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Scaled models for predicting buckling of delaminated orthotropic beam-plates
Composite Structures 90 (2009) 87–91
Contents lists available at ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Scaled models for predicting buckling of delaminated orthotropic beam-plates Jalil Rezaeepazhand a,*, Michael R. Wisnom b a b
Department of Mechanical Engineering, Ferdowsi University of Mashhad, P.O. Box 91775-1111, Mashhad, Iran Advanced Composites Centre for Innovation and Science, University of Bristol, Bristol BS8 1TR, UK
a r t i c l e
i n f o
Article history: Available online 20 February 2009 Keywords: Buckling Delamination Scaling laws Similarity
a b s t r a c t This study investigates the applicability of scaled models for predicting the buckling behavior of delaminated composites. Such a study is important since it provides the necessary scaling laws, and the factors which affect the accuracy of the scale models. Employment of similitude theory to establish similarity among structural systems can save considerable expense and time, provided that the proper scaling laws are found and validated. In this study a number of parametric studies are performed. The limitations and acceptable intervals of all parameters and corresponding scale factors are investigated. Particular emphasis is placed on the case of delamination buckling of orthotropic plates. Both complete and partial similarities are discussed. This study indicates that models with a different delamination size, depth and number of delaminations than those of the prototype are capable of predicting buckling loads of the prototypes with good accuracy. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Thin-walled beams and plates of various constructions find wide uses as primary structural elements in simple and complex structural configurations. Such composites are attractive because of their high stiffness and/or strength to weight ratios. The enormous design flexibility of advanced composites is obtained because of the large number of design parameters. Due to complexity and lack of complete design based information, any new design must be extensively evaluated numerically and experimentally until it achieves the necessary reliability and performance. However, the experimental evaluation of composite structures is costly and time consuming. Consequently, it is extremely cost effective if available experimental data of a specific structure can be used to predict the behavior of a group of similar systems. Damage tolerance and residual strength after impact of composite structures are very important issues in aerospace engineering. A number of damage assessment methods have been suggested for laminated composites containing impact damage. Delamination buckling theory and equivalent inclusions are two residual strength models which are used for assessing damage of impacted composite structures. It has been shown [1], for various low impact energy levels, thicknesses and stacking sequences, that delamination buckling theory is in good agreement with experimental data. For the past two decades, a significant amount of research has been conducted on the buckling behavior of delaminated compos* Corresponding author. Tel.: +98 915 3114093; fax: +98 511 8455247. E-mail address: [email protected] (J. Rezaeepazhand). 0263-8223/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2009.02.005
ite laminates. Using numerical, experimental and analytical methods, valuable information on the effect of different parameters, such as delamination shape, location, and number of delaminations on buckling of delaminated beams, plates and shells has been presented. A brief description and a historical review of this subject are presented by Simitses [2]. A few recent examples are given in [3–8]. None of the investigations has been focused on establishing similarity and scaling for the buckling response of composite laminates with embedded delaminations. The main objective of this study is to demonstrate the applicability of similitude theory in designing scaled down models for predicting the buckling behavior of delaminated laminated beamplates subjected to uni-axial compression, and hence to justify the use of such scaled models to investigate the behavior of full sized structures. The method presented herein has already been shown to be effective in the design of scaled down models for buckling and vibration response of laminated composite shells [9,10] and panels [11,12]. Scaling laws provide a relationship among similar structures and can be used to extrapolate the available experimental/ computed data of one structure, herein called the model, into design information for another similar structure which is called the prototype. Both complete and partial similarities are discussed. In the present studies, the material behavior is assumed to be linearly elastic. Therefore, scale effects due to damage or other forms of non-linearity are not present or have similar effect on the prototype and its models. Furthermore, it is assumed that the prototype and its scaled down models are without imperfections or the imperfections are such that they have the same effect on both structures.
J. Rezaeepazhand, M.R. Wisnom / Composite Structures 90 (2009) 87–91
2. Governing equation and boundary conditions A one-dimensional model of the laminated plate is employed similar to the one used by Numayer [7]. This model assumes the plates are homogeneous and orthotropic, the material behavior is linearly elastic, and delaminations exists and grow along a plane parallel to the reference plane. The beam and delamination configuration shown in Fig. 1 is adopted in the present study, which is similar to the configuration considered by most of the previous studies [2]. The plate is fixed along its loaded edges, and has length L, thickness H and width W. Delaminations are symmetric about the longitudinal centerline of the beam. Thin local delamination buckling occurs when the instability is initiated by the buckling of the delaminated layer. The governing differential equation for each sublaminate is expressed as [7]: 4
d wi 4 dxi
þ Pi L2
A11 B11 A11 D11
2 i
2
d wi 2
dxi
¼0
ð1Þ
where i = 1, 2, 3, . . . are the regions in the delaminated beam, A11 and D11 are the stiffness matrixes of each sublaminate and Pi the buckling load of each sublaminate, expressed as a fraction of applied compressive load. 3. Scaling laws for delaminated beam Eq. (1) represents the response behavior of both the prototype and its models. Performing the necessary transformation [11], and applying similitude theory, yields the following scaling laws for orthotropic laminated beam-plates. 2 kpi ¼ k2 L ka kD11
ð2Þ
xprototype xmodel
where kx ¼ denotes the scale factor of parameter x. It is necessary to state that, the presented form for arranging the scaling laws is not unique. However, previous experience of establishing scaling laws for laminated plates and shells [9,10] strongly recommended this type of presentation. Since Pi = P hi/H, the resulting scaling law for applied compressive force P corresponding to local buckling of the sublaminate Pi is: 2 2 kP ¼ kH k2 kh a kL kQ ii
ð3Þ
So far, the necessary scaling laws for buckling of delaminated beamplates have been established. In the following sections, the possibility of the existence of different models is discussed. The procedure consists of systematically observing the effect of each parameter and corresponding scaling laws. Then acceptable intervals and limitations for these parameters and scaling laws are discussed. In each case, a set of valid scaling factors and corresponding response scaling laws that accurately predict the response of prototypes from models is introduced. The definition of the computed values for the prototype or models, are the computed buckling loads which are given in published works [3,4]. However, the predicted values
of buckling loads of the prototype are those given by projecting the computed buckling load of the model using the resulting scaling laws. The normalized buckling load is the buckling load of the delaminated plate divided by that of the undelaminated case. Model and prototype have the same material properties or kQii = 1. 4. Complete similarity The necessary condition for complete similarity between the model and its prototype is that all scaling laws are satisfied simultaneously [11]. Note that this condition is a necessary but not a sufficient condition for complete similarity. In order to achieve complete similarity, models and prototypes must buckle in a similar mode shape. In the present case, for thin delaminations (hi < 0.4H) and for long delaminations (ai > 0.3L), the buckling modes are usually local [6–8]. Hence, complete similarity can be achieved even for laminates with different delamination lengths if the model and prototype have the same buckling mode. Fig. 2 presents the normalized buckling load predicted using Eq. (3) for a model with the same number of plies, material properties, dimensions, and delamination position (h/H = 0.2) as the prototypes compared with the directly computed results. However, the delamination length of the model, (a/L)m = 0.5, is different from those of the prototypes (0.2 < (a/L)p < 1.0). It is clear that the behavior of a wide range of prototype delamination lengths can be accurately predicted from the scaling equation using a model with intermediate a/L. However, the result for a/L = 0.2 is not as accurate as other prototypes. This particular result may be attributed to the difference in buckling mode shapes of the model and prototype. The model is buckled in a local mode however, for small a/L, the laminate tends to buckle in a global or mixed mode. This behavior violates the complete similarity conditions. Similar results are presented in Fig. 3, where the delamination is located at h/H = 0.5. Two models with a/L = 0.5 and 0.8 are used to predict the buckling behavior of prototypes with different delamination lengths (0.3 < (a/L)p < 1.0). These results somewhat differ from the results for h/H = 0.2 (Fig. 2). As shown in Fig. 3, the model with larger delamination (a/L = 0.8) is predicting the prototype behavior with better accuracy than the one with a/L = 0.5. Neither model is as accurate as the models in Fig. 2. Again this behavior may be attributed to dissimilarity between the model and prototype buckling mode shapes. When delamination is located close to the mid-plane of the laminate, the mode is more likely to be influenced by global rather than local buckling, and the change in mode shape means that the scaling laws are more prone to distortion as delamination length varies than laminates with delaminations close to the surface. This also explains why normalized
1 Normalized Buckling Load
88
h/H=0.2 0.8
Prototype Ref[3] Predicted from model : a/L=0.5
0.6 0.4
0.2
0 0.2
0.4
0.6
0.8
1
a/L
Fig. 1. Geometries of a beam with multiple delaminations.
Fig. 2. Buckling loads predicted from Eq. (3) using model with a/L = 0.5 (h/H = 0.2) compared with directly computed prototype buckling loads.
89
J. Rezaeepazhand, M.R. Wisnom / Composite Structures 90 (2009) 87–91
2
Normalized Buckling Load
h/H=0.5 1.6
Prototype Ref[3] Predicted from model: a/L=0.5
1.2
Predicted from model : a/L=0.8 0.8
0.4
0 0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
a/L Fig. 3. Predicted and directly computed buckling loads of the prototypes when models with a/L = 0.5, 0.8 are used (h/H = 0.5).
Normalized Buckling Load
1.6
h1/H=0.5 & h2/H=0.75 1.2
Prototype Ref[3] Predicted from model : a/L=0.5 Predicted from model : a/L=0.8
0.8
except one are kept identical to those of the prototype. Then, the effect of a single relaxation for a wide range of each parameter is investigated. Different possibilities of distorted models, i.e. distortion in number of plies, geometrical parameters, and material properties can be considered. In this case of buckling behavior of delaminated beam-plates, distortion in buckling mode shape, delamination size and position, and number of delamination are considered. Fig. 5 presents computed and predicted normalized buckling loads for prototypes with two equal length (a/L = 0.5) delaminations at h1/H = 0.5, and h2/H = 0.75. Models and prototypes have the same number of plies, material properties, and dimensions. However, laminates with single delaminations are used as models. One model has h/H = 0.2 and that of the other one is equal to h/H = 0.5. Models with wide range of delamination lengths (0.2 < (a/L)m < 1.0) are considered. As can be seen from Fig. 5, models with h/H = 0.2 are considerably more accurate than the models with the delamination at the mid-plane (h/H = 0.5). Once again, the importance of delamination near the surface in buckling of delaminate plates can be seen here. The effects of distortion in delamination length for models with single delaminations of different lengths on the accuracy of predictions for prototypes with two delaminations are illustrated in Fig. 6. All models have a near the surface delamination (h/ H = 0.2). The percent of discrepancy is defined as:
% Discrepancy ¼ 0.4
Pprototype Ppredicted 100 Pprototype
0.28
0 0.6 a/L
0.8
1
Fig. 4. Comparison of predicted and directly computed buckling loads of the prototypes with double delaminations when model with a/L = 0.5 is used.
buckling loads greater than 1 are predicted at small a/L. The scaling equation assumes an extrapolation of the local buckling behavior, whereas in fact it switches to being more dominated by global buckling. Fig. 4 presents computed and predicted normalized buckling loads for laminates with multiple delaminations. All laminates have two equal length delaminations at h1/H = 0.5, and h2/ H = 0.75. Models and prototypes have the same number of plies, material properties, dimensions, number of delaminations, and delamination position. Two models with a/L = 0.5 and 0.8 are used to predict the buckling behavior of prototypes with different delamination lengths (0.2 < (a/L)p < 1.0). Similarly to Fig. 2, the models are capable of predicting the buckling loads of the prototype with good accuracy. In this case, because the second delamination is not larger than the one closer to the surface, the second delamination does not have a significant effect on buckling, and the laminates behave similarly to ones with a single delamination at the same position [6]. However, again this breaks down at very short a/L as the mode is more influenced by global buckling with two delaminations present.
Normalized Buckling Load
0.4
0.21
0.14 Prototype(h1/H=0.5 & h2/H=0.75) Ref[3] 0.07
Predicted from model: h/H=0.2 Predicted from model: h/H=0.5
0 0.2
0.4
0.6
0.8
1
(a/L) m Fig. 5. Comparison of predicted and directly computed buckling loads of the prototypes with double delaminations a/L = 0.5 when single delaminated model is used.
12 model : a/L = 0.3
8
model : a/L = 0.5
% Discrepancy
0.2
4
model : a/L = 0.8
0 -4
5. Partial similarity -8
Often complete similarity is difficult to achieve. When at least one of the similarity conditions cannot be satisfied, only partial similarity is achieved. Relaxations in the relationship between two systems cause model behavior to be different from that of the prototype. Since each variable has a different influence on the response of the system, the resulting similarity conditions have different influences. In each case here, all of the model parameters
-12 0.1
0.3
0.5
0.7
0.9
1.1
a/L Fig. 6. Discrepancy between predicted and directly computed buckling loads of the prototypes with two delaminations and different a/L, model with single delamination, h/H = 0.2.
J. Rezaeepazhand, M.R. Wisnom / Composite Structures 90 (2009) 87–91
Both models with large delamination length (a/L = 0.5 and 0.8) have approximately the same accuracy. The results seem to be slightly more accurate for the model with a/L = 0.5 than the other two models for prototypes with small delamination length. However, for prototypes with large delamination length, the model with a/L = 0.8 is better. As a further example to demonstrate the wider applicability of the approach, Fig. 7 presents results for 16 layer unidirectional carbon/epoxy laminates with different multiple delaminations as considered by Hwang and Liu [4]. They considered a single delaminated laminate as type IA. Type IB is a laminate with a long delamination close to the surface and three short delaminations with equal size (ashort = 0.5a) and distance beneath it. Type IC, as shown in Fig. 8, is a laminate with a long delamination near the surface and three shorter delaminations with different size which create a triangular shape multiple delamination [4]. The long delaminations have h/H = 0.125, whilst the shorter ones have h/H = 0.25, 0.375 and 0.5. Fig. 7 presents the discrepancy between the predicted buckling loads of multiple delaminated laminates IB and IC when three single delaminations of type IA with (a/L)m = 0.3, 0.5, and 0.7 are used as models compared with the directly computed results reported in [4]. As demonstrated by Fig. 7, all models predict the buckling load of the prototype with less than 1% discrepancy. As shown in Fig. 3, when a delamination is located close to the mid-plane of the laminate, the resulting scaling laws are sensitive to distortion of delamination length. In order to reduce this sensitivity, introducing another distortion may improve the performance of the models. Fig. 9 presents the predicted and computed buckling loads of prototypes with different delamination lengths. Models and prototypes have the same number of plies, material properties, and dimensions. However, one model has h/H = 0.2 and the other one has the same h/H as those of the prototypes. Both models have identical a/L = 0.5. It is clear that, introducing the second distortion in the models, improves the predictability. Since each variable has a different influence on the response of the system, introducing the right type distortion improves the prediction of the models [9,10].
1.5 model : a/L = 0.3
% Discrepancy
1
model : a/L=0.5
0.5
model : a/L=0.7
1.6 Normalized Buckling Load
90
Prototype : h/H=0.5 Ref[3] Predicted from model : a/L=0.5 , h/H=0.5
1.2
Predicted from model : a/L=0.5 , h/H=0.2 0.8
0.4
0 0.4
0.5
0.6
0.7 a/L
0.8
0.9
1
Fig. 9. Predicted and directly computed buckling loads of the prototypes when models with different h/H are used.
6. Discussion An analytical investigation was undertaken to assess the applicability of scaled down models in evaluating delaminated composites. Employment of structural similitude to establish similarity among structural systems can save considerable expense and time, provided that the proper scaling laws are found and validated. Scaling laws provide the relationship between a full scale structure (prototype) and its scale models and can be used to extrapolate the experimental data of a scale model to provide design information for a prototype. In this study, the accuracy of predicted buckling response of the prototype by various models is presented. The limitations and acceptable intervals for some parameters and corresponding scale factors are presented. The results presented herein indicate that, for buckling response of delaminated beamplates, based on structural similitude, a set of scaling laws can be found to develop design rules for small scale models. By ply-level scaling of the prototype a wide range of models can be found which lead to excellent accuracy. When the model and its prototype have the same material properties, models with a different delamination length and location can be found, which are able to predict the behavioral response of the prototype with good accuracy. For models with the same material properties and stacking sequence as the prototype, distorted models with different number of delaminations, delamination length and depth than those of the prototype can predict the behavior of the prototype as long as the model and the prototype have the same mode shape.
0
7. Conclusions
-0.5 -1 -1.5 0.2
0.3
0.4
0.5
0.6
0.7
0.8
a/L Fig. 7. Discrepancy between predicted and directly computed buckling loads of the prototypes with four delaminations from Hwang and Liu [4] based on models with single delaminations of different a/L.
This study presents the applicability of structural similitude in predicting the elastic buckling behavior of unidirectional delaminated beam-plates. Based on the direct use of the governing equations, the necessary similarity conditions are presented. Accuracy of the models with both complete and partial similarity is investigated. The results presented herein indicate that, for delamination buckling of unidirectional laminates, based on structural simili-
Fig. 8. Three different multiple delaminations which considered by Hwang and Liu [4].
J. Rezaeepazhand, M.R. Wisnom / Composite Structures 90 (2009) 87–91
tude, a set of scaling laws can be found which may be used to established design rules for similar scaled models. As demonstrated, a wide range of models are capable of predicting the buckling loads of prototypes with good accuracy, indicating that scaled tests may be used effectively to investigate the buckling response of structures with delaminations. Some recommendations for future research include the extension of the present work to micro-level scaling, delamination growth and fatigue of the delaminated structures. References [1] Nyman T, Bredberg A, Schon J. Equivalent damage and residual strength for impact damaged composite structures. J Reinf Plast Compos 2000;19(6):428–48. [2] Simitses GJ. Delamination buckling of flat laminates. In: Marshal IH, Turvey GJ, editors. Buckling and post buckling of composite plates. Chapman & Hall: London; 1994. [3] Cappello F, Tumino D. Numerical analysis of composite plates with multiple delaminations subjected to uniaxial buckling load. Compos Sci Technol 2006;66:264–72.
91
[4] Hwang S, Liu G. Buckling behavior of composite laminate with multiple delaminations under uniaxial compression. Compos Struct 2001;53:235–43. [5] Zhou G, Rivera LA. Investigation for the reduction of in-plane compressive strength in preconditioned thin composite panels. J Compos Mater 2005;30(5):391–422. [6] Tafreshi A, Oswald T. Global buckling behavior and local damage propagation in composite plates with embedded delaminations. Int J Pressure Vessel Piping 2003;80:9–20. [7] Numayer KS, Hadda RH. Analytical solution of buckling of beams with two delaminations. Mech Compos Mater Struct 2001;8:283–97. [8] Sekine H, Hu N, Kouchakzadeh MA. Buckling analysis of elliptically delaminated composite laminates with consideration of partial closure of delamination. J Compos Mater 2000;34(7):551–72. [9] Simitses GJ, Rezaeepazhand J, Sierakowski RL. Scaled models for laminated cylindrical shells subjected to external pressure. Mech Compos Mater Struct 1997;4:267–80. [10] Rezaeepazhand J, Simitses GJ, Starnes Jr JH. Scale models for laminated cylindrical shells subjected to axial compression. Compos Struct 1996;34(4):361–9. [11] Rezaeepazhand J, Simitses GJ, Starnes Jr JH. Design of scaled down models for stability of laminated plates. AIAA J 1995;33(3):515–9. [12] Simitses GJ, Rezaeepazhand J. Structural similitude and scaling laws for crossply laminated plates. J Thermoplast Compos Mater 1995;8:240–51.
Contents lists available at ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Scaled models for predicting buckling of delaminated orthotropic beam-plates Jalil Rezaeepazhand a,*, Michael R. Wisnom b a b
Department of Mechanical Engineering, Ferdowsi University of Mashhad, P.O. Box 91775-1111, Mashhad, Iran Advanced Composites Centre for Innovation and Science, University of Bristol, Bristol BS8 1TR, UK
a r t i c l e
i n f o
Article history: Available online 20 February 2009 Keywords: Buckling Delamination Scaling laws Similarity
a b s t r a c t This study investigates the applicability of scaled models for predicting the buckling behavior of delaminated composites. Such a study is important since it provides the necessary scaling laws, and the factors which affect the accuracy of the scale models. Employment of similitude theory to establish similarity among structural systems can save considerable expense and time, provided that the proper scaling laws are found and validated. In this study a number of parametric studies are performed. The limitations and acceptable intervals of all parameters and corresponding scale factors are investigated. Particular emphasis is placed on the case of delamination buckling of orthotropic plates. Both complete and partial similarities are discussed. This study indicates that models with a different delamination size, depth and number of delaminations than those of the prototype are capable of predicting buckling loads of the prototypes with good accuracy. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Thin-walled beams and plates of various constructions find wide uses as primary structural elements in simple and complex structural configurations. Such composites are attractive because of their high stiffness and/or strength to weight ratios. The enormous design flexibility of advanced composites is obtained because of the large number of design parameters. Due to complexity and lack of complete design based information, any new design must be extensively evaluated numerically and experimentally until it achieves the necessary reliability and performance. However, the experimental evaluation of composite structures is costly and time consuming. Consequently, it is extremely cost effective if available experimental data of a specific structure can be used to predict the behavior of a group of similar systems. Damage tolerance and residual strength after impact of composite structures are very important issues in aerospace engineering. A number of damage assessment methods have been suggested for laminated composites containing impact damage. Delamination buckling theory and equivalent inclusions are two residual strength models which are used for assessing damage of impacted composite structures. It has been shown [1], for various low impact energy levels, thicknesses and stacking sequences, that delamination buckling theory is in good agreement with experimental data. For the past two decades, a significant amount of research has been conducted on the buckling behavior of delaminated compos* Corresponding author. Tel.: +98 915 3114093; fax: +98 511 8455247. E-mail address: [email protected] (J. Rezaeepazhand). 0263-8223/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2009.02.005
ite laminates. Using numerical, experimental and analytical methods, valuable information on the effect of different parameters, such as delamination shape, location, and number of delaminations on buckling of delaminated beams, plates and shells has been presented. A brief description and a historical review of this subject are presented by Simitses [2]. A few recent examples are given in [3–8]. None of the investigations has been focused on establishing similarity and scaling for the buckling response of composite laminates with embedded delaminations. The main objective of this study is to demonstrate the applicability of similitude theory in designing scaled down models for predicting the buckling behavior of delaminated laminated beamplates subjected to uni-axial compression, and hence to justify the use of such scaled models to investigate the behavior of full sized structures. The method presented herein has already been shown to be effective in the design of scaled down models for buckling and vibration response of laminated composite shells [9,10] and panels [11,12]. Scaling laws provide a relationship among similar structures and can be used to extrapolate the available experimental/ computed data of one structure, herein called the model, into design information for another similar structure which is called the prototype. Both complete and partial similarities are discussed. In the present studies, the material behavior is assumed to be linearly elastic. Therefore, scale effects due to damage or other forms of non-linearity are not present or have similar effect on the prototype and its models. Furthermore, it is assumed that the prototype and its scaled down models are without imperfections or the imperfections are such that they have the same effect on both structures.
J. Rezaeepazhand, M.R. Wisnom / Composite Structures 90 (2009) 87–91
2. Governing equation and boundary conditions A one-dimensional model of the laminated plate is employed similar to the one used by Numayer [7]. This model assumes the plates are homogeneous and orthotropic, the material behavior is linearly elastic, and delaminations exists and grow along a plane parallel to the reference plane. The beam and delamination configuration shown in Fig. 1 is adopted in the present study, which is similar to the configuration considered by most of the previous studies [2]. The plate is fixed along its loaded edges, and has length L, thickness H and width W. Delaminations are symmetric about the longitudinal centerline of the beam. Thin local delamination buckling occurs when the instability is initiated by the buckling of the delaminated layer. The governing differential equation for each sublaminate is expressed as [7]: 4
d wi 4 dxi
þ Pi L2
A11 B11 A11 D11
2 i
2
d wi 2
dxi
¼0
ð1Þ
where i = 1, 2, 3, . . . are the regions in the delaminated beam, A11 and D11 are the stiffness matrixes of each sublaminate and Pi the buckling load of each sublaminate, expressed as a fraction of applied compressive load. 3. Scaling laws for delaminated beam Eq. (1) represents the response behavior of both the prototype and its models. Performing the necessary transformation [11], and applying similitude theory, yields the following scaling laws for orthotropic laminated beam-plates. 2 kpi ¼ k2 L ka kD11
ð2Þ
xprototype xmodel
where kx ¼ denotes the scale factor of parameter x. It is necessary to state that, the presented form for arranging the scaling laws is not unique. However, previous experience of establishing scaling laws for laminated plates and shells [9,10] strongly recommended this type of presentation. Since Pi = P hi/H, the resulting scaling law for applied compressive force P corresponding to local buckling of the sublaminate Pi is: 2 2 kP ¼ kH k2 kh a kL kQ ii
ð3Þ
So far, the necessary scaling laws for buckling of delaminated beamplates have been established. In the following sections, the possibility of the existence of different models is discussed. The procedure consists of systematically observing the effect of each parameter and corresponding scaling laws. Then acceptable intervals and limitations for these parameters and scaling laws are discussed. In each case, a set of valid scaling factors and corresponding response scaling laws that accurately predict the response of prototypes from models is introduced. The definition of the computed values for the prototype or models, are the computed buckling loads which are given in published works [3,4]. However, the predicted values
of buckling loads of the prototype are those given by projecting the computed buckling load of the model using the resulting scaling laws. The normalized buckling load is the buckling load of the delaminated plate divided by that of the undelaminated case. Model and prototype have the same material properties or kQii = 1. 4. Complete similarity The necessary condition for complete similarity between the model and its prototype is that all scaling laws are satisfied simultaneously [11]. Note that this condition is a necessary but not a sufficient condition for complete similarity. In order to achieve complete similarity, models and prototypes must buckle in a similar mode shape. In the present case, for thin delaminations (hi < 0.4H) and for long delaminations (ai > 0.3L), the buckling modes are usually local [6–8]. Hence, complete similarity can be achieved even for laminates with different delamination lengths if the model and prototype have the same buckling mode. Fig. 2 presents the normalized buckling load predicted using Eq. (3) for a model with the same number of plies, material properties, dimensions, and delamination position (h/H = 0.2) as the prototypes compared with the directly computed results. However, the delamination length of the model, (a/L)m = 0.5, is different from those of the prototypes (0.2 < (a/L)p < 1.0). It is clear that the behavior of a wide range of prototype delamination lengths can be accurately predicted from the scaling equation using a model with intermediate a/L. However, the result for a/L = 0.2 is not as accurate as other prototypes. This particular result may be attributed to the difference in buckling mode shapes of the model and prototype. The model is buckled in a local mode however, for small a/L, the laminate tends to buckle in a global or mixed mode. This behavior violates the complete similarity conditions. Similar results are presented in Fig. 3, where the delamination is located at h/H = 0.5. Two models with a/L = 0.5 and 0.8 are used to predict the buckling behavior of prototypes with different delamination lengths (0.3 < (a/L)p < 1.0). These results somewhat differ from the results for h/H = 0.2 (Fig. 2). As shown in Fig. 3, the model with larger delamination (a/L = 0.8) is predicting the prototype behavior with better accuracy than the one with a/L = 0.5. Neither model is as accurate as the models in Fig. 2. Again this behavior may be attributed to dissimilarity between the model and prototype buckling mode shapes. When delamination is located close to the mid-plane of the laminate, the mode is more likely to be influenced by global rather than local buckling, and the change in mode shape means that the scaling laws are more prone to distortion as delamination length varies than laminates with delaminations close to the surface. This also explains why normalized
1 Normalized Buckling Load
88
h/H=0.2 0.8
Prototype Ref[3] Predicted from model : a/L=0.5
0.6 0.4
0.2
0 0.2
0.4
0.6
0.8
1
a/L
Fig. 1. Geometries of a beam with multiple delaminations.
Fig. 2. Buckling loads predicted from Eq. (3) using model with a/L = 0.5 (h/H = 0.2) compared with directly computed prototype buckling loads.
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2
Normalized Buckling Load
h/H=0.5 1.6
Prototype Ref[3] Predicted from model: a/L=0.5
1.2
Predicted from model : a/L=0.8 0.8
0.4
0 0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
a/L Fig. 3. Predicted and directly computed buckling loads of the prototypes when models with a/L = 0.5, 0.8 are used (h/H = 0.5).
Normalized Buckling Load
1.6
h1/H=0.5 & h2/H=0.75 1.2
Prototype Ref[3] Predicted from model : a/L=0.5 Predicted from model : a/L=0.8
0.8
except one are kept identical to those of the prototype. Then, the effect of a single relaxation for a wide range of each parameter is investigated. Different possibilities of distorted models, i.e. distortion in number of plies, geometrical parameters, and material properties can be considered. In this case of buckling behavior of delaminated beam-plates, distortion in buckling mode shape, delamination size and position, and number of delamination are considered. Fig. 5 presents computed and predicted normalized buckling loads for prototypes with two equal length (a/L = 0.5) delaminations at h1/H = 0.5, and h2/H = 0.75. Models and prototypes have the same number of plies, material properties, and dimensions. However, laminates with single delaminations are used as models. One model has h/H = 0.2 and that of the other one is equal to h/H = 0.5. Models with wide range of delamination lengths (0.2 < (a/L)m < 1.0) are considered. As can be seen from Fig. 5, models with h/H = 0.2 are considerably more accurate than the models with the delamination at the mid-plane (h/H = 0.5). Once again, the importance of delamination near the surface in buckling of delaminate plates can be seen here. The effects of distortion in delamination length for models with single delaminations of different lengths on the accuracy of predictions for prototypes with two delaminations are illustrated in Fig. 6. All models have a near the surface delamination (h/ H = 0.2). The percent of discrepancy is defined as:
% Discrepancy ¼ 0.4
Pprototype Ppredicted 100 Pprototype
0.28
0 0.6 a/L
0.8
1
Fig. 4. Comparison of predicted and directly computed buckling loads of the prototypes with double delaminations when model with a/L = 0.5 is used.
buckling loads greater than 1 are predicted at small a/L. The scaling equation assumes an extrapolation of the local buckling behavior, whereas in fact it switches to being more dominated by global buckling. Fig. 4 presents computed and predicted normalized buckling loads for laminates with multiple delaminations. All laminates have two equal length delaminations at h1/H = 0.5, and h2/ H = 0.75. Models and prototypes have the same number of plies, material properties, dimensions, number of delaminations, and delamination position. Two models with a/L = 0.5 and 0.8 are used to predict the buckling behavior of prototypes with different delamination lengths (0.2 < (a/L)p < 1.0). Similarly to Fig. 2, the models are capable of predicting the buckling loads of the prototype with good accuracy. In this case, because the second delamination is not larger than the one closer to the surface, the second delamination does not have a significant effect on buckling, and the laminates behave similarly to ones with a single delamination at the same position [6]. However, again this breaks down at very short a/L as the mode is more influenced by global buckling with two delaminations present.
Normalized Buckling Load
0.4
0.21
0.14 Prototype(h1/H=0.5 & h2/H=0.75) Ref[3] 0.07
Predicted from model: h/H=0.2 Predicted from model: h/H=0.5
0 0.2
0.4
0.6
0.8
1
(a/L) m Fig. 5. Comparison of predicted and directly computed buckling loads of the prototypes with double delaminations a/L = 0.5 when single delaminated model is used.
12 model : a/L = 0.3
8
model : a/L = 0.5
% Discrepancy
0.2
4
model : a/L = 0.8
0 -4
5. Partial similarity -8
Often complete similarity is difficult to achieve. When at least one of the similarity conditions cannot be satisfied, only partial similarity is achieved. Relaxations in the relationship between two systems cause model behavior to be different from that of the prototype. Since each variable has a different influence on the response of the system, the resulting similarity conditions have different influences. In each case here, all of the model parameters
-12 0.1
0.3
0.5
0.7
0.9
1.1
a/L Fig. 6. Discrepancy between predicted and directly computed buckling loads of the prototypes with two delaminations and different a/L, model with single delamination, h/H = 0.2.
J. Rezaeepazhand, M.R. Wisnom / Composite Structures 90 (2009) 87–91
Both models with large delamination length (a/L = 0.5 and 0.8) have approximately the same accuracy. The results seem to be slightly more accurate for the model with a/L = 0.5 than the other two models for prototypes with small delamination length. However, for prototypes with large delamination length, the model with a/L = 0.8 is better. As a further example to demonstrate the wider applicability of the approach, Fig. 7 presents results for 16 layer unidirectional carbon/epoxy laminates with different multiple delaminations as considered by Hwang and Liu [4]. They considered a single delaminated laminate as type IA. Type IB is a laminate with a long delamination close to the surface and three short delaminations with equal size (ashort = 0.5a) and distance beneath it. Type IC, as shown in Fig. 8, is a laminate with a long delamination near the surface and three shorter delaminations with different size which create a triangular shape multiple delamination [4]. The long delaminations have h/H = 0.125, whilst the shorter ones have h/H = 0.25, 0.375 and 0.5. Fig. 7 presents the discrepancy between the predicted buckling loads of multiple delaminated laminates IB and IC when three single delaminations of type IA with (a/L)m = 0.3, 0.5, and 0.7 are used as models compared with the directly computed results reported in [4]. As demonstrated by Fig. 7, all models predict the buckling load of the prototype with less than 1% discrepancy. As shown in Fig. 3, when a delamination is located close to the mid-plane of the laminate, the resulting scaling laws are sensitive to distortion of delamination length. In order to reduce this sensitivity, introducing another distortion may improve the performance of the models. Fig. 9 presents the predicted and computed buckling loads of prototypes with different delamination lengths. Models and prototypes have the same number of plies, material properties, and dimensions. However, one model has h/H = 0.2 and the other one has the same h/H as those of the prototypes. Both models have identical a/L = 0.5. It is clear that, introducing the second distortion in the models, improves the predictability. Since each variable has a different influence on the response of the system, introducing the right type distortion improves the prediction of the models [9,10].
1.5 model : a/L = 0.3
% Discrepancy
1
model : a/L=0.5
0.5
model : a/L=0.7
1.6 Normalized Buckling Load
90
Prototype : h/H=0.5 Ref[3] Predicted from model : a/L=0.5 , h/H=0.5
1.2
Predicted from model : a/L=0.5 , h/H=0.2 0.8
0.4
0 0.4
0.5
0.6
0.7 a/L
0.8
0.9
1
Fig. 9. Predicted and directly computed buckling loads of the prototypes when models with different h/H are used.
6. Discussion An analytical investigation was undertaken to assess the applicability of scaled down models in evaluating delaminated composites. Employment of structural similitude to establish similarity among structural systems can save considerable expense and time, provided that the proper scaling laws are found and validated. Scaling laws provide the relationship between a full scale structure (prototype) and its scale models and can be used to extrapolate the experimental data of a scale model to provide design information for a prototype. In this study, the accuracy of predicted buckling response of the prototype by various models is presented. The limitations and acceptable intervals for some parameters and corresponding scale factors are presented. The results presented herein indicate that, for buckling response of delaminated beamplates, based on structural similitude, a set of scaling laws can be found to develop design rules for small scale models. By ply-level scaling of the prototype a wide range of models can be found which lead to excellent accuracy. When the model and its prototype have the same material properties, models with a different delamination length and location can be found, which are able to predict the behavioral response of the prototype with good accuracy. For models with the same material properties and stacking sequence as the prototype, distorted models with different number of delaminations, delamination length and depth than those of the prototype can predict the behavior of the prototype as long as the model and the prototype have the same mode shape.
0
7. Conclusions
-0.5 -1 -1.5 0.2
0.3
0.4
0.5
0.6
0.7
0.8
a/L Fig. 7. Discrepancy between predicted and directly computed buckling loads of the prototypes with four delaminations from Hwang and Liu [4] based on models with single delaminations of different a/L.
This study presents the applicability of structural similitude in predicting the elastic buckling behavior of unidirectional delaminated beam-plates. Based on the direct use of the governing equations, the necessary similarity conditions are presented. Accuracy of the models with both complete and partial similarity is investigated. The results presented herein indicate that, for delamination buckling of unidirectional laminates, based on structural simili-
Fig. 8. Three different multiple delaminations which considered by Hwang and Liu [4].
J. Rezaeepazhand, M.R. Wisnom / Composite Structures 90 (2009) 87–91
tude, a set of scaling laws can be found which may be used to established design rules for similar scaled models. As demonstrated, a wide range of models are capable of predicting the buckling loads of prototypes with good accuracy, indicating that scaled tests may be used effectively to investigate the buckling response of structures with delaminations. Some recommendations for future research include the extension of the present work to micro-level scaling, delamination growth and fatigue of the delaminated structures. References [1] Nyman T, Bredberg A, Schon J. Equivalent damage and residual strength for impact damaged composite structures. J Reinf Plast Compos 2000;19(6):428–48. [2] Simitses GJ. Delamination buckling of flat laminates. In: Marshal IH, Turvey GJ, editors. Buckling and post buckling of composite plates. Chapman & Hall: London; 1994. [3] Cappello F, Tumino D. Numerical analysis of composite plates with multiple delaminations subjected to uniaxial buckling load. Compos Sci Technol 2006;66:264–72.
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[4] Hwang S, Liu G. Buckling behavior of composite laminate with multiple delaminations under uniaxial compression. Compos Struct 2001;53:235–43. [5] Zhou G, Rivera LA. Investigation for the reduction of in-plane compressive strength in preconditioned thin composite panels. J Compos Mater 2005;30(5):391–422. [6] Tafreshi A, Oswald T. Global buckling behavior and local damage propagation in composite plates with embedded delaminations. Int J Pressure Vessel Piping 2003;80:9–20. [7] Numayer KS, Hadda RH. Analytical solution of buckling of beams with two delaminations. Mech Compos Mater Struct 2001;8:283–97. [8] Sekine H, Hu N, Kouchakzadeh MA. Buckling analysis of elliptically delaminated composite laminates with consideration of partial closure of delamination. J Compos Mater 2000;34(7):551–72. [9] Simitses GJ, Rezaeepazhand J, Sierakowski RL. Scaled models for laminated cylindrical shells subjected to external pressure. Mech Compos Mater Struct 1997;4:267–80. [10] Rezaeepazhand J, Simitses GJ, Starnes Jr JH. Scale models for laminated cylindrical shells subjected to axial compression. Compos Struct 1996;34(4):361–9. [11] Rezaeepazhand J, Simitses GJ, Starnes Jr JH. Design of scaled down models for stability of laminated plates. AIAA J 1995;33(3):515–9. [12] Simitses GJ, Rezaeepazhand J. Structural similitude and scaling laws for crossply laminated plates. J Thermoplast Compos Mater 1995;8:240–51.