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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL.

40, 51—60 (1997)

IMPROVED TRANSVERSE SHEAR STRESSES IN COMPOSITE FINITE ELEMENTS BASED ON FIRST ORDER SHEAR DEFORMATION THEORY R. ROLFES AND K. ROHWER

German Aerospace Research Establishment (D¸R), Institute of Structural Mechanics, P.O. Box 32 67, D-38022 Braunschweig, Germany

SUMMARY A method for calculating improved transverse shear stresses in laminated composite plates, which bases on the first-order shear deformation theory is developed. In contrast to many recently established methods, either higher-order lamination theories or layerwise theories, it is easily applicable to finite elements, since only C0-continuity is necessary and the numerical effort is low. The basic idea is to calculate the transverse shear stresses directly from the transverse shear forces by neglecting the influence of the membrane forces and assuming two cylindrical bending modes. Shear correction factors are no longer required, since the transverse shear stiffnesses are also provided. Numerical examples for symmetric cross-ply and antisymmetric angle-ply laminates show the superiority of the method against using shear correction factors. Furthermore, results obtained with MSC/NASTRAN, which uses a similar but simplified approach, are surpassed. KEY WORDS :

fiber composite; laminate; shear deformation theory; transverse shear stiffness; interlaminar shear; finite shell elements

1. INTRODUCTION The failure of composites, especially under impact loading, has been subject to intensive research since many years. It turned out, that important failure modes like delamination are driven by transverse shear as well as normal stresses. Hence, modern failure criteria for composite materials account for the full stress tensor.1 However, determining the full state of stress in composite structures is much more involved than in homogeneous isotropic materials. The workhorses for the stress analysis of composite plates have been the Classical Lamination Theory (CLT) and the Whitney—Pagano Theory,2 the latter also known as the First-Order Shear Deformation Theory (FSDT). In the framework of finite element analysis the FSDT is much more popular, since only C0- instead of C1- continuity is necessary. Transverse shear stresses can be calculated from the FSDT using the material law. However, the assumption of a constant shear angle in thickness direction results in layerwise constant stresses. This is quite unsatisfactory, since the condition of vanishing transverse shear stresses at the boundaries is not fulfilled and the interlaminar shear stresses at the interfaces cannot be assigned a distinct value. Both deficiencies can be avoided by applying the equilibrium conditions according to Pryor and Barker.3 This procedure utilizes the first derivatives of the membrane stresses w.r.t the membrane co-ordinates. Pryor and Barker choose shape functions for the displacements (u, v, w) and the transverse shear strains. Thus, they need cubic shape functions for w in order to calculate the required stress

CCC 0029—5981/97/010051—10 ( 1997 by John Wiley & Sons, Ltd.

Received 16 January 1995 Revised 23 January 1996

52

R. ROLFES AND K. ROHWER

derivatives (element with 28 degrees of freedom). The order can be reduced by one if the rotations instead of the transverse shear strains are selected as variables. However, still quadratic shape functions must be employed. Furthermore, comparisons with exact analytical solutions revealed, that the resulting transverse shear stresses from the FSDT are not always satisfactory (e.g. Reference 17). Byun and Kapania4 suggested a postprocessing using global interpolating functions for smoothing the membrane displacements. By that means they achieved good transverse shear and even transverse normal stresses. However, the global functions make the method geometry dependent and therefore impractical for general finite element codes. A variety of higher order lamination theories has been proposed during the last decade in order to improve the transverse shear stress calculation (e.g. References 5—9). Based on some of those, Manjunatha and Kant10,11 and Kant and Menon12 calculated transverse stresses by use of the equilibrium conditions. They achieved good finite element results, however, due to the higher order displacement derivatives involved, they needed biquadratic and bicubic shape functions for evaluating the transverse shear and normal stresses, respectively. Engblom and Ochoa13,14 modified the displacement field and received satisfactory results for transverse shear and normal stresses in elements with only 32 (FSDT) and 40 (theory with quadratic in-plane and constant out-of-plane displacements) degrees of freedom. However, the equilibrium equations lead to an initial value problem for the transverse stresses, which does not allow the simultaneous fulfillment of the boundary conditions at the top and bottom surface. An analytical trade-off of higher order theories carried out by Rohwer15 showed, that for finite element analyses the FSDT is the best compromise between accuracy and effort. This holds for plate slendernesses down to five. Most of the theories presume C1-continuity,5~7 which practically excludes finite element application. Others need twice as much functional degrees of freedom compared to the FSDT9 or provide no better results than the FSDT.8 More recently developed are the so-called layerwise theories (e.g. References 16—18), which use piecewise polynomial distributions (zig-zag function) of the membrane displacements in thickness direction and provide a very good approximation of the transverse shear stresses. If no constraints for the zig-zag functions are introduced, the number of functional degrees of freedom depends on the number of layers. This theory can be easily translated into finite elements, however, leads to a computational effort in the range of a full 3D-analysis.18~19 Thus, the practical application is limited to detail investigations. The number of functional degrees of freedom can be drastically reduced by a priori ensuring continuity of the transverse shear stresses.17,21 Corresponding finite elements need C1-continuity or base on mixed formulations.22 However, the problem of stability and convergence of mixed finite elements is not completely solved up to now. Still relying on the FSDT, there is a potential for improving the transverse shear stress approximation by enhancing the transverse shear stiffnesses. Usually some kind of shear correction factor is used, but Wittrick23 proved that for orthotropic material it is impossible to choose effective shear moduli independent of the displacement mode. Assuming two cylindrical bending modes Rohwer24 calculated improved transverse shear stiffnesses and showed in an analytical study15 that they provide reasonable transverse shear stresses. The present paper investigates the accuracy of the method within the framework of finite elements. Furthermore, the transverse shear stresses are calculated from the transverse shear forces. In contrast to directly applying the equilibrium of forces, this procedure allows for simultaneous fulfillment of both boundary conditions and saves one order of derivation of the shape functions. The latter is especially important for finite element analyses, since it allows the calculation of transverse shear stresses already in elements with not more than 20 degrees of freedom. The numerical results are compared with MSC/NASTRAN calculations.

COMPOSITE FINITE ELEMENTS BASED ON FIRST ORDER SHEAR DEFORMATION THEORY

53

2. THEORY The Whitney—Pagano lamination theory is based on the kinematical assumption of Mindlin and Reissner u u 0 0 u" v " v #z 0 x (1) 0 y w w 0 0 where u , v , w are the displacements of the reference surface in x,y and z direction, respectively, 0 0 0 and 0 and 0 are the rotations of the cross-section. Employing finite elements shape functions x y are chosen for the displacements (u , v , w ) and the rotations. The transverse shear strains (or 0 0 0 shear angles) are obtained by the strain—displacement relations

C D C

D

c w #0 x xz " ,x (2) w #0 c y ,y yz The straightforward way of calculating the transverse shear stresses uses the material law, which provides c"

(3) s(k)"G(k)c z G(k) contains the shear moduli of the kth lamina. Integration over the laminates thickness gives the transverse shear forces R"Hc

(4)

with shear stiffnesses n H" + G(k)a(k) (5) k/1 The values provided by (5) are too large, since they are based on the assumption of a constant shear angle in thickness direction and also do not satisfy the transverse shear stresses vanishing at the boundaries. Usually, they are reduced by a shear correction factor of e.g. 2/3 or 5/6. However, none of these factors is generally applicable.23 A better physical foundation has the equilibrium approach by Rohwer,24 who calculates transverse shear stiffnesses according to a distinct displacement mode (e.g. cylindrical bending). Moreover the method also provides the transverse shear stresses. The equilibrium of forces in x- and y-direction solved with respect to the transverse shear stresses reads

C D

P A

B C

D

q(k) f/z p(k) #q(k) g (x, y) xy,y df# 1 x,x s " xz "! (6) z p(k) #q(k) q(k) g (x, y) xy,x yz f/0 y,y 2 where the co-ordinate f starts at one of the laminate surfaces. g and g are determined from the 1 2 boundary condition at f"0, usually they vanish. Using the material law for the kth lamina, p x r " p "Q1 (k) (e0#zj) (7) m y q xy where Q1 (k) are the reduced stiffnesses of the kth lamina and e0 and j denote the laminate strains and curvatures, respectively, yields f/z

P f/0 (B1 Q1 (k)(e0,x#zj,x )#B2 Q1 (k) (e0,y#zj,y )) df

s "! z

(8)

54

R. ROLFES AND K. ROHWER

B and B are Boolean matrices of the form 1 2

C C

1 0 0 B " 1 0 0 1 and

0 0 1 B " 2 0 1 0

D D

(9)

(10)

and shoud not be mixed up with the coupling matrix B of the laminate. The relation (8) could directly be used for determining the transverse shear stresses. However, due to the appearance of strain derivatives, second derivatives of the shape functions (for u , v , 0 0 0 , 0 ) would have to be evaluated. Thus at least quadratic shape functions would have to be x y chosen or a smoothing procedure according to Byun and Kapania4 would become necessary. To circumvent this, the derivatives of the laminate’s strains are replaced—under certain conditions— by the transverse shear forces. Using the elasticity law of the laminate

C D C DC D N A B " M BT D

e0 j

(11)

where A, D and B are the membrane, bending and coupling stiffnesses, respectively, and M x M" M y M xy

(12)

and N x (13) N" N y N xy the laminate strains can be expressed by the moments if the membrane forces N are neglected. This assumption is reasonable, since the influence of the membrane forces on the transverse shear stresses is very small. It follows that e0"!A~1Bj

(14)

j"D*~1M

(15)

D*"(D!BTA~1B)

(16)

and where Inserting equations (14)—(16) into the equilibrium condition as formulated in (8), provides transverse shear stresses only depending on the moment derivatives w.r.t. x and y, s "!B F(z)M !B F(z)M z 1 ,x 2 ,y

(17)

F(z)"(a(z)A~1B!b(z))D*~1

(18)

The matrix F(z) reads where a(z) and b(z) are partial membrane and coupling stiffnesses of the laminate, respectively, f/z

P f/0 Q1 df

a(z)"

(19)

COMPOSITE FINITE ELEMENTS BASED ON FIRST ORDER SHEAR DEFORMATION THEORY

55

and f/z

Pf/0 Q1 f df

b(z)"

(20)

Further reduction of equation (17) is only possible when making additional significant assumptions. These are the displacement modes mentioned above. Assuming cylindrical bending around the x-axis yields M " ,x

M x,x 0 0

(21)

around the y-axis provides 0 M " M y,y ,y 0

(22)

Then, the derivatives of the moments can be related to the shear forces via R "!M xz x,x

(23)

R "!M yz y,y

(24)

C

(25)

and

which finally results in F F s " 11 32 z F F 31 22

DC D R xz R yz

or s "f(z) R (26) z This relation automatically fulfills the second boundary condition. Inserting z"h (h denotes the thickness of the laminate) into equations (19) and (20) yields a(h)"A

(27)

b(h)"B

(28)

and

This in turn introduced into equation (18) provides F(h)"0

(29)

and with equation (26) s (h)"0 (30) z From relation (26) it is only a simple step to an equation for improved shear stiffnesses when regarding the complementary transverse shear energy. Formulated in shear stresses it is 1 I" # 2

P sTG~1s dz

(31)

56

R. ROLFES AND K. ROHWER

expressed in shear forces it reads I "1 RT H3 ~1 R # 2

(32)

Introducing equation (26) into equation (31) and comparing with equation (32) provides the expression for the improved transverse shear stiffnesses based on the equilibrium approach,

CP

H3 "

fTG~1 f dz

D

~1

(33)

The sequence of calculation steps is as follows. First, the shear stiffnesses are calculated from equation (33), then equation (4) is used to determine the transverse shear forces, and equation (26) finally provides the transverse shear stresses. Of course, this is not a pure application of the equilibrium conditions (6), since the material law is involved via relation (4). However, using the improved shear stiffnesses, equation (4) provides good transverse shear forces and, what is the big advantage of the method, only first derivatives of the shape functions are needed (conf. equation (2)). The MSC/NASTRAN approach is principally very similar. However, much more rigorous simplifying assumptions are made. The membrane shear stresses in equation (6) are neglected and the material law (7) is greatly simplified by assuming that the global co-ordinate system is the principal system for each lamina (only correct for cross-ply laminates) and the Poisson ratio is zero. The subsequent numerical examples show that these very rigorous assumptions nevertheless provide reasonable results for cross-ply laminates, but can hardly model angle-ply laminates. 3. NUMERICAL EXAMPLES The calculations were carried out using QUAD8 elements of MSC/NASTRAN version 67·5. The improved transverse shear stiffnesses (conf. equation (33)) were determined using the preprocessor PRIMEL.25 The values were introduced into MSC/NASTRAN, which carried out the finite element analysis and provided the transverse shear forces. These were processed by a self-written postprocessor in order to evaluate the transverse shear stresses according to equation (26). The values, which are subsequently denoted as MSC/NASTRAN results are calculated using the simplified MSC/NASTRAN equilibrium approach for both, the transverse shear stiffnesses (equation (33)) and the transverse shear stresses (equation (26)). The origin of the x-, y-co-ordinate system is in the center of the plate throughout all examples. 3.1. ¹hree-layered cross-ply laminate The first example to be regarded is a simply supported quadratic three-layered (0/90/0) plate with slenderness a/h"4 under sinusoidal transverse load. The material properties were chosen as E "138 GPa L E "5·52 GPa T l "0·25 LT G "2·76 GPa LT G "1·104 GPa TT

(34)

COMPOSITE FINITE ELEMENTS BASED ON FIRST ORDER SHEAR DEFORMATION THEORY

57

Figure 1 depicts the transverse shear stress (q ) distribution over the plate thickness at the center xz point of one edge (x"a/2, y"0). None of the approximations can reflect, that the maximum stress occurs in the facing layers and not in the core layer. However, the authors’ result is closest to the exact solution. Especially, there is a significant improvement compared to the FSDTcalculations with different shear correction factors. MSC/NASTRAN only provides values at the layer interfaces. They are better than the FSDT-results with shear correction factors, since MSC/NASTRAN uses an equilibrium approach for the transverse shear stiffnesses and stresses, but somewhat inferior to the authors’ results. This is most probably due to the simplifications introduced by MSC/NASTRAN (see above). The comparative solutions (elasticity, CLT, FSDT with shear correction factors) are taken from di Sciuva.17 He also provides a very good solution based on a layerwise theory, which is not reflected here.

3.2. Four-layered cross-ply laminate As the second example serves the four-layered plate (0/90/90/0) which has been analytically investigated by Rohwer.24 The material properties are E "138 GPa L E "9·3 GPa T l "0·30 LT

(35)

l "0·50 TT G "4·6 GPa LT

Figure 1. Transverse shear stress q in a (0/90/0)—laminate at x"a/2, y"0; plate slenderness a/h"4; WP n: numerical xz solution with FSDT (Whitney—Pagano) and shear correction factor n; 3D: elasticity solution; MSC: MSC/NASTRAN; RR: present solution

58

R. ROLFES AND K. ROHWER

Again the load is of sinusoidal shape. The plate thickness is kept to unity while the edge lengths are varying with a fixed ratio of a D xx " (36) b 4 D yy where D and D are the bending stiffnesses in x- and y-direction, respectively. In Figures 2 xx yy and 3 the present method is compared to the elasticity solution and the analytical FSDT-solution with improved transverse shear stiffnesses (conf. equation (33)). Again the center points of the edges are regarded. The authors’ numerical solution is comparable to the analytical FSDTsolution. While the results for q are somewhat better, q is a little inferior. This holds especially xz yz for the value at the interface between the 90°-layers. Rohwer24 has stated that the FSDT with improved shear stiffnesses is the best compromise between accuracy and effort from the analytical

S

Figure 2. Transverse shear stress q in a (0/90/90/0)—laminate at x"a/2, y"0; plate slenderness a/h"5; WP: xz analytical solution with FSDT (Whitney—Pagano); 3D: elasticity solution; MSC: MSC/NASTRAN; RR: present solution

Figure 3. Transverse shear stress q in a (0/90/90/0)—laminate at x"0, y"b/2; plate slenderness a/h"5; WP: yz analytical solution with FSDT (Whitney—Pagano); 3D: elasticity solution; MSC: MSC/NASTRAN; RR: present solution

COMPOSITE FINITE ELEMENTS BASED ON FIRST ORDER SHEAR DEFORMATION THEORY

59

Figure 4. Transverse shear stress q in a (#45/!45)—laminate at x"17a/36, y"a/36; plate slenderness a/h"6; CLT: xz numerical solution with CLT; CS: semi-analytical solution of Chaudhuri/Seide; MSC: MSC/NASTRAN; RR: present solution

point of view. It can now be concluded, that the numerical solution with finite elements provides results of the same quality as the analytical solution. For this example the MSC/NASTRAN values are very close to the present method. 3.3. Antisymmetric angle-ply laminate An antisymmetric (#45/!45) laminate with engineering constants of E "276 GPa L E "6·9 GPa T l "0·25 (37) LT G "3·4 GPa LT G "1·4 GPa TT is the last example. The semi-analytical approach of Chaudhuri/Seide26 is taken as the yard stick. Figure 4 shows q for the quadratic plate under uniform load at x"17a/36, y"a/36. The xz present method very well approximates the exact solution. Both, the maximum and the value at the layer interface show an error of less than 8 per cent. It is not surprising, that the CLT is far inferior. But also MSC/NASTRAN shows a significantly larger deviation from the exact solution at the interface (28 per cent). This can be traced back to the underlying assumption of a cross-ply laminate, which certainly cannot model the stated problem. 4. CONCLUSIONS A simple method of improving the transverse shear stress results within finite element calculations based on the FSDT has been demonstrated. The basic idea consists in directly calculating the transverse shear stresses from the transverse shear forces. For that purpose, the influence of the membrane forces on the transverse shear stresses was neglected and cylindrical bending displacement modes were assumed. The method also provides improved transverse shear stiffnesses.

60

R. ROLFES AND K. ROHWER

Thus, the selection of an appropriate shear correction factor is no longer necessary. In contrast to the usual method, not only equilibrium conditions, but also the material law for the transverse shear forces were used. The numerical results surpass the usual FSDT stress results based on shear correction factors and equilibrium conditions. The general purpose code MSC/NASTRAN uses a similar but simplified method. Comparisons revealed that the proposed method is more accurate especially for angle-ply laminates. REFERENCES 1. A. Puck, ‘Ein Bruchkriterium gibt die Richtung an’, Kunststoffe, 82, 607—610 (1992). 2. N. J. Pagano, ‘Exact solutions for rectangular bidirectional composites and sandwich plates’, J. Compos. Mater., 4, 20—34 (1970). 3. C. W. Pryor and R. M. Barker, ‘A finite element analysis including transverse shear effects for application to laminated plates’, AIAA J., 9, 912—917 (1971). 4. C. Byun and R. K. Kapania, ‘Prediction of interlaminar stresses in laminated plates using global orthogonal interpolation polynomials’, AIAA J., 30, 2740—2749 (1992). 5. N. R. Senthilnathan, S. P. Lim, K. H. Lee and S. T. Chow, ‘Buckling of shear-deformable plates’, AIAA J., 25, 1268—1271 (1987). 6. M. V. V. Murthy, ‘An improved transverse shear deformation theory for laminated anisotropic plates’, NASA ¹echnical Paper, Vol. 1903 (1981). 7. J. N. Reddy, ‘A simple higher-order theory for laminated composite plates’, J. Appl. Mech., 51, 745—752 (1984). 8. Y. W. Kwon and J. E. Akin, ‘Analysis of layered composite plates using high-order deformation theory’, Comput. Struct., 27, 619—623 (1987). 9. B. N. Pandya and T. Kant, ‘Flexural analysis of laminated composites using refined higher-order C0 plate bending elements’, Comp. Methods Appl. Mech. Eng., 66, 173—198 (1988). 10. B. S. Manjunatha and T. Kant, ‘On evaluation of transverse stresses in layered symmetric composite and sandwich laminates under flexure’, Eng. Comput., 10, 499—518 (1993). 11. T. Kant and B. S. Manjunatha, ‘On accurate estimation of transverse stresses in multilayered laminates’, Comput. Struct., 50, 351—365 (1994). 12. T. Kant and M. P. Menon, ‘A finite element-difference computational model for stress analysis of layered composite cylindrical shells’, Finite Elements Anal. Des., 14, 55—71 (1993). 13. J. J. Engblom and O. O. Ochoa, ‘Through-the thickness stress predictions for laminated plates of advanced composite materials’, Int. j. numer. methods eng., 21, 1759—1776 (1985). 14. J. J. Engblom and O. O. Ochoa, ‘Finite element formulation including interlaminar stress calculations’, Comput. Struct., 23, 241—249 (1986). 15. K. Rohwer, ‘Application of higher order theories to the bending analysis of layered composite plates’, Int. J. Solids Struct., 29, 105—119 (1992). 16. J. N. Reddy, ‘A generalization of two-dimensional theories of laminated composite plates’, Commun. appl. numer. methods, 3, 173—180 (1987). 17. M. di Sciuva, ’Bending, vibration and buckling of simply supported thick multilayered orthotropic plates: an evaluation of a new displacement model’, J. Sound »ibration, 105, 425—442 (1986). 18. R. A. Chaudhuri, ‘An equilibrium method for prediction of transverse shear stresses in a thick laminated plate’, Comput. Struct., 23, 139—146 (1986). 19. D. H. Robbins and J. N. Reddy, ‘Modelling of thick composites using a layerwise laminate theory’, Int. j. numer. methods eng., 36, 655—677 (1993). 20. J. N. Reddy, E. J. Barbero and J. L. Teply, ‘A plate bending element based on a generalized laminate plate theory’, Int. j. numer. methods eng., 28, 2275—2292 (1989). 21. He Ling-Hui, ‘A linear theory of laminated shells accounting for continuity of displacements and transverse shear stresses at layer interfaces’, Int. J. Solids Struct., 31, 613—627 (1994). 22. Y. K. Cheung and Shenglin Di, ‘Analysis of laminated composite plates by hybrid stress isoparametric element’, Int. J. Solids Struct., 30, 2843—2857 (1993). 23. W. H. Wittrick, ‘Analytical, three-dimensional elasticity solutions to some plate problems, and some observations on Mindlin plate theory’, Int. J. Solids Struct., 23, 441—464 (1987). 24. K. Rohwer, ‘Improved transverse shear stiffnesses for layered finite elements’, DF»¸R-FB 88-32, Braunschweig (1988). 25. B. Geier and U. Renken, ‘PRIMEL, ein Pra¨prozessor fu¨r Laminatberechnungen, Einfu¨hrung und Benutzeranleitung’, DF»¸R-Mitteilung 88-14. 26. R. A. Chaudhuri and P. Seide, ‘An approximate semi-analytical method for prediction of interlaminar shear stresses in an arbitrarily laminated thick plate’, Comput. Struct., 25, 627—636 (1987).

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40, 51—60 (1997)

IMPROVED TRANSVERSE SHEAR STRESSES IN COMPOSITE FINITE ELEMENTS BASED ON FIRST ORDER SHEAR DEFORMATION THEORY R. ROLFES AND K. ROHWER

German Aerospace Research Establishment (D¸R), Institute of Structural Mechanics, P.O. Box 32 67, D-38022 Braunschweig, Germany

SUMMARY A method for calculating improved transverse shear stresses in laminated composite plates, which bases on the first-order shear deformation theory is developed. In contrast to many recently established methods, either higher-order lamination theories or layerwise theories, it is easily applicable to finite elements, since only C0-continuity is necessary and the numerical effort is low. The basic idea is to calculate the transverse shear stresses directly from the transverse shear forces by neglecting the influence of the membrane forces and assuming two cylindrical bending modes. Shear correction factors are no longer required, since the transverse shear stiffnesses are also provided. Numerical examples for symmetric cross-ply and antisymmetric angle-ply laminates show the superiority of the method against using shear correction factors. Furthermore, results obtained with MSC/NASTRAN, which uses a similar but simplified approach, are surpassed. KEY WORDS :

fiber composite; laminate; shear deformation theory; transverse shear stiffness; interlaminar shear; finite shell elements

1. INTRODUCTION The failure of composites, especially under impact loading, has been subject to intensive research since many years. It turned out, that important failure modes like delamination are driven by transverse shear as well as normal stresses. Hence, modern failure criteria for composite materials account for the full stress tensor.1 However, determining the full state of stress in composite structures is much more involved than in homogeneous isotropic materials. The workhorses for the stress analysis of composite plates have been the Classical Lamination Theory (CLT) and the Whitney—Pagano Theory,2 the latter also known as the First-Order Shear Deformation Theory (FSDT). In the framework of finite element analysis the FSDT is much more popular, since only C0- instead of C1- continuity is necessary. Transverse shear stresses can be calculated from the FSDT using the material law. However, the assumption of a constant shear angle in thickness direction results in layerwise constant stresses. This is quite unsatisfactory, since the condition of vanishing transverse shear stresses at the boundaries is not fulfilled and the interlaminar shear stresses at the interfaces cannot be assigned a distinct value. Both deficiencies can be avoided by applying the equilibrium conditions according to Pryor and Barker.3 This procedure utilizes the first derivatives of the membrane stresses w.r.t the membrane co-ordinates. Pryor and Barker choose shape functions for the displacements (u, v, w) and the transverse shear strains. Thus, they need cubic shape functions for w in order to calculate the required stress

CCC 0029—5981/97/010051—10 ( 1997 by John Wiley & Sons, Ltd.

Received 16 January 1995 Revised 23 January 1996

52

R. ROLFES AND K. ROHWER

derivatives (element with 28 degrees of freedom). The order can be reduced by one if the rotations instead of the transverse shear strains are selected as variables. However, still quadratic shape functions must be employed. Furthermore, comparisons with exact analytical solutions revealed, that the resulting transverse shear stresses from the FSDT are not always satisfactory (e.g. Reference 17). Byun and Kapania4 suggested a postprocessing using global interpolating functions for smoothing the membrane displacements. By that means they achieved good transverse shear and even transverse normal stresses. However, the global functions make the method geometry dependent and therefore impractical for general finite element codes. A variety of higher order lamination theories has been proposed during the last decade in order to improve the transverse shear stress calculation (e.g. References 5—9). Based on some of those, Manjunatha and Kant10,11 and Kant and Menon12 calculated transverse stresses by use of the equilibrium conditions. They achieved good finite element results, however, due to the higher order displacement derivatives involved, they needed biquadratic and bicubic shape functions for evaluating the transverse shear and normal stresses, respectively. Engblom and Ochoa13,14 modified the displacement field and received satisfactory results for transverse shear and normal stresses in elements with only 32 (FSDT) and 40 (theory with quadratic in-plane and constant out-of-plane displacements) degrees of freedom. However, the equilibrium equations lead to an initial value problem for the transverse stresses, which does not allow the simultaneous fulfillment of the boundary conditions at the top and bottom surface. An analytical trade-off of higher order theories carried out by Rohwer15 showed, that for finite element analyses the FSDT is the best compromise between accuracy and effort. This holds for plate slendernesses down to five. Most of the theories presume C1-continuity,5~7 which practically excludes finite element application. Others need twice as much functional degrees of freedom compared to the FSDT9 or provide no better results than the FSDT.8 More recently developed are the so-called layerwise theories (e.g. References 16—18), which use piecewise polynomial distributions (zig-zag function) of the membrane displacements in thickness direction and provide a very good approximation of the transverse shear stresses. If no constraints for the zig-zag functions are introduced, the number of functional degrees of freedom depends on the number of layers. This theory can be easily translated into finite elements, however, leads to a computational effort in the range of a full 3D-analysis.18~19 Thus, the practical application is limited to detail investigations. The number of functional degrees of freedom can be drastically reduced by a priori ensuring continuity of the transverse shear stresses.17,21 Corresponding finite elements need C1-continuity or base on mixed formulations.22 However, the problem of stability and convergence of mixed finite elements is not completely solved up to now. Still relying on the FSDT, there is a potential for improving the transverse shear stress approximation by enhancing the transverse shear stiffnesses. Usually some kind of shear correction factor is used, but Wittrick23 proved that for orthotropic material it is impossible to choose effective shear moduli independent of the displacement mode. Assuming two cylindrical bending modes Rohwer24 calculated improved transverse shear stiffnesses and showed in an analytical study15 that they provide reasonable transverse shear stresses. The present paper investigates the accuracy of the method within the framework of finite elements. Furthermore, the transverse shear stresses are calculated from the transverse shear forces. In contrast to directly applying the equilibrium of forces, this procedure allows for simultaneous fulfillment of both boundary conditions and saves one order of derivation of the shape functions. The latter is especially important for finite element analyses, since it allows the calculation of transverse shear stresses already in elements with not more than 20 degrees of freedom. The numerical results are compared with MSC/NASTRAN calculations.

COMPOSITE FINITE ELEMENTS BASED ON FIRST ORDER SHEAR DEFORMATION THEORY

53

2. THEORY The Whitney—Pagano lamination theory is based on the kinematical assumption of Mindlin and Reissner u u 0 0 u" v " v #z 0 x (1) 0 y w w 0 0 where u , v , w are the displacements of the reference surface in x,y and z direction, respectively, 0 0 0 and 0 and 0 are the rotations of the cross-section. Employing finite elements shape functions x y are chosen for the displacements (u , v , w ) and the rotations. The transverse shear strains (or 0 0 0 shear angles) are obtained by the strain—displacement relations

C D C

D

c w #0 x xz " ,x (2) w #0 c y ,y yz The straightforward way of calculating the transverse shear stresses uses the material law, which provides c"

(3) s(k)"G(k)c z G(k) contains the shear moduli of the kth lamina. Integration over the laminates thickness gives the transverse shear forces R"Hc

(4)

with shear stiffnesses n H" + G(k)a(k) (5) k/1 The values provided by (5) are too large, since they are based on the assumption of a constant shear angle in thickness direction and also do not satisfy the transverse shear stresses vanishing at the boundaries. Usually, they are reduced by a shear correction factor of e.g. 2/3 or 5/6. However, none of these factors is generally applicable.23 A better physical foundation has the equilibrium approach by Rohwer,24 who calculates transverse shear stiffnesses according to a distinct displacement mode (e.g. cylindrical bending). Moreover the method also provides the transverse shear stresses. The equilibrium of forces in x- and y-direction solved with respect to the transverse shear stresses reads

C D

P A

B C

D

q(k) f/z p(k) #q(k) g (x, y) xy,y df# 1 x,x s " xz "! (6) z p(k) #q(k) q(k) g (x, y) xy,x yz f/0 y,y 2 where the co-ordinate f starts at one of the laminate surfaces. g and g are determined from the 1 2 boundary condition at f"0, usually they vanish. Using the material law for the kth lamina, p x r " p "Q1 (k) (e0#zj) (7) m y q xy where Q1 (k) are the reduced stiffnesses of the kth lamina and e0 and j denote the laminate strains and curvatures, respectively, yields f/z

P f/0 (B1 Q1 (k)(e0,x#zj,x )#B2 Q1 (k) (e0,y#zj,y )) df

s "! z

(8)

54

R. ROLFES AND K. ROHWER

B and B are Boolean matrices of the form 1 2

C C

1 0 0 B " 1 0 0 1 and

0 0 1 B " 2 0 1 0

D D

(9)

(10)

and shoud not be mixed up with the coupling matrix B of the laminate. The relation (8) could directly be used for determining the transverse shear stresses. However, due to the appearance of strain derivatives, second derivatives of the shape functions (for u , v , 0 0 0 , 0 ) would have to be evaluated. Thus at least quadratic shape functions would have to be x y chosen or a smoothing procedure according to Byun and Kapania4 would become necessary. To circumvent this, the derivatives of the laminate’s strains are replaced—under certain conditions— by the transverse shear forces. Using the elasticity law of the laminate

C D C DC D N A B " M BT D

e0 j

(11)

where A, D and B are the membrane, bending and coupling stiffnesses, respectively, and M x M" M y M xy

(12)

and N x (13) N" N y N xy the laminate strains can be expressed by the moments if the membrane forces N are neglected. This assumption is reasonable, since the influence of the membrane forces on the transverse shear stresses is very small. It follows that e0"!A~1Bj

(14)

j"D*~1M

(15)

D*"(D!BTA~1B)

(16)

and where Inserting equations (14)—(16) into the equilibrium condition as formulated in (8), provides transverse shear stresses only depending on the moment derivatives w.r.t. x and y, s "!B F(z)M !B F(z)M z 1 ,x 2 ,y

(17)

F(z)"(a(z)A~1B!b(z))D*~1

(18)

The matrix F(z) reads where a(z) and b(z) are partial membrane and coupling stiffnesses of the laminate, respectively, f/z

P f/0 Q1 df

a(z)"

(19)

COMPOSITE FINITE ELEMENTS BASED ON FIRST ORDER SHEAR DEFORMATION THEORY

55

and f/z

Pf/0 Q1 f df

b(z)"

(20)

Further reduction of equation (17) is only possible when making additional significant assumptions. These are the displacement modes mentioned above. Assuming cylindrical bending around the x-axis yields M " ,x

M x,x 0 0

(21)

around the y-axis provides 0 M " M y,y ,y 0

(22)

Then, the derivatives of the moments can be related to the shear forces via R "!M xz x,x

(23)

R "!M yz y,y

(24)

C

(25)

and

which finally results in F F s " 11 32 z F F 31 22

DC D R xz R yz

or s "f(z) R (26) z This relation automatically fulfills the second boundary condition. Inserting z"h (h denotes the thickness of the laminate) into equations (19) and (20) yields a(h)"A

(27)

b(h)"B

(28)

and

This in turn introduced into equation (18) provides F(h)"0

(29)

and with equation (26) s (h)"0 (30) z From relation (26) it is only a simple step to an equation for improved shear stiffnesses when regarding the complementary transverse shear energy. Formulated in shear stresses it is 1 I" # 2

P sTG~1s dz

(31)

56

R. ROLFES AND K. ROHWER

expressed in shear forces it reads I "1 RT H3 ~1 R # 2

(32)

Introducing equation (26) into equation (31) and comparing with equation (32) provides the expression for the improved transverse shear stiffnesses based on the equilibrium approach,

CP

H3 "

fTG~1 f dz

D

~1

(33)

The sequence of calculation steps is as follows. First, the shear stiffnesses are calculated from equation (33), then equation (4) is used to determine the transverse shear forces, and equation (26) finally provides the transverse shear stresses. Of course, this is not a pure application of the equilibrium conditions (6), since the material law is involved via relation (4). However, using the improved shear stiffnesses, equation (4) provides good transverse shear forces and, what is the big advantage of the method, only first derivatives of the shape functions are needed (conf. equation (2)). The MSC/NASTRAN approach is principally very similar. However, much more rigorous simplifying assumptions are made. The membrane shear stresses in equation (6) are neglected and the material law (7) is greatly simplified by assuming that the global co-ordinate system is the principal system for each lamina (only correct for cross-ply laminates) and the Poisson ratio is zero. The subsequent numerical examples show that these very rigorous assumptions nevertheless provide reasonable results for cross-ply laminates, but can hardly model angle-ply laminates. 3. NUMERICAL EXAMPLES The calculations were carried out using QUAD8 elements of MSC/NASTRAN version 67·5. The improved transverse shear stiffnesses (conf. equation (33)) were determined using the preprocessor PRIMEL.25 The values were introduced into MSC/NASTRAN, which carried out the finite element analysis and provided the transverse shear forces. These were processed by a self-written postprocessor in order to evaluate the transverse shear stresses according to equation (26). The values, which are subsequently denoted as MSC/NASTRAN results are calculated using the simplified MSC/NASTRAN equilibrium approach for both, the transverse shear stiffnesses (equation (33)) and the transverse shear stresses (equation (26)). The origin of the x-, y-co-ordinate system is in the center of the plate throughout all examples. 3.1. ¹hree-layered cross-ply laminate The first example to be regarded is a simply supported quadratic three-layered (0/90/0) plate with slenderness a/h"4 under sinusoidal transverse load. The material properties were chosen as E "138 GPa L E "5·52 GPa T l "0·25 LT G "2·76 GPa LT G "1·104 GPa TT

(34)

COMPOSITE FINITE ELEMENTS BASED ON FIRST ORDER SHEAR DEFORMATION THEORY

57

Figure 1 depicts the transverse shear stress (q ) distribution over the plate thickness at the center xz point of one edge (x"a/2, y"0). None of the approximations can reflect, that the maximum stress occurs in the facing layers and not in the core layer. However, the authors’ result is closest to the exact solution. Especially, there is a significant improvement compared to the FSDTcalculations with different shear correction factors. MSC/NASTRAN only provides values at the layer interfaces. They are better than the FSDT-results with shear correction factors, since MSC/NASTRAN uses an equilibrium approach for the transverse shear stiffnesses and stresses, but somewhat inferior to the authors’ results. This is most probably due to the simplifications introduced by MSC/NASTRAN (see above). The comparative solutions (elasticity, CLT, FSDT with shear correction factors) are taken from di Sciuva.17 He also provides a very good solution based on a layerwise theory, which is not reflected here.

3.2. Four-layered cross-ply laminate As the second example serves the four-layered plate (0/90/90/0) which has been analytically investigated by Rohwer.24 The material properties are E "138 GPa L E "9·3 GPa T l "0·30 LT

(35)

l "0·50 TT G "4·6 GPa LT

Figure 1. Transverse shear stress q in a (0/90/0)—laminate at x"a/2, y"0; plate slenderness a/h"4; WP n: numerical xz solution with FSDT (Whitney—Pagano) and shear correction factor n; 3D: elasticity solution; MSC: MSC/NASTRAN; RR: present solution

58

R. ROLFES AND K. ROHWER

Again the load is of sinusoidal shape. The plate thickness is kept to unity while the edge lengths are varying with a fixed ratio of a D xx " (36) b 4 D yy where D and D are the bending stiffnesses in x- and y-direction, respectively. In Figures 2 xx yy and 3 the present method is compared to the elasticity solution and the analytical FSDT-solution with improved transverse shear stiffnesses (conf. equation (33)). Again the center points of the edges are regarded. The authors’ numerical solution is comparable to the analytical FSDTsolution. While the results for q are somewhat better, q is a little inferior. This holds especially xz yz for the value at the interface between the 90°-layers. Rohwer24 has stated that the FSDT with improved shear stiffnesses is the best compromise between accuracy and effort from the analytical

S

Figure 2. Transverse shear stress q in a (0/90/90/0)—laminate at x"a/2, y"0; plate slenderness a/h"5; WP: xz analytical solution with FSDT (Whitney—Pagano); 3D: elasticity solution; MSC: MSC/NASTRAN; RR: present solution

Figure 3. Transverse shear stress q in a (0/90/90/0)—laminate at x"0, y"b/2; plate slenderness a/h"5; WP: yz analytical solution with FSDT (Whitney—Pagano); 3D: elasticity solution; MSC: MSC/NASTRAN; RR: present solution

COMPOSITE FINITE ELEMENTS BASED ON FIRST ORDER SHEAR DEFORMATION THEORY

59

Figure 4. Transverse shear stress q in a (#45/!45)—laminate at x"17a/36, y"a/36; plate slenderness a/h"6; CLT: xz numerical solution with CLT; CS: semi-analytical solution of Chaudhuri/Seide; MSC: MSC/NASTRAN; RR: present solution

point of view. It can now be concluded, that the numerical solution with finite elements provides results of the same quality as the analytical solution. For this example the MSC/NASTRAN values are very close to the present method. 3.3. Antisymmetric angle-ply laminate An antisymmetric (#45/!45) laminate with engineering constants of E "276 GPa L E "6·9 GPa T l "0·25 (37) LT G "3·4 GPa LT G "1·4 GPa TT is the last example. The semi-analytical approach of Chaudhuri/Seide26 is taken as the yard stick. Figure 4 shows q for the quadratic plate under uniform load at x"17a/36, y"a/36. The xz present method very well approximates the exact solution. Both, the maximum and the value at the layer interface show an error of less than 8 per cent. It is not surprising, that the CLT is far inferior. But also MSC/NASTRAN shows a significantly larger deviation from the exact solution at the interface (28 per cent). This can be traced back to the underlying assumption of a cross-ply laminate, which certainly cannot model the stated problem. 4. CONCLUSIONS A simple method of improving the transverse shear stress results within finite element calculations based on the FSDT has been demonstrated. The basic idea consists in directly calculating the transverse shear stresses from the transverse shear forces. For that purpose, the influence of the membrane forces on the transverse shear stresses was neglected and cylindrical bending displacement modes were assumed. The method also provides improved transverse shear stiffnesses.

60

R. ROLFES AND K. ROHWER

Thus, the selection of an appropriate shear correction factor is no longer necessary. In contrast to the usual method, not only equilibrium conditions, but also the material law for the transverse shear forces were used. The numerical results surpass the usual FSDT stress results based on shear correction factors and equilibrium conditions. The general purpose code MSC/NASTRAN uses a similar but simplified method. Comparisons revealed that the proposed method is more accurate especially for angle-ply laminates. REFERENCES 1. A. Puck, ‘Ein Bruchkriterium gibt die Richtung an’, Kunststoffe, 82, 607—610 (1992). 2. N. J. Pagano, ‘Exact solutions for rectangular bidirectional composites and sandwich plates’, J. Compos. Mater., 4, 20—34 (1970). 3. C. W. Pryor and R. M. Barker, ‘A finite element analysis including transverse shear effects for application to laminated plates’, AIAA J., 9, 912—917 (1971). 4. C. Byun and R. K. Kapania, ‘Prediction of interlaminar stresses in laminated plates using global orthogonal interpolation polynomials’, AIAA J., 30, 2740—2749 (1992). 5. N. R. Senthilnathan, S. P. Lim, K. H. Lee and S. T. Chow, ‘Buckling of shear-deformable plates’, AIAA J., 25, 1268—1271 (1987). 6. M. V. V. Murthy, ‘An improved transverse shear deformation theory for laminated anisotropic plates’, NASA ¹echnical Paper, Vol. 1903 (1981). 7. J. N. Reddy, ‘A simple higher-order theory for laminated composite plates’, J. Appl. Mech., 51, 745—752 (1984). 8. Y. W. Kwon and J. E. Akin, ‘Analysis of layered composite plates using high-order deformation theory’, Comput. Struct., 27, 619—623 (1987). 9. B. N. Pandya and T. Kant, ‘Flexural analysis of laminated composites using refined higher-order C0 plate bending elements’, Comp. Methods Appl. Mech. Eng., 66, 173—198 (1988). 10. B. S. Manjunatha and T. Kant, ‘On evaluation of transverse stresses in layered symmetric composite and sandwich laminates under flexure’, Eng. Comput., 10, 499—518 (1993). 11. T. Kant and B. S. Manjunatha, ‘On accurate estimation of transverse stresses in multilayered laminates’, Comput. Struct., 50, 351—365 (1994). 12. T. Kant and M. P. Menon, ‘A finite element-difference computational model for stress analysis of layered composite cylindrical shells’, Finite Elements Anal. Des., 14, 55—71 (1993). 13. J. J. Engblom and O. O. Ochoa, ‘Through-the thickness stress predictions for laminated plates of advanced composite materials’, Int. j. numer. methods eng., 21, 1759—1776 (1985). 14. J. J. Engblom and O. O. Ochoa, ‘Finite element formulation including interlaminar stress calculations’, Comput. Struct., 23, 241—249 (1986). 15. K. Rohwer, ‘Application of higher order theories to the bending analysis of layered composite plates’, Int. J. Solids Struct., 29, 105—119 (1992). 16. J. N. Reddy, ‘A generalization of two-dimensional theories of laminated composite plates’, Commun. appl. numer. methods, 3, 173—180 (1987). 17. M. di Sciuva, ’Bending, vibration and buckling of simply supported thick multilayered orthotropic plates: an evaluation of a new displacement model’, J. Sound »ibration, 105, 425—442 (1986). 18. R. A. Chaudhuri, ‘An equilibrium method for prediction of transverse shear stresses in a thick laminated plate’, Comput. Struct., 23, 139—146 (1986). 19. D. H. Robbins and J. N. Reddy, ‘Modelling of thick composites using a layerwise laminate theory’, Int. j. numer. methods eng., 36, 655—677 (1993). 20. J. N. Reddy, E. J. Barbero and J. L. Teply, ‘A plate bending element based on a generalized laminate plate theory’, Int. j. numer. methods eng., 28, 2275—2292 (1989). 21. He Ling-Hui, ‘A linear theory of laminated shells accounting for continuity of displacements and transverse shear stresses at layer interfaces’, Int. J. Solids Struct., 31, 613—627 (1994). 22. Y. K. Cheung and Shenglin Di, ‘Analysis of laminated composite plates by hybrid stress isoparametric element’, Int. J. Solids Struct., 30, 2843—2857 (1993). 23. W. H. Wittrick, ‘Analytical, three-dimensional elasticity solutions to some plate problems, and some observations on Mindlin plate theory’, Int. J. Solids Struct., 23, 441—464 (1987). 24. K. Rohwer, ‘Improved transverse shear stiffnesses for layered finite elements’, DF»¸R-FB 88-32, Braunschweig (1988). 25. B. Geier and U. Renken, ‘PRIMEL, ein Pra¨prozessor fu¨r Laminatberechnungen, Einfu¨hrung und Benutzeranleitung’, DF»¸R-Mitteilung 88-14. 26. R. A. Chaudhuri and P. Seide, ‘An approximate semi-analytical method for prediction of interlaminar shear stresses in an arbitrarily laminated thick plate’, Comput. Struct., 25, 627—636 (1987).

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