corotational nonlinear dynamic analysis of … · a triangular ﬂat shell element implemented...

Proceedings of PACAM XICopyright c© 2009 by ABCM

11th Pan-American Congress of Applied Mechanics - PACAM XIJanuary 04-08, 2010, Foz do Iguaçu, PR, Brazil

COROTATIONAL NONLINEAR DYNAMIC ANALYSIS OFLAMINATED COMPOSITE SHELL STRUCTURES

Felipe Schaedler de Almeida, [email protected] Miguel Awruch, [email protected]

Graduate Program in Civil Engineering, Federal University of Rio Grande do Sul, Av. Osvaldo Aranha, 99, 90035-190 PortoAlegre, RS, Brazil

Abstract. The dynamic analysis of laminated composite shell structures is performed using a simple displacement-based 18-degree-of-freedom triangular flat shell element, obtained by the superposition of a membrane element anda plate element. The membrane element is based on the assumed natural deviatoric strain formulation (ANDES),having corner drilling degrees of freedom and optimal in-plane bending response. The plate element employs theTimoshenko’s laminated composite beam function to define the deflections and rotations on the element boundaries.This formulation provides first-order shear flexibility to the element and naturally avoids shear-locking problemsas thin shells are analyzed. The geometrically nonlinear behavior of the structures is achieved by the elementindependent corotational formulation (EICR) together with a consistent treatment of finite rotations. An energyconserving procedure for the time-integration of the nonlinear dynamic equations is also included. Finally, twoexamples are presented to show that the algorithm is able to solve highly nonlinear dynamic problems.

Keywords: Laminated composite materials, Shell structures, Nonlinear dynamics

1. INTRODUCTION

Laminated composites as structural materials has becoming a very important aspect in structural design. This isreflected in the growing amount of researches devoted to this subject. One of the main interesting research fields for theengineering community is those related to the computational modeling and analysis of laminated composite structures.Correct prediction of the mechanical behavior is an essential condition for the introduction of laminated composites incritical structural applications.

Laminates are very attractive due to their high mechanical performance and to the possibility of tailoring their proper-ties to specific application by adjustments of fiber orientation and stacking sequence. Considerable modifications on theplate or shell stiffness can be obtained by the adoption of different lamination sequences, as will be shown latter. Fur-thermore, the ply ortotropicity leads the laminate mechanical behavior to be more complex than for isotropic materials,with the rising of coupling deformation modes and high shear deformation sensitivity. Together to the uncommon mate-rial behavior, other sophistications may also be considered in computational analysis in order to correctly determine theresponse of laminated composite structures. Geometric nonlinearity is one of the aspects to be taken into account for theanalysis improvement, since large displacements and rotations may be observed in many structural application. Dynamicsis other point frequently present in structural designs and must be carefully studied to provide realistic predictions.

This work aims to present a numerical study on the dynamic analyses of laminated composite shells considering geo-metrically nonlinear effects. The algorithm employed for the analysis is composed by a linear composite shell element as-sociated with the element independent corotational formulation (EICR) (Nour-Omid and Rankin, 1991), to account for thegeometric nonlinearity with finite rotations, and the approximately energy-conserving corotational procedure (AECCP)(Crisfield, 1997) to perform the dynamic analysis. Theoretical aspects are briefly discussed in the next section, followedby a section with numerical studies on the nonlinear dynamic analysis of shells.

2. COROTATIONAL NONLINEAR DYNAMIC ANALYSIS OF SHELLS

The construction of the present algorithm is achieved by first implementing a shell element for the linear static analysisand subsequently applying the approximately energy-conserving corotational procedure (AECCP), based on the elementindependent corotational formulation (EICR), to expand the element capabilities to the nonlinear dynamic analysis ofshells.

A triangular flat shell element implemented based on static linear formulation is obtained by the superposition of three-noded membrane and plate elements, as shown in Fig. 1, resulting in a 18-dof element. The membrane element used inthe composition of the shell element is given by Felippa (2003). This element is based on the assumed natural deviatoricstrain formulation (ANDES) and possesses two in-plane translations and one drilling rotation per node. Although theoriginal formulation is developed to give optimal response for pure in-plane bending of isotropic membranes, the elementapplication is expanded here for the analysis of laminated composite materials, following the recommendations given in



the above reference work. The plate element adopted to fill the bending and transverse shearing stiffness of the shellelement is given by Zhang and Kim (2005). This element takes into account laminated composite sections and firstorder shear flexibility by using the Timoshenko’s laminated composite beam function to interpolate in-plane rotations andtransverse displacements in the element sides. The absence of shear-locking problems is inherited by the plate elementfrom the Timoshenko’s laminated beam function, which provides a unified formulation for the analysis of thin and thickplates and shells without any stabilization scheme.

Figure 1. Membrane + plate degrees of freedom = shell degrees of freedom

The extension of the finite element for linear static analysis to the geometrically nonlinear analysis is obtained bythe application of the EICR formulation (Nour-Omid and Rankin, 1991). Basically, the variation of local displacementsδpl (in the element corotational system) is related to the variation of the global displacements δp by a transformationmatrix Λ, as given in Eq.(1). As finite rotations are considered in the EICR formulation, the nodal rotations are definedby tensors R, represented by 3 × 3 matrices, while the rotation variations are defined by a spin vector ω (Felippa andHaugen, 2005). The transformation matrix, given in Eq.(2), is composed by three matrices: H which transform the spinvector ω (containing variations of the rotations) in the additive rotations (θx,θy and θz ), P which extract the deformationpart of the rotations and displacements (ommiting rigid rotations and displacements) and T which transform the globalvariable into local variables. The formulation for the transformation matrices and the update of rotations are given inmany reference works on corortational formulation as Felippa and Haugen (2005), Crisfield (1997) and Nour-Omid andRankin (1991).

δpl = Λδp (1)

Λ = HPT (2)

Considering the linear relation between the local displacements pil and the local nodal forces qil given by the linearstiffness matrix Kl, and the local-global displacement transformation given in Eq.(1), it is possible to express the globalnodal forces in terms of local displacements by the Eq.(3), which is derived by equating the virtual work in the local and inthe global reference systems. The tangent stiffness matrix Kt is defined by the variation of the internal forces with respectto the global displacements, as given in Eq.(4) (Felippa and Haugen, 2005), where ΛtKlΛ is called material stiffnessmatrix and Ktσ is called geometric stiffness matrix.

qil = Λtqil = ΛtKlpil (3)

δqi = Λtδqil + δΛtqil =(ΛtKlΛ + Ktσ (qil)

)δp = Ktδp (4)

The AECCP was developed by Crisfield (1997) as an approximation to the corotational mid-point dynamic algorithm(Crisfield and Shi, 1994), introduced in order to conserve the full energy of the mechanical system. Mid-point algorithmsare constructed by equating the change of total momentum of the system to the impulse of internal and external forcesacting on the system during the time step (Crisfield et al., 1997). The equilibrium equation is given by Eq.(5), where,∆K, ∆P and ∆φ are the kinetic, external and strain energy increments in the time step, respectivelly; qmass,m, qe,m andqi,m are the mid-point inertial, external and internal nodal element forces, respectively. The mid-point vectors form anequivalent force vector g∗m that may vanish as the numerical integration reaches convergence, conserving approximatelythe total energy of the system.

∆K + ∆P + ∆φ ∼=(qtmass,m − qte,m + qti,m

)∆p = g∗tm∆p = 0 (5)

The mid-point inertial vector is defined for a specific element by Eq.(6), where ∆t is the time increment, M is theelement mass matrix and p is the nodal velocities vector, which contains both translational global velocities and, body



attached angular velocities. The relation between spatial and body attached angular velocities is given for the vector p byR∗e,n+1, which is a block diagonal matrix defined in Eq.(7) (Crisfield et al., 1997). Following Crisfield et al. (1997), in

this work the mass matrix is built direcly in the global reference system by interpolating global velocities.

qmass,m =1

∆t(R∗e,n+1Mpn+1 −R∗

e,nMpn)

(6)

R∗e,n = diag

[I R1,n I R2,n I R3,n

](7)

The external and internal forces acting on the structures between the time n and n+1 are represented by their mid-pointvalues, given in Eq.(8) and Eq.(9), respectively.

qe,m =qe,n+1 + qe,n

2(8)

qi,m =(

Λn+1 + Λn

2

)t qi,n+1 + qi,n2

(9)

Like for the static cortational formulation (Eq.(4)), an equivalent tangent stiffness matrix is obtained by the variationof the equivalent force vector g∗m with respect to the global displacements. The equivalent tangent stiffness matrix isformed by the inertial contribution, given in Eq.(10) and the static contribution, given in Eq.(11). No contribution tothe equivalent stiffness is originated by the external forces as long as they are conservatives. The complete derivationof Kmass are given in Crisfield (1997) for beam elements; however, as only terms related to nodes are varied, the sameformulation can be applied for triangular shell elements with the addition of one node with respect to the beam element.

δqmas,m =1

∆t(δR∗

e,n+1Mpn+1 + R∗e,n+1Mδpn+1

)= Kmas (10)

δqi,m =(

Λn+1 + Λn

2

)tδqi,n+1

2+(δΛn+1

2

)t qi,n+1 + qi,n2

=

[(Λn+1 + Λn

2

)t Kl

2Λn + Ktσ

(qi,n+1 + qi,n

2

)]δp

(11)

The procedure to integrate the equilibrium equations (Eq. (5)) in time is given by Crisfield (1997), where initially apredictor step is used, followed by corrector iterations. The predictor step is necessary since pn+1, pn+1 and qi,n+1 arenot known at the beginning of the integration in a new time step.

3. EXAMPLES

In this section two examples of the solution of nonlinear dynamic problems are presented.

3.1 Nonlinear transient response of cylindrical composite shells

The nonlinear transient analysis of three specially laminated composite shells are present in this example. The ge-ometry of the simply supported spherical panel is presented in Fig. 2a, where R = 5m and a = 0.5m. Two valuesfor the total thickness of the shell (h) are considered, being their values given by the geometrical relations a/h = 100and a/h = 50. The laminates under consideration are a bending stiff laminate [0◦2/ ± 30◦]s, a quasi-isotropic laminate[0◦/ ± 45◦/90◦]s, and a torsion stiff laminate [±45◦/ ∓ 45◦]s. These laminates are made using the T300/5208 CFRP(carbon fiber reinforced polymer) composite, with the following properties (Kundu and Sinha, 2006): E11 = 181 GPa,E22 = 10.3 GPa, G12 = G13 = 7.17 GPa, G23 = 3.58 GPa, ν12 = 0.28 and ρ = 1600 kg/m3. All plies of each laminatehave the same thickness. The structure is loaded by a suddenly applied step internal pressure of 1.0× 104N/m2.

The normalized vertical displacement of the panel center (uz/h) obtained in this work using a mesh with 16 × 16rectangles with two triangles each one (corresponding to 512 elements and 289 nodes) to describe the whole shell and atime step increment ∆t = 1.0 × 10−4s, are presented in Fig.2b, Fig.2c and Fig.2d for the bending stiff, quasi-isotropicand torsion stiff laminates, respectively. The time increment and the number of nodes are equal to those adopted by Kunduand Sinha (2006), which performed the analyses using a nine-noded isoparametric composite shell element consideringfirst order shear deformation and developed in curvilinear coordinates. The reference work is based on the total lagrangianapproach and the Newmark method has been used for the time integration.



Figure 2. Simply supported spherical shell: (a) Geometry (b) Response for the bending stiff laminate [0◦2/ ± 30◦]s (c)Response for the quasi-isotropic laminate [0◦/± 45◦/90◦]s (d) Response for the torsion stiff laminate [±45◦/∓ 45◦]s

Results obtained by the present work compare very well with those given by the reference, as can be seen in Figures2b-d. Only a tiny difference in the period and amplitude of the panel center vertical displacement is observed for thelaminates with ratio a/h = 100. The ability of the present element in capture the change in stiffness due to differentstaking sequences is demonstrated in this example by the variety of laminates considered.

3.2 Motion of a short cylinder

This is a classical example, first presented by Simo and Tarnow (1994), and widely used to demonstrate the abilityof the proposed formulations in solving problems with large motion (displacements and rotations) for long-term com-putations. Due to the lack of reference examples dealing with dynamic large motion analysis of laminated compositestructures in the literature, this example is adopted in the present work to demonstrate that the implemented algorithmalso posses the mentioned characteristics, in spite of the fact that an isotropic material is considered.

The geometry of the short cylinder is defined by the diameter D = 15, the height H = 3 and the thickness h = 0.02.The material characteristics are: Young’s modulus E = 2 × 108, Poisson’s ratio ν = 0.5 and mass density ρ = 1.The loading conditions applied for the nodes located in positions described by the angles 0◦, 90◦, 180◦ and 360◦ takenanticlockwise from the x axis are given in Fig.3, as well as the amplitude function f(t). Like in the reference work (Branket al., 2003), a mesh with 28× 3 retangles with two triangles each one (corresponding to 168 triangular elements and 84nodes) is used to model the shell. However, the maximum time step increment that lead to correct results for this work is∆t = 5.0 × 10−2s, which is half the time step used by Brank et al. (2003), but 2.5 times bigger than the time step usedby Simo and Tarnow (1994).

The displacements of the point A, shown in Fig.3, initially located at (D2 , 0, 0), are presented in Fig.4a, where theresults obtained in this work are compared to those given by Brank et al. (2003). Very good agreement is observed. Figure4b, shows velocity results for the same point. As the response obtained in the reference work is quite noisy it couldnot be reproduced here, but the mean value of the velocities agree well with those obtained by the present work. Theenergy conservative property, inherent of the method adopted in the present work, is demonstrated in Fig. 4c, where the



Figure 3. Short cylinder geometry and load condition.

kinetic energy (K) and strain potential energy (P) are plotted. Again, the results obtained here compare very well withthose presented by Brank et al. (2003). Finally, Fig. 4d illustrates the motion of the cylinder by depicting a sequence ofdeformed shapes, without any magnification of the actual deformations.

Figure 4. Short cylinder: (a) Displacements of point A (b) Velocities of point A (c) Kinetic and strain potential energies(d) Sequence of deformed shapes

4. CONCLUSIONS

The two examples demonstrated that the implemented algorithm is able to correctly perform nonlinear dynamic analy-sis of composite structures. In the first example three different laminates where studied, showing that the finite element isable to reproduce the change of stiffness due to specific stacking sequences in each laminte. The second example showedthat problems with large displacements and rotations can aslo be solved by the algorithm. The stability of the method,based on the conservation of the total energy, was demonstrated in the second example by plotting kinetic and strain



energy.

5. ACKNOWLEDGEMENTS

The authors wish to thank the Brazilian agencies CNPq and CAPES for their financial support.

6. REFERENCES

Brank, B., Korelc, J., and Ibrahimbegovic, A. 2003. Dynamics and time-stepping schemes for elastic shells undergoingfinite rotations. Computers and Structures, Vol.81(12):1193 – 1210.

Crisfield, M., Galvanetto, U., and Jelenic, G. 1997. Dynamics of 3-d co-rotational beams. Computational Mechanics,Vol.20(6):507 – 519.

Crisfield, M. and Shi, J. 1994. Co-rotational element/time-integration strategy for non-linear dynamics. InternationalJournal for Numerical Methods in Engineering, Vol.37(11):1897 – 1913.

Crisfield, M. A. 1997. Non-linear Finite Element Analysis of Solid and Structures - Vol2: Advanced Topics. Wiley.

Felippa, C. and Haugen, B. 2005. A unified formulation of small-strain corotational finite elements: I. theory. ComputerMethods in Applied Mechanics and Engineering, Vol.194(21-24):2285 – 2335.

Felippa, C. A. 2003. A study of optimal membrane triangles with drilling freedoms. Computer Methods in AppliedMechanics and Engineering, Vol.192(16-18):2125 – 2168.

Kundu, C. and Sinha, P. 2006. Nonlinear transient analysis of laminated composite shells. Journal of Reinforced Plasticsand Composites, Vol.25(11):1129 – 1147.

Nour-Omid, B. and Rankin, C. C. 1991. Finite rotation analysis and consistent linearization using projectors. ComputerMethods in Applied Mechanics and Engineering, Vol.93(3):353 – 384.

Simo, J. and Tarnow, N. 1994. New energy and momentum conserving algorithm for the non-linear dynamics of shells.International Journal for Numerical Methods in Engineering, Vol.37(15):2527 – 2549.

Zhang, Y. and Kim, K. 2005. A simple displacement-based 3-node triangular element for linear and geometrically non-linear analysis of laminated composite plates. Computer Methods in Applied Mechanics and Engineering, Vol.194(45-47):4607 – 4632.

7. RESPONSIBILITY NOTICE

The authors are the only responsible for the printed material included in this paper.

corotational nonlinear dynamic analysis of … · a triangular ﬂat shell element implemented...

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