Deflection characteristics of cross ply laminates under different loading conditions
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This document summarizes research on the deflection characteristics of cross-ply laminated composite plates under different loading conditions. A layerwise model is used to estimate deflection for 7-ply and 6-ply laminates with sinusoidal, uniformly distributed, and center point loads. Numerical solutions are obtained using MATLAB. Results from the layerwise model are compared to classical laminate plate theory and higher order shear deformation theory, showing good agreement. Parametric studies examine the effects of plate geometry, layering, and material properties on deflection. Tables and figures present sample maximum deflection values and compare the models.
![Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
30 – 31, December 2014, Ernakulam, India
31
DEFLECTION CHARACTERISTICS OF CROSS PLY
LAMINATES UNDER DIFFERENT LOADING
CONDITIONS
Sruthy K.1
, Shashi Dharan2
, Lija M. Paul3
, Subha K4
1, 3
(Civil Engineering, Sree Narayana Gurukulam College of Engineering, Kadayiruppu, Ernakulam, India)
2, 4
(Civil Engineering, NSS College of Engineering, Palakkad, India)
ABSTRACT
Laminated composite plates are nowadays widely used in engineering practice, especially in space, automobile
and civil applications. Composite plates are one of the most important structural elements that were studied by many
researchers in the last 6 or 7 decades. The present study estimates the deflection of cross ply laminated composite plates
under sinusoidal, uniformly distributed and centre point load. A layerwise model proposed by Savithri is used for this
study. Savithri's model considers different displacement function for each layer, which is made possible by incorporating
the Heaviside unit step function. The equations are generated using Principle of virtual work. All numerical solutions are
obtained using originally coded MATLAB programs. The validity of present theory is verified by comparing the results
with those of Classical Laminate Plate Theory and Higher order Shear Deformation Theory. The effects of plate aspect
ratio, number of layers and material properties on the deflection are studied.
Keywords: Classical Laminate Plate Theory, Heaviside Unit Step Function, Higher Order Shear Deformation Theory,
Laminated Composite Plates, Layerwise Model.
1. INTRODUCTION
Composite materials are those formed by combining two or more materials on a macroscopic scale such that
they have better engineering properties than conventional materials. Composite materials are finding increasing
applications in varied field of engineering like aerospace, rotorcraft, automobiles, ship building and other modern
industries. This is largely because many composite materials exhibits high strength-to-weight and stiffness-to-weight
ratios, which make them ideally suited for use in weight sensitive structures. In addition, many of these materials are
more corrosion resistant, high damping resistance, temperature resistant and have low thermal coefficient of expansion
than metals.
In order to obtain accurate result of plate response characteristics a number of contributions based on three
dimensional elasticity theory have been made to analyse laminated composite plates by Pagano[1], Pagano and
Hatfield[2] etc.
Many modeling approaches are available for the analysis of laminated plates. They are Classical Laminated
Plate Theory (CLPT), Higher Order Shear Deformable Theory (HSDT), Layerwise Theory (LWT). Among the many
equivalent single layer theories (ESL), the third order shear deformation theory of Reddy[3] is the most widely accepted
model in the study of laminates. This model cannot represent shear stress continuity at the interfaces and zigzag nature of
the displacement field. To improve the accuracy of transverse shear stress prediction, layer wise theories have been
INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND
TECHNOLOGY (IJCIET)
ISSN 0976 – 6308 (Print)
ISSN 0976 – 6316(Online)
Volume 5, Issue 12, December (2014), pp. 31-44
© IAEME: www.iaeme.com/Ijciet.asp
Journal Impact Factor (2014): 7.9290 (Calculated by GISI)
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IJCIET
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![Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
30 – 31, December 2014, Ernakulam, India
32
proved to be very promising technique. Several Layerwise Model for Laminated Plates have been presented by Mau[4],
Di Seiuva [5], Ren [6] etc.
Researchers like Bhimaraddi, A[7], have worked on the vibration analysis of composite plate. Research work
was done on the buckling analysis Shukla, K., Nath, Y., Kreuzer, E., and Kumar, K[8], flexural analysis subjected to
sinusoidal loading, thermal loading etc Y. M. Ghugal, S. K. Kulkarni [9].
The goal of this work is to present a comparison between the deflection characteristics of cross ply laminates
under different loading conditions (sinusoidal, uniformly distributed and centre point load) and to present a parametric
study. All numerical solutions are obtained using the originally coded MATLAB program. Useful programming
procedures of different finite element models in MATLAB can be found in books of Rudra Pratap [10].
2. PROBLEM FORMULATION
The laminate under consideration is composed of a finite number N of orthotropic layers of constant thickness
h. The plate is referred to a standard Cartesian coordinate system x,y, and z and is subjected to an arbitrary transverse
load q(x,y) on the top surface z=-h/2. The axes of material symmetry are parallel to the plate axes. The material of the
plate is assumed to be linearly elastic and deformations are assumed to be small.
The displacement field taken for the layerwise theory [11] is of the form
(1)
(2)
(3)
where p(z) = z - and H(z) is the Heaviside unit function defined as
H(z- ) = (4)
the summation is extended over N-1 interfaces and Zi is the Z coordinate of the ith
interface, and ri and ti are function of
the material properties of the laminate under consideration.
2.1. Strain Displacement Relations
The normal and shear strain components can be expressed as
(5)
(6)
(7)
(8)
(9)](https://image.slidesharecdn.com/deflectioncharacteristicsofcrossplylaminatesunderdifferentloadingconditions-141216075830-conversion-gate02/85/Deflection-characteristics-of-cross-ply-laminates-under-different-loading-conditions-2-320.jpg)
![Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
30 – 31, December 2014, Ernakulam, India
43
0 10 20 30 40 50 60 70 80 90 100
0
5
10
15
20
25
30
Side to thickness ratio, a/h
Deflection,w
Maximum Cental deflection as a function of side to thickness ratio,CPL,(Model3)
0/90/0
0/90/0/90/0
0/90/0/90/0/90/0
0/90/0/90/0/90/0/90/0
Fig 21: Maximum Central Deflection as a function of side to thickness ratio (symmetric) under CPL
0 10 20 30 40 50 60 70 80 90 100
0
5
10
15
20
25
30
Side to thickness ratio, a/h
Deflection,w
Maximum Cental deflection as a function of side to thickness ratio,CPL,(Model3)
0/90
0/90/0/90
0/90/0/90/0/90
0/90/0/90/0/90/0/90
0/90/0/90/0/90/0/90/0/90
Fig 22: Maximum Central Deflection as a function of side to thickness ratio (antisymmetric) under CPL
4. CONCLUSION
The present work is focused on the deflection of cross ply laminates under different loading conditions. It also
discusses the effect of side to thickness ratio, aspect ratio material anisotropy and number of layers on the deflections.
Higher order Shear Deformation Theory considers the effect of thickness, but a single displacement function is used for
the analysis throughout the thickness. Its results are better than Classical laminate plate theory. Layerwise theory
presented here considers different displacement function for each layer, which is made possible by incorporating the
Heaviside unit step function. Layerwise theory is in close agreement with HSDT under all loading conditions. When
compared with 3D exact solution, it is seen that percentage error in deflection of 5ply laminate under sinusoidal loading
is very small in LWT. Similar trend is observed for other loading conditions.
REFERENCES
[1] Pagano N. J, Exact Solution for Rectangular Bidirectional Composite and Sand Witch Plates, Journal of
Composite Materials, 4, 1970, 20-34.
[2] Pagano N. J and Hatfield S. J., Elastic Behaviour of Multilayered Bidirectional Composites, AIAA Journal,
10, 931-933.
[3] Reddy J.N. , A Simple Higher order Theory for Laminated Composite Plates, Journal of Applilied Mechanics,
51, 1984, 745-752.
[4] Mau S T., A Refined Laminated Plate Theory, Journal of Applilied Mechanics, 40, 1973, 606-607.
[5] Di Sciuva, M., Bending , Vibration and Buckling of Simply supported Thick Multilayered Orthotropic Plates :
An Evaluation of a new displacement model, Journal of Sound and Vibration, 105, 1986, 425- 442.
[6] Ren J G., A new Theory of Laminated Plates, Composite Science and Technology, 26, 1986, 225-239.](https://image.slidesharecdn.com/deflectioncharacteristicsofcrossplylaminatesunderdifferentloadingconditions-141216075830-conversion-gate02/85/Deflection-characteristics-of-cross-ply-laminates-under-different-loading-conditions-13-320.jpg)
![Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
30 – 31, December 2014, Ernakulam, India
44
[7] Bhimaraddi, A. Nonlinear Free Vibration of Laminated Composite Plates, Journal of Engineering Mechanics.,
118(1), 1992, 174–189.
[8] Shukla, K., Nath, Y., Kreuzer, E., and Kumar, K ,Buckling of Laminated Composite Rectangular Plates.,
Journal of Aerospace Engineering,18(4), 2005, 215–223.
[9] P. M. Mohite and C. S. Upadhyay, A study of various modeling approaches for analysis of laminated
composite plates, Comput. Methods Appl. Mech. Engrg., 114, 2005, 307-378.
[10] Rudra Prathap, Getting started with MATLAB, Oxford press, 2010.
[11] Savithri S., Linear and Non linear analysis of Thick Homogeneous and Laminated Plates, PhD Thesis,
Department of Mathemaics, IIT Madras, 1991.
[12] Autar K Kaw, Mechanics of Composite Materials, Taylor & Francis, 2006.
[13] Ansari Fatima-uz-Zehra and S.B. Shinde, “Flexural Analysis of Thick Beams using Single Variable Shear
Deformation Theory”, International Journal of Civil Engineering & Technology (IJCIET), Volume 3, Issue 2,
2012, pp. 292 - 304, ISSN Print: 0976 – 6308, ISSN Online: 0976 – 6316.](https://image.slidesharecdn.com/deflectioncharacteristicsofcrossplylaminatesunderdifferentloadingconditions-141216075830-conversion-gate02/85/Deflection-characteristics-of-cross-ply-laminates-under-different-loading-conditions-14-320.jpg)
Deflection characteristics of cross ply laminates under different loading conditions
- 1. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14) 30 – 31, December 2014, Ernakulam, India 31 DEFLECTION CHARACTERISTICS OF CROSS PLY LAMINATES UNDER DIFFERENT LOADING CONDITIONS Sruthy K.1 , Shashi Dharan2 , Lija M. Paul3 , Subha K4 1, 3 (Civil Engineering, Sree Narayana Gurukulam College of Engineering, Kadayiruppu, Ernakulam, India) 2, 4 (Civil Engineering, NSS College of Engineering, Palakkad, India) ABSTRACT Laminated composite plates are nowadays widely used in engineering practice, especially in space, automobile and civil applications. Composite plates are one of the most important structural elements that were studied by many researchers in the last 6 or 7 decades. The present study estimates the deflection of cross ply laminated composite plates under sinusoidal, uniformly distributed and centre point load. A layerwise model proposed by Savithri is used for this study. Savithri's model considers different displacement function for each layer, which is made possible by incorporating the Heaviside unit step function. The equations are generated using Principle of virtual work. All numerical solutions are obtained using originally coded MATLAB programs. The validity of present theory is verified by comparing the results with those of Classical Laminate Plate Theory and Higher order Shear Deformation Theory. The effects of plate aspect ratio, number of layers and material properties on the deflection are studied. Keywords: Classical Laminate Plate Theory, Heaviside Unit Step Function, Higher Order Shear Deformation Theory, Laminated Composite Plates, Layerwise Model. 1. INTRODUCTION Composite materials are those formed by combining two or more materials on a macroscopic scale such that they have better engineering properties than conventional materials. Composite materials are finding increasing applications in varied field of engineering like aerospace, rotorcraft, automobiles, ship building and other modern industries. This is largely because many composite materials exhibits high strength-to-weight and stiffness-to-weight ratios, which make them ideally suited for use in weight sensitive structures. In addition, many of these materials are more corrosion resistant, high damping resistance, temperature resistant and have low thermal coefficient of expansion than metals. In order to obtain accurate result of plate response characteristics a number of contributions based on three dimensional elasticity theory have been made to analyse laminated composite plates by Pagano[1], Pagano and Hatfield[2] etc. Many modeling approaches are available for the analysis of laminated plates. They are Classical Laminated Plate Theory (CLPT), Higher Order Shear Deformable Theory (HSDT), Layerwise Theory (LWT). Among the many equivalent single layer theories (ESL), the third order shear deformation theory of Reddy[3] is the most widely accepted model in the study of laminates. This model cannot represent shear stress continuity at the interfaces and zigzag nature of the displacement field. To improve the accuracy of transverse shear stress prediction, layer wise theories have been INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND TECHNOLOGY (IJCIET) ISSN 0976 – 6308 (Print) ISSN 0976 – 6316(Online) Volume 5, Issue 12, December (2014), pp. 31-44 © IAEME: www.iaeme.com/Ijciet.asp Journal Impact Factor (2014): 7.9290 (Calculated by GISI) www.jifactor.com IJCIET ©IAEME
- 2. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14) 30 – 31, December 2014, Ernakulam, India 32 proved to be very promising technique. Several Layerwise Model for Laminated Plates have been presented by Mau[4], Di Seiuva [5], Ren [6] etc. Researchers like Bhimaraddi, A[7], have worked on the vibration analysis of composite plate. Research work was done on the buckling analysis Shukla, K., Nath, Y., Kreuzer, E., and Kumar, K[8], flexural analysis subjected to sinusoidal loading, thermal loading etc Y. M. Ghugal, S. K. Kulkarni [9]. The goal of this work is to present a comparison between the deflection characteristics of cross ply laminates under different loading conditions (sinusoidal, uniformly distributed and centre point load) and to present a parametric study. All numerical solutions are obtained using the originally coded MATLAB program. Useful programming procedures of different finite element models in MATLAB can be found in books of Rudra Pratap [10]. 2. PROBLEM FORMULATION The laminate under consideration is composed of a finite number N of orthotropic layers of constant thickness h. The plate is referred to a standard Cartesian coordinate system x,y, and z and is subjected to an arbitrary transverse load q(x,y) on the top surface z=-h/2. The axes of material symmetry are parallel to the plate axes. The material of the plate is assumed to be linearly elastic and deformations are assumed to be small. The displacement field taken for the layerwise theory [11] is of the form (1) (2) (3) where p(z) = z - and H(z) is the Heaviside unit function defined as H(z- ) = (4) the summation is extended over N-1 interfaces and Zi is the Z coordinate of the ith interface, and ri and ti are function of the material properties of the laminate under consideration. 2.1. Strain Displacement Relations The normal and shear strain components can be expressed as (5) (6) (7) (8) (9)
- 3. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14) 30 – 31, December 2014, Ernakulam, India 33 (10) 2.2. Stress - Strain Relationship The stress strain relationship for any layer can be written as, = (11) and = (12) where are plane stress reduced elastic constants in the material axes of the layer under consideration. 2.3. Governing Equations Principle of virtual work is used to derive the equations of motion appropriate for the displacement field and constitutive equations. We have (13) 2.4. Equilibrium Equations + = 0 (14) + = 0 (15) + 2 + = -q (16) =0 (17) (18) where the stress resultant are defined as ( = (19) ( = (20) ( = (21)
- 4. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14) 30 – 31, December 2014, Ernakulam, India 34 ( = (22) ( = (23) Stress resultants are related to the strain by the relations (Aij, Bij, Cij ) = (24) (Dij, Eij, Fij ) = (25) (26) for i, j = 1,2,6 and ( = (27) for i, j = 4,5 2.5. Bending Analysis 2.5.1 Solution Approach To obtain deflections for symmetric and anti-symmetric cross-ply laminates SS-1 boundary conditions are used. The boundary conditions are At x=0 and x=a, v0 = v1 = wo = Nx =Px = 0 At y=0 and y=b, u0 = u1 = wo = Ny =Py = 0 The displacement boundary condition are satisfied by assuming the following shape or displacements = (28) = (29) = (30) = (31) = (32) We assume that the load can be expanded in the series form as q(x,y) = (33) where, Qmn = (34)
- 5. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14) 30 – 31, December 2014, Ernakulam, India 35 Qmn = q0 and m=n=1 for sinusoidal transverse load (35) Qmn = - and m,n =1,3,5,…… for uniformly distributed load (36) Qmn = and m, n = 1,2,3…. for centre point load (37) 3. NUMERICAL EXAMPLES Seven ply (0/90/0/90/0/90/0) and six ply (0/90/0/90/0/90) laminates are analysed under different loading conditions. The layers in both direction are of equal thickness and the edges of different length (b/a=0.5,1,1.5,2,2.5,3,3.5,4,4.5,5) and E1/E2 ratio varies from 5 to 40. The lamina properties are assumed to be E1=25 X 106 psi, E2= 106 psi, G12 = G13 = 0.5 X 106 psi, G23 = 0.2 X 106 psi, ϒ12 = ϒ13 = 0.25 Table 1: Maximum deflection for (0/90/0/90/0/90) laminate for various types of loading (hi= h/6) a/h Source SINUSOIDAL UDL CPL 2 LWT(Present) 5.0979 7.8420 27.6352 HSDT 4.4709 6.7900 26.3994 4 LWT(Present) 1.7760 2.6878 11.1829 HSDT 1.5411 2.3243 10.3492 10 LWT(Present) 0.6802 1.0545 4.0458 HSDT 0.6382 0.9929 3.7705 20 LWT(Present) 0.5167 0.8165 2.6013 HSDT 0.5060 0.8010 2.5037 50 LWT(Present) 0.4706 0.7498 2.1163 HSDT 0.4688 0.7474 2.0972 100 LWT(Present) 0.4640 0.7403 2.0406 HSDT 0.4635 0.7397 2.0356 CLPT 0.4617 0.7371 2.0147 Table 2: Maximum Deflection For (0/90/0/90/0/90/0) Laminate For Various Types Of Loading (h1=h3=h5=h7=h/8, h2 =h4= h6=h/6) a/h Source SINUSOIDAL UDL CPL 2 LWT(Present) 5.2092 7.9545 29.4786 HSDT 4.5395 6.8274 29.0102 4 LWT(Present) 1.7679 2.6644 11.6122 HSDT 1.5353 2.3078 10.9322 10 LWT(Present) 0.6554 1.0139 4.0098 HSDT 0.6142 0.9535 3.7380 20 LWT(Present) 0.4880 0.7699 2.4939 HSDT 0.4774 0.7546 2.3959 50 LWT(Present) 0.4404 0.7009 1.9934 HSDT 0.4387 0.6984 1.9746 100 LWT(Present) 0.4335 0.6910 1.9162 HSDT 0.4331 0.6904 1.9113 CLPT 0.4312 0.6877 1.8899
- 6. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14) 30 – 31, December 2014, Ernakulam, India 36 Table 3: Comparison Of Maximum Deflection For (0/90/0/90/0) Laminate For Sinusoidal Loading (h1=h3=h5=h/6, h2 =h4=h/4) a/h Source SINUSOIDAL % error 2 Exact 5.2949 - LWT(Present) 5.2512 0.825 HSDT 4.7911 9.51 4 Exact 1.8505 - LWT(Present) 1.8151 1.91 HSDT 1.5949 13.81 10 Exact 0.6771 - LWT(Present) 0.6689 1.21 HSDT 0.6269 7.41 20 Exact 0.4938 - LWT(Present) 0.4919 0.38 HSDT 0.4809 2.61 50 Exact 0.4412 - LWT(Present) 0.4410 0.05 HSDT 0.4392 0.45 100 Exact 0.4338 - LWT(Present) 0.4337 0.02 HSDT 0.4332 0.14 CLPT 0.4312 - 0 10 20 30 40 50 60 70 80 90 100 0 1 2 3 4 5 6 Side to thickness ratio, a/h Deflection,w Maximum Cental deflection as a function of side to thickness ratio,0/90/0/90/0/90/0,Sinusoidal Model1 Model2 Model3 Fig 1: Maximum central deflection as a function of side to thickness ratio (0/90/0/90/0/90/0) under sinusoidal loading 0 10 20 30 40 50 60 70 80 90 100 0 1 2 3 4 5 6 Side to thickness ratio, a/h Deflection,w Maximum Cental deflection as a function of side to thickness ratio,0/90/0/90/0/90,Sinusoidal Model1 Model2 Model3 Fig 2: Maximum central deflection as a function of side to thickness ratio (0/90/0/90/0/90) under sinusoidal loading
- 7. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14) 30 – 31, December 2014, Ernakulam, India 37 From the above figures (Fig 1 and Fig 2) it is observed that CLPT under predicts the deflection. Similar trend is observed for UDL and CPL and are plotted in Fig.7, Fig.8, Fig.13 and Fig.14 respectively. 5 10 15 20 25 30 35 40 0 0.5 1 1.5 2 2.5 3 E1/E2 Deflection,w Maximum Cental deflection as a function of E1/E2,0/90/0/90/0/90/0,Sinusoidal Model1 Model2(a/h=4) Model3(a/h=4) Fig 3: Maximum central deflection as a function of e1/e2 ratio (0/90/0/90/0/90/0) under sinusoidal loading 5 10 15 20 25 30 35 40 0 0.5 1 1.5 2 2.5 3 E1/E2 Deflection,w Maximum Cental deflection as a function of E1/E2,0/90/0/90/0/90,Sinusoidal Model1 Model2(a/h=4) Model3(a/h=4) Fig 4: Maximum central deflection as a function of e1/e2 ratio (0/90/0/90/0/90) under sinusoidal loading The above figures (Fig 3 and Fig 4) show the effect of material anisotropy on the deflection. The results predicted by HSDT and Layerwise model are close. The dependence of the coupling effect on the modulus ratio is illustrated. Similar trend is observed for UDL and CPL and are plotted in Fig 9, Fig 10, Fig 15 and Fig 16 respectively. 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 b/a Deflection,w Maximum Cental deflection as a function of b/a,0/90/0/90/0/90/0,sinusoidal Model1 Model2(a/h=4) Model3(a/h=4) Fig 5: Maximum central deflection as a function of b/a ratio (0/90/0/90/0/90/0) under sinusoidal loading
- 8. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14) 30 – 31, December 2014, Ernakulam, India 38 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 b/a Deflection,w Maximum Cental deflection as a function of b/a,0/90/0/90/0/90,sinusoidal Model1 Model2(a/h=4) Model3(a/h=4) Fig 6: Maximum central deflection as a function of b/a ratio (0/90/0/90/0/90) under sinusoidal loading From Fig 5 and Fig 6 it is noted that as the aspect ratio increases, deflection also increases. There is hardly any change in deflection, once the aspect ratio reaches a value of 2.5. Similar trend is observed for UDL are plotted in Fig 11 and Fig 12 respectively. 0 10 20 30 40 50 60 70 80 90 100 0 1 2 3 4 5 6 7 8 Side to thickness ratio, a/h Deflection,w Maximum Cental deflection as a function of side to thickness ratio,0/90/0/90/0/90/0,UDL Model1 Model2 Model3 ansys Fig 7: Maximum central deflection as a function of side to thickness ratio (0/90/0/90/0/90/0) under UDL 0 10 20 30 40 50 60 70 80 90 100 0 1 2 3 4 5 6 7 8 Side to thickness ratio, a/h Deflection,w Maximum Cental deflection as a function of side to thickness ratio,0/90/0/90/0/90,UDL Model1 Model2 Model3 Fig 8: Maximum central deflection as a function of side to thickness ratio (0/90/0/90/0/90) under UDL
- 9. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14) 30 – 31, December 2014, Ernakulam, India 39 5 10 15 20 25 30 35 40 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 E1/E2 Deflection,w Maximum Cental deflection as a function of E1/E2,0/90/0/90/0/90/0,UDL Model1 Model2(a/h=4) Model3(a/h=4) Fig 9: Maximum central deflection as a function of e1/e2 ratio (0/90/0/90/0/90/0) under UDL 5 10 15 20 25 30 35 40 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 E1/E2 Deflection,w Maximum Cental deflection as a function of E1/E2,0/90/0/90/0/90,UDL Model1 Model2(a/h=4) Model3(a/h=4) Fig 10: Maximum central deflection as a function of e1/e2 ratio (0/90/0/90/0/90) under UDL 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 b/a Deflection,w Maximum Cental deflection as a function of b/a,0/90/0/90/0/90/0,UDL Model1 Model2(a/h=4) Model3(a/h=4) Fig 11: Maximum central deflection as a function of b/a ratio (0/90/0/90/0/90/0) under UDL
- 10. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14) 30 – 31, December 2014, Ernakulam, India 40 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 b/a Deflection,w Maximum Cental deflection as a function of b/a,0/90/0/90/0/90,UDL Model1 Model2(a/h=4) Model3(a/h=4) Fig 12: Maximum central deflection as a function of b/a ratio (0/90/0/90/0/90) under UDL 0 10 20 30 40 50 60 70 80 90 100 0 5 10 15 20 25 30 Side to thickness ratio, a/h Deflection,w Maximum Cental deflection as a function of side to thickness ratio,0/90/0/90/0/90/0,Centre Point Load Model1 Model2 Model3 Fig 13: Maximum central deflection as a function of side to thickness ratio (0/90/0/90/0/90/0) under CPL 0 10 20 30 40 50 60 70 80 90 100 0 5 10 15 20 25 30 Side to thickness ratio, a/h Deflection,w Maximum Cental deflection as a function of side to thickness ratio,0/90/0/90/0/90,Centre Point Load Model1 Model2 Model3 Fig 14: Maximum central deflection as a function of side to thickness ratio (0/90/0/90/0/90) under cpl
- 11. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14) 30 – 31, December 2014, Ernakulam, India 41 5 10 15 20 25 30 35 40 0 2 4 6 8 10 12 14 16 18 20 E1/E2 Deflection,w Maximum Cental deflection as a function of E1/E2,0/90/0/90/0/90/0,Centre Point Load Model1 Model2(a/h=4) Model3(a/h=4) Fig 15: Maximum central deflection as a function of e1/e2 ratio (0/90/0/90/0/90/0) under CPL 5 10 15 20 25 30 35 40 0 2 4 6 8 10 12 14 16 18 20 E1/E2 Deflection,w Maximum Cental deflection as a function of E1/E2,0/90/0/90/0/90,Centre Point Load Model1 Model2(a/h=4) Model3(a/h=4) Fig 16: Maximum central deflection as a function of e1/e2 ratio (0/90/0/90/0/90) under CPL 0 10 20 30 40 50 60 70 80 90 100 0 1 2 3 4 5 6 Side to thickness ratio, a/h Deflection,w Maximum Cental deflection as a function of side to thickness ratio,Sinusoidal,(Model3) 0/90/0 0/90/0/90/0 0/90/0/90/0/90/0 0/90/0/90/0/90/0/90/0 Fig 17: Maximum Central Deflection as a function of side to thickness ratio (symmetric) under Sinusoidal load
- 12. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14) 30 – 31, December 2014, Ernakulam, India 42 0 10 20 30 40 50 60 70 80 90 100 0 1 2 3 4 5 6 Side to thickness ratio, a/h Deflection,w Maximum Cental deflection as a function of side to thickness ratio,Sinusoidal,(Model3) 0/90 0/90/0/90 0/90/0/90/0/90 0/90/0/90/0/90/0/90 0/90/0/90/0/90/0/90/0/90 Fig 18: Maximum Central Deflection as a function of side to thickness ratio (anisymmetric) under Sinusoidal load It is noted from the above figures (Fig 17 and Fig 18) that, as the number of ply increases, deflection decreases. Similar in the case of UDL and CPL (Fig 19 to Fig 22). 0 10 20 30 40 50 60 70 80 90 100 0 1 2 3 4 5 6 7 8 9 Side to thickness ratio, a/h Deflection,w Maximum Cental deflection as a function of side to thickness ratio,UDL,(Model3) 0/90/0 0/90/0/90/0 0/90/0/90/0/90/0 0/90/0/90/0/90/0/90/0 Fig 19: Maximum Central Deflection as a function of side to thickness ratio (symmetric) under UDL 0 10 20 30 40 50 60 70 80 90 100 0 1 2 3 4 5 6 7 8 Side to thickness ratio, a/h Deflection,w Maximum Cental deflection as a function of side to thickness ratio,UDL,(Model3) 0/90 0/90/0/90 0/90/0/90/0/90 0/90/0/90/0/90/0/90 0/90/0/90/0/90/0/90/0/90 Fig 20: Maximum Central Deflection as a function of side to thickness ratio (antisymmetric) under UDL
- 13. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14) 30 – 31, December 2014, Ernakulam, India 43 0 10 20 30 40 50 60 70 80 90 100 0 5 10 15 20 25 30 Side to thickness ratio, a/h Deflection,w Maximum Cental deflection as a function of side to thickness ratio,CPL,(Model3) 0/90/0 0/90/0/90/0 0/90/0/90/0/90/0 0/90/0/90/0/90/0/90/0 Fig 21: Maximum Central Deflection as a function of side to thickness ratio (symmetric) under CPL 0 10 20 30 40 50 60 70 80 90 100 0 5 10 15 20 25 30 Side to thickness ratio, a/h Deflection,w Maximum Cental deflection as a function of side to thickness ratio,CPL,(Model3) 0/90 0/90/0/90 0/90/0/90/0/90 0/90/0/90/0/90/0/90 0/90/0/90/0/90/0/90/0/90 Fig 22: Maximum Central Deflection as a function of side to thickness ratio (antisymmetric) under CPL 4. CONCLUSION The present work is focused on the deflection of cross ply laminates under different loading conditions. It also discusses the effect of side to thickness ratio, aspect ratio material anisotropy and number of layers on the deflections. Higher order Shear Deformation Theory considers the effect of thickness, but a single displacement function is used for the analysis throughout the thickness. Its results are better than Classical laminate plate theory. Layerwise theory presented here considers different displacement function for each layer, which is made possible by incorporating the Heaviside unit step function. Layerwise theory is in close agreement with HSDT under all loading conditions. When compared with 3D exact solution, it is seen that percentage error in deflection of 5ply laminate under sinusoidal loading is very small in LWT. Similar trend is observed for other loading conditions. REFERENCES [1] Pagano N. J, Exact Solution for Rectangular Bidirectional Composite and Sand Witch Plates, Journal of Composite Materials, 4, 1970, 20-34. [2] Pagano N. J and Hatfield S. J., Elastic Behaviour of Multilayered Bidirectional Composites, AIAA Journal, 10, 931-933. [3] Reddy J.N. , A Simple Higher order Theory for Laminated Composite Plates, Journal of Applilied Mechanics, 51, 1984, 745-752. [4] Mau S T., A Refined Laminated Plate Theory, Journal of Applilied Mechanics, 40, 1973, 606-607. [5] Di Sciuva, M., Bending , Vibration and Buckling of Simply supported Thick Multilayered Orthotropic Plates : An Evaluation of a new displacement model, Journal of Sound and Vibration, 105, 1986, 425- 442. [6] Ren J G., A new Theory of Laminated Plates, Composite Science and Technology, 26, 1986, 225-239.
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