniformly loaded clamped circular plate
0.01610 0.01611 0.01612 0.01675 0.01881 0.02233
0.01575 0.01576 0.01577 0.01643 0.01856 0.02212
0.01563 0.01563 0.01563 0.01634 0.01848 0.02437
Analytical solution 12 elements 48 elements [11]
[11] ARS-Q12
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0.001 0.01 0.02 0.10 0.20 0.30
h=R
0.04821 0.04821 0.04824 0.04889 0.30092 0.67191
3 elements
TURE12
0.05917 0.05917 0.05914 0.05990 0.04374 0.05543
0.06176 0.06176 0.06178 0.06247 0.05467 0.06317
3.05 3.06 3.06 3.03 17.86 9.92
0.06446 0.06441 0.06435 0.06464 0.06606 0.07003
12 elements 27 elements Error % 3 elements (27 elements)
TURE24
0.06151 0.06354 0.06372 0.06443 0.06654 0.07016
0.06242 0.06363 0.06370 0.06441 0.06656 0.07010
2.00 0.12 0.04 0.01 0.00 0.04
12 elements 27 elements Error % (27 elements)
Table 9 Central displacements wc ðD=qR4 Þ for uniformly loaded simply supported circular plate
0.06305 0.06306 0.06308 0.06370 0.06577 0.06928
0.06354 0.06355 0.06357 0.06422 0.06635 0.06991
0.06370 0.06371 0.06373 0.06442 0.06656 0.07013
Analytical solution 12 elements 48 elements [11]
[11] ARS-Q12
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Clamped cir- 0.0894 cular plate Simply sup- 0.2026 ported circular plate
0.0828
0.2073
0.0821
0.2065
0.2069
0.0826 0.2056
0.0816
TURE12 TURE24 SAP2000 ARS-Q12 MITC4 (27 ele(27 ele- (27 ele- (48 ele- (48 elements) ments) ments) ments) ments) [11] [11]
h=R ¼ 0:02
0.2068
0.0824
DKMQ (48 elements) [11]
0.2064
0.0812
Exact solution (Batoz and Dahn [11])
0.1675
0.0644
0.2073
0.0823
0.2071
0.0828
TURE12 TURE24 ARS-Q12 (27 ele(27 ele(48 elements) ments) ments) [11]
h=R ¼ 0:20
Table 10 Central moment ðMc =qR2 Þ for clamped circular plate and for simply supported circular plate subjected to uniform load q
0.2056
0.0816
MITC4 (48 elements) [11]
0.2072
0.0828
DKMQ (48 elements) [11]
0.2064
0.0812
Exact solution (Batoz and Dahn [11])
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Fig. 7. Central displacement convergence test for uniformly loaded clamped circular plate with h=R ¼ 0:02.
Fig. 8. Central displacement convergence test for uniformly loaded clamped circular plate with h=R ¼ 0:20.
Fig. 9. The displacement distributions along the radius of the uniformly loaded clamped circular plates for h=R ¼ 0:02.
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Fig. 10. The displacement distributions along the radius of the uniformly loaded clamped circular plate for h=R ¼ 0:20.
Fig. 11. Razzaque’s skew plate (2 2 mesh).
the theoretical solution given by Razzaque [11] as shown in Table 11. On the contrary, error ratio of TURE24 element, even with (2 2) mesh 0.0755%, is less than error ratio of TURE12 element. 5.4. Example 4: locking test Two typical examples are taken to test the shear locking response of the present elements. The first one is a square plate, which is subjected to a uniform load or a concentrated load at the center and idealized with two different boundary conditions, i.e. a simply supported and a clamped boundary. The second one is a circular plate, simply supported or clamped boundary, and subjected to uniform
TURE12
0.6787 0.7665 – 0.7859 –
Mesh
2 2 4 4 6 6 8 8 12 12
0.7939 0.7974 – 0.7981 –
TURE24
0.7727 0.7907 – 0.7949 –
SAP2000
0.6666 0.7691 0.7829 0.7876 0.7909
0.3976 0.6737 0.7371 0.7610 0.7785
0.6666 0.7695 0.7829 0.7876 0.7908
0.812 0.792 – 0.794 –
[17] C8
DKMQ
ARS-Q12
MITC4
[11]
Table 11 Central displacement wc ð100D=qL4 Þ for Razzaque’s skew plate
0.806 0.769 – 0.788 –
S8
0.385 0.753 – 0.759 –
H4 (lumped)
0.551 0.710 – 0.769 –
H4 (consistent)
0.7945
Razzaque [11]
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TURE12
72.203 89.453 – 94.358 –
Mesh
2 2 4 4 6 6 8 8 12 12
96.625 97.375 – 96.297 –
TURE24
95.781 95.937 – 96.015 –
SAP2000
92.46 95.95 96.09 96.05 96.02
37.90 77.60 87.10 90.90 93.70
92.20 96.00 96.10 96.10 96.00
92.82 94.83 – 95.79 –
C8
DKMQ
ARS-Q12
MITC4
[17]
[11]
Table 12 Central moment ð103 My =qL2 Þ for Razzaque’s skew plate
91.57 91.29 – 95.99 –
S8
41.62 79.49 – 91.48 –
H4 (lumped)
59.16 83.62 – 92.53 –
H4 (consistent)
95.89
Razzaque [11]
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Fig. 12. Central displacement convergence test for Razzaque skew plate with h=a ¼ 0:01.
Table 13 Central displacement wc ð100D=qa4 Þ locking test for uniformly loaded clamped square plate h=a
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.20
TURE12
TURE24
Full integration
Reduced Selective Full inteintegration reduced gration integration
Reduced Selective integration reduced integration
0.0132 0.0406 0.0661 0.0853 0.0993 0.1100 0.1186 0.1260 0.1327 0.1396 0.2196
0.1269 0.1277 0.1289 0.1307 0.1329 0.1355 0.1386 0.1424 0.1460 0.1511 0.2106
0.1268 0.1276 0.1288 0.1303 0.1328 0.1355 0.1386 0.1418 0.1460 0.1499 0.2198
0.1264 0.1272 0.1284 0.1302 0.1323 0.1350 0.1382 0.1418 0.1460 0.1499 0.2106
0.1255 0.1278 0.1289 0.1305 0.1326 0.1352 0.1386 0.1418 0.1460 0.1499 0.2198
0.1267 0.1275 0.1288 0.1303 0.1328 0.1355 0.1386 0.1418 0.1460 0.1499 0.2198
Analytical solution
Thin plate solution (Timoshenko and WoinowskyKrieger [15]): 0.1265
0.1499 [11] 0.2167 [11]
loading. The geometric and material properties used are the same as Examples 1 and 2. In the locking test, a symmetric quadrant of the square plate and the circular plate are modelled with various thickness ratios by using the 8 8 finite element meshes for the former plate and 27 element meshes for the latter, respectively. The results from the Mindlin plate bending analysis using 4-noded (TURE12) and 8-noded (TURE24) elements with full integration (FI), reduced integration (RI), and selective reduced integration (SRI) are compared with those of the analytical solution. The integral rules used in the examples are given in Table 1. The solutions were obtained for the boundary conditions and loading types mentioned above for the square or circular plates; however, the results are given in Tables 13 and 14 and Figs. 13–16 for only one type, i.e. a uniform loading under
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Table 14 Central displacement wc ðD=qR4 Þ locking test for uniformly loaded clamped circular plate h=R
0.001 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.20
TURE12
TURE24
Full integration
Reduced integration
Selective reduced integration
Full Reduced Selective integration integration reduced integration
0:1678 105 0:16607 105 0:64359 105 0:13765 104 0:22916 104 0:33138 104 0:43813 104 0:54418 104 0:64703 104 0:74435 104 0:83563 104 0.01832
0.01384 0.01407 0.01447 0.01478 0.01499 0.01518 0.01533 0.01548 0.01573 0.01577 0.01593 0.01813
0.01489 0.01493 0.01503 0.01515 0.01527 0.01539 0.01552 0.01564 0.01578 0.01592 0.01607 0.01816
0.01463 0.01539 0.01546 0.01553 0.01562 0.01571 0.01581 0.01592 0.01603 0.01617 0.01630 0.01849
0.01563 0.01563 0.01565 0.01569 0.01574 0.01580 0.01588 0.01597 0.01608 0.01621 0.01635 0.01849
0.01563 0.01563 0.01565 0.01569 0.01574 0.01580 0.01588 0.01597 0.01608 0.01621 0.01635 0.01849
Analytical solution [11]
Thin plate solution 0.01563
0.01634 0.01848
clamped boundary, because the results found for the other conditions are similar with the first one. Shear locking problem, in the TURE12 element, which started from assumed thin plate limit h=a or h=R 0:10, were observed very clearly and prevented by using selective reduced integration as shown in Tables 13 and 14. On the other hand, it was found that the shear locking problem in the TURE24 element is not as evident as that in the TURE12 element and the thin plate transition limit can be reduced to 0.05 for this element. Besides, the full integration rule has provided good results even in thin plate application for TURE24 element. Consequently, prevention of shear locking in the TURE24 element is much easier than that in the TURE12 element.
Fig. 13. Occurrence of shear locking for uniformly loaded clamped square plate.
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Fig. 14. Prevention of shear locking for uniformly loaded clamped square plate.
Fig. 15. Occurrence of shear locking for uniformly loaded clamped circular plate.
Fig. 16. Prevention of shear locking for uniformly loaded clamped circular plate.
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It can be concluded that the selective reduced integration rule should be used for h=a 0:10 for TURE12 element and for h=a ¼ 0:05 0:06 for TURE24 element, respectively. But for h=a > 0:10, the full integration rule must be employed definitely for both TURE12 and TURE24 elements. In addition, occurrence of shear locking with full integration and prevention of it with selective reduced integration are plotted with the corresponding analytical solutions in Figs. 13–16 for both square and circular plates subjected to uniform loading with the clamped boundary condition. It can be said that the tested elements with the selective reduced integration rule give an optimal performance.
6. Conclusion The approach of proposed two finite element solutions, which are 4-noded and 8-noded Mindlin plate bending elements, presented in the analysis of both moderately thick and thin plates. Convergence of these elements is discussed for different mesh densities and different h=a ratios. As can be seen from the examples presented, both the displacements and the moments obtained with the present elements are in close agreement with the results of previous investigations for the moderately thick plates as well as for thin plates. It has been noted that the element (TURE24) with relatively few elements has better and more rapid convergence characteristics than the other element (TURE12). But these elements have the problem of shear locking under the thin plate limit. However, the selective reduced integration rule can be successfully used to overcome this problem for thin plates, and 8-noded finite element gives better results than 4-noded one.
References [1] Reissner E. The effect of transverse shear deformation on the bending of elastic plates. Journal of Applied Mechanics 1945;12:69–77. [2] Mindlin RD. Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. Journal of Applied Mechanics 1951;18:31–8. [3] Hughes TJR, Taylor RL, Kanoknukulchai W. A simple and efficient finite element for plate bending. International Journal for Numerical Methods in Engineering 1977;11:1529–43. [4] Hrabok MM, Hrudey TM. A review and catalogue of plate bending finite elements. Computers and Structures 1984;19:479–95. [5] Bergan PG, Wang X. Quadrilateral plate bending elements with shear deformations. Computers and Structures 1984;19:25–34. [6] Hinton E, Huang HC. A family of quadrilateral Mindlin plate elements with substitute shear strain fields. Computers and Structures 1985;23:409–31. [7] Liu I, Keih T, Lin C. A conforming quadrilateral plate bending element with shear deformations. Computers and Structures 1995;56:93–100. [8] Zienkiewicz OC, Taylor RL, Too JM. Reduced integration technique in general analysis of plates and shells. International Journal for Numerical Methods in Engineering 1971;3:275–90. [9] Pawsey SF, Clough RW. Improved numerical integration of thick shell finite elements. International Journal for Numerical Methods in Engineering 1971;3:575–86.
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