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Nonlinear axisymmetric dynamic buckling of laminated angle-ply composite spherical caps
Composite Structures 59 (2003) 89–97 www.elsevier.com/locate/compstruct
Nonlinear axisymmetric dynamic buckling of laminated angle-ply composite spherical caps M. Ganapathi *, S.S. Gupta, B.P. Patel Institute of Armament Technology, Girinagar, Pune 411 025, India
Abstract Here, the nonlinear axisymmetric dynamic behavior of clamped laminated angle-ply composite spherical caps under suddenly applied loads of infinite duration is studied. The formulation is based on first-order shear deformation theory and it includes the inplane and rotary inertia effects. Geometric nonlinearity is introduced in the formulation using von KarmanÕs strain–displacement relations. The governing equations obtained are solved employing the NewmarkÕs integration technique coupled with a modified Newton–Raphson iteration scheme. The load corresponding to a sudden jump in the maximum average displacement in the time history of the shell structure is taken as the dynamic buckling pressure. The performance of the present model is validated against the available analytical/three-dimensional finite element solutions. The effect of shell geometrical parameter and ply angle on the axisymmetric dynamic buckling load of shallow spherical shells is brought out. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Dynamic snap through; Axisymmetric; Angle-ply; Spherical caps; Nonlinear response
1. Introduction Nonlinear dynamic analysis of spherical shells have been studied extensively because of their wide application in aerospace and other structures. Although most of the available works are related to axisymmetric dynamic behavior of isotropic spherical shells, few studies are made confined to single-layered orthotropic spherical shells subjected to a step pressure load of infinite duration. The increased use of composite materials in the construction of structural components has further led to a recent surge of interest among researchers, in particular, in the failure of composite shells under dynamic situation. The analysis of isotropic shallow spherical shells has been carried out by Budiansky and Roth [1], Simitses [2], Haung [3], Stephens and Fulton [4], Ball and Burt [5], and Stricklin and Martinez [6]. Budiansky and Roth [1] have employed the Galerkin method whereas Simitses [2], adopted Ritz–Galerkin procedure. A finite difference scheme has been introduced in the method of * Corresponding author. Tel.: +91-20-4389550; fax: +91-204389509. E-mail addresses: mganapathi@rediffmail.com, [email protected] (M. Ganapathi).
solution by Haung [3], Stephens and Fultton [4], and Ball and Burt [5] while Stricklin and Martinez [6] utilized more efficient finite element procedure. The effect of geometric imperfections on the dynamic buckling load, by employing buckling criterion based on the displacement response, is investigated by Kao and Perrone [7], and Kao [8] based on finite difference method whereas recently Saigal et al. [9] and Yang and Liaw [10] analyzed using finite element technique. Lock et al. [11] have carried out an experimental study on buckling of shells. The limited studies available on axisymmetric dynamic buckling of single-layer orthotropic shallow spherical shells are based on classical lamination theory [12–15], except the work of Ganapathi and Varadan [16]. Alwar and Shekhar Reddy [12], and Dumir et al. [14] have examined the problem using the method of orthogonal collocation whereas Chao and Lin [15] have obtained the critical loads based on the finite difference scheme, including the influence of geometric imperfection. Due to the low transverse shear moduli of advanced composite materials relative to their in-plane moduli, transverse shear deformation could be significant even in thin composite structures compared to homogeneous isotropic materials. Hence, it is more appropriate to analyze the dynamics of composite structures including shear deformation and rotary inertia.
0263-8223/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 2 ) 0 0 2 2 7 - 1
90
M. Ganapathi et al. / Composite Structures 59 (2003) 89–97
However, to the authorsÕ knowledge, work on the axisymmetric dynamic buckling behavior of laminated angle-ply composite spherical shells under externally applied pressure is not commonly available yet in the literature, and such study is immensely useful to the designers while optimizing the designs of composite shell structures under dynamic loading. Here, a three-noded shear flexible axisymmetric curved shell element developed based on the field-consistency principle [17] is employed to analyze the axisymmetric dynamic buckling of laminated composite spherical caps under externally applied pressure load. Geometric nonlinearity is assumed in the present study using von KarmanÕs strain–displacement relations. In addition, the formulation includes in-plane and rotary inertia effects. The nonlinear governing equations derived are solved employing NewmarkÕs numerical integration method in conjunction with the modified Newton–Raphson iteration scheme. The dynamic buckling pressure is taken as the pressure corresponding to a sudden jump in the maximum average displacement in the time history of the shell structure [1,18]. The present formulation is validated considering isotropic and crossply cases for which solutions are available. Numerical results are presented for laminated composite spherical caps with different values for geometrical parameter and ply angle.
9 ou0 w > þ > > > os R > = u0 sin / w cos / L fep g ¼ þ > > r r > > > > > > > > v sin / ov 0 0 ; : þ r os 9 8 obs ou0 > > þ > > > > > > os Ros > > = < bs sin / u0 sin / feb g ¼ þ > > r Rr > > > > > > > > ov cos / ob b sin / h h ; : 0 þ r os r os 8
2 9 9 8 > 1 ow > ow > > > > > > = = < < bs þ 2 os os NL fes g ¼ ; fep g ¼ > > > 0 > ; > : b v0 cos / > > > ; : h r 0 8 > > > > > <
ð3Þ where r, R and / are the radius of the parallel circle, radius of the meridional circle and angle made by the tangent at any point in the middle-surface of the shell with the axis of revolution. If fN g represents the stress resultants (Nss , Nhh , Nsh ) and fMg the moment resultants (Mss , Mhh , Msh ), one can relate these to membrane strains fep gð¼ feLp g þ feNL p gÞ and bending strains feb g through the constitutive relations as fN g ¼ ½A fep g þ ½B feb g and fMg ¼ ½B fep g þ ½D feb g ð4Þ
2. Formulation An axisymmetric laminated composite shell of revolution is considered with the coordinates s, h and z along the meridional, circumferential and radial/thickness directions, respectively. The displacements u, v, w at a point (s; h; z) from the median surface are expressed as functions of mid-plane displacements u0 , v0 and w, and independent rotations bs and bh of the meridional and hoop sections, respectively, as
where ½A , ½D and ½B are extensional, bending and bending-extensional coupling stiffness coefficients matrices of the composite laminate. Similarly, the transverse shear force {Q} representing the quantities (Qsz , Qhz ) are related to the transverse shear strains fes g through the constitutive relation as ð5Þ
fQg ¼ ½E fes g
where t is the time. Using von KarmanÕs assumption for moderately large deformation, GreenÕs strains can be written in terms of middle-surface deformations as, NL L zeb ep ep feg ¼ þ þ ð2Þ es 0 0
where ½E is the transverse shear stiffness coefficients matrix of the laminate. For a composite laminate of thickness h, consisting of N layers with stacking angles /i (i ¼ 1; . . . ; N ) and layers thickness hi (i ¼ 1; . . . ; N ), the necessary expressions to compute the stiffness coefficients, available in the literature [20] are used here. The potential energy functional U(d) is given by, Z h
T T 1 U ðdÞ ¼ ep ½A ep þ ep ½B feb g 2 A
þ feb gT ½B ep þ feb gT ½D feb g Z i þ fes gT ½E fes g dA qwdA ð6Þ
where, the membrane strains feLp g, bending strains feb g , shear strains fes g and nonlinear in-plane strains feNL p g in the Eq. (3) are written as [19]
where d is the vector of degrees of freedom associated to the displacement field in a finite element discretisation and q is the applied external pressure load.
uðs; z; tÞ ¼ u0 ðs; tÞ þ zbs ðs; tÞ vðs; z; tÞ ¼ v0 ðs; tÞ þ zbh ðs; tÞ wðs; z; tÞ ¼ wðs; tÞ
ð1Þ
A
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The kinetic energy of the shell is given by Z h i 1 ð7Þ T ðdÞ ¼ p u_ 20 þ v_ 20 þ w_ 2 þ I b_ 2s þ b_ 2h dA 2 A R h=2 R h=2 where p ¼ h=2 qdz and I ¼ h=2 qz2 dz and q is the mass density. The dot over the variable denotes derivative with respect to time. Following the procedure given in the work of Rajasekaran et al. [21], the potential energy functional U given in Eq. (6) is rewritten as T
U ðdÞ ¼ fdg ð1=2Þ½K þ ½ð1=6Þ½N 1ðdÞ þ ð1=12Þ T
½N 2ðdÞ fdg þ fdg fF g
ð8Þ
where ½K is the linear stiffness matrix, ½N 1 and ½N 2 are nonlinear stiffness matrices linearly and quadratically dependent on the field variables, respectively and {F} is the load vector. Substituting Eqs. (7) and (8) in LagrangeÕs equation of motion, the governing equation for the shell are obtained as 1 1 € ½M fdg þ ½K þ ½ N 1ðdÞ þ ½ N 2ðdÞ fdg ¼ fF g ð9Þ 2 3 where ½M is the mass matrix. Eq. (9) is solved using the implicit method, as mentioned by Subbaraj and Dokainish [22]. In this method, equilibrium conditions are considered at the same time step for which solution is sought. If the solution is known at time t and one wishes to obtain the displacements, etc., at time t þ Dt, then the equilibrium equations considered at time t þ Dt are given as ½M fd€gtþDt þ ½½N ðdÞ fdg tþDt ¼ fF gtþDt
ð10Þ
where fd€gtþDt and fdgtþDt are the vectors of the nodal accelerations and displacements at time t þ Dt, respectively. ½½N ðdÞ fdg tþDt is the internal force vector at time t þ Dt and is given as ½½N ðdÞ fdg tþDt ¼ ð½½K þ ð1=2Þ½N 1ðdÞ þ ð1=3Þ ½N 2ðdÞ fdgÞtþDt
The nonlinear equations obtained by the above procedure are solved by the NewmarkÕs numerical integration method. Equilibrium is achieved for each time step through modified Newton–Raphson iteration until the convergence criteria suggested by Bergan and Clough [23] are satisfied within the specific tolerance limit of less than one percent. 3. Dynamic buckling criteria Criteria for the static buckling of axisymmetric shallow spherical shells are well defined, whereas it is not so for the dynamic case. It requires the evaluation of the transient response of the shell for different load amplitudes. However, the dynamic buckling criterion suggested by Budiansky and Roth [1] is generally accepted because the results obtained by various investigators by different numerical techniques using the criterion are in reasonable agreement with each other. This criterion is based on the plots of the peak nondimensional average displacement in the time history of the structure with respect to the amplitude of the pressure load (e.g. inserted figure in Fig. 1). The average displacement D is defined as Ra rwdr D ¼ R0a rZdr 0 The numerator is the volume generated by the shell deformation and the denominator corresponds to the original volume under the spherical cap. Z is the height of a point on the middle-surface of the shell measured from the base. There is a load range where a sharp jump in peak average displacement occurs for a small change in load magnitude. The inflection point of the loaddeflection curve is considered as the dynamic buckling load.
ð11Þ
In developing equations for the implicit integration, the internal forces ½N ðdÞ fdg at the time t þ Dt is written in terms of the internal forces at time t, by using the tangent stiffness approach, as ½½N ðdÞ fdg tþDt ¼ ½½N ðdÞ fdg t þ ½KT ðdÞ t fDdg
91
ð12Þ
where ½KT ðdÞ t ¼ ½½K þ ½N 1 þ ½N 2 is the tangent stiffness matrix and fDdg ¼ fdgtþDt fdgt . Substituting Eq. (12) into Eq. (10), one obtains the governing equation at t þ Dt as ½M fd€gtþDt þ ½KT ðdÞ t fDdg ¼ fF gtþDt ½½N ðdÞ fdg t ð13Þ To improve the solution accuracy and to avoid numerical instabilities, it is necessary to employ iteration within each time, thus maintaining the equilibrium.
4. Element description The laminated axisymmetric three-noded curved shell element used here is a C 0 continuous shear flexible one and has five nodal degrees of freedoms. If the interpolation functions for three-noded element are used directly to interpolate the five field variables u0 , v0 , w, bs and bh in deriving the transverse shear and membrane strains, the element will lock and show oscillations in the shear and membrane stresses. Field consistency requires that the membrane and transverse shear strains must be interpolated in a consistent manner. Thus, bs term in the expression for fes g given in Eq. (4) has to be consistent with field function ow=os as shown in the works of Prathap and Ramesh Babu [17]. Similarly the w and (u0 , v0 ) terms in the expression of feLp g (first and third strain components) have to be
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Fig. 1. Average displacement versus nondimensional time for isotropic spherical cap (k ¼ 7).
consistent with the field functions ou0 =os and ov0 =os, respectively. This is achieved by using the field redistributed substitute shape functions to interpolate those specific terms that must be consistent as described by Prathap and Ramesh Babu [17]. The element derived in this fashion behaves very well for both thick and thin situations, and permits the greater flexibility in the choice of integration order for the energy terms. It has good convergence and has no spurious rigid modes.
dynamic analysis available in the literature, the critical time step of a conditionally stable finite difference scheme [24,25] is introduced as a guide and a convergence study was conducted to select a time step which yields a stable and accurate solution. The material properties assumed in the present analysis are
5. Results and discussion
where E, G and m are YoungÕs modulus, shear modulus and PoissonÕs ratio. Subscripts L and T are the longitudinal and transverse directions respectively with respect to the fibres. All the layers are of equal thickness. The ply angles are measured with respect to the meridional axis. Results of nondimensional dynamic pressure, Pcr , are presented for isotropic, orthotropic, cross- and angle-ply laminates for different values of the geometrical parameter k. Pcr and k are defined as
2 4 1 h qa 2 1=2 Pcr ¼ 3 1 m 8 H Eh4
1=2 1=4 H k ¼ 2 3 1 m2 h
The present study is concerned with the axisymmetric dynamic buckling behavior of laminated angle-ply composite spherical caps. Since the finite element used here is based on the field consistency approach, an exact integration is employed to evaluate all the strain energy terms. The shear correction factor, which is required in a first-order theory to account for the variation of transverse shear stresses, is taken as 5/6. For the present analysis, based on progressive mesh refinement, 15 element idealization is found to be adequate in modeling the spherical caps. The initial conditions for obtaining the nonlinear dynamic response are assumed as zero values for the displacements and velocities. From the dynamic response curves, the load amplitudes and the corresponding maximum average displacements are obtained for applying the buckling criteria. The constants a and b (controlling parameters for stability and accuracy of the solution) in the NewmarkÕs integration are taken as 0.5 and 0.25, which correspond to the unconditionally stable scheme in the linear analysis. Since there is no estimate of the time step for the nonlinear
isotropic case: E ¼ 210 GPa, m ¼ 0:3, q ¼ 7800 kg/m3 orthotropic case: EL =ET ¼ 25:0, GLT =ET ¼ 0:5, GTT = ET ¼ 0:2, mLT ¼ 0:25, ET ¼ 1 GPa, q ¼ 1500 kg/m3
Here, H, a are the central shell rise and base radius, respectively. E, m correspond to isotropic case. For the chosen shell parameter and lamination scheme, the dynamic buckling study is conducted for step loading of infinite duration. The length of response calculation 1=2 time sð¼ ðEh2 =12ð1 m2 Þqa4 Þ tÞ in the present study is varied between 1 and 2 with the criterion that in the neighborhood of the buckling, s is large enough to allow
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deflection-time curves to develop fully. The time step selected, based on the convergence study, is Ds ¼ 0:002. The value selected for s and Ds is of the same order as that of Refs. [5,8,15]. Before proceeding for the dynamic buckling characteristics of laminated cases, the formulation developed herein is validated against the dynamic buckling of isotropic spherical shells subjected to uniform external pressure of infinite duration. The nonlinear axisymmetric dynamic response history with time for the geometric shell parameter k ¼ 7 is shown in Fig. 1 for different externally applied pressure. Further, using such plots, the variation of maximum average displacement with applied load obtained for k ¼ 7, is also highlighted in Fig. 1 for predicting the critical load. The critical dynamic pressures calculated for various geometrical parameter values are presented in Fig. 2 along with those of available analytical/numerical results [3,4,12,16] and they are, in general, found to be in good agreement. The investigation for dynamic buckling of two- and eight-layered cross-ply composite spherical caps is carried out for different geometrical parameters. The dynamic responses evaluated due to different load levels for two-layered cap (k ¼ 7, 0°/90°) are shown in Fig. 3. From such plot, the relationship between the maximum average displacement and applied pressure load obtained is also depicted in Fig. 3. It is seen that there is a sudden jump in the value of the average displacement when the external pressure reaches the value
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Pcr ¼ 0:02946 for the two-layered cross-ply spherical cap. The dynamic buckling values obtained in this manner are brought out in Fig. 4. To check the efficacy of the present formulation, the results obtained here for two-layered cross-ply case are verified against the solutions evaluated using 3D solid elements (ANSYS 5.6 Software) and they are, in general, found to be in good agreement as seen from Fig. 4. It is revealed here that the predicted axisymmetric dynamic critical loads for two-layered cross-ply spherical caps are less for very shallow shells compared to those of eight-layered crossply case and it depends, for the deep spherical caps, on the coupling due to bending and stretching, and the geometrical shell parameter values. However, it can be further inferred that the variation of the buckling load with geometrical parameter is less for two-layered case in comparison with those of eight-layered one. Moreover, it can be seen that the buckling behavior/trend of laminated shells with multi-layers, is qualitatively similar to that of isotropic case. Next, the influence of ply angle with various geometrical shell parameter, k on the dynamic critical pressure of two- and eight-layered composite spherical cases is examined. The nonlinear transient response history with time, evaluated for eight-layered (15°/ 15°)4 shell with k ¼ 8, along with the dynamic snapthrough behaviors is highlighted in Fig. 5. The buckling characteristics of angle-ply case are qualitatively similar to those of isotropic and cross-ply spherical caps. The
Fig. 2. Comparision of axisymmetric nondimensional critical load for isotropic spherical cap.
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Fig. 3. Average displacement versus nondimensional time for two-layered cross-ply (0°/90°) spherical cap (k ¼ 7).
Fig. 4. Nondimensional axisymmetric critical load versus shell geometry parameter for two-layered (0°/90°) and eight-layered (0°/90°)4 cross-ply spherical cap.
critical dynamic buckling loads calculated for various ply angle cases, based on such plots (Fig. 5), are depicted in Fig. 6 for eight-layered case. It is observed from this figure that, for higher ply angle considered here, the critical load is low and the rate of change in the critical load value with geometrical parameter k is very less. It
can be further inferred that, for lower angle-ply spherical caps, change in value of the buckling load with geometrical shell parameter is more compared to that of higher ply angle cases. The significance of bendingstretching coupling introduced due to the lay-up is also brought out in Fig. 7, considering two-layered angle-ply
M. Ganapathi et al. / Composite Structures 59 (2003) 89–97
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Fig. 5. Average displacement versus nondimensional time for two-layered angle-ply (15°/)15°) spherical cap (k ¼ 8).
Fig. 6. Nondimensional axisymmetric critical load versus shell geometry parameter for eight-layered angle-ply ðh= hÞ4 spherical cap.
spherical shells. It can be viewed from Figs. 6 and 7 that for two-layered case, the critical loads are low and the variation of buckling characteristics is, in general, very mild compared to those of eight-layered laminates. It is also observed from these figures that, for the shallow
region, the maximum dynamic critical pressure predicted among the angle-ply cases considered here, corresponds to shell with ply angle, h ¼ 30° and, for the moderately deep shell region, the lay-up with 15° yields the maximum values.
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Fig. 7. Nondimensional axisymmetric critical load versus shell geometry parameter for two-layered angle-ply (h= h) spherical cap.
6. Conclusions Axisymmetric dynamic buckling analysis of clamped spherical caps, made up of laminated composite materials, subjected to externally applied pressure, has been investigated through transient dynamic response. A three-noded axisymmetric curved shell element based on field consistency principle has been employed for this purpose. Numerical results obtained here for an isotropic case are found to be in fairly good agreement with the previous findings and also with the solutions obtained using three-dimensional finite element model. From the detailed study, it is observed that the lowest dynamic critical dynamic buckling pressure of a spherical cap significantly depends on its geometrical parameter, ply angle and lay-ups. It is hoped that this study will be useful for the designers/engineers while designing/optimizing the composite spherical shell structures under externally applied dynamic loads.
References [1] Budiansky B, Roth RS. Axisymmetric dynamic buckling of clamped shallow spherical shells. NASA TND-510 1962:597– 609. [2] Simitses GJ. Axisymmetric dynamic snap-through buckling of shallow spherical caps. Am Inst Aeronaut Austronaut J 1967;5:1019–21. [3] Haung NC. Axisymmetric dynamic snap-through of elastic clamped shallow shell. Am Inst Aeronaut Austronaut J 1969; 7:215–20.
[4] Stephens WB, Fulton RE. Axisymmetric static and dynamic buckling of spherical caps due to centrally distributed pressure. Am Inst Aeronaut Austronaut J 1969;7:2120–6. [5] Ball RE, Burt JA. Dynamic buckling of shallow spherical shells. ASME J Appl Mech 1973;41:411–6. [6] Stricklin JA, Martinez JE. Dynamic buckling of clamped spherical cap under step pressure loadings. Am Inst Aeronaut Austronaut J 1969;7:1212–3. [7] Kao R, Perrone N. Dynamic buckling of axisymmetic spherical caps with initial imperfection. Comput Struct 1978;9:463–73. [8] Kao R. Nonlinear dynamic buckling of spherical caps with initial imperfection. Comput Struct 1980;12:49–63. [9] Saigal S, Yang TY, Kapania RK. Dynamic buckling of imperfection sensistive shell structures. J Aircr 1987;24:718–24. [10] Yang TY, Liaw DG. Elastic–plastic dynamic buckling of thin shell finite elements with asymmetric imperfections. Am Inst Aeronaut Austronaut J 1988;25:479–85. [11] Lock MH, Okubo S, Whitter JS. Experiments on the snapping of a shallow dome under a step pressure load. Am Inst Aeronaut Austronaut J 1968;6:1320–6. [12] Alwar RS, Sekhar Reddy B. Dynamic buckling of isotropic and orthotropic shallow spherical caps with circular hole. Int J Mech Sci 1979;21:681–8. [13] Ganapathi M, Varadan TK. Dynamic buckling of orthotropic shallow spherical shells. Comput Struct 1982;15:517–20. [14] Dumir PC, Gandhi ML, Nath Y. Axisymmetric static and dynamic buckling of orthotropic shallow spherical caps with flexible supports. Acta Mechanica 1984;52:93–106. [15] Chao CC, Lin IS. Static and dynamic snap-through of orthotropic spherical caps. Compos Struct 1990;14:281–301. [16] Ganapathi M, Varadan TK. Dynamic buckling of laminated anisotropic spherical caps. ASME J Appl Mech 1995;62:13–9. [17] Prathap G, Ramesh Babu C. A field-consistent three-noded quadratic curved axisymmetric shell element. Int J Numer Methods Eng 1986;23:711–23. [18] Simitses GJ. Dynamic Stability of Suddenly Loaded Structures. New York: Springer-Verlag; 1989.
M. Ganapathi et al. / Composite Structures 59 (2003) 89–97 [19] Kraus H. Thin Elastic Shells. New York: John Wiley; 1967. [20] Jones RM. Mechanics of Composite Materials. New York: McGraw-Hill; 1975. [21] Rajasekaran S, Murray DW. Incremental finite element matrices. ASCE J Struct Div 1973;99:2423–38. [22] Subbaraj K, Dokainish MA. A survey of direct time-integration methods in computational structural dynamics II: Implicit methods. Comput Struct 1989;32:1387–401.
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[23] Bergan PG, Clough RW. Convergence criteria for iterative process. Am Inst Aeronaut Austronaut J 1972;10:1107–8. [24] Leech JN. Stability of finite difference equations for the transient response of a flat plate. Am Inst Aeronaut Austronaut J 1965;3:1772–3. [25] Tsui TY, Tong P. Stability of transient solution of moderately thick plate by finite difference method. Am Inst Aeronaut Austronaut J 1971;9:2062–3.
Nonlinear axisymmetric dynamic buckling of laminated angle-ply composite spherical caps M. Ganapathi *, S.S. Gupta, B.P. Patel Institute of Armament Technology, Girinagar, Pune 411 025, India
Abstract Here, the nonlinear axisymmetric dynamic behavior of clamped laminated angle-ply composite spherical caps under suddenly applied loads of infinite duration is studied. The formulation is based on first-order shear deformation theory and it includes the inplane and rotary inertia effects. Geometric nonlinearity is introduced in the formulation using von KarmanÕs strain–displacement relations. The governing equations obtained are solved employing the NewmarkÕs integration technique coupled with a modified Newton–Raphson iteration scheme. The load corresponding to a sudden jump in the maximum average displacement in the time history of the shell structure is taken as the dynamic buckling pressure. The performance of the present model is validated against the available analytical/three-dimensional finite element solutions. The effect of shell geometrical parameter and ply angle on the axisymmetric dynamic buckling load of shallow spherical shells is brought out. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Dynamic snap through; Axisymmetric; Angle-ply; Spherical caps; Nonlinear response
1. Introduction Nonlinear dynamic analysis of spherical shells have been studied extensively because of their wide application in aerospace and other structures. Although most of the available works are related to axisymmetric dynamic behavior of isotropic spherical shells, few studies are made confined to single-layered orthotropic spherical shells subjected to a step pressure load of infinite duration. The increased use of composite materials in the construction of structural components has further led to a recent surge of interest among researchers, in particular, in the failure of composite shells under dynamic situation. The analysis of isotropic shallow spherical shells has been carried out by Budiansky and Roth [1], Simitses [2], Haung [3], Stephens and Fulton [4], Ball and Burt [5], and Stricklin and Martinez [6]. Budiansky and Roth [1] have employed the Galerkin method whereas Simitses [2], adopted Ritz–Galerkin procedure. A finite difference scheme has been introduced in the method of * Corresponding author. Tel.: +91-20-4389550; fax: +91-204389509. E-mail addresses: mganapathi@rediffmail.com, [email protected] (M. Ganapathi).
solution by Haung [3], Stephens and Fultton [4], and Ball and Burt [5] while Stricklin and Martinez [6] utilized more efficient finite element procedure. The effect of geometric imperfections on the dynamic buckling load, by employing buckling criterion based on the displacement response, is investigated by Kao and Perrone [7], and Kao [8] based on finite difference method whereas recently Saigal et al. [9] and Yang and Liaw [10] analyzed using finite element technique. Lock et al. [11] have carried out an experimental study on buckling of shells. The limited studies available on axisymmetric dynamic buckling of single-layer orthotropic shallow spherical shells are based on classical lamination theory [12–15], except the work of Ganapathi and Varadan [16]. Alwar and Shekhar Reddy [12], and Dumir et al. [14] have examined the problem using the method of orthogonal collocation whereas Chao and Lin [15] have obtained the critical loads based on the finite difference scheme, including the influence of geometric imperfection. Due to the low transverse shear moduli of advanced composite materials relative to their in-plane moduli, transverse shear deformation could be significant even in thin composite structures compared to homogeneous isotropic materials. Hence, it is more appropriate to analyze the dynamics of composite structures including shear deformation and rotary inertia.
0263-8223/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 2 ) 0 0 2 2 7 - 1
90
M. Ganapathi et al. / Composite Structures 59 (2003) 89–97
However, to the authorsÕ knowledge, work on the axisymmetric dynamic buckling behavior of laminated angle-ply composite spherical shells under externally applied pressure is not commonly available yet in the literature, and such study is immensely useful to the designers while optimizing the designs of composite shell structures under dynamic loading. Here, a three-noded shear flexible axisymmetric curved shell element developed based on the field-consistency principle [17] is employed to analyze the axisymmetric dynamic buckling of laminated composite spherical caps under externally applied pressure load. Geometric nonlinearity is assumed in the present study using von KarmanÕs strain–displacement relations. In addition, the formulation includes in-plane and rotary inertia effects. The nonlinear governing equations derived are solved employing NewmarkÕs numerical integration method in conjunction with the modified Newton–Raphson iteration scheme. The dynamic buckling pressure is taken as the pressure corresponding to a sudden jump in the maximum average displacement in the time history of the shell structure [1,18]. The present formulation is validated considering isotropic and crossply cases for which solutions are available. Numerical results are presented for laminated composite spherical caps with different values for geometrical parameter and ply angle.
9 ou0 w > þ > > > os R > = u0 sin / w cos / L fep g ¼ þ > > r r > > > > > > > > v sin / ov 0 0 ; : þ r os 9 8 obs ou0 > > þ > > > > > > os Ros > > = < bs sin / u0 sin / feb g ¼ þ > > r Rr > > > > > > > > ov cos / ob b sin / h h ; : 0 þ r os r os 8
2 9 9 8 > 1 ow > ow > > > > > > = = < < bs þ 2 os os NL fes g ¼ ; fep g ¼ > > > 0 > ; > : b v0 cos / > > > ; : h r 0 8 > > > > > <
ð3Þ where r, R and / are the radius of the parallel circle, radius of the meridional circle and angle made by the tangent at any point in the middle-surface of the shell with the axis of revolution. If fN g represents the stress resultants (Nss , Nhh , Nsh ) and fMg the moment resultants (Mss , Mhh , Msh ), one can relate these to membrane strains fep gð¼ feLp g þ feNL p gÞ and bending strains feb g through the constitutive relations as fN g ¼ ½A fep g þ ½B feb g and fMg ¼ ½B fep g þ ½D feb g ð4Þ
2. Formulation An axisymmetric laminated composite shell of revolution is considered with the coordinates s, h and z along the meridional, circumferential and radial/thickness directions, respectively. The displacements u, v, w at a point (s; h; z) from the median surface are expressed as functions of mid-plane displacements u0 , v0 and w, and independent rotations bs and bh of the meridional and hoop sections, respectively, as
where ½A , ½D and ½B are extensional, bending and bending-extensional coupling stiffness coefficients matrices of the composite laminate. Similarly, the transverse shear force {Q} representing the quantities (Qsz , Qhz ) are related to the transverse shear strains fes g through the constitutive relation as ð5Þ
fQg ¼ ½E fes g
where t is the time. Using von KarmanÕs assumption for moderately large deformation, GreenÕs strains can be written in terms of middle-surface deformations as, NL L zeb ep ep feg ¼ þ þ ð2Þ es 0 0
where ½E is the transverse shear stiffness coefficients matrix of the laminate. For a composite laminate of thickness h, consisting of N layers with stacking angles /i (i ¼ 1; . . . ; N ) and layers thickness hi (i ¼ 1; . . . ; N ), the necessary expressions to compute the stiffness coefficients, available in the literature [20] are used here. The potential energy functional U(d) is given by, Z h
T T 1 U ðdÞ ¼ ep ½A ep þ ep ½B feb g 2 A
þ feb gT ½B ep þ feb gT ½D feb g Z i þ fes gT ½E fes g dA qwdA ð6Þ
where, the membrane strains feLp g, bending strains feb g , shear strains fes g and nonlinear in-plane strains feNL p g in the Eq. (3) are written as [19]
where d is the vector of degrees of freedom associated to the displacement field in a finite element discretisation and q is the applied external pressure load.
uðs; z; tÞ ¼ u0 ðs; tÞ þ zbs ðs; tÞ vðs; z; tÞ ¼ v0 ðs; tÞ þ zbh ðs; tÞ wðs; z; tÞ ¼ wðs; tÞ
ð1Þ
A
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The kinetic energy of the shell is given by Z h i 1 ð7Þ T ðdÞ ¼ p u_ 20 þ v_ 20 þ w_ 2 þ I b_ 2s þ b_ 2h dA 2 A R h=2 R h=2 where p ¼ h=2 qdz and I ¼ h=2 qz2 dz and q is the mass density. The dot over the variable denotes derivative with respect to time. Following the procedure given in the work of Rajasekaran et al. [21], the potential energy functional U given in Eq. (6) is rewritten as T
U ðdÞ ¼ fdg ð1=2Þ½K þ ½ð1=6Þ½N 1ðdÞ þ ð1=12Þ T
½N 2ðdÞ fdg þ fdg fF g
ð8Þ
where ½K is the linear stiffness matrix, ½N 1 and ½N 2 are nonlinear stiffness matrices linearly and quadratically dependent on the field variables, respectively and {F} is the load vector. Substituting Eqs. (7) and (8) in LagrangeÕs equation of motion, the governing equation for the shell are obtained as 1 1 € ½M fdg þ ½K þ ½ N 1ðdÞ þ ½ N 2ðdÞ fdg ¼ fF g ð9Þ 2 3 where ½M is the mass matrix. Eq. (9) is solved using the implicit method, as mentioned by Subbaraj and Dokainish [22]. In this method, equilibrium conditions are considered at the same time step for which solution is sought. If the solution is known at time t and one wishes to obtain the displacements, etc., at time t þ Dt, then the equilibrium equations considered at time t þ Dt are given as ½M fd€gtþDt þ ½½N ðdÞ fdg tþDt ¼ fF gtþDt
ð10Þ
where fd€gtþDt and fdgtþDt are the vectors of the nodal accelerations and displacements at time t þ Dt, respectively. ½½N ðdÞ fdg tþDt is the internal force vector at time t þ Dt and is given as ½½N ðdÞ fdg tþDt ¼ ð½½K þ ð1=2Þ½N 1ðdÞ þ ð1=3Þ ½N 2ðdÞ fdgÞtþDt
The nonlinear equations obtained by the above procedure are solved by the NewmarkÕs numerical integration method. Equilibrium is achieved for each time step through modified Newton–Raphson iteration until the convergence criteria suggested by Bergan and Clough [23] are satisfied within the specific tolerance limit of less than one percent. 3. Dynamic buckling criteria Criteria for the static buckling of axisymmetric shallow spherical shells are well defined, whereas it is not so for the dynamic case. It requires the evaluation of the transient response of the shell for different load amplitudes. However, the dynamic buckling criterion suggested by Budiansky and Roth [1] is generally accepted because the results obtained by various investigators by different numerical techniques using the criterion are in reasonable agreement with each other. This criterion is based on the plots of the peak nondimensional average displacement in the time history of the structure with respect to the amplitude of the pressure load (e.g. inserted figure in Fig. 1). The average displacement D is defined as Ra rwdr D ¼ R0a rZdr 0 The numerator is the volume generated by the shell deformation and the denominator corresponds to the original volume under the spherical cap. Z is the height of a point on the middle-surface of the shell measured from the base. There is a load range where a sharp jump in peak average displacement occurs for a small change in load magnitude. The inflection point of the loaddeflection curve is considered as the dynamic buckling load.
ð11Þ
In developing equations for the implicit integration, the internal forces ½N ðdÞ fdg at the time t þ Dt is written in terms of the internal forces at time t, by using the tangent stiffness approach, as ½½N ðdÞ fdg tþDt ¼ ½½N ðdÞ fdg t þ ½KT ðdÞ t fDdg
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ð12Þ
where ½KT ðdÞ t ¼ ½½K þ ½N 1 þ ½N 2 is the tangent stiffness matrix and fDdg ¼ fdgtþDt fdgt . Substituting Eq. (12) into Eq. (10), one obtains the governing equation at t þ Dt as ½M fd€gtþDt þ ½KT ðdÞ t fDdg ¼ fF gtþDt ½½N ðdÞ fdg t ð13Þ To improve the solution accuracy and to avoid numerical instabilities, it is necessary to employ iteration within each time, thus maintaining the equilibrium.
4. Element description The laminated axisymmetric three-noded curved shell element used here is a C 0 continuous shear flexible one and has five nodal degrees of freedoms. If the interpolation functions for three-noded element are used directly to interpolate the five field variables u0 , v0 , w, bs and bh in deriving the transverse shear and membrane strains, the element will lock and show oscillations in the shear and membrane stresses. Field consistency requires that the membrane and transverse shear strains must be interpolated in a consistent manner. Thus, bs term in the expression for fes g given in Eq. (4) has to be consistent with field function ow=os as shown in the works of Prathap and Ramesh Babu [17]. Similarly the w and (u0 , v0 ) terms in the expression of feLp g (first and third strain components) have to be
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Fig. 1. Average displacement versus nondimensional time for isotropic spherical cap (k ¼ 7).
consistent with the field functions ou0 =os and ov0 =os, respectively. This is achieved by using the field redistributed substitute shape functions to interpolate those specific terms that must be consistent as described by Prathap and Ramesh Babu [17]. The element derived in this fashion behaves very well for both thick and thin situations, and permits the greater flexibility in the choice of integration order for the energy terms. It has good convergence and has no spurious rigid modes.
dynamic analysis available in the literature, the critical time step of a conditionally stable finite difference scheme [24,25] is introduced as a guide and a convergence study was conducted to select a time step which yields a stable and accurate solution. The material properties assumed in the present analysis are
5. Results and discussion
where E, G and m are YoungÕs modulus, shear modulus and PoissonÕs ratio. Subscripts L and T are the longitudinal and transverse directions respectively with respect to the fibres. All the layers are of equal thickness. The ply angles are measured with respect to the meridional axis. Results of nondimensional dynamic pressure, Pcr , are presented for isotropic, orthotropic, cross- and angle-ply laminates for different values of the geometrical parameter k. Pcr and k are defined as
2 4 1 h qa 2 1=2 Pcr ¼ 3 1 m 8 H Eh4
1=2 1=4 H k ¼ 2 3 1 m2 h
The present study is concerned with the axisymmetric dynamic buckling behavior of laminated angle-ply composite spherical caps. Since the finite element used here is based on the field consistency approach, an exact integration is employed to evaluate all the strain energy terms. The shear correction factor, which is required in a first-order theory to account for the variation of transverse shear stresses, is taken as 5/6. For the present analysis, based on progressive mesh refinement, 15 element idealization is found to be adequate in modeling the spherical caps. The initial conditions for obtaining the nonlinear dynamic response are assumed as zero values for the displacements and velocities. From the dynamic response curves, the load amplitudes and the corresponding maximum average displacements are obtained for applying the buckling criteria. The constants a and b (controlling parameters for stability and accuracy of the solution) in the NewmarkÕs integration are taken as 0.5 and 0.25, which correspond to the unconditionally stable scheme in the linear analysis. Since there is no estimate of the time step for the nonlinear
isotropic case: E ¼ 210 GPa, m ¼ 0:3, q ¼ 7800 kg/m3 orthotropic case: EL =ET ¼ 25:0, GLT =ET ¼ 0:5, GTT = ET ¼ 0:2, mLT ¼ 0:25, ET ¼ 1 GPa, q ¼ 1500 kg/m3
Here, H, a are the central shell rise and base radius, respectively. E, m correspond to isotropic case. For the chosen shell parameter and lamination scheme, the dynamic buckling study is conducted for step loading of infinite duration. The length of response calculation 1=2 time sð¼ ðEh2 =12ð1 m2 Þqa4 Þ tÞ in the present study is varied between 1 and 2 with the criterion that in the neighborhood of the buckling, s is large enough to allow
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deflection-time curves to develop fully. The time step selected, based on the convergence study, is Ds ¼ 0:002. The value selected for s and Ds is of the same order as that of Refs. [5,8,15]. Before proceeding for the dynamic buckling characteristics of laminated cases, the formulation developed herein is validated against the dynamic buckling of isotropic spherical shells subjected to uniform external pressure of infinite duration. The nonlinear axisymmetric dynamic response history with time for the geometric shell parameter k ¼ 7 is shown in Fig. 1 for different externally applied pressure. Further, using such plots, the variation of maximum average displacement with applied load obtained for k ¼ 7, is also highlighted in Fig. 1 for predicting the critical load. The critical dynamic pressures calculated for various geometrical parameter values are presented in Fig. 2 along with those of available analytical/numerical results [3,4,12,16] and they are, in general, found to be in good agreement. The investigation for dynamic buckling of two- and eight-layered cross-ply composite spherical caps is carried out for different geometrical parameters. The dynamic responses evaluated due to different load levels for two-layered cap (k ¼ 7, 0°/90°) are shown in Fig. 3. From such plot, the relationship between the maximum average displacement and applied pressure load obtained is also depicted in Fig. 3. It is seen that there is a sudden jump in the value of the average displacement when the external pressure reaches the value
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Pcr ¼ 0:02946 for the two-layered cross-ply spherical cap. The dynamic buckling values obtained in this manner are brought out in Fig. 4. To check the efficacy of the present formulation, the results obtained here for two-layered cross-ply case are verified against the solutions evaluated using 3D solid elements (ANSYS 5.6 Software) and they are, in general, found to be in good agreement as seen from Fig. 4. It is revealed here that the predicted axisymmetric dynamic critical loads for two-layered cross-ply spherical caps are less for very shallow shells compared to those of eight-layered crossply case and it depends, for the deep spherical caps, on the coupling due to bending and stretching, and the geometrical shell parameter values. However, it can be further inferred that the variation of the buckling load with geometrical parameter is less for two-layered case in comparison with those of eight-layered one. Moreover, it can be seen that the buckling behavior/trend of laminated shells with multi-layers, is qualitatively similar to that of isotropic case. Next, the influence of ply angle with various geometrical shell parameter, k on the dynamic critical pressure of two- and eight-layered composite spherical cases is examined. The nonlinear transient response history with time, evaluated for eight-layered (15°/ 15°)4 shell with k ¼ 8, along with the dynamic snapthrough behaviors is highlighted in Fig. 5. The buckling characteristics of angle-ply case are qualitatively similar to those of isotropic and cross-ply spherical caps. The
Fig. 2. Comparision of axisymmetric nondimensional critical load for isotropic spherical cap.
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Fig. 3. Average displacement versus nondimensional time for two-layered cross-ply (0°/90°) spherical cap (k ¼ 7).
Fig. 4. Nondimensional axisymmetric critical load versus shell geometry parameter for two-layered (0°/90°) and eight-layered (0°/90°)4 cross-ply spherical cap.
critical dynamic buckling loads calculated for various ply angle cases, based on such plots (Fig. 5), are depicted in Fig. 6 for eight-layered case. It is observed from this figure that, for higher ply angle considered here, the critical load is low and the rate of change in the critical load value with geometrical parameter k is very less. It
can be further inferred that, for lower angle-ply spherical caps, change in value of the buckling load with geometrical shell parameter is more compared to that of higher ply angle cases. The significance of bendingstretching coupling introduced due to the lay-up is also brought out in Fig. 7, considering two-layered angle-ply
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Fig. 5. Average displacement versus nondimensional time for two-layered angle-ply (15°/)15°) spherical cap (k ¼ 8).
Fig. 6. Nondimensional axisymmetric critical load versus shell geometry parameter for eight-layered angle-ply ðh= hÞ4 spherical cap.
spherical shells. It can be viewed from Figs. 6 and 7 that for two-layered case, the critical loads are low and the variation of buckling characteristics is, in general, very mild compared to those of eight-layered laminates. It is also observed from these figures that, for the shallow
region, the maximum dynamic critical pressure predicted among the angle-ply cases considered here, corresponds to shell with ply angle, h ¼ 30° and, for the moderately deep shell region, the lay-up with 15° yields the maximum values.
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Fig. 7. Nondimensional axisymmetric critical load versus shell geometry parameter for two-layered angle-ply (h= h) spherical cap.
6. Conclusions Axisymmetric dynamic buckling analysis of clamped spherical caps, made up of laminated composite materials, subjected to externally applied pressure, has been investigated through transient dynamic response. A three-noded axisymmetric curved shell element based on field consistency principle has been employed for this purpose. Numerical results obtained here for an isotropic case are found to be in fairly good agreement with the previous findings and also with the solutions obtained using three-dimensional finite element model. From the detailed study, it is observed that the lowest dynamic critical dynamic buckling pressure of a spherical cap significantly depends on its geometrical parameter, ply angle and lay-ups. It is hoped that this study will be useful for the designers/engineers while designing/optimizing the composite spherical shell structures under externally applied dynamic loads.
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