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RETRACTED: Buckling behavior of the multiple delaminated composite plates under shear and axial compression
Computational Materials Science 64 (2012) 173–178
Contents lists available at SciVerse ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Buckling behavior of the multiple delaminated composite plates under shear and axial compression Ionel Chirica ⇑, Elena-Felicia Beznea ‘‘Dunarea de Jos’’ University of Galati, 47 Domneasca Str., Galati 800008, Romania
a r t i c l e
i n f o
Article history: Received 31 October 2011 Received in revised form 13 March 2012 Accepted 15 March 2012 Available online 17 April 2012 Keywords: Composite laminated plates Double delamination Shear and axial buckling
a b s t r a c t In the paper, a methodology to evaluate the influence of the double delaminations on the buckling and postbuckling behavior of the composite plates under shear and compression loading, used in ship hull structure, is presented. The parametric calculus was done for various positions of delaminations and seven values of loading ratios. The numerical and experimental tests on the delaminated plates are presented. Composite plates containing a multiple number of delaminations are analyzed with respect to combined boundary shear and axial compression buckling. Finally, the graphical results for one position of the delamination through the thickness and two cases of position of delaminations in the case of double delaminated plate and seven values of loading ratio are presented. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Fiber-reinforced composite materials have been increasingly used over the past few decades in a variety of applications in which a fairly high ratio of stiffness/strength to weight is required. Composite panels are generic structural elements in weight sensitive structural for marine applications. Laminated composite panels, which are anisotropic, are gaining popularity in structural applications such as ship hulls, decks, ship and offshore superstructures. These panels are becoming increasingly used in structural marine applications due to their high specific stiffness and specific strength [1,2]. The use of laminated composites provides flexibility to tailor different properties of the structural elements to achieve the stiffness and strength characteristics. However, these materials are prone to a wide range of defects and damage that can cause significant reductions in stiffness and strength [3,4]. Ship structure plates are subjected to any combination of in plane, out of plane and shear loads during application. Due to the geometry and general load of the ship hull, buckling is one of the most important failure criteria. Delamination reduces the elastic buckling load of the laminated composite structures and leads to global structural failure at loads below the design level. Laminated plates which were subjected to impact loading are known to contain multiple delaminations positioned throughthe-thickness of the laminate. These delaminations can be approximated as circular and often oblong and their orientation matches ⇑ Corresponding author. Tel.: +40 722 383282; fax: +40 236 461353. E-mail address: [email protected] (I. Chirica). 0927-0256/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2012.03.032
the orientation of the adjacent composite layers which lie further away from the point of impact [5]. It is also known, that delaminations cause reduction of the buckling load of laminated structures [6,7] and therefore the load carrying capacity of laminate structures which might be subjected to impact loading, such as ship hull structures, could be substantially reduced as well. Delamination may arise during manufacturing (e.g., incomplete wetting, air entrapment) or during service (e.g., low velocity impact, cargo strikes on the ship deck). They may not be visible or barely visible on the surface, since they are embedded within the composite structures. The presence of in-plane loading may cause buckling of stiffened panels and an accurate knowledge of critical buckling load and mode shapes are essential for reliable and lightweight structural design. In [8] in order to investigate the effects of multiple delaminations’ width on the buckling loads of the simply supported woven steel-reinforced thermoplastic-laminated composite plates, having the strip and lateral delamination regions between each layers, 3-D finite elements models have been established. The critical delamination representing the maximum delamination sizes and tolerate value in a structure without decreasing compressive strength has been obtained from the evaluation of the buckling analyses. The results show that the buckling loads values are increasing since the orientation angles are increasing. In [9–11] the compressive behavior of composite laminates with multiple through the width delaminations is investigated analytically based on the CLPT (classical laminated plate theory) and its formulation developed on the basis of the Rayleigh Ritz approximation technique. Also the three dimensional finite element analysis is performed using ANSYS 5.4 general purpose
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commercial software and the results are compared with those obtained by the analytical model. The optimizing of laminated composite plates taking into account the shear buckling load is the object of the studies performed in [12,13]. In [14–16] the studies are focused on the glass-fiberreinforced polymer (GFRP) girders elements that are sensitive to shear stresses in the case of an in-plane biaxial compression-tension buckling problem, according to the rotated stress field theory. In [17,18] the studies dealing with postbuckling behavior of laminated composite plates under the combination of in-plane shear, compression and lateral loading using FEM analysis are done. In [19] the loss of compressive stiffness of a thin square plate buckled in shear is calculated by means of a finite element program developed for this purpose. Two cases are considered, first a perfectly flat plate, and second a plate with a transversal imperfection having the amplitude equal to 10% of the plate thickness. Buckling behavior of multiple delaminated plates subjected to shear and compression loading has been studied in [20–23] by means of the finite element analysis. The studies show several trends in the effect of number positions on surface and throughthe-thickness position of delaminations upon the buckling response. These findings could be used for evaluation of reliability of laminate structures which might be subjected to impact loading, such as ship hull structures. This paper addresses the effects of the delaminations on the combined shear and axial buckling and postbuckling behavior of rectangular plates made of advanced composite materials. The analysis includes the effects of delaminations positions, loading and boundary conditions, results of numerical and experimental tests. Some overall important findings of these studies are that plates having delaminations can buckle at loads higher than the buckling loads for corresponding perfect plates and can exhibit substantial postbuckling load-carrying capability. The presence of in-plane loading may cause buckling of stiffened panels. An accurate knowledge of critical buckling load and mode shapes are essential for reliable and lightweight structural design.
2. Shear and axial buckling theory of orthotropic plates Limit point buckling is an instability in which the load–displacement curve reaches a maximum and then exhibits negative stiffness and releases strain energy. During limit point buckling there are no sudden changes in the equilibrium path; however, if load is continuously increased then the structure may jump or ‘‘snap’’ to another point on the load–deflection curve. For this reason, this type of instability is often called ‘‘snap-through’’ buckling, because the structure snaps to a new equilibrium position. A limit point is characterized by the load–frequency curve passing through a frequency of zero with a zero slope. The load–deflection curve also has a zero slope at the point of maximum load (limit point). Buckling analysis of a plate may be divided into three parts [24]: – Classical buckling analysis. – Difficult classical effects. – Non-classical phenomena. The classical buckling analysis is a generalization of the Euler buckling for beams. The awkward effects in classical buckling analysis are in connection with vibrations, shear deformations, springs, non-homogeneities and variable thicknesses, nonlinear relations between
stresses and strains. Non-classical buckling analysis involves considerations such as imperfections, non-elastic behavior of the material, dynamic effects of the loading, the fact that the in-plane loading is not in the initial point of the plate. Finally, we have to remark that no plate is initially perfect and if initial deviation (from flatness or symmetry) exists, no clear buckling phenomenon may be identifying. The deviations of the plate from the flatness and symmetry are usually called imperfections (initial transversal imperfections, delaminations) as it will be treated in the following chapters. The following cases (in analytical, numerical and experimental ways) are presented: compressive buckling, shear buckling, mixed compressive and shear buckling. The results (for linear and nonlinear model) are presented as variation of the buckling loads function of maximum transversal displacement (buckling and postbuckling behavior). The state of equilibrium of a plate deformed by forces acting in the plane of the middle surface is unique and the equilibrium is stable if the forces are sufficiently small. If, while maintaining the distribution of forces constant at the edge of the plate, the forces are increased in magnitude, there may arise a time when the basic form of equilibrium ceases to be unique and stable and other forms become possible, which are characterized by the curvatures of the middle surface. For symmetrically laminated cross-ply plates there is no coupling between bending and twisting. The equation of the deflected surface of symmetrically laminated plates for combined shear and uniaxial loading is:
D11
@4w @4w @4w @2w @2w þ 2ðD þ D Þ þ D N 2N ¼0 12 66 22 x xy @x4 @x2 @y2 @y4 @x2 @x@y ð1Þ
where D11, D12, D22, D66 are the orthotropic plate stiffnesses, calculated according to the equation:
Dij ¼
n 1P Q k ðz3 z3k1 Þ 3 k¼1 ij k
ð2Þ
where Q kij is the rigidity coefficient from the Hooke’s law written for the k-th ply. The thickness and position of every k ply can be calculated from the equation:
tk ¼ zk zk1
ð3Þ
and
zk ¼ zk1 þ
tk 2
ð4Þ
Linear buckling of beams, membranes and plates has been studied extensively. A linearized stability analysis is convenient from a mathematical viewpoint but quite restrictive in practical applications. What is needed is a capability for determining the nonlinear load–deflection behavior of a structure. Considerable effort has also been expended on this problem and two approaches have evolved: class-I methods, which are incremental in nature and do not necessarily satisfy equilibrium; and class-II methods, which are self-correcting and tend to stay on the true equilibrium path [5]. Historically, class-I was the first finite element approach to solving geometrically nonlinear problems [6]. In this method the load is applied as a sequence of sufficiently small increments so that the structure can be assumed to respond linearly during each increment. For solving geometrically and material nonlinear problems, the load is applied as a sequence of sufficiently small increments so
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that the structure can be assumed to respond linearly during each increment. For each increment of load, increments of displacements and corresponding increments of stress and strain are computed. These incremental quantities are used to compute various corrective stiffness matrices (variously termed geometric, initial stress, and initial strain matrices) which serve to take into account the deformed geometry of the structure. A subsequent increment of load is applied and the process is continued until the desired number of load increments has been applied. The net effect is to solve a sequence of linear problems wherein the stiffness properties are recomputed based on the current geometry prior to each load increment. The solution procedure takes the following mathematical form
ðK þ KI Þi1 Ddi ¼ DQ
ð5Þ
where K is the linear stiffness matrix; KI is an incremental stiffness matrix based upon displacements at load step i-1; Ddi is the increment of displacement due to the i-th load increment; DQ is the increment of load applied. The plate material is damaged according to a specific criterion. The buckling load determination may use the Tsai-Wu failure criterion in the case if the general buckling does not occur till the first-ply failure occurring. In this case, the buckling load is considered as the in-plane load corresponding to the first-ply failure occurring. The Tsai-Wu failure criterion provides the mathematical relation for strength under combined stresses. The strength of the laminated composite can be based on the strength of individual plies within the laminate. In addition, the failure of plies can be successive as the applied load increases. There may be a first ply failure followed by other ply failures until the last ply fails, denoting the ultimate failure of the laminate. The failure criterion is used to calculate a failure index (FI) from the computed stresses and user-supplied material strengths. The failure indices are computed for all layers in each element of your model. During post processing, it is possible to plot failure indices of the mesh for any layer. The Tsai-Wu failure criterion is commonly used for orthotropic materials with unequal tensile and compressive strengths. The failure index is computed using the following equation [1,6].
FI ¼ F 1 r1 þ F 2 r2 þ F 11 r21 þ F 22 r22 þ F 66 r26 þ 2F 12 r1 r2
ð6Þ
where F1 ¼
1 RT1
1 RC1
;
F 11 ¼
1 RT1 RC1
; F2 ¼
1 RT2
1 RC2
;
F 22 ¼
1 RT2 RC2
;
F 66 ¼
1 R212
: ð7Þ
The coefficient F12, represents the parameter of interaction between r1 and r2. This coefficient can be determined by various ways: by a mechanical biaxial test or by stochastic based methods, using experimental data [25]. As an alternative Tsai and Hahn [2,26] suggested that F12 be estimated from the following relationship
F 12
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0:5 F 11 F 22
until the moment of the first-ply failure occurring, the in-plane load corresponding to this moment is considered as the buckling load. Three types of failure status are determined in each layer of each element: – OK
FI < 1
ð10Þ
– Fail 1 in the case:
jr1 j < RT1 and jr1 j < RC1
ð11Þ
– Fail 2 in the case:
jr1 j P RT1 and jr1 j P RC1
ð12Þ
3. Numerical analysis of shear and axial buckling of composite plates In the study, the numerical analysis was carried out using licensed finite element package software COSMOS/M. There are several ways in which the panel can be modelled for the delamination analysis. For the present study, a 3-D model with 3-node SHELL3L composite element of COSMOS/M is used (Fig. 2). In the area of the delamination the panel is divided into two sub-laminates by a hypothetical plane containing the delamination. The delamination model has been developed by using the surface-to-surface contact option. In COSMOS/M the delaminated regions were modelled by two layers of elements with coincident but separate nodes and section definitions to model offsets from the common reference plane. The direction of the axial loading is considered along the symmetry geometrical axis of the plate. As it is seen in Fig. 1, in this paper four cases of double delaminated plate are studied. The shape of each delamination is considered as circular one having the diameter of 20 mm. For each case, the various positions of the double delamination in the stack are considered. For the same position the delaminations are placed between the same layers, according to Table 2. In Fig. 2 an example of plate mesh is presented. The square plates (320 320 mm) are made of E-glass/polyester concerning 16 biaxial layers having the orthotropic directions and thickness (4.96 mm) according to Table 1. Topological code of the plate is [02/45/902/45/02]s. The material characteristics, determined in experimental tests (using stretching machine and strain gauges) are:
y Case 4 p
q
Case 2 p
ð8Þ
In Eq. (7), the parameters RCi ; RTi are the compressive strength and tensile strength in the material in longitudinal direction (i = 1) and transversal direction (i = 2). The parameter R12 is in-plane shear strength in the material 1–2 plane. The failure index in calculated in each ply of each element. In the ply where failure index is
FI > 1
x
ð9Þ
the first-ply failure occurs, according to the Tsai-Wu criterion. In the next steps, the tensile and compressive properties of this element are reduced by the failure index. If the buckling did not appeared
Case 1
q
Case 3
Fig. 1. Positions of double delamination in the plate.
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Fig. 3. Boundary conditions.
Fig. 2. Plate mesh in the Case 1.
Table 1 Composite stack. Macro-layer No.
Number of layers
Fiber direction (°)
Thick-ness (mm)
1 2 3 4 5 6 7 8 9 10
2 1 2 1 2 2 1 2 1 2
0 45 90 45 0 0 45 90 45 0
0.62 0.31 0.62 0.31 0.62 0.62 0.31 0.62 0.31 0.62
Ex ¼ 38:6 GPa; Ey ¼ 8:27 GPa; Ez ¼ 8:27 GPa; Gxy ¼ 4:14 GPa; Gxz ¼ 4:14 GPa; Gyz ¼ 4:6 GPa;
can be considered, as is presented in Table 1. Taking into consideration there are 10 macro-layers, only positions placed between two neighbors macro-layers, from 1 to 6 respectively are studies. The boundary conditions, shown in Fig. 3 (where the corresponding d.o.f., denoted with u for displacement and r for rotation are considered to be equal to zero) are described on the plate sides. On the outline of the delaminated area the continuity condition for the displacements are imposed. Initially, the both sublaminate plates are considered to be in contact. Between the delaminated layers the contact finite elements (GAP elements) are used. The loading was considered to be a combination between axial compression (p) and shear (q) acting on the plate sides as it is shown in Fig. 1. The combination is determined by the loading ratio k = q/p. For the loading ratio, 6 values were considered: 0 (pure axial compression); 0.2; 0.4; 0.6; 0.8; 1.0 and 1 (pure shearing). In Figs. 4 and 5, the variation of the axial loading p (q for pure shearing) versus the magnitude of the transversal displacement are presented for all loading ratios, for two cases of delaminated plates (cases 1 and 3) and positions of delaminations between macro-layers 1 and 2 (Case a). For each case, according to the changing in the slope of the curves, the range of the buckling load values (determined by graphical method, by drawing a tangent line in the point of suddenly changing the curve slope) is shown.
lxy ¼ 0:3; lyz ¼ 0:42; lxz ¼ 0:3; RTx ¼ 1:062 GPa;RCx ¼ 0:610 GPa; RTy ¼ 0:031 GPa; RCy ¼ 0:118 GPa; Rxy ¼ 0:72 GPa: For the same case of double delaminated plate, the delaminations are considered to be placed in the same plane, between two neighbor macro-layers (MLs), as it is presented in Table 2. A macro-layer is a group of layers having the same characteristics (direction of fibers versus direction of axial compression). Due to the position in the stack of the layers, a number of 10 macro-layers
Table 2 Position of delaminations in all 4 cases of double delaminations. Macro-layer No.
Case a
1 2 3 4 5 6 7 8 9 10
1 ML 9 ML
b 2 ML 8 ML
c
3 ML 7 ML
d
4 ML 6 ML
e
5 ML 5 ML
Fig. 4. Case 1-a: 0.124 MPa < pcr < 0.414 MPa.
I. Chirica, E.-F. Beznea / Computational Materials Science 64 (2012) 173–178
177
Fig. 6. Experimental rig.
Fig. 5. Case 3-a: 0.105 MPa < pcr < 0.527 MPa.
Table 3 Buckling load (MPa) for all plates in the Case a. k
0 0.2 0.4 0.6 0.8 1.0 1
Fail type
Fail Fail Fail Fail Fail Fail Fail
1 1 1 1 1 1 1
Case 1-a
2-a
3-a
4-a
Exp.
0.872 0.872 0.872 0.653 0.653 0.435 0.043
0.872 0.872 0.872 0.653 0.653 0.435 0.043
0.872 0.872 0.872 0.653 0.653 0.435 0.043
0.872 0.872 0.872 0.653 0.653 0.435 0.043
0.79 0.86 0.85 0.68 0.61 0.45 0.04
In Table 3, the values of ultimate strength (buckling load) determined by Tsai-Wu criterion for the Case 1-a are presented. As it is seen, the first failure for all cases occurs for the tension case. It is possible to simultaneously occur other failures for compression stresses. For ultimate strength only the first failure occurred in the layers is interested. This can be so tension and compression stress. For the loading ratio, k, less than 0.5, the buckling load has the same value, being equal to 0.872 MPa. For 0.5 < k < 1, the buckling loads have the same value, but in these cases the value is equal to 0.653 MPa. For k P 1 the buckling load is decreasing rapidly from 0.435 MPa to 0.043 MPa (that is the value of k toward infinity for shear buckling). 4. Experimental analysis In order to analyse the response of each plate in Case 1, the displacement obtained experimental in the point from the middle of the plate versus numerical load–deflection curves were studied. In these curves, it is possible to identify the main changes in the behavior of the plate. Moreover, the changes in the behavior of the plate experimental curves are compared with that obtained by the numerical model (see Table 3).
The experiments for determination of buckling and postbuckling behavior of the double delaminated plates with various positions of delaminations was made with an experimental rig, shown in Fig. 6. The rig comprises a stretching machine, strain gauges equipment for stress determination, displacement transducers (LVDT) for maximum transversal displacement measurement and a bolted joint system, specially built for this type of experiments. The bolted joint system is transforming the axial force into an axial compressive force and a shear force system that are applied on the sides of the plate. For each test, the tensile force of the stretching machine was increased step by step with an increment DF = 2 kN. The significance of failure status from second column in Table 3 is described in Eq. (11). According to the measurements results, in Table 3 (last column) the values of the buckling load are presented only for the plate 1-a. Experimental results obtained from the strain gauges and the displacement transducers can be analyzed and compared with the numerical ones. Twenty strain gauges were used to measure the stresses occurred in the layer from the plate’s surface. The value of the experimental stress from the last column in Table 3 corresponds to the same value of transversal displacement obtained in numerical analysis and experimental tests for the plate 1-a.
5. Discussion and conclusions The numerical results obtained from the tests and further analysis of the experimental and numerical results, allow to reach certain conclusions related to the behavior of double delaminated composite plate under combined axial and shear loads. The phenomenon study requires a stress analysis in order to improve the evaluation of the structural response of the delaminated plate. The plate is subjected to a combined stress state in the presence of a nonlinear material and two delaminations placed in various positions. The numerical model could be used to compare the experimental test results obtained for the 20 plates. Figs. 4 and 5 show, in terms of deflection and stress evolution in the middle point of the panel, these different structural responses, for various loading ratios. The comparative analysis has enabled us to confirm the behavior of multiple delaminated plates under combined axial and shear load. As it is shown in Figs. 4 and 5, the value of the point placed on the middle of the plate (wcr) corresponding to the buckling load can be determined in numerical calculus. In the tested plates, since the load performed by the stretching machine was increased, the transversal displacement of the point
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placed on the middle of the plate (wm) was registered. When the value of wm reached the value corresponding to the buckling load obtained by numerical analysis, wcr, the experiment was stopped. The value of the load corresponding to wcr is considered to be experimental buckling load and it can be named ultimate stress. The results obtained by experimental tests have been also used in order to set up suitable numerical models able to interpret correctly the response of the delaminated composite plates. Acknowledgment The work has been performed in the scope of the FP7 Project Move It! – Project No. 285405. References [1] H. Altenbach, J. Altenbach, W. Kissing (Eds.), Mechanics of Composite Structural Elements, Springer, 2001. [2] D.F. Adams, L.A. Carlsson, R.B. Pipes (Eds.), Experimental Characterization of Advanced Composite materials, Taylor & Francis Group, 2003. [3] T. Sadowski, L. Marsavina, N. Peride, E.M. Craciun, Comp. Mater. Sci. 46 (3) (2009) 687–693. [4] T. Sadowski, L. Marsavina, E.M. Craciun, M. Knec, Comp. Mater. Sci. 52 (1) (2012) 231–235. [5] G.J. Thurley, I.H. Marshall (Eds.), Buckling and Postbuckling of Composite Plates, Chapman & Hall, London, 1995.
[6] S.A. Ambarcumyan, Theory of Anisotropic Plates: Strength, Stability, and Vibrations, Hemispher Publishing, Washington, 1991. [7] C. Kassapoglou, Design and Analysis of Composite Structures: With Applications to Aerospace Structures, John Wiley & Sons Ltd., 2010. [8] M. Zor, J. Reinf. Plast. Compos. 28 (2009) 305–316. [9] H.R. Ovesy, M. Kharazi, J. Eng. Mech. 137 (2) (2011) 87–100. [10] M. Kharazi, H.R. Ovesy, Iran. Aerospace Soc., JAST 5 (1) (2008) 13–22. [11] M. Kharazi, H.R. Ovesy, Thin-Wall. Struct. 46 (7–9) (2008) 939–946. [12] L. Papadopoulos, C. Kassapoglou, Compos. A: Appl. Sci. Manuf. 38 (5) (2007) 1425–1430. [13] J.L. Grenestedt, Struct. Multidiscip. Optim. 3 (2) (1991) 115–120. [14] D.D. Manshadi, P. Anastasios, A.P. Vassilopoulos, T. Keller, J. Compos. Constr. 15 (3) (2011) 431–440. [15] J. Loughlan, Thin-Wall. Struct. 34 (2) (1999) 97–114. [16] F. Stan, A.V. Munteanu, C. Fetecau, Rev Mater Plast 48 (1) (2011) 1–6. [17] S.C. Han, S.Y. Lee, G. Rus, Int. J. Solids Struct. 43 (2006) 5713–5735. [18] U.K. Mallela, A. Upadhyay, Thin-Wall. Struct. 44 (2006) 354–361. [19] A. Rothwell, H. Allahyari, J. Strain. Anal. Eng. Des. 15 (4) (1980) 209–215. [20] F. Cappello, D. Tumino, Compos. Sci. Technol., Exp. Tech. Des. Compos. Mater. 66 (2) (2006) 264–272. [21] H. Huang, G.A. Kardomateas, Int. J. Solids Struct. 35 (13) (1998) 1355–1362. [22] S.F. Hwang, G.H. Liu, Compos. Struct. 53 (2) (2001) 235–243. [23] D.Y. Park, S.Y. Chang, S.S. Yhim, KSCE J. Civ. Eng. 10 (6) (2006) 17–30. [24] E.F. Beznea, I. Chirica, Advances in composite materials – ecodesign and analysis, in: Brahim Attaf (Ed.) INTECH, 2011, pp. 383–408. [25] P. Clouston, L. Frank, J.D. Barrett, J. Mater. Civ. Eng. (1998) 112–116. [26] S.W. Tsai, H.T. Hahn, Introduction to Composite Materials, Technomic, Lancaster, PA, 1980.
Contents lists available at SciVerse ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Buckling behavior of the multiple delaminated composite plates under shear and axial compression Ionel Chirica ⇑, Elena-Felicia Beznea ‘‘Dunarea de Jos’’ University of Galati, 47 Domneasca Str., Galati 800008, Romania
a r t i c l e
i n f o
Article history: Received 31 October 2011 Received in revised form 13 March 2012 Accepted 15 March 2012 Available online 17 April 2012 Keywords: Composite laminated plates Double delamination Shear and axial buckling
a b s t r a c t In the paper, a methodology to evaluate the influence of the double delaminations on the buckling and postbuckling behavior of the composite plates under shear and compression loading, used in ship hull structure, is presented. The parametric calculus was done for various positions of delaminations and seven values of loading ratios. The numerical and experimental tests on the delaminated plates are presented. Composite plates containing a multiple number of delaminations are analyzed with respect to combined boundary shear and axial compression buckling. Finally, the graphical results for one position of the delamination through the thickness and two cases of position of delaminations in the case of double delaminated plate and seven values of loading ratio are presented. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Fiber-reinforced composite materials have been increasingly used over the past few decades in a variety of applications in which a fairly high ratio of stiffness/strength to weight is required. Composite panels are generic structural elements in weight sensitive structural for marine applications. Laminated composite panels, which are anisotropic, are gaining popularity in structural applications such as ship hulls, decks, ship and offshore superstructures. These panels are becoming increasingly used in structural marine applications due to their high specific stiffness and specific strength [1,2]. The use of laminated composites provides flexibility to tailor different properties of the structural elements to achieve the stiffness and strength characteristics. However, these materials are prone to a wide range of defects and damage that can cause significant reductions in stiffness and strength [3,4]. Ship structure plates are subjected to any combination of in plane, out of plane and shear loads during application. Due to the geometry and general load of the ship hull, buckling is one of the most important failure criteria. Delamination reduces the elastic buckling load of the laminated composite structures and leads to global structural failure at loads below the design level. Laminated plates which were subjected to impact loading are known to contain multiple delaminations positioned throughthe-thickness of the laminate. These delaminations can be approximated as circular and often oblong and their orientation matches ⇑ Corresponding author. Tel.: +40 722 383282; fax: +40 236 461353. E-mail address: [email protected] (I. Chirica). 0927-0256/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2012.03.032
the orientation of the adjacent composite layers which lie further away from the point of impact [5]. It is also known, that delaminations cause reduction of the buckling load of laminated structures [6,7] and therefore the load carrying capacity of laminate structures which might be subjected to impact loading, such as ship hull structures, could be substantially reduced as well. Delamination may arise during manufacturing (e.g., incomplete wetting, air entrapment) or during service (e.g., low velocity impact, cargo strikes on the ship deck). They may not be visible or barely visible on the surface, since they are embedded within the composite structures. The presence of in-plane loading may cause buckling of stiffened panels and an accurate knowledge of critical buckling load and mode shapes are essential for reliable and lightweight structural design. In [8] in order to investigate the effects of multiple delaminations’ width on the buckling loads of the simply supported woven steel-reinforced thermoplastic-laminated composite plates, having the strip and lateral delamination regions between each layers, 3-D finite elements models have been established. The critical delamination representing the maximum delamination sizes and tolerate value in a structure without decreasing compressive strength has been obtained from the evaluation of the buckling analyses. The results show that the buckling loads values are increasing since the orientation angles are increasing. In [9–11] the compressive behavior of composite laminates with multiple through the width delaminations is investigated analytically based on the CLPT (classical laminated plate theory) and its formulation developed on the basis of the Rayleigh Ritz approximation technique. Also the three dimensional finite element analysis is performed using ANSYS 5.4 general purpose
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commercial software and the results are compared with those obtained by the analytical model. The optimizing of laminated composite plates taking into account the shear buckling load is the object of the studies performed in [12,13]. In [14–16] the studies are focused on the glass-fiberreinforced polymer (GFRP) girders elements that are sensitive to shear stresses in the case of an in-plane biaxial compression-tension buckling problem, according to the rotated stress field theory. In [17,18] the studies dealing with postbuckling behavior of laminated composite plates under the combination of in-plane shear, compression and lateral loading using FEM analysis are done. In [19] the loss of compressive stiffness of a thin square plate buckled in shear is calculated by means of a finite element program developed for this purpose. Two cases are considered, first a perfectly flat plate, and second a plate with a transversal imperfection having the amplitude equal to 10% of the plate thickness. Buckling behavior of multiple delaminated plates subjected to shear and compression loading has been studied in [20–23] by means of the finite element analysis. The studies show several trends in the effect of number positions on surface and throughthe-thickness position of delaminations upon the buckling response. These findings could be used for evaluation of reliability of laminate structures which might be subjected to impact loading, such as ship hull structures. This paper addresses the effects of the delaminations on the combined shear and axial buckling and postbuckling behavior of rectangular plates made of advanced composite materials. The analysis includes the effects of delaminations positions, loading and boundary conditions, results of numerical and experimental tests. Some overall important findings of these studies are that plates having delaminations can buckle at loads higher than the buckling loads for corresponding perfect plates and can exhibit substantial postbuckling load-carrying capability. The presence of in-plane loading may cause buckling of stiffened panels. An accurate knowledge of critical buckling load and mode shapes are essential for reliable and lightweight structural design.
2. Shear and axial buckling theory of orthotropic plates Limit point buckling is an instability in which the load–displacement curve reaches a maximum and then exhibits negative stiffness and releases strain energy. During limit point buckling there are no sudden changes in the equilibrium path; however, if load is continuously increased then the structure may jump or ‘‘snap’’ to another point on the load–deflection curve. For this reason, this type of instability is often called ‘‘snap-through’’ buckling, because the structure snaps to a new equilibrium position. A limit point is characterized by the load–frequency curve passing through a frequency of zero with a zero slope. The load–deflection curve also has a zero slope at the point of maximum load (limit point). Buckling analysis of a plate may be divided into three parts [24]: – Classical buckling analysis. – Difficult classical effects. – Non-classical phenomena. The classical buckling analysis is a generalization of the Euler buckling for beams. The awkward effects in classical buckling analysis are in connection with vibrations, shear deformations, springs, non-homogeneities and variable thicknesses, nonlinear relations between
stresses and strains. Non-classical buckling analysis involves considerations such as imperfections, non-elastic behavior of the material, dynamic effects of the loading, the fact that the in-plane loading is not in the initial point of the plate. Finally, we have to remark that no plate is initially perfect and if initial deviation (from flatness or symmetry) exists, no clear buckling phenomenon may be identifying. The deviations of the plate from the flatness and symmetry are usually called imperfections (initial transversal imperfections, delaminations) as it will be treated in the following chapters. The following cases (in analytical, numerical and experimental ways) are presented: compressive buckling, shear buckling, mixed compressive and shear buckling. The results (for linear and nonlinear model) are presented as variation of the buckling loads function of maximum transversal displacement (buckling and postbuckling behavior). The state of equilibrium of a plate deformed by forces acting in the plane of the middle surface is unique and the equilibrium is stable if the forces are sufficiently small. If, while maintaining the distribution of forces constant at the edge of the plate, the forces are increased in magnitude, there may arise a time when the basic form of equilibrium ceases to be unique and stable and other forms become possible, which are characterized by the curvatures of the middle surface. For symmetrically laminated cross-ply plates there is no coupling between bending and twisting. The equation of the deflected surface of symmetrically laminated plates for combined shear and uniaxial loading is:
D11
@4w @4w @4w @2w @2w þ 2ðD þ D Þ þ D N 2N ¼0 12 66 22 x xy @x4 @x2 @y2 @y4 @x2 @x@y ð1Þ
where D11, D12, D22, D66 are the orthotropic plate stiffnesses, calculated according to the equation:
Dij ¼
n 1P Q k ðz3 z3k1 Þ 3 k¼1 ij k
ð2Þ
where Q kij is the rigidity coefficient from the Hooke’s law written for the k-th ply. The thickness and position of every k ply can be calculated from the equation:
tk ¼ zk zk1
ð3Þ
and
zk ¼ zk1 þ
tk 2
ð4Þ
Linear buckling of beams, membranes and plates has been studied extensively. A linearized stability analysis is convenient from a mathematical viewpoint but quite restrictive in practical applications. What is needed is a capability for determining the nonlinear load–deflection behavior of a structure. Considerable effort has also been expended on this problem and two approaches have evolved: class-I methods, which are incremental in nature and do not necessarily satisfy equilibrium; and class-II methods, which are self-correcting and tend to stay on the true equilibrium path [5]. Historically, class-I was the first finite element approach to solving geometrically nonlinear problems [6]. In this method the load is applied as a sequence of sufficiently small increments so that the structure can be assumed to respond linearly during each increment. For solving geometrically and material nonlinear problems, the load is applied as a sequence of sufficiently small increments so
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that the structure can be assumed to respond linearly during each increment. For each increment of load, increments of displacements and corresponding increments of stress and strain are computed. These incremental quantities are used to compute various corrective stiffness matrices (variously termed geometric, initial stress, and initial strain matrices) which serve to take into account the deformed geometry of the structure. A subsequent increment of load is applied and the process is continued until the desired number of load increments has been applied. The net effect is to solve a sequence of linear problems wherein the stiffness properties are recomputed based on the current geometry prior to each load increment. The solution procedure takes the following mathematical form
ðK þ KI Þi1 Ddi ¼ DQ
ð5Þ
where K is the linear stiffness matrix; KI is an incremental stiffness matrix based upon displacements at load step i-1; Ddi is the increment of displacement due to the i-th load increment; DQ is the increment of load applied. The plate material is damaged according to a specific criterion. The buckling load determination may use the Tsai-Wu failure criterion in the case if the general buckling does not occur till the first-ply failure occurring. In this case, the buckling load is considered as the in-plane load corresponding to the first-ply failure occurring. The Tsai-Wu failure criterion provides the mathematical relation for strength under combined stresses. The strength of the laminated composite can be based on the strength of individual plies within the laminate. In addition, the failure of plies can be successive as the applied load increases. There may be a first ply failure followed by other ply failures until the last ply fails, denoting the ultimate failure of the laminate. The failure criterion is used to calculate a failure index (FI) from the computed stresses and user-supplied material strengths. The failure indices are computed for all layers in each element of your model. During post processing, it is possible to plot failure indices of the mesh for any layer. The Tsai-Wu failure criterion is commonly used for orthotropic materials with unequal tensile and compressive strengths. The failure index is computed using the following equation [1,6].
FI ¼ F 1 r1 þ F 2 r2 þ F 11 r21 þ F 22 r22 þ F 66 r26 þ 2F 12 r1 r2
ð6Þ
where F1 ¼
1 RT1
1 RC1
;
F 11 ¼
1 RT1 RC1
; F2 ¼
1 RT2
1 RC2
;
F 22 ¼
1 RT2 RC2
;
F 66 ¼
1 R212
: ð7Þ
The coefficient F12, represents the parameter of interaction between r1 and r2. This coefficient can be determined by various ways: by a mechanical biaxial test or by stochastic based methods, using experimental data [25]. As an alternative Tsai and Hahn [2,26] suggested that F12 be estimated from the following relationship
F 12
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0:5 F 11 F 22
until the moment of the first-ply failure occurring, the in-plane load corresponding to this moment is considered as the buckling load. Three types of failure status are determined in each layer of each element: – OK
FI < 1
ð10Þ
– Fail 1 in the case:
jr1 j < RT1 and jr1 j < RC1
ð11Þ
– Fail 2 in the case:
jr1 j P RT1 and jr1 j P RC1
ð12Þ
3. Numerical analysis of shear and axial buckling of composite plates In the study, the numerical analysis was carried out using licensed finite element package software COSMOS/M. There are several ways in which the panel can be modelled for the delamination analysis. For the present study, a 3-D model with 3-node SHELL3L composite element of COSMOS/M is used (Fig. 2). In the area of the delamination the panel is divided into two sub-laminates by a hypothetical plane containing the delamination. The delamination model has been developed by using the surface-to-surface contact option. In COSMOS/M the delaminated regions were modelled by two layers of elements with coincident but separate nodes and section definitions to model offsets from the common reference plane. The direction of the axial loading is considered along the symmetry geometrical axis of the plate. As it is seen in Fig. 1, in this paper four cases of double delaminated plate are studied. The shape of each delamination is considered as circular one having the diameter of 20 mm. For each case, the various positions of the double delamination in the stack are considered. For the same position the delaminations are placed between the same layers, according to Table 2. In Fig. 2 an example of plate mesh is presented. The square plates (320 320 mm) are made of E-glass/polyester concerning 16 biaxial layers having the orthotropic directions and thickness (4.96 mm) according to Table 1. Topological code of the plate is [02/45/902/45/02]s. The material characteristics, determined in experimental tests (using stretching machine and strain gauges) are:
y Case 4 p
q
Case 2 p
ð8Þ
In Eq. (7), the parameters RCi ; RTi are the compressive strength and tensile strength in the material in longitudinal direction (i = 1) and transversal direction (i = 2). The parameter R12 is in-plane shear strength in the material 1–2 plane. The failure index in calculated in each ply of each element. In the ply where failure index is
FI > 1
x
ð9Þ
the first-ply failure occurs, according to the Tsai-Wu criterion. In the next steps, the tensile and compressive properties of this element are reduced by the failure index. If the buckling did not appeared
Case 1
q
Case 3
Fig. 1. Positions of double delamination in the plate.
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Fig. 3. Boundary conditions.
Fig. 2. Plate mesh in the Case 1.
Table 1 Composite stack. Macro-layer No.
Number of layers
Fiber direction (°)
Thick-ness (mm)
1 2 3 4 5 6 7 8 9 10
2 1 2 1 2 2 1 2 1 2
0 45 90 45 0 0 45 90 45 0
0.62 0.31 0.62 0.31 0.62 0.62 0.31 0.62 0.31 0.62
Ex ¼ 38:6 GPa; Ey ¼ 8:27 GPa; Ez ¼ 8:27 GPa; Gxy ¼ 4:14 GPa; Gxz ¼ 4:14 GPa; Gyz ¼ 4:6 GPa;
can be considered, as is presented in Table 1. Taking into consideration there are 10 macro-layers, only positions placed between two neighbors macro-layers, from 1 to 6 respectively are studies. The boundary conditions, shown in Fig. 3 (where the corresponding d.o.f., denoted with u for displacement and r for rotation are considered to be equal to zero) are described on the plate sides. On the outline of the delaminated area the continuity condition for the displacements are imposed. Initially, the both sublaminate plates are considered to be in contact. Between the delaminated layers the contact finite elements (GAP elements) are used. The loading was considered to be a combination between axial compression (p) and shear (q) acting on the plate sides as it is shown in Fig. 1. The combination is determined by the loading ratio k = q/p. For the loading ratio, 6 values were considered: 0 (pure axial compression); 0.2; 0.4; 0.6; 0.8; 1.0 and 1 (pure shearing). In Figs. 4 and 5, the variation of the axial loading p (q for pure shearing) versus the magnitude of the transversal displacement are presented for all loading ratios, for two cases of delaminated plates (cases 1 and 3) and positions of delaminations between macro-layers 1 and 2 (Case a). For each case, according to the changing in the slope of the curves, the range of the buckling load values (determined by graphical method, by drawing a tangent line in the point of suddenly changing the curve slope) is shown.
lxy ¼ 0:3; lyz ¼ 0:42; lxz ¼ 0:3; RTx ¼ 1:062 GPa;RCx ¼ 0:610 GPa; RTy ¼ 0:031 GPa; RCy ¼ 0:118 GPa; Rxy ¼ 0:72 GPa: For the same case of double delaminated plate, the delaminations are considered to be placed in the same plane, between two neighbor macro-layers (MLs), as it is presented in Table 2. A macro-layer is a group of layers having the same characteristics (direction of fibers versus direction of axial compression). Due to the position in the stack of the layers, a number of 10 macro-layers
Table 2 Position of delaminations in all 4 cases of double delaminations. Macro-layer No.
Case a
1 2 3 4 5 6 7 8 9 10
1 ML 9 ML
b 2 ML 8 ML
c
3 ML 7 ML
d
4 ML 6 ML
e
5 ML 5 ML
Fig. 4. Case 1-a: 0.124 MPa < pcr < 0.414 MPa.
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177
Fig. 6. Experimental rig.
Fig. 5. Case 3-a: 0.105 MPa < pcr < 0.527 MPa.
Table 3 Buckling load (MPa) for all plates in the Case a. k
0 0.2 0.4 0.6 0.8 1.0 1
Fail type
Fail Fail Fail Fail Fail Fail Fail
1 1 1 1 1 1 1
Case 1-a
2-a
3-a
4-a
Exp.
0.872 0.872 0.872 0.653 0.653 0.435 0.043
0.872 0.872 0.872 0.653 0.653 0.435 0.043
0.872 0.872 0.872 0.653 0.653 0.435 0.043
0.872 0.872 0.872 0.653 0.653 0.435 0.043
0.79 0.86 0.85 0.68 0.61 0.45 0.04
In Table 3, the values of ultimate strength (buckling load) determined by Tsai-Wu criterion for the Case 1-a are presented. As it is seen, the first failure for all cases occurs for the tension case. It is possible to simultaneously occur other failures for compression stresses. For ultimate strength only the first failure occurred in the layers is interested. This can be so tension and compression stress. For the loading ratio, k, less than 0.5, the buckling load has the same value, being equal to 0.872 MPa. For 0.5 < k < 1, the buckling loads have the same value, but in these cases the value is equal to 0.653 MPa. For k P 1 the buckling load is decreasing rapidly from 0.435 MPa to 0.043 MPa (that is the value of k toward infinity for shear buckling). 4. Experimental analysis In order to analyse the response of each plate in Case 1, the displacement obtained experimental in the point from the middle of the plate versus numerical load–deflection curves were studied. In these curves, it is possible to identify the main changes in the behavior of the plate. Moreover, the changes in the behavior of the plate experimental curves are compared with that obtained by the numerical model (see Table 3).
The experiments for determination of buckling and postbuckling behavior of the double delaminated plates with various positions of delaminations was made with an experimental rig, shown in Fig. 6. The rig comprises a stretching machine, strain gauges equipment for stress determination, displacement transducers (LVDT) for maximum transversal displacement measurement and a bolted joint system, specially built for this type of experiments. The bolted joint system is transforming the axial force into an axial compressive force and a shear force system that are applied on the sides of the plate. For each test, the tensile force of the stretching machine was increased step by step with an increment DF = 2 kN. The significance of failure status from second column in Table 3 is described in Eq. (11). According to the measurements results, in Table 3 (last column) the values of the buckling load are presented only for the plate 1-a. Experimental results obtained from the strain gauges and the displacement transducers can be analyzed and compared with the numerical ones. Twenty strain gauges were used to measure the stresses occurred in the layer from the plate’s surface. The value of the experimental stress from the last column in Table 3 corresponds to the same value of transversal displacement obtained in numerical analysis and experimental tests for the plate 1-a.
5. Discussion and conclusions The numerical results obtained from the tests and further analysis of the experimental and numerical results, allow to reach certain conclusions related to the behavior of double delaminated composite plate under combined axial and shear loads. The phenomenon study requires a stress analysis in order to improve the evaluation of the structural response of the delaminated plate. The plate is subjected to a combined stress state in the presence of a nonlinear material and two delaminations placed in various positions. The numerical model could be used to compare the experimental test results obtained for the 20 plates. Figs. 4 and 5 show, in terms of deflection and stress evolution in the middle point of the panel, these different structural responses, for various loading ratios. The comparative analysis has enabled us to confirm the behavior of multiple delaminated plates under combined axial and shear load. As it is shown in Figs. 4 and 5, the value of the point placed on the middle of the plate (wcr) corresponding to the buckling load can be determined in numerical calculus. In the tested plates, since the load performed by the stretching machine was increased, the transversal displacement of the point
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placed on the middle of the plate (wm) was registered. When the value of wm reached the value corresponding to the buckling load obtained by numerical analysis, wcr, the experiment was stopped. The value of the load corresponding to wcr is considered to be experimental buckling load and it can be named ultimate stress. The results obtained by experimental tests have been also used in order to set up suitable numerical models able to interpret correctly the response of the delaminated composite plates. Acknowledgment The work has been performed in the scope of the FP7 Project Move It! – Project No. 285405. References [1] H. Altenbach, J. Altenbach, W. Kissing (Eds.), Mechanics of Composite Structural Elements, Springer, 2001. [2] D.F. Adams, L.A. Carlsson, R.B. Pipes (Eds.), Experimental Characterization of Advanced Composite materials, Taylor & Francis Group, 2003. [3] T. Sadowski, L. Marsavina, N. Peride, E.M. Craciun, Comp. Mater. Sci. 46 (3) (2009) 687–693. [4] T. Sadowski, L. Marsavina, E.M. Craciun, M. Knec, Comp. Mater. Sci. 52 (1) (2012) 231–235. [5] G.J. Thurley, I.H. Marshall (Eds.), Buckling and Postbuckling of Composite Plates, Chapman & Hall, London, 1995.
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