Analysis of thin plate structures using the absolute nodal coordinate formulation

345 Analysis of thin plate structures using the absolute nodal coordinate formulation K Dufva 1 and A A Shabana 2 1 Department of Mechanical Engineering, Lappeenranta University of echnology, Lappeenranta, Finland 2 Department of Mechanical Engineering, University of Illinois at Chicago, Chicago, Illinois, USA he manuscript was received on 10 January 2005 and was accepted after revision for publication on 9 May 2005. DOI: 10.1243/146441905X50678 Abstract: he absolute nodal coordinate formulation can be used in multibody system applications where the rotation and deformation within the finite element are large and where there is a need to account for geometrical non-linearities. In this formulation, the gradients of the global positions are used as nodal coordinates and no rotations are interpolated over the finite element. For thin plate and shell elements, the plane stress conditions can be applied and only gradients obtained by differentiation with respect to the element mid-surface spatial parameters need to be defined. his automatically reduces the number of element degrees of freedoms, eliminates the high frequencies due to the oscillations of some gradient components along the element thickness, and as a result makes the plate element computationally more efficient. In this paper, the performance of a thin plate element based on the absolute nodal coordinate formulation is investigated. he lower dimension plate element used in this investigation allows for an arbitrary rigid body displacement and large deformation within the element. he element leads to a constant mass matrix and zero Coriolis and centrifugal forces. he performance of the element is compared with other plate elements previously developed using the absolute nodal coordinate formulation. It is shown that the finite element used in this investigation is much more efficient when compared with previously proposed elements in the case of thin structures. Numerical examples are presented in order to demonstrate the use of the formulation developed in this paper and the computational advantages gained from using the thin plate element. he thin plate element examined in this study can be efficiently used in many applications including modelling of paper materials, belt drives, rotor dynamics, and tyres. Keywords: large deformation, thin plate elements, absolute nodal coordinate formulation, multibody applications 1 INRODUCION he finite element method is often used to solve the deformation problems in many multibody system applications. he large rotations that characterize the body motion in such systems cannot be Corresponding author: Department of Mechanical Engineering (M/C 251), University of Illinois at Chicago, 2031 Engineering Research Facility, 842 West aylor Street, Chicago, Illinois 60607-7022, USA. accurately described using incremental finite element formulations that employ linearization and infinitesimal rotations as nodal coordinates. In contrast, methods based on the large rotation vector approach do not lead to a unique rotation field and suffer from the problem of coordinate redundancy and energy drift. he problems associated with the use of the incremental procedures and large rotation vector formulations in multibody system applications are discussed in the literature [1, 2]. When the deformations are small, the floating frame of JMBD2 # IMechE 2005 Proc. IMechE Vol. 219 Part K: J. Multi-body Dynamics

346 K Dufva and A A Shabana reference formulation that employs assumed mode shapes is commonly used in multibody system applications. For large deformation problems, the absolute nodal coordinate formulation has been used in many applications [3, 4]. In the absolute nodal coordinate formulation, the bodies are discretized using finite elements as in the classical finite element approach, but gradients of absolute position vectors instead of rotations are used as nodal coordinates. Using this coordinate description, one obtains a constant mass matrix for the finite element and zero Coriolis and centrifugal forces. he stiffness matrix, on the contrary, is highly non-linear function of the element nodal coordinates [5]. Several plate elements based on the absolute nodal coordinate formulation have been proposed in the literature for solving large deformation problems [2, 6, 7]. he results presented in some of these investigations were experimentally verified and demonstrated that the proposed plate and shell elements can be effectively and efficiently used in solving large deformation problems in multibody system applications in which the components undergo finite rotations. he success of using the proposed plate elements based on the absolute nodal coordinate formulation in the analysis of the large rotation and deformation of very flexible bodies can be attributed to many factors, which can be summarized as follows. he absolute nodal coordinate formulation can be implemented in the framework of a non-incremental solution procedure; the rotation field is uniquely defined, the mass matrix is constant, and the Coriolis and centrifugal forces are equal to zero, and in this formulation, there is no restriction on the amount of rotation or deformation within the finite element. Although the absolute nodal coordinate formulation has been successfully used in the analysis of very flexible bodies, numerical difficulties are encountered when the multibody system includes very thin and stiff components. It was observed that for thin and stiff structures, the oscillations of some of the gradient components along the element thickness introduce very high frequencies that make the absolute nodal coordinate formulation less efficient; in some extreme cases, the solution can only be obtained using implicit integration methods. he purpose of this study is to examine the performance of a computationally efficient reduced-order finite plate element for thin and stiff structures [7]. Formulation of the element elastic forces is based on the classical approach where plate bending and plane stress conditions are applied. Previous absolute nodal coordinate formulation plate and shell elements are based on the element local coordinate system and plane stress assumption or a general continuum mechanics approach [2]. When a general continuum mechanics approach is applied, no assumptions are made regarding the cross-section deformation as in the case when the classical plate theories are used. his approach can be used when the ratio of the element thickness to its length is high. In the case of thin and stiff structures, numerical problems due to deformation of the cross-section can be encountered. he reduced-order element used in this investigation does not suffer from the aforementioned numerical problems and some computer models based on this reduced-order element can be more than 100 times faster than models that are based on the elements that employ full parameterization. Generalization of the plate elements used in the finite element formulations to the absolute nodal coordinate formulation is proposed in reference [7, 8]. Kirchhoff plate theory that does not account for the shear deformation is assumed and the elements developed in these investigations are well suited for thin plate applications. Different descriptions for the element elastic forces in longitudinal and transverse directions have also been proposed. wo-dimensional interpolation functions are used and owing to the need for second derivatives, 48 degrees of freedom were required [7]. In the work of Dmitrochenko and Pogorelov [7], the use of different set of shape functions to obtain an element with 36 degrees of freedom without second derivatives was discussed without providing the exact element formulation or investigating numerically its performance. In the reduced-order thin plate element examined in this study, two-dimensional shape functions are used to formulate an element with 36 degrees of freedom. he performance of the new element is tested and the obtained results are used to compare this element with previously proposed plate elements based on the absolute nodal coordinate formulation. It is shown that the proposed low-order element does not suffer from the high oscillation problem which results from the variation of the gradients used in the previously proposed elements along the element thickness. On the basis of the results obtained in this investigation that demonstrates that the thin element formulation can be more than 100 times faster than the elements that employ full parameterization, this thin element formulation can be efficiently used in other applications including rotor dynamics, paper modelling, belt drives, and tyres. 2 ELEMEN KINEMAICS In this section, both the reduced order 36-degreesof-freedom and the 48-degrees-of-freedom plate elements are reviewed in order to explain the basic differences between the two elements. he rigid body modes of the reduced-order element are also checked in this section. Proc. IMechE Vol. 219 Part K: J. Multi-body Dynamics JMBD2 # IMechE 2005

Analysis of thin plate structures 347 2.1 Complete parameterization In previous investigations, a higher-order plate element with 48 degrees of freedom was proposed [2]. his higher-order element, which does not restrict the modes of deformation of the element crosssection, can be used to relax the assumptions of classical plate and shell theories. Figure 1 shows the global and local coordinates used to define the absolute position and gradient coordinates in the absolute nodal coordinate formulation. he global position vector r of the material point P on the plate element can be defined using the element shape functions and the nodal coordinate vector as follows r ¼ S(x, y, z)e (1) where S is the element shape function matrix expressed in terms of the element spatial coordinates x, y,andz and e is the vector of nodal coordinates that consist of nodal positions and slope coordinates. hese positions and slope coordinates are absolute variables defined in global inertial frame. In the element proposed by Mikkola and Shabana [2], the shape functions are functions of the three spatial element parameters x, y, andz. his representation allows for the deformation of the element crosssection. For a node n, the nodal coordinate vector is defined as follows e n @rn @r n ¼ n @r n r (2) @x @y @z Using Fig. 1, the total vector of the element nodal coordinates can be written as follows e ¼ e A e B e C e D (3) For plate and shell elements, it is not, in general, necessary to ensure the continuity of the coordinates at the element interface. his problem was addressed in reference [2] by proposing two different shape function matrices, S A and S B. he shape function matrix S A, which is obtained using incomplete fourth-order interpolation polynomials, does not guarantee the continuity of the coordinates at the element interface. In order to ensure element compatibility, the shape function matrix S B was proposed and used to define an element that satisfies the convergence requirements and is capable of describing the rigid body motion and the case of constant strain. he first shape function matrix, S A, on the contrary, does not meet the requirements for completeness and monotonic convergence. he numerical results obtained using several examples, however, show that the results obtained using the two shape function matrices are in good agreement [2]. Both shape functions lead to an element with 48 degrees of freedom. 2.2 hin plate For thin plates, the deformation of the element along the thickness direction can be neglected. his leads to reduced set of deformation modes as the displacement field of the element becomes dependent on the spatial coordinates x and y only. In this case, the position vector gradients obtained by differentiation with respect to z are not considered as nodal coordinates, leading to a reduced-order element with 36 degrees of freedom. he normal of the midsurface of the plate can always be defined using cross product of the vectors r x and r y, with subscripts x and y referring to partial derivatives with respect to these coordinates. Shape functions can be directly obtained from the shape function matrix S A by omitting the components that depend on the z coordinate. he obtained shape function matrix that depends only on the coordinates x and y is referred to as S C and is presented in Appendix 2. For the reduced-order element, the element nodal coordinate vector at node n is defined as follows e n @rn @r n ¼ n r (4) @x @y he reduced element is of the non-conforming type and the continuity of the gradients at the interface between adjoined elements is not ensured. Note that with the element that employs full parameterization and nine gradient coordinates for each node, continuity of the displacement gradients can be ensured at the mid-surface interfaces when using the shape function matrix S B. Fig. 1 Plate element dimensions and coordinates 2.3 Rigid body motion In the remainder of this section, the capability of the reduced-order element to represent an arbitrary rigid body motion is demonstrated. o this end, a general three-dimensional displacement that can be expressed in terms of a translation of a reference point and three JMBD2 # IMechE 2005 Proc. IMechE Vol. 219 Part K: J. Multi-body Dynamics

348 K Dufva and A A Shabana rotations u x, u y,andu z about the element x, y, andz axes, respectively, is considered. For a general threedimensional rotation, the global position vector of an arbitrary point on the plate element due to rotations only can be written as follows u ¼ Au b (5) where u b ¼ x y 0 is the position vector of an arbitrary point on the mid-surface defined in the element coordinate system. he rotation matrix A in terms of the three Euler angles is written as follows 2 cos u z cos u y sin u z cos u x þ cos u z sin u y sin u x 6 A ¼ 4 sin u z cos u y cos u z cos u x þ sin u z sin u y sin u x sin u y cos u y sin u x sin u z sin u x þ cos u z sin u y cos u x 3 cos u z sin u x þ sin u z sin u y cos u 7 x 5 cos u y cos u x Columns of the rotation matrix, A, are denoted, respectively, as i, j, and k. From equation 5, it follows that the global position of point P due to an arbitrary rotational motion can be written as 2 x cos u z cos u y þ y( sin u z cos u x þ cos u z sin u y sin u x ) u ¼ x sin u z cos u y þ y( cos u z cos u x 6 4 þ sin u z sin u y sin u x ) sin u y x þ y cos u y cos u x 3 7 5 (6) he global position of the arbitrary point as the result of a general rigid body displacement can be written as follows r ¼ R þ Au b (7) where R is the global position vector of the origin of the element coordinate system. he vector of nodal coordinates at node n as the result of the rigid body motion is as follows e n ¼ R n i j (8) where R n is the global position of the node that can be defined using equation (7). In the absolute nodal coordinate formulation, the location of the arbitrary point on the plate element using equations (1) and (8) can be obtained as r ¼ S(x, y)e 2 3 R 1 þ x cos u z cos u y þ y( sin u z cos u x þ cos u z sin u y sin u x ) ¼ R 2 þ x sin u z cos u y þ y( cos u z cos u x 6 þ sin u z sin u y sin u x ) 7 4 5 R 3 sin u y x þ y cos u y cos u x (9) Using equations (1), (6), and (8), it is clear that the reduced-order element can describe exact rigid body motion when absolute positions and slopes are used as nodal coordinates. 3 FORMULAION OF HE ELASIC FORCES In this section, the formulations of the elastic forces for the high- and reduced-order elements are presented. Special formulation needs to be used in the case of the reduced-order element because not all the position vector gradients are present. 3.1 General formulation he Lagrangian strain tensor can be written as 1 m ¼ 1 2 (J J I) (10) where subscript m is used to indicate the matrix form of the Lagrangian strains and J is the matrix of the position vector gradients defined as J ¼ @r @j ¼ 2 3 @r 1 @r 1 @r 1 @j 1 @j 2 @j 3 @r 2 @r 2 @r 2 @j 1 @j 2 @j 3 6 7 4 @r 3 @r 3 @r 3 5 @j 1 @j 2 @j 3 (11) In this equation, ¼ r ¼ r 1 r 2 r 3 j ¼ j1 j 2 j 3 Se0 (12) where e 0 is the vector of nodal coordinates in the initial undeformed configuration. Using equation (10), the strain vector can be defined as 1 ¼ 1 11 1 22 1 33 1 12 1 13 1 23 ¼ 1 2 ½(r j 1 r j1 1) (r j 2 r j2 1) (r j 3 r j3 1) 2r j 1 r j2 r j 1 r j3 2r j 2 r j3 Š (13) If the finite element spatial coordinates are denoted by the vector x given in terms of its components as ¼½x1 x ¼ x y z x 2 x 3 Š (14) one can then define the following transformation @x @j ¼ @j 1 ¼ a 1 a 2 a 3 @x (15) Proc. IMechE Vol. 219 Part K: J. Multi-body Dynamics JMBD2 # IMechE 2005

Analysis of thin plate structures 349 It follows from chain rule of differentiation that r ji ¼ @r @j i ¼ @r @x a i ¼ r x a i (16) Using this equation, the vector of Lagrangian strains can be written in terms of gradients defined in the element coordinate system as follows 1 ¼ 1 2 (a 1 r x r xa 1 1) (a 2 r x r xa 2 1) (a 3 r x r xa 3 1) 2a 1 r x r xa 2 2a 1 r x r xa 3 2a 2 r x r xa 3 (17) he derivatives of the strain components with respect to the element coordinates enter in the formulation of the elastic forces and need to be evaluated. Note that r x a i ¼ a 1i S x1 þ a 2i S x2 þ a 3i S x3 e ¼ Dei e (18) where D ei ¼ a 1i S x1 þ a 2i S x2 þ a 3i S x3 and S xi ¼ @S=@x i. It follows that (19) @(r x a i ) ¼ D ei (20) @e In terms of D ei, the strain vector can be written as follows 1 ¼ 1 2 ½e D e1 D e1e 1 e D e2 D e2e 1 e D e3 D e3e 1 2e D e1 D e2e 2e D e1 D e3e 2e D e2 D e3eš (21) herefore, the derivatives of the strain components with respect to the vector of nodal coordinates can be defined as 2 e D e1 D 3 e1 e D e2 D e2 @1 @e ¼ e D e3 D e3 e (D e1 D e2 þ D e2 D e1) 6 e (D e1 D e3 þ D e3 D 7 4 e1) 5 e (D e2 D e3 þ D e3 D e2) (22) his matrix is linear in the vector of nodal coordinates e, whereas the strain vector is a quadratic function of e. he vector of elastic forces can be obtained using the virtual work, which can be written as ð dw e ¼ 1 Ed1jJjdV (23) V where E is the matrix of elastic coefficients, and V is the volume. he second Piola Kirchoff stress tensor is used to obtain this expression. Using equations (21) and (22), the virtual work of the elastic forces can be written as follows dw e ¼ Q kde (24) where the vector of element generalized elastic forces Q k is defined as ð Q k ¼ V @1 E1jJjdV (25) @e he strain vector 1 can be written as 1 ¼ 1 J þ I m (26) where 1 J ¼ 1 2 ½a 1 r x r xa 1 a 2 r x r xa 2 a 3 r x r xa 3 2a 1 r x r xa 2 2a 1 r x r xa 3 2a 2 r x r xa 3 Š (27) and I m ¼ 1 2 1 1 1 0 0 0 (28) Using the definitions in the preceding two equations and denoting EI m ¼ E m, one can write the vector of elastic forces in the following form ð Q k ¼ V @1 ð E1 J jjjdv þ @e V @1 E m jjjdv (29) @e It is clear that the vector of elastic forces is a nonlinear function of the nodal coordinates. 3.2 hin plate formulation For the thin plate based on the absolute nodal coordinate formulation, the vector of the element elastic forces can be derived using the strain energy function. In this study, the plane stress conditions are assumed for the membrane stiffness, whereas the bending stiffness of the element is accounted for using the curvature of the element mid-plane [6, 7]. In the general continuum mechanics approach previously presented in this section, the plane stress assumptions are not made and for thin plates, the use of this general approach leads to high numerical stiffness because of the oscillation of some gradient components along the element thickness. Element local coordinate systems may also be used to define the element elastic forces [6], but the use of such local frame does not simplify the formulation and the element stiffness matrix remains highly nonlinear. Furthermore, the use of an element local coordinate system requires the use of non-linear JMBD2 # IMechE 2005 Proc. IMechE Vol. 219 Part K: J. Multi-body Dynamics

350 K Dufva and A A Shabana strain displacement relation in the case of large deformation. Non-linear Green Lagrange strain measure is employed in order to account for geometrically non-linear behaviour and to ensure zero strain under rigid body motion. Strain components are expressed using gradients obtained by differentiation with respect to global spatial coordinates as defined by equation (11). If the element coordinate system is assumed to be initially parallel to the global coordinate system, the transformation matrix of equation (15) is the identity matrix. In this article, only initially undeformed elements are considered. he Lagrangian strain tensor can be defined using the matrix of position vector gradients J as follows 1 m ¼ 1 2 (J J I) (30) where I is the 33 identity matrix. For plane stress conditions, the stresses in the plate thickness direction are assumed to be zero and the strains in this direction are expressed as function of the strains at the element mid-surface. he strain vector at the mid-surface can be obtained from equation (30) as follows 1 ¼ 1 m11 1 m22 21 m12 ¼ 1 xx 1 yy 21 xy (31) where 1 xx and 1 yy are the normal strain components in x and y directions, respectively, and 1 xy is the shear strain. In order to account for the bending stiffness, the correct definition for the element midsurface curvature needs to be defined. o define the curvature in terms of the gradient vectors, the following relations are used [7] K xx ¼ r xx n knk 3, K yy ¼ r yy n knk 3, K xy ¼ r xy n knk 3 (32) where n is the normal to the element mid-surface obtained as n ¼ r x r y. If the plate element is initially curved, the strain energy density function must be integrated with respect to the undeformed curved reference configuration. Relation between volumes in the uncurved reference and initially curved configuration can be defined using constant transformation and can be expressed as follows V o ¼ @x @j V (33) where V o is the volume of the element in the initially curved configuration and V is volume in the uncurved configuration. he strain energy of the element can now be written as follows [9] U ¼ 1 ð 1 E1 dv o þ 1 ð k Ek dv o (34) 2 V o 2 V o where E is the matrix of elastic coefficients obtained using the plane stress conditions [10], and the strain vector is defined using equation (31). he vector of curvatures multiplied by constant thickness of. the element defines k ¼ z k xx k yy 2k xy In problems where the membrane stresses are dominant, the effect of the curvature can be neglected. In the absolute nodal coordinate formulation, the vector of elastic forces can be obtained as follows Q k ¼ @U (35) @e where Q k is the vector of the element elastic forces. 4 EQUAIONS OF MOION he use of equation (1) for thin plates leads to a constant mass matrix and zero Coriolis and centrifugal forces. Differentiating equation (1) with respect to time, the element kinetic energy is obtained as follows ¼ 1 2 ð V o r_r _r dv o (36) where _r is the velocity vector of an arbitrary point on the plate, r the material density, and V o the element initial volume. Equation (36) leads to a constant mass matrix defined as follows ð M ¼ rs C S C dv o V o (37) he mass matrix remains constant under an arbitrary large displacement. he equations of motion can be written in terms of the constant mass matrix M, the non-linear element nodal forces Q k, and the applied external nodal forces, Q e, as follows [2] Më þ Q k ¼ Q e (38) where ë is the vector of nodal accelerations. As the mass matrix is constant, one can use the Cholesky coordinates to define an identity inertia matrix as discussed in previous publications [11]. Proc. IMechE Vol. 219 Part K: J. Multi-body Dynamics JMBD2 # IMechE 2005

Analysis of thin plate structures 351 5 NUMERICAL EXAMPLES he performance of the low-order element is demonstrated in this section using numerical examples. he results obtained are compared with the results obtained using the element proposed by Mikkola and Shabana [2]. he improved accuracy as the result of using the low-order element for thin plates employing the definition of the curvature is first demonstrated using a static problem. he reduced-order element with shape function matrix S C is referred to in this section as Model I and the full parameterization element with all nine gradient coordinates is referred to as Model II. A simple cantilever plate structure is subjected to two force components at the free end as shown in Fig. 2. he cantilever structure has dimensions of 0.5, 0.15, and 0.001 m, respectively, for the length l, width w, and thickness h. he material is assumed steel with Young s modulus of 2.07 10 11 Pa and Poisson s ratio of 0.3. he applied force F is 30 N. ests were made with 1, 2, 4, 8, and 16 elements. Displacements of the free end tip are shown in Fig. 3. It is clear from the results presented in this figure that the low-order element has better accuracy and convergence characteristics because of the use of the curvature definition in the thin plate model. Smaller number of elements is required for convergence when compared with the higherorder element. he performance of the low-order element in dynamic analysis is examined using a flexible pendulum. Results obtained using the same pendulum are reported in previous publications [2, 7]. Young s modulus of the material is assumed 1.0 10 5 Pa, material density is 7810 kg/m 3, and Poisson s ratio is 0.3. he pendulum has a length of 0.3 m, a width of 0.3 m, and a thickness of 0.01 m. he pendulum is simulated for 0.3 s using 1, 2 2,4 2, and 8 2 elements. Boundary conditions for the pendulum are applied at the corner node. In order to obtain the constraints for the spherical joint, all translation displacements of the node are fixed while rotations are free. his is accomplished using constraint equations for the position coordinates of the node. he equations Fig. 3 ip displacement as a function of the element numbers (, Model I; V, Model II) of motion are solved using the software Matlab and the integrator ode23tb. Simulation times for the pendulum example with a thickness of 0.01 m are presented in able 1. he results are obtained using an Intel Pentium 4 PC with 2.8 GHz processor. Figure 4 shows the results obtained using the two models for the position of the point A of the pendulum. For thin structures, the element with all nine gradients leads to numerical problems. he relation between the element thickness and the computation time is studied using the same pendulum example using the 2 2 element model. he element thickness is varied from 0.01 to 0.0005 m. he simulation times for the two models are presented in able 2. It is clear from the results presented in this table that the simulation time for the low-order element does not appreciably change as the element thickness decreases, whereas the simulation time for the higher-order element significantly increases as the element thickness decreases. It is important to note that as the thickness decreases, the low-order element model is more than 100 times faster than the higher-order element. Figure 5 shows the able 1 Simulation times as a function of the element numbers Number of elements Ratio of simulation time Simulation time (s) Model I Model II Fig. 2 hin cantilever plate 1 4.0 2 8 2 2 6.0 7 42 4 2 6.6 133 885 8 2 5.2 8340 43 700 JMBD2 # IMechE 2005 Proc. IMechE Vol. 219 Part K: J. Multi-body Dynamics

352 K Dufva and A A Shabana Fig. 4 Positions of the end point A and the centre point B of the pendulum (, Model I;, Model II) position coordinates of points A and B as predicted by the two models for thickness 0.01 m, whereas the deformed shape of the flexible pendulum with this thickness is depicted in Fig. 6. Note that, because of the symmetry, the X and Y coordinates are the same in the result presented in Fig. 5. he deformed shapes are depicted using the 1 and 8 2 element models. he obtained results are also found to be in good agreement with those obtained by Dmitrochenko and Pogorelov [7]. Bending moments and shear forces can be obtained from the stress distribution by integration over the element thickness. hese moments and forces can be calculated after the finite element solution for displacements is obtained. Generally, in the finite element analysis, the stresses are obtained accurately only at integration points and can be able 2 hickness (m) Simulation times as a function of the plate thickness Ratio of simulation time Simulation time (s) Model I Model II 0.01 6 7 42 0.001 58.8 8 470 0.0005 141.3 8 1130 interpolated over the element. Bending moments and shear forces are then obtained from stress distribution or directly at the integration points. 6 SUMMARY AND CONCLUSIONS Eliminating some of the gradient components that exhibit high frequency oscillations along the thickness direction can enhance the performance of plate and shell elements used in the absolute nodal coordinate formulation. Elimination of these gradient components that do not significantly affect the solution in the case of thin plates leads to more efficient reduced-order element that also has better convergence characteristics. wo-dimensional shape functions are used to define four-node element with 36 degrees of freedom. he use of these shape functions allows eliminating systematically the gradients defined by differentiation with respect to the spatial coordinate along the thickness direction. In the reduced-order element, the bending stiffness is accounted for using the mid-surface curvature. As pointed out, the use of the mid-surface curvature also improves the element accuracy and convergence properties. he good convergence characteristic of the reduced-order element is demonstrated in the case Proc. IMechE Vol. 219 Part K: J. Multi-body Dynamics JMBD2 # IMechE 2005

Analysis of thin plate structures 353 Fig. 5 Position coordinates of the points A and B (, Model I;, Model II) Fig. 6 Deformed shape of the pendulum JMBD2 # IMechE 2005 Proc. IMechE Vol. 219 Part K: J. Multi-body Dynamics

354 K Dufva and A A Shabana of large displacement static problems. A thin cantilever plate was used to demonstrate the numerical efficiency and better convergence characteristic of the reduced-order element. It was shown that the elimination of the third gradient vector can have a significant effect on improving the element performance. he difference in CPU time between models based on the high- and low-order elements was reported for different number of elements and thickness values. he computation time was found to be highly dependent on the element thickness and dramatically increases when the element thickness decreases. It was demonstrated that as the element thickness decreases, the models based on the loworder element can be more than 100 times faster than the models based on elements that employ full parameterization. It is important to note that other factors such as the integration method can have a strong influence on the overall computational time. he results show that the low-order element is computationally more efficient and owing to its better convergence characteristics, fewer elements are required when compared with the 48 degreesof-freedom element that employs the general continuum mechanics approach in the formulation of the elastic forces. ACKNOWLEDGEMEN his research was supported, in part, by the US Army Research Office. REFERENCES 1 Shabana, A. A. and Mikkola, A. M. On the use of the degenerate plate and the absolute nodal coordinate formulations in multibody system applications. J. Sound Vib., 2003, 259(2), 481 489. 2 Mikkola, A. M. and Shabana, A. A. A non-incremental finite element procedure for the analysis of large deformation of plates and shells in mechanical system applications. Multibody Syst. Dyn., 2003, 9(3), 283 309. 3 Yoo, W. S., Lee, J. H., Park, S. J., Sohn, J. H., Dmitrochenko, O. N., and Pogorelov, D. Y. Large oscillations of a thin cantilever beam: physical experiments and simulation using the absolute nodal coordinate formulation. Nonlinear Dyn., 2003, 34(1 2), 3 29. 4 Von Dombrowski, S. Analysis of large flexible body deformation in multibody systems using absolute coordinates. Multibody Syst. Dyn., 2002, 8(4), 409 432. 5 Garcia-Vallejo, D., Mayo, J., Escalona, J. L., and Dominguez, J. Efficient evaluation of the elastic forces and the Jacobian in the absolute nodal coordinate formulation. Nonlinear Dyn., 2004, 35(4), 313 329. 6 Shabana, A. A. and Christensen, A. P. hree dimensional absolute nodal coordinate formulation: plate problem. Int. J. Numer. Meth. Eng., 1997, 40, 2775 2790. 7 Dmitrochenko, O. N. and Pogorelov, D. Y. Generalization of plate finite elements for absolute nodal coordinate formulation. Multibody Syst. Dyn., 2003, 10, 17 43. 8 Yoo, W. S., Lee, J. H., Park, S. J., Sohn, J. H., Pogorelov, D. Y., and Dmitrochenko, O. N. Large deflection analysis of a thin plate: computer simulations and experiments. Multibody Syst. Dyn., 2004, 11, 185 208. 9 Chia, C. Y. Nonlinear analysis of plates, 1980 (McGraw- Hill, Inc., USA). 10 Bathe, K. J. Finite element procedures, 1996 ( Prentice- Hall Inc., New Jersey). 11 Yakoub, R. Y. and Shabana, A. A. Use of Cholesky coordinates and the absolute nodal coordinate formulation in the computer simulation of flexible multibody systems. Nonlinear Dyn., 1999, 20, 267 282. APPENDIX 1 Notation A rotation matrix in terms of the three Euler angles e vector of nodal coordinates e 0 vector of nodal coordinates in the initial configuration e n vector of nodal coordinates for a node (n ¼ A, B, C, D) E matrix of the material elastic coefficients i, j, k columns of the rotation matrix A I identity matrix J matrix of the position vector gradients M mass matrix of the element n normal vector of the element mid-surface Q e external nodal forces Q k vector of the element elastic forces r global position vector of a point r n global position vector of a node n r a vector of partial derivatives of the position vector with respect to (a ¼ x, y, z) r aa vector of second partial derivatives of the position vector with respect to (a ¼ x, y) R global position vector of the origin of the element coordinate system R n global position vector of the node due to rigid body motion S a matrix of the element shape functions (a ¼ A, B, C) kinetic energy of the element u global position vector of an arbitrary point on the element due to rigid body rotations Proc. IMechE Vol. 219 Part K: J. Multi-body Dynamics JMBD2 # IMechE 2005

Analysis of thin plate structures 355 u b U V o V W x position vector of a point on the mid-surface in the element coordinate system strain energy of the element volume of the element in the initial curved configuration volume of the element in the uncurved configuration virtual work of the element elastic forces vector of the element spatial coordinates x, y, z spatial coordinates of the element X, Y, Z global inertial coordinates 1 xy shear strain component 1 xx, 1 yy normal strain components in x and y directions 1 strain vector obtained from Lagrangian strain tensor 1 m Lagrangian strain tensor k xx, k yy curvatures of the element mid-surface in x and y directions k xy curvature of the element mid-surface due to torsion k h vector of curvatures dimensionless element coordinate in y direction u a Euler angles (a ¼ x, y, z) r material density j position vector of the material points in the current configuration j dimensionless element coordinate in x direction j a APPENDIX 2 components of the global position vector of the point in the current configuration, (a ¼ 1, 2, 3) he shape function matrix S C used to develop the low-order element presented in this investigation is written as follows S C ¼½S 1 I S 2 I S 3 I S 4 I S 5 I S 6 I S 7 I S 8 I S 9 I S 10 I S 11 I S 12 I Š where I is a 33 identity matrix and shape functions are as follows S 1 ¼ (j 1)(h 1)(2h 2 h þ 2j 2 j 1) S 2 ¼ lj(j 1) 2 (h 1) S 3 ¼ wh(h 1) 2 (j 1) S 4 ¼ j(2h 2 h 3j þ 2j 2 )(h 1) S 5 ¼ lj 2 (j 1)(h 1) S 6 ¼ wjh(h 1) 2 S 7 ¼ jh(1 3j 3h þ 2h 2 þ 2j 2 ) S 8 ¼ 1j 2 h(j 1) S 9 ¼ wjh 2 (h 1) S 10 ¼ h(j 1)(2j 2 j 3h þ 2h 2 ) S 11 ¼ ljh(j 1) S 12 ¼ wh 2 (j 1) 2 (h 1) where j ¼ j=l, h ¼ y=w. JMBD2 # IMechE 2005 Proc. IMechE Vol. 219 Part K: J. Multi-body Dynamics