# Using MATLAB to Design and Analyse Composite Laminates

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1 Engineering, 2010, 2, doi: /eng Published Online November 2010 ( Using MATLAB to Design and Analyse Composite Laminates Abstract Avinash Ramsaroop, Krishnan Kanny * Department of Mechanical Engineering, Durban University of Technology, Durban, South Africa Received July 3, 2010; revised August 5, 2010; accepted August 19, 2010 This work deals with the generation of MATLAB script files that assists the user in the design of a composite laminate to operate within safe conditions. The inputs of the program are the material properties, material limits and loading conditions. Equations based on Hooke s Law for two-dimensional composites were used to determine the global and local stresses and strains on each layer. Failure analysis of the structure was performed via the Tsai-Wu failure theory. The output of the program is the optimal number of fibre layers required for the composite laminate, as well as the orientation of each layer. Keywords: Composite Laminates, Hooke s Law, Fibre Angle, MATLAB, Composite Design 1. Introduction Composites are etremely versatile materials and may be tailored to suit any function. They have found uses from the aerospace industry to common everyday applications. However, one drawback of these materials is tedious design processes [1]. Therefore, in an attempt to reduce this time consuming phase, it was decided to develop a computer program that assisted the user in designing a composite structure. The program needed to perform the necessary calculations in the fraction of the time it would take if done using conventional techniques. Conventional methods for designing composite structures involve the use of Hooke s law for two-dimensional unidirectional composites. Equations relating the stresses and strains in these materials have been developed and are available from various tets [2-4]. These equations, however, are limited to flat unidirectional laminates. The procedure to follow is quite laborious. The material properties, material limits, number of fibre layers, and the fibre orientation and thickness of each layer as well as the loading conditions need to be known. These quantities are then used in the governing equations and numerous matri computations are required. The outputs from these equations are the global and local stresses and strains of each layer. The local stresses on each layer are then compared to the allowable material limits and, if they eceed these limits, the entire process needs to be repeated with new angles and, if necessary, more fibre layers. If a composite structure has to be designed to optimally carry the applied loads, the manual calculations would take an etremely long time. Designers usually use standard fibre angles in their design to overcome the time factor because eact fibre orientation is diffic [5]. As a res composite structures are usually overdesigned as incorrect fibre angles may mean more fibre layers which in turn leads to increased material costs. Many researchers [6-9] have attempted to reduce material costs by reducing the laminate thickness but these were based on fied fibre angles. Certain regions of a composite structure, such as holes and other high stress zones, may need more reinforcement than others. Only these areas would require more fibre layers. However, with the conventional approach to design, the entire structure would be laid up to compensate for the high stress concentrations which increases the material costs. Therefore an approach was needed where the calculations can be performed easily and automatically, and where only the regions with high stresses have etra reinforcement. A logical solution was to write computer programs to perform the computations. MATLAB, a numerical computing package, was used as a basis for the programs as it is more than sufficient to handle the numerous matri computations [10-14]. It may also be interfaced with programs written in other languages, such as C, C++, and FORTRAN [13-14]. Basically, two programs were generated and each program consisted of a main script file and various functions. The first program was written according to the conventional methods for designing composite structures. The purpose of this program was to determine whether its

2 A. RAMSAROOP ET AL. 905 functions operated efficiently and performed calculations correctly as these would be used for the second program. Ress from this program were compared to manually calculated eamples. The second program used a different approach where the number of inputs were reduced to just the material properties, material limits and applied loading conditions. The outputs of this program are the number of layers and the fibre angle of each layer. These programs are discussed in further detail after the equations governing the stress-strain relationships are eamined. 2. Hooke s Law for Two-Dimensional Composite Structures Figure 1 shows a schematic representation of a composite lamina. The direction along the fibre ais is designated 1. The direction transverse to the fibre ais but in the plane of the lamina is designated 2. The direction transverse to both the fibre ais and the plane of the lamina (out of page) is designated 3. This direction is not shown in the figure as it only becomes necessary in three-dimensional cases. The 1-2 co-ordinate system can be considered to be local co-ordinates based on the fibre direction. However this system is inadequate as fibres can be placed at various angles with respect to each other and the structure. Therefore a new co-ordinate system needs to be defined that takes into account the angle the fibre makes with its surroundings. This new system is referred to as global co-ordinates (-y system) and is related to the local co-ordinates (1-2 system) by the angle θ. A composite material is not isotropic and therefore its stresses and strains cannot be related by the simple Hooke s Law ( E ). This law has to be etended to two-dimensions and redefined for the local and global co-ordinate systems [2]. The res is Equations (1) and (2). Figure 1. Global co-ordinate system in relation to local co-ordinate system. Q Q Q12 Q Q Q11 Q12 Q 16 y Q12 Q22 Q26 y y Q16 Q26 Q 66 y where σ 1,2 are the normal stresses in directions 1 and 2; τ 12 is the shear stress in the 1-2 plane; ε 1,2 are the normal strains in directions 1 and 2; γ 12 is the shear strain in the 1-2 plane; [Q] is the reduced stiffness matri; σ,y are the normal stresses in directions and y; τ y is the shear stress in the -y plane; ε,y are the normal strains in directions and y; γ y is the shear strain in the -y plane; [ Q ] is the transformed reduced stiffness matri. The elements of the [Q] matri in Equation (1) are dependent on the material constants and may be calculated using Equation (3). E E E Q, Q, Q, Q G (1) (2) (3) where E 1,2 are Young s modulus in directions 1 and 2; G 12 is the shear modulus in the 1-2 plane; ν 12,21 are Poisson s ratios in the 1-2 and 2-1 planes. The [ Q ] matri in Equation (2) may be determined by Equation (4). 1 1 Q T QRTR (4) where [T] is the transformation matri; [R] is the Reuter matri. These matrices are given by: 2 2 c s 2sc T s c 2sc and R (5) 2 2 sc sc c s where c = cosθ; s = sinθ. The local stresses and strains in Equation (1) are related to the global stresses and strains in Equation (2) by Equation (6) y T 2 and 2 RTR y y y Equations (1) to (6) are used to determine the stresses and strains for a single composite layer. Since composites are mi-layered entities, equations for this case must also be set up. The res is Equation (7). 0 N A B M B D (7) (6)

3 906 A. RAMSAROOP ET AL. where N is the vector of resant forces; M is the vector of resant moments; ε 0 is the vector of the mid-plane strains; κ is the vector of mid-plane curvatures. Vectors ε and κ are related to the global co-ordinates by Equation (8). 0 0 y y z y (8) 0 y y y where z is an arbitrary distance from the mid-plane. The [A], [B], and [D] matrices in Equation (7) are known as the etensional, coupling, and bending stiffness matrices, respectively. The elements of these matrices may be determined from Equations (9) to (11). n 1 A Q h h i 1, 2, 3 j 1, 2, 3 (9) ij ij k k k 1 k n 1 B Q h h i 1, 2, 3 j 1, 2, 3 (10) ij ij k k 2 k 1 k n 1 Dij Qij hk hk i 1, 2, 3 j 1, 2, 3 (11) 3 k 1 k where n is the number of layers; Qij is the i-th, k j-th element of the [ Q ] matri of the k-th layer; h k is the distance of the top or bottom of the k-th layer from the mid-plane of the composite. Figure 2 illustrates how to determine the distance h k from the mid-plane. The final aspect involved in the design of a composite structure is the failure analysis. There are various failure theories; however, the Tsai-Wu failure criterion is only one that closely correlates with eperimental data. This failure theory is given by Equation (12). H H H H H H 2H (12) The parameters for the Tsai-Wu failure criterion are given by Equation (13) H1 ; H2 ; H6 0 T C T C H11 ; H22 ; H66 T C T C H12 T C T C (13) T where 12, are the imate tensile stresses in direction 1 and 2; 12, are the imate compressive C stresses in direction 1 and 2; 12 is the imate shear stress in the 1-2 plane. In order to better facilitate the use of this failure theory, each stress component of Equation (12) was miplied by a variable [2]. This variable is referred to as the strength ratio (SR) and combining this in Equation (12) resed in Equation (14). ( H11 H22 H612) SR (14) ( H111 H222 H H1212 ) SR 1 The purpose of SR is to directly determine by what ratio the local stresses must be increased or decreased to avoid failure. This also directly relates to the applied forces. The criterion for SR is that it can only be positive. If SR is less than 1, then failure occurs because it means that the loading needs to decrease to avoid failure. A SR value of 1 implies that the composite structure is perfectly suited for the applied loading conditions. A value of greater than 1 means that the structure is more than capable of carrying the applied loading and that the loading may also be increased. For eample, a SR value of 1.5 means that the loading may be increased up to 50% without failure occurring. Figure 2. Locations of layers in a composite structure [2].

5 908 A. RAMSAROOP ET AL. thickness of each layer and/or the number of layers. The choice of values may be uncertain and values would have to be assumed. Furthermore, it would also be an uncertainty on which values to change. Much time may be spent in achieving a structure that would not fail. On the other hand, if failure does not occur, the structure may prove to be overdesigned in which case the manufacturing costs of the structure may increase as unnecessary material will be used. To overcome these problems, a new script file was generated. Table 1. Material properties, limits and loading conditions for graphite/epoy composite eample in Kaw [2]. Material Properties Material Limits Forces Moments T C T C E 1 E 2 G ν (GPa) (GPa) (GPa) 12 (MPa) (MPa) (MPa) (MPa) (MPa) N M Table 4.1 Global Strains (m/m) in Eample 4.3 Ply # Position ε ε y γ y Top (10 8 ) (10 6 ) (10 6 ) 1 (0 o ) Middle (10 7 ) (10 6 ) (10 6 ) Bottom (10 7 ) (10 6 ) (10 6 ) Top (10 7 ) (10 6 ) (10 6 ) 2 (30 o ) Middle (10 7 ) (10 6 ) (10 7 ) Bottom (10 7 ) (10 6 ) (10 7 ) Top (10 7 ) (10 6 ) (10 7 ) 3 ( 45 o ) Middle (10 7 ) (10 6 ) (10 6 ) Bottom (10 7 ) (10 6 ) (10 6 ) Figure 3. Table from eample in Kaw [2] showing global strains. Table 4.2 Global Stresses (Pa) in Eample 4.3 Ply # Position y y Top (10 4 ) (10 4 ) (10 4 ) 1 (0 o ) Middle (10 4 ) (10 4 ) (10 4 ) Bottom (10 4 ) (10 4 ) (10 4 ) Top (10 4 ) (10 4 ) (10 4 ) 2 (30 o ) Middle (10 5 ) (10 4 ) (10 4 ) Bottom (10 5 ) (10 4 ) (10 4 ) Top (10 5 ) (10 5 ) (10 5 ) 3 ( 45 o ) Middle (10 4 ) (10 4 ) (10 4 ) Bottom (10 4 ) (10 4 ) (10 4 ) Figure 4. Table from eample in Kaw [2] showing global stresses. Table 4.3 Local Strains (m/m) in Eample 4.3 Ply # Position Top (10 8 ) (10 6 ) (10 6 ) 1 (0 o ) Middle (10 7 ) (10 6 ) (10 6 ) Bottom (10 7 ) (10 6 ) (10 6 ) Top (10 7 ) (10 6 ) (10 6 ) 2 (30 o ) Middle (10 7 ) (10 6 ) (10 6 ) Bottom (10 6 ) (10 6 ) (10 6 ) Top (10 6 ) (10 6 ) (10 6 ) 3 ( 45 o ) Middle (10 7 ) (10 6 ) (10 6 ) Bottom (10 7 ) (10 6 ) (10 7 ) Figure 5. Table from eample in Kaw [2] showing local strains.

6 A. RAMSAROOP ET AL. 909 Table 4.4 Local Stresses (Pa) in Eample 4.3 Ply # Position Top (10 4 ) (10 4 ) (10 4 ) 1 (0 o ) Middle (10 4 ) (10 4 ) (10 4 ) Bottom (10 4 ) (10 4 ) (10 4 ) Top (10 4 ) (10 4 ) (10 4 ) 2 (30 o ) Middle (10 5 ) (10 4 ) (10 4 ) Bottom (10 5 ) (10 4 ) (10 4 ) Top (10 5 ) (10 4 ) (10 4 ) 3 ( 45 o ) Middle (10 4 ) (10 4 ) (10 3 ) Bottom (10 4 ) (10 4 ) (10 3 ) Figure 6. Table from eample in Kaw [2] showing local stresses. Figure 7. Global strains from Conventional Approach program with Table 1 as inputs. Figure 8. Global stresses from Conventional Approach program with Table 1 as inputs.

7 910 A. RAMSAROOP ET AL. Figure 9. Local strains from Conventional Approach program with Table 1 as inputs. Figure 10. Local stresses from Conventional Approach program with Table 1 as inputs Fibre-Angle-Output Program Initially it was decided that this program should take the applied loading conditions, material properties and material limits as inputs. The program was to assume that the number of layers equals the number of input forces, and that the fibre angle of each layer equals the direction of each force. It would then perform all the necessary matri calculations as in the Conventional Approach. The global and local stresses and strains would be determined as before and failure analysis with the Tsai-Wu criterion may be performed. However, the assumptions made above cannot be used in a real situation as it may res in an overdesign of a composite structure as in the case of the conventional approach. Hence a program was written that does not make any assumptions but rather builds the laminate up one layer at a time. Basically this program varies the fibre angles to achieve a composite structure that is perfectly suited to carry the applied loading conditions. The

8 A. RAMSAROOP ET AL. 911 Figure 11. SR ratios from Conventional Approach program with Table 1 as inputs. outputs of the program are the number of layers and the fibre angle of each layer. The program asks the user to input the following: material properties material limits loading conditions The [Q] matri and the Tsai-Wu parameters can be calculated immediately as these do not vary with fibre angle but rather with material properties and limits. The program begins with one layer at an angle of 0 and computes h k, as well as the [T], [ Q ], [A], [B] and [D] matrices. Thereafter the mid-plane strains and curvatures, and, the global and local stresses and strains are calculated. The Tsai-Wu failure theory is applied and a value for SR is obtained for each layer. The program then analyzes the SR values and confirms whether it is in a certain range. The lower limit of this range is 1 as any value below this would mean failure. The upper limit in this range is 1.2 and this is to avoid overdesigning. If the SR value for each layer is between 1 and 1.2, then the design of the composite structure is complete. On the other hand, if the SR value for any of the layers is out of this range, the program varies that particular fibre angle to obtain a SR value between 1 and 1.2. However, due to the angle change all the matri calculations have to be redone. A new SR value is obtained for each layer and this is compared to the old values. If the new SR value for a particular layer is higher than the old SR value, then the angle of that layer is changed to the new one. For eample, after the initial calculation a layer with fibre angle of 30 has a SR value of 0.8. This is not acceptable and the program changes the angle to obtain a better SR value. Say at 35 the SR value is 0.9, the program will then change the angle of the 30 layer to 35. If at 35 the SR value was 0.7, then the program will retain the 30 angle. The program continues to change angles and compare old and new SR values until the SR value of each layer is between 1 and 1.2, in which case the design is completed. However, if this range cannot be achieved, a new layer is added on. The program starts from the beginning and recalculates the various matrices. SR values are obtained and the comparison between new and old values resumes. New layers will be added on until the SR value for each layer is between 1 and 1.2. The program then outputs the number of layers and the fibre angle of each layer. Figure 12 illustrates the procedure the script file follows to obtain the desired outputs. The Fibre-Angle-Output program was subjected to several runs. The ress, namely the number of layers and the fibre angles, needed to be verified. Therefore these were used as inputs in the Conventional Approach program. The Tsai-Wu failure theory was used and in each case there was no failure. As an eample, consider the graphite/epoy composite laminate eamined earlier. The material properties, material limits and loading conditions for this laminate were given in Table 1. The SR values for three layers with fibre angles of 0, 30 and 45 were shown in Figure 11. The values in Table 1

9 912 A. RAMSAROOP ET AL Obtain material properties and allowable limits Calculate [Q], [S] matrices Parameters for Tsai-Wu 5 4 Enter magnitude and angle of each force Resant forces and moments 6 Calculate all parameters: [T], [[Q Q ], h, z, z, A, A, B, B, D D mid-plane strains and curvatures global and local stresses and strains 12 Add new layer NO 11 Use Tsai-Wu failure criterion and find SR 10 Change angles YES Can SR be improved 9 NO 7 Use Tsai-Wu failure criterion and find SR 8 Is SR within acceptable limits YES 13 Output angles Figure 12. Flowchart describing the Fibre-Angle-Output program. were input into the Fibre-Angle-Output program. The output of the program is shown in Figure 13. It can be seen from Figure 13 that the number of layers is the same as when the eample was analysed by the Conventional Approach program; however, the fibre angles are different. Here the required angles are 49, 79 and 49 instead of the previous 0, 30 and 45. Further the SR values are all between 1 and 1.2 indicating no failure or overdesign. The new fibre angles (49, 79 and 49 ) along with the material properties, material limits and loading conditions in Table 1 were then input into the Conventional Approach program. The resing SR values were eactly the same as in Figure 13. Further, the local stresses of the new laminate, shown in Figure 14, was compared with the local stresses of the earlier laminate, shown in Figure 10. It can be seen that the local stresses in Figure 14 are constant through the thickness of each layer. This shows that the stresses were evenly distributed and not varied through the thickness of each layer. Previously the SR value on the first layer was less than 1 (Figure 11) and lower than the other two layers indicating earliest failure. However with the new layup, the SR value has increased to a value greater than one. This means that the first layer is now capable of carrying a higher load and this is evident by the values for the local stresses. The previous value at the top of the first layer was 33.5 MPa (Figure 10), and the new value is 75.7 MPa (Figure 14). This clearly demonstrates the higher load bearing capability of the new laminate. Similar trends may be seen from the other positions and from the other two layers.

10 A. RAMSAROOP ET AL. 913 Figure 13. Output of Fibre-Angle-Output program with Table 1 as input. Figure 14. Local stresses for laminate with angles of 49, 79, and 49. It was then decided to increase the loading conditions of the 49, 79 and 49 laminate to determine whether an increase in load will res in failure. The applied loads were increased by 10% and the resing SR values are shown in Figure 15. It can clearly be seen that all the SR values for all the layers were less than 1. An increase in loading resed in failure and therefore it may be deduced that the Fibre-Angle-Output program optimally designed the composite laminate for the loading conditions in Table 1. This further implies a saving in material cost as there would be no etra material used because of overdesigning.

12 A. RAMSAROOP ET AL. 915 Figure 16. Fringe plots from Patran showing stress components for 0, 30, and 45 laminate. Figure 17. Fringe plots from Patran showing stress components for 49, 79, and 49 laminate.

13 916 A. RAMSAROOP ET AL. as the fibre angle of each layer was determined. These outputs were input into the Conventional Approach program in order to verify the ress. In each case the resing composite structure did not fail. The eample used showed that the Fibre-Angle-Output program optimally designed the composite laminate for the given loading conditions. Finite element analyses were conducted to verify this conclusion. The use of these programs will greatly reduce the design time of composite structures as the numerous computations are completed in a fraction of the time that it would take if done manually. The designer merely has to run the Fibre-Angle-Output program, and view the number of layers required and the fibre angle for each layer. The program will also aid in reducing material costs. Presently the program is limited to flat unidirectional structures but work is underway to etend it to three dimensions. There is no limit on the type of composite material as far as the matri and fibre reinforcement is concerned. 5. References [1] S. K. Mazumdar, Composite Manufacturing: Materials, Product, and Process Engineering, CRC Press LLC, Boca Raton, [2] A. K. Kaw, Mechanics of Composite Materials, CRC Press LLC, Boca Raton, [3] C. T. Herakovich, Mechanics of Fibrous Materials, John Wiley & Sons Inc., Hoboken, [4] R. M. Christensen, Mechanics of Composite Materials, Dover Publications, New York, [5] M. Akbulut and F. O. Sonmez, Optimum Design of Composite Laminates for Minimum Thickness, Computers and Structures, Vol. 86, No , 2008, pp [6] K. Sivakumar, N. G. R. Iyengar and K. Deb, Optimum Design of Laminated Composite Plates with Cutouts Using a Genetic Algorithm, Composite Structures, Vol. 42, No. 3, 1998, pp [7] S. A. Barakat and G. A. Abu-Farsakh, The Use of an Energy-Based Criterion to Determine Optimum Configurations of Fibrous Composites, Composite Science and Technology, Vol. 59, No. 12, 1999, pp [8] P. Kere, M. Lyly and J. Koski, Using Micriterion Optimization for Strength Design of Composite Laminates, Composite Structures, Vol. 62, No. 3-4, 2003, pp [9] C. H. Park, W. I. Lee, W. S. Han and A. Vautrin, Weight Minimization of Composite Laminated Plates with Miple Constraints, Composite Science and Technology, Vol. 63, No. 7, 2003, pp [10] MediaWiki, MATLAB, wiki/matlab [11] R. V. Dukkipati, MATLAB for Mechanical Engineers, New Age International (P) Limited, New Delhi, [12] G. Z. Voyiadjis and P. I. Kattan, Mechanics of Composite Materials with MATLAB, Springer Netherlands, Dordrecht, [13] A. Gilat, MATLAB: An Introduction with Applications, John Wiley & Sons Inc., Hoboken, [14] B. D. Hahn and D. T. Valentine, Essential MATLAB for Engineers and Scientists, Butterworth-Heinemann, Rome, 2007.

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