Analysis of thin plate structures using the absolute nodal coordinate formulation
|
|
- Jesse Holt
- 5 years ago
- Views:
Transcription
1 345 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 DOI: / X50678 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 , 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
2 346 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
3 Analysis of thin plate structures 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 n ¼ n @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 n ¼ n 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
4 348 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 (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 ) 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 ¼ (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 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 1 ¼ a 1 a 2 a (15) Proc. IMechE Vol. 219 Part K: J. Multi-body Dynamics JMBD2 # IMechE 2005
5 Analysis of thin plate structures 349 It follows from chain rule of differentiation that r 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 i. It follows that x a i ) ¼ D ei 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 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 E1jJjdV 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 (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 ð E1 J jjjdv E m jjjdv 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
6 350 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 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 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
7 Analysis of thin plate structures 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 m, respectively, for the length l, width w, and thickness h. he material is assumed steel with Young s modulus of 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 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 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 JMBD2 # IMechE 2005 Proc. IMechE Vol. 219 Part K: J. Multi-body Dynamics
8 352 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 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
9 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
10 354 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), 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), 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), Von Dombrowski, S. Analysis of large flexible body deformation in multibody systems using absolute coordinates. Multibody Syst. Dyn., 2002, 8(4), 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), Shabana, A. A. and Christensen, A. P. hree dimensional absolute nodal coordinate formulation: plate problem. Int. J. Numer. Meth. Eng., 1997, 40, Dmitrochenko, O. N. and Pogorelov, D. Y. Generalization of plate finite elements for absolute nodal coordinate formulation. Multibody Syst. Dyn., 2003, 10, 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, 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, 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
11 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
12
COORDINATE FORMULATION
NEER ENGI LARGE DISPLACEMENT ANALYSIS OF SHELL STRUCTURES USING THE ABSOLUTE NODAL COORDINATE FORMULATION Mechanical Engineering Technical Report ME-TR-6 DATA SHEET Title: Large Displacement Analysis of
More informationGeometry-dependent MITC method for a 2-node iso-beam element
Structural Engineering and Mechanics, Vol. 9, No. (8) 3-3 Geometry-dependent MITC method for a -node iso-beam element Phill-Seung Lee Samsung Heavy Industries, Seocho, Seoul 37-857, Korea Hyu-Chun Noh
More informationMODAL DERIVATIVES BASED REDUCTION METHOD FOR FINITE DEFLECTIONS IN FLOATING FRAME
Modal derivatives based reduction method for finite deflections in floating frame 11th World Congress on Computational Mechanics (WCCM XI) 5th European Conference on Computational Mechanics (ECCM V) 6th
More informationINTEGRATION OF THE EQUATIONS OF MOTION OF MULTIBODY SYSTEMS USING ABSOLUTE NODAL COORDINATE FORMULATION
INTEGRATION OF THE EQUATIONS OF MOTION OF MULTIBODY SYSTEMS USING ABSOLUTE NODAL COORDINATE FORMULATION Grzegorz ORZECHOWSKI *, Janusz FRĄCZEK * * The Institute of Aeronautics and Applied Mechanics, The
More informationUNCONVENTIONAL FINITE ELEMENT MODELS FOR NONLINEAR ANALYSIS OF BEAMS AND PLATES
UNCONVENTIONAL FINITE ELEMENT MODELS FOR NONLINEAR ANALYSIS OF BEAMS AND PLATES A Thesis by WOORAM KIM Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the
More informationA consistent dynamic finite element formulation for a pipe using Euler parameters
111 A consistent dynamic finite element formulation for a pipe using Euler parameters Ara Arabyan and Yaqun Jiang Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, AZ 85721,
More informationA study of moderately thick quadrilateral plate elements based on the absolute nodal coordinate formulation
Multibody Syst Dyn (2014) 31:309 338 DOI 10.1007/s11044-013-9383-6 A study of moderately thick quadrilateral plate elements based on the absolute nodal coordinate formulation Marko K. Matikainen Antti
More informationMethods of Analysis. Force or Flexibility Method
INTRODUCTION: The structural analysis is a mathematical process by which the response of a structure to specified loads is determined. This response is measured by determining the internal forces or stresses
More informationThe Absolute Nodal Coordinate Formulation
The Absolute Nodal Coordinate Formulation ANCF Antonio Recuero Dan Negrut May 27, 2016 Abstract This white paper describes the fundamentals of the nonlinear nite element theory used to implement ANCF nite
More informationExample 3.7 Consider the undeformed configuration of a solid as shown in Figure 3.60.
162 3. The linear 3-D elasticity mathematical model The 3-D elasticity model is of great importance, since it is our highest order hierarchical model assuming linear elastic behavior. Therefore, it provides
More informationMITOCW MITRES2_002S10linear_lec07_300k-mp4
MITOCW MITRES2_002S10linear_lec07_300k-mp4 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources
More informationStructural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian
Structural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian ahmadian@iust.ac.ir Dynamic Response of MDOF Systems: Mode-Superposition Method Mode-Superposition Method:
More informationChapter 5 Structural Elements: The truss & beam elements
Institute of Structural Engineering Page 1 Chapter 5 Structural Elements: The truss & beam elements Institute of Structural Engineering Page 2 Chapter Goals Learn how to formulate the Finite Element Equations
More informationME751 Advanced Computational Multibody Dynamics
ME751 Advanced Computational Multibody Dynamics November 2, 2016 Antonio Recuero University of Wisconsin-Madison Quotes of the Day The methods which I set forth do not require either constructions or geometrical
More informationCOMPARISON OF TWO MODERATELY THICK PLATE ELEMENTS BASED ON THE ABSOLUTE NODAL COORDINATE FORMULATION
MULTIBODY DYNAMICS 2009, ECCOMAS Thematic Conference K. Arczewski, J. Frączek, M. Wojtyra (eds.) Warsaw, Poland, 29 June 2 July 2009 COMPARISON OF TWO MODERATELY THICK PLATE ELEMENTS BASED ON THE ABSOLUTE
More informationChapter 12 Plate Bending Elements. Chapter 12 Plate Bending Elements
CIVL 7/8117 Chapter 12 - Plate Bending Elements 1/34 Chapter 12 Plate Bending Elements Learning Objectives To introduce basic concepts of plate bending. To derive a common plate bending element stiffness
More informationMeasurement of deformation. Measurement of elastic force. Constitutive law. Finite element method
Deformable Bodies Deformation x p(x) Given a rest shape x and its deformed configuration p(x), how large is the internal restoring force f(p)? To answer this question, we need a way to measure deformation
More informationMATERIAL PROPERTIES. Material Properties Must Be Evaluated By Laboratory or Field Tests 1.1 INTRODUCTION 1.2 ANISOTROPIC MATERIALS
. MARIAL PROPRIS Material Properties Must Be valuated By Laboratory or Field ests. INRODUCION he fundamental equations of structural mechanics can be placed in three categories[]. First, the stress-strain
More informationInternational Journal of Advanced Engineering Technology E-ISSN
Research Article INTEGRATED FORCE METHOD FOR FIBER REINFORCED COMPOSITE PLATE BENDING PROBLEMS Doiphode G. S., Patodi S. C.* Address for Correspondence Assistant Professor, Applied Mechanics Department,
More informationMechanics PhD Preliminary Spring 2017
Mechanics PhD Preliminary Spring 2017 1. (10 points) Consider a body Ω that is assembled by gluing together two separate bodies along a flat interface. The normal vector to the interface is given by n
More informationDiscrete Analysis for Plate Bending Problems by Using Hybrid-type Penalty Method
131 Bulletin of Research Center for Computing and Multimedia Studies, Hosei University, 21 (2008) Published online (http://hdl.handle.net/10114/1532) Discrete Analysis for Plate Bending Problems by Using
More informationCHAPTER 14 BUCKLING ANALYSIS OF 1D AND 2D STRUCTURES
CHAPTER 14 BUCKLING ANALYSIS OF 1D AND 2D STRUCTURES 14.1 GENERAL REMARKS In structures where dominant loading is usually static, the most common cause of the collapse is a buckling failure. Buckling may
More informationEsben Byskov. Elementary Continuum. Mechanics for Everyone. With Applications to Structural Mechanics. Springer
Esben Byskov Elementary Continuum Mechanics for Everyone With Applications to Structural Mechanics Springer Contents Preface v Contents ix Introduction What Is Continuum Mechanics? "I Need Continuum Mechanics
More informationBAR ELEMENT WITH VARIATION OF CROSS-SECTION FOR GEOMETRIC NON-LINEAR ANALYSIS
Journal of Computational and Applied Mechanics, Vol.., No. 1., (2005), pp. 83 94 BAR ELEMENT WITH VARIATION OF CROSS-SECTION FOR GEOMETRIC NON-LINEAR ANALYSIS Vladimír Kutiš and Justín Murín Department
More information4.5 The framework element stiffness matrix
45 The framework element stiffness matri Consider a 1 degree-of-freedom element that is straight prismatic and symmetric about both principal cross-sectional aes For such a section the shear center coincides
More informationNONLINEAR CONTINUUM FORMULATIONS CONTENTS
NONLINEAR CONTINUUM FORMULATIONS CONTENTS Introduction to nonlinear continuum mechanics Descriptions of motion Measures of stresses and strains Updated and Total Lagrangian formulations Continuum shell
More informationACCURATE MODELLING OF STRAIN DISCONTINUITIES IN BEAMS USING AN XFEM APPROACH
VI International Conference on Adaptive Modeling and Simulation ADMOS 213 J. P. Moitinho de Almeida, P. Díez, C. Tiago and N. Parés (Eds) ACCURATE MODELLING OF STRAIN DISCONTINUITIES IN BEAMS USING AN
More informationME751 Advanced Computational Multibody Dynamics
ME751 Advanced Computational Multibody Dynamics October 24, 2016 Antonio Recuero University of Wisconsin-Madison Quote of the Day If a cluttered desk is a sign of a cluttered mind, of what, then, is an
More informationCable-Pulley Interaction with Dynamic Wrap Angle Using the Absolute Nodal Coordinate Formulation
Proceedings of the 4 th International Conference of Control, Dynamic Systems, and Robotics (CDSR'17) Toronto, Canada August 21 23, 2017 Paper No. 133 DOI: 10.11159/cdsr17.133 Cable-Pulley Interaction with
More informationUsing MATLAB and. Abaqus. Finite Element Analysis. Introduction to. Amar Khennane. Taylor & Francis Croup. Taylor & Francis Croup,
Introduction to Finite Element Analysis Using MATLAB and Abaqus Amar Khennane Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup, an informa business
More informationLecture 15 Strain and stress in beams
Spring, 2019 ME 323 Mechanics of Materials Lecture 15 Strain and stress in beams Reading assignment: 6.1 6.2 News: Instructor: Prof. Marcial Gonzalez Last modified: 1/6/19 9:42:38 PM Beam theory (@ ME
More informationFinite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 11
Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Module - 01 Lecture - 11 Last class, what we did is, we looked at a method called superposition
More information202 Index. failure, 26 field equation, 122 force, 1
Index acceleration, 12, 161 admissible function, 155 admissible stress, 32 Airy's stress function, 122, 124 d'alembert's principle, 165, 167, 177 amplitude, 171 analogy, 76 anisotropic material, 20 aperiodic
More informationNONLINEAR VIBRATIONS OF ROTATING 3D TAPERED BEAMS WITH ARBITRARY CROSS SECTIONS
COMPDYN 2013 4 th ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering M. Papadrakakis, V. Papadopoulos, V. Plevris (eds.) Kos Island, Greece, 12 14 June
More informationTruss Structures: The Direct Stiffness Method
. Truss Structures: The Companies, CHAPTER Truss Structures: The Direct Stiffness Method. INTRODUCTION The simple line elements discussed in Chapter introduced the concepts of nodes, nodal displacements,
More informationThe Finite Element Method for Solid and Structural Mechanics
The Finite Element Method for Solid and Structural Mechanics Sixth edition O.C. Zienkiewicz, CBE, FRS UNESCO Professor of Numerical Methods in Engineering International Centre for Numerical Methods in
More informationFLEXIBILITY METHOD FOR INDETERMINATE FRAMES
UNIT - I FLEXIBILITY METHOD FOR INDETERMINATE FRAMES 1. What is meant by indeterminate structures? Structures that do not satisfy the conditions of equilibrium are called indeterminate structure. These
More informationGeometric nonlinear formulation for curved beams with varying curvature
THEORETICAL & APPLIED MECHANICS LETTERS 2, 636 212) Geometric nonlinear formulation for curved beams with varying curvature Keqi Pan, a) and Jinyang Liu b) School of Naval Architecture, Ocean and Civil
More informationIraq Ref. & Air. Cond. Dept/ Technical College / Kirkuk
International Journal of Scientific & Engineering Research, Volume 6, Issue 4, April-015 1678 Study the Increasing of the Cantilever Plate Stiffness by Using s Jawdat Ali Yakoob Iesam Jondi Hasan Ass.
More informationA *69>H>N6 #DJGC6A DG C<>C::G>C<,8>:C8:H /DA 'D 2:6G - ( - ) +"' ( + -"( (' (& -+" % '('%"' +"-2 ( -!"',- % )% -.C>K:GH>IN D; AF69>HH>6,-+
The primary objective is to determine whether the structural efficiency of plates can be improved with variable thickness The large displacement analysis of steel plate with variable thickness at direction
More information1 Introduction IPICSE-2016
(06) DOI: 0.05/ matecconf/06860006 IPICSE-06 Numerical algorithm for solving of nonlinear problems of structural mechanics based on the continuation method in combination with the dynamic relaxation method
More informationBENCHMARK LINEAR FINITE ELEMENT ANALYSIS OF LATERALLY LOADED SINGLE PILE USING OPENSEES & COMPARISON WITH ANALYTICAL SOLUTION
BENCHMARK LINEAR FINITE ELEMENT ANALYSIS OF LATERALLY LOADED SINGLE PILE USING OPENSEES & COMPARISON WITH ANALYTICAL SOLUTION Ahmed Elgamal and Jinchi Lu October 07 Introduction In this study: I) The response
More informationVibration of Thin Beams by PIM and RPIM methods. *B. Kanber¹, and O. M. Tufik 1
APCOM & ISCM -4 th December, 23, Singapore Vibration of Thin Beams by PIM and RPIM methods *B. Kanber¹, and O. M. Tufik Mechanical Engineering Department, University of Gaziantep, Turkey. *Corresponding
More informationThe CR Formulation: BE Plane Beam
6 The CR Formulation: BE Plane Beam 6 Chapter 6: THE CR FORMUATION: BE PANE BEAM TABE OF CONTENTS Page 6. Introduction..................... 6 4 6.2 CR Beam Kinematics................. 6 4 6.2. Coordinate
More informationCRITERIA FOR SELECTION OF FEM MODELS.
CRITERIA FOR SELECTION OF FEM MODELS. Prof. P. C.Vasani,Applied Mechanics Department, L. D. College of Engineering,Ahmedabad- 380015 Ph.(079) 7486320 [R] E-mail:pcv-im@eth.net 1. Criteria for Convergence.
More informationMechanics of Materials II. Chapter III. A review of the fundamental formulation of stress, strain, and deflection
Mechanics of Materials II Chapter III A review of the fundamental formulation of stress, strain, and deflection Outline Introduction Assumtions and limitations Axial loading Torsion of circular shafts
More information3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1
Math Problem a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b- True or false and why 1. if A is
More informationDynamic Model of a Badminton Stroke
ISEA 28 CONFERENCE Dynamic Model of a Badminton Stroke M. Kwan* and J. Rasmussen Department of Mechanical Engineering, Aalborg University, 922 Aalborg East, Denmark Phone: +45 994 9317 / Fax: +45 9815
More informationStructural Dynamics Lecture 4. Outline of Lecture 4. Multi-Degree-of-Freedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations.
Outline of Multi-Degree-of-Freedom Systems Formulation of Equations of Motions. Newton s 2 nd Law Applied to Free Masses. D Alembert s Principle. Basic Equations of Motion for Forced Vibrations of Linear
More informationBending of Simply Supported Isotropic and Composite Laminate Plates
Bending of Simply Supported Isotropic and Composite Laminate Plates Ernesto Gutierrez-Miravete 1 Isotropic Plates Consider simply a supported rectangular plate of isotropic material (length a, width b,
More informationNonlinear bending analysis of laminated composite stiffened plates
Nonlinear bending analysis of laminated composite stiffened plates * S.N.Patel 1) 1) Dept. of Civi Engineering, BITS Pilani, Pilani Campus, Pilani-333031, (Raj), India 1) shuvendu@pilani.bits-pilani.ac.in
More informationFinite Element Method in Geotechnical Engineering
Finite Element Method in Geotechnical Engineering Short Course on + Dynamics Boulder, Colorado January 5-8, 2004 Stein Sture Professor of Civil Engineering University of Colorado at Boulder Contents Steps
More informationLecture 8. Stress Strain in Multi-dimension
Lecture 8. Stress Strain in Multi-dimension Module. General Field Equations General Field Equations [] Equilibrium Equations in Elastic bodies xx x y z yx zx f x 0, etc [2] Kinematics xx u x x,etc. [3]
More informationMulti Linear Elastic and Plastic Link in SAP2000
26/01/2016 Marco Donà Multi Linear Elastic and Plastic Link in SAP2000 1 General principles Link object connects two joints, i and j, separated by length L, such that specialized structural behaviour may
More information7. Hierarchical modeling examples
7. Hierarchical modeling examples The objective of this chapter is to apply the hierarchical modeling approach discussed in Chapter 1 to three selected problems using the mathematical models studied in
More informationGeometrically exact beam dynamics, with and without rotational degree of freedom
ICCM2014 28-30 th July, Cambridge, England Geometrically exact beam dynamics, with and without rotational degree of freedom *Tien Long Nguyen¹, Carlo Sansour 2, and Mohammed Hjiaj 1 1 Department of Civil
More informationCHAPTER 4 DESIGN AND ANALYSIS OF CANTILEVER BEAM ELECTROSTATIC ACTUATORS
61 CHAPTER 4 DESIGN AND ANALYSIS OF CANTILEVER BEAM ELECTROSTATIC ACTUATORS 4.1 INTRODUCTION The analysis of cantilever beams of small dimensions taking into the effect of fringing fields is studied and
More informationFIXED BEAMS IN BENDING
FIXED BEAMS IN BENDING INTRODUCTION Fixed or built-in beams are commonly used in building construction because they possess high rigidity in comparison to simply supported beams. When a simply supported
More informationContinuum Mechanics and the Finite Element Method
Continuum Mechanics and the Finite Element Method 1 Assignment 2 Due on March 2 nd @ midnight 2 Suppose you want to simulate this The familiar mass-spring system l 0 l y i X y i x Spring length before/after
More informationUniversity of Groningen
University of Groningen Nature-inspired microfluidic propulsion using magnetic actuation Khaderi, S. N.; Baltussen, M. G. H. M.; Anderson, P. D.; Ioan, D.; den Toonder, J.M.J.; Onck, Patrick Published
More informationDynamic and buckling analysis of FRP portal frames using a locking-free finite element
Fourth International Conference on FRP Composites in Civil Engineering (CICE8) 22-24July 8, Zurich, Switzerland Dynamic and buckling analysis of FRP portal frames using a locking-free finite element F.
More informationNonlinear Theory of Elasticity. Dr.-Ing. Martin Ruess
Nonlinear Theory of Elasticity Dr.-Ing. Martin Ruess geometry description Cartesian global coordinate system with base vectors of the Euclidian space orthonormal basis origin O point P domain of a deformable
More informationTheories of Straight Beams
EVPM3ed02 2016/6/10 7:20 page 71 #25 This is a part of the revised chapter in the new edition of the tetbook Energy Principles and Variational Methods in pplied Mechanics, which will appear in 2017. These
More informationMIXED RECTANGULAR FINITE ELEMENTS FOR PLATE BENDING
144 MIXED RECTANGULAR FINITE ELEMENTS FOR PLATE BENDING J. N. Reddy* and Chen-Shyh-Tsay School of Aerospace, Mechanical and Nuclear Engineering, University of Oklahoma, Norman, Oklahoma The paper describes
More information3D Elasticity Theory
3D lasticity Theory Many structural analysis problems are analysed using the theory of elasticity in which Hooke s law is used to enforce proportionality between stress and strain at any deformation level.
More informationA HIGHER-ORDER BEAM THEORY FOR COMPOSITE BOX BEAMS
A HIGHER-ORDER BEAM THEORY FOR COMPOSITE BOX BEAMS A. Kroker, W. Becker TU Darmstadt, Department of Mechanical Engineering, Chair of Structural Mechanics Hochschulstr. 1, D-64289 Darmstadt, Germany kroker@mechanik.tu-darmstadt.de,
More informationComputational non-linear structural dynamics and energy-momentum integration schemes
icccbe 2010 Nottingham University Press Proceedings of the International Conference on Computing in Civil and Building Engineering W Tizani (Editor) Computational non-linear structural dynamics and energy-momentum
More informationPlates and Shells: Theory and Computation. Dr. Mostafa Ranjbar
Plates and Shells: Theory and Computation Dr. Mostafa Ranjbar Outline -1-! This part of the module consists of seven lectures and will focus on finite elements for beams, plates and shells. More specifically,
More informationShape Optimization of Revolute Single Link Flexible Robotic Manipulator for Vibration Suppression
15 th National Conference on Machines and Mechanisms NaCoMM011-157 Shape Optimization of Revolute Single Link Flexible Robotic Manipulator for Vibration Suppression Sachindra Mahto Abstract In this work,
More informationCoupled thermo-structural analysis of a bimetallic strip using the absolute nodal coordinate formulation
Coupled thermo-structural analysis of a bimetallic strip using the absolute nodal coordinate formulation Gregor Čepon, Blaž Starc, Blaž Zupančič, Miha Boltežar May 16, 2017 Cite as: Gregor Čepon, Blaž
More informationFORMULATION OF THE INTERNAL STRESS EQUATIONS OF PINNED PORTAL FRAMES PUTTING AXIAL DEFORMATION INTO CONSIDERATION
FORMUATION OF THE INTERNA STRESS EQUATIONS OF PINNED PORTA FRAMES PUTTING AXIA DEFORMATION INTO CONSIDERATION Okonkwo V. O. B.Eng, M.Eng, MNSE, COREN.ecturer, Department of Civil Engineering, Nnamdi Azikiwe
More informationJEPPIAAR ENGINEERING COLLEGE
JEPPIAAR ENGINEERING COLLEGE Jeppiaar Nagar, Rajiv Gandhi Salai 600 119 DEPARTMENT OFMECHANICAL ENGINEERING QUESTION BANK VI SEMESTER ME6603 FINITE ELEMENT ANALYSIS Regulation 013 SUBJECT YEAR /SEM: III
More informationUNIT IV FLEXIBILTY AND STIFFNESS METHOD
SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road 517583 QUESTION BANK (DESCRIPTIVE) Subject with Code : SA-II (13A01505) Year & Sem: III-B.Tech & I-Sem Course & Branch: B.Tech
More informationBHAR AT HID AS AN ENGIN E ERI N G C O L L E G E NATTR A MPA LL I
BHAR AT HID AS AN ENGIN E ERI N G C O L L E G E NATTR A MPA LL I 635 8 54. Third Year M E C H A NICAL VI S E M ES TER QUE S T I ON B ANK Subject: ME 6 603 FIN I T E E LE ME N T A N A L YSIS UNI T - I INTRODUCTION
More informationDynamics. describe the relationship between the joint actuator torques and the motion of the structure important role for
Dynamics describe the relationship between the joint actuator torques and the motion of the structure important role for simulation of motion (test control strategies) analysis of manipulator structures
More informationReview of Strain Energy Methods and Introduction to Stiffness Matrix Methods of Structural Analysis
uke University epartment of Civil and Environmental Engineering CEE 42L. Matrix Structural Analysis Henri P. Gavin Fall, 22 Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods
More informationLarge deflection analysis of planar solids based on the Finite Particle Method
yuying@uiuc.edu 10 th US National Congress on Computational Mechanics Large deflection analysis of planar solids based on the Finite Particle Method 1, 2 Presenter: Ying Yu Advisors: Prof. Glaucio H. Paulino
More informationRigid Pavement Mechanics. Curling Stresses
Rigid Pavement Mechanics Curling Stresses Major Distress Conditions Cracking Bottom-up transverse cracks Top-down transverse cracks Longitudinal cracks Corner breaks Punchouts (CRCP) 2 Major Distress Conditions
More informationBasic Energy Principles in Stiffness Analysis
Basic Energy Principles in Stiffness Analysis Stress-Strain Relations The application of any theory requires knowledge of the physical properties of the material(s) comprising the structure. We are limiting
More informationANALYSIS OF HIGHRISE BUILDING STRUCTURE WITH SETBACK SUBJECT TO EARTHQUAKE GROUND MOTIONS
ANALYSIS OF HIGHRISE BUILDING SRUCURE WIH SEBACK SUBJEC O EARHQUAKE GROUND MOIONS 157 Xiaojun ZHANG 1 And John L MEEK SUMMARY he earthquake response behaviour of unframed highrise buildings with setbacks
More informationCO-ROTATIONAL DYNAMIC FORMULATION FOR 2D BEAMS
COMPDYN 011 ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering M. Papadrakakis, M. Fragiadakis, V. Plevris (eds.) Corfu, Greece, 5-8 May 011 CO-ROTATIONAL
More informationDEVELOPMENT OF A CONTINUUM PLASTICITY MODEL FOR THE COMMERCIAL FINITE ELEMENT CODE ABAQUS
DEVELOPMENT OF A CONTINUUM PLASTICITY MODEL FOR THE COMMERCIAL FINITE ELEMENT CODE ABAQUS Mohsen Safaei, Wim De Waele Ghent University, Laboratory Soete, Belgium Abstract The present work relates to the
More informationExercise: concepts from chapter 5
Reading: Fundamentals of Structural Geology, Ch 5 1) Study the oöids depicted in Figure 1a and 1b. Figure 1a Figure 1b Figure 1. Nearly undeformed (1a) and significantly deformed (1b) oöids with spherulitic
More informationAccepted Manuscript. R.C. Batra, J. Xiao S (12) Reference: COST Composite Structures. To appear in:
Accepted Manuscript Finite deformations of curved laminated St. Venant-Kirchhoff beam using layerwise third order shear and normal deformable beam theory (TSNDT) R.C. Batra, J. Xiao PII: S0263-8223(12)00486-2
More informationFINITE ELEMENT ANALYSIS OF ARKANSAS TEST SERIES PILE #2 USING OPENSEES (WITH LPILE COMPARISON)
FINITE ELEMENT ANALYSIS OF ARKANSAS TEST SERIES PILE #2 USING OPENSEES (WITH LPILE COMPARISON) Ahmed Elgamal and Jinchi Lu October 07 Introduction In this study, we conduct a finite element simulation
More informationUNIT- I Thin plate theory, Structural Instability:
UNIT- I Thin plate theory, Structural Instability: Analysis of thin rectangular plates subject to bending, twisting, distributed transverse load, combined bending and in-plane loading Thin plates having
More informationStatic & Dynamic. Analysis of Structures. Edward L.Wilson. University of California, Berkeley. Fourth Edition. Professor Emeritus of Civil Engineering
Static & Dynamic Analysis of Structures A Physical Approach With Emphasis on Earthquake Engineering Edward LWilson Professor Emeritus of Civil Engineering University of California, Berkeley Fourth Edition
More informationLinear and Nonlinear Dynamics of a Turbine Blade in Presence of an. Underplatform Damper with Friction
Linear and Nonlinear Dynamics of a Turbine Blade in Presence of an Underplatform Damper with Friction BY STEFANO MICHELIS Laurea, Politenico di Torino, Torino, Italy, 2012 THESIS Submitted as partial fulfillments
More informationFinite element modelling of structural mechanics problems
1 Finite element modelling of structural mechanics problems Kjell Magne Mathisen Department of Structural Engineering Norwegian University of Science and Technology Lecture 10: Geilo Winter School - January,
More informationSEMM Mechanics PhD Preliminary Exam Spring Consider a two-dimensional rigid motion, whose displacement field is given by
SEMM Mechanics PhD Preliminary Exam Spring 2014 1. Consider a two-dimensional rigid motion, whose displacement field is given by u(x) = [cos(β)x 1 + sin(β)x 2 X 1 ]e 1 + [ sin(β)x 1 + cos(β)x 2 X 2 ]e
More informationGeneral elastic beam with an elastic foundation
General elastic beam with an elastic foundation Figure 1 shows a beam-column on an elastic foundation. The beam is connected to a continuous series of foundation springs. The other end of the foundation
More informationVIBRATION ANALYSIS OF EULER AND TIMOSHENKO BEAMS USING DIFFERENTIAL TRANSFORMATION METHOD
VIBRATION ANALYSIS OF EULER AND TIMOSHENKO BEAMS USING DIFFERENTIAL TRANSFORMATION METHOD Dona Varghese 1, M.G Rajendran 2 1 P G student, School of Civil Engineering, 2 Professor, School of Civil Engineering
More informationGeneric Strategies to Implement Material Grading in Finite Element Methods for Isotropic and Anisotropic Materials
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Engineering Mechanics Dissertations & Theses Mechanical & Materials Engineering, Department of Winter 12-9-2011 Generic
More informationCOPYRIGHTED MATERIAL. Index
Index A Admissible function, 163 Amplification factor, 36 Amplitude, 1, 22 Amplitude-modulated carrier, 630 Amplitude ratio, 36 Antinodes, 612 Approximate analytical methods, 647 Assumed modes method,
More informationLecture M1 Slender (one dimensional) Structures Reading: Crandall, Dahl and Lardner 3.1, 7.2
Lecture M1 Slender (one dimensional) Structures Reading: Crandall, Dahl and Lardner 3.1, 7.2 This semester we are going to utilize the principles we learnt last semester (i.e the 3 great principles and
More informationDynamic analysis of railway bridges by means of the spectral method
Dynamic analysis of railway bridges by means of the spectral method Giuseppe Catania, Silvio Sorrentino DIEM, Department of Mechanical Engineering, University of Bologna, Viale del Risorgimento, 436 Bologna,
More informationDispersion relation for transverse waves in a linear chain of particles
Dispersion relation for transverse waves in a linear chain of particles V. I. Repchenkov* It is difficult to overestimate the importance that have for the development of science the simplest physical and
More informationResearch Article A Hybrid Interpolation Method for Geometric Nonlinear Spatial Beam Elements with Explicit Nodal Force
Mathematical Problems in Engineering Volume 216, Article ID 898676, 16 pages http://dx.doi.org/1.1155/216/898676 Research Article A Hybrid Interpolation Method for Geometric Nonlinear Spatial Beam Elements
More informationInterpolation Functions for General Element Formulation
CHPTER 6 Interpolation Functions 6.1 INTRODUCTION The structural elements introduced in the previous chapters were formulated on the basis of known principles from elementary strength of materials theory.
More informationNumerical simulation of the coil spring and investigation the impact of tension and compression to the spring natural frequencies
Numerical simulation of the coil spring and investigation the impact of tension and compression to the spring natural frequencies F. D. Sorokin 1, Zhou Su 2 Bauman Moscow State Technical University, Moscow,
More information