Design guidelines for ply drop-off in laminated composite structures

A. Mukherjee* , B. Varughese

Department of Civil Engineering, Indian Institute of Technology, Powai, Bombay 400 076, India

Abstract

The present investigation aims at developing a few guidelines for the design of tapered laminated composites. The tapering in laminatedcomposites is introduced by terminating (dropping-off) plies at different locations. The main objective in designing a drop-off is to reducestress concentration. At present some thumb rules are used to design the drop-off. In this paper, guidelines have been developed by studyingthe effect of important parameters that determine the strength of the laminate. The numerical study shows that some of the thumb rules used atpresent are rather conservative and may be relaxed to an extent.

Keywords: A. Laminates; B. Delamination; B. Stress concentrations

1. Introduction

Conventional metallic materials for the design of struc-tural components like wing, fin, etc. of an aircraft are inhib-ited by their high densities resulting in a reduction ofpayload. The evolution of laminated composite materialshas opened new vistas in the development of aircraft indus-try. The laminated composites offer a weight reductionthereby increasing the payload. A unique feature of lami-nated composites is layered construction. The structuralcomponents are made by laying plies over one another sothat the required thickness of the laminate is achieved. Thelaminate made in this manner is flat since the thickness ofeach ply is uniform throughout. In practice, however, taper-ing of laminates is necessary in many structural compo-nents. The tapering is introduced by terminating plies atdifferent locations. This is known as ply drop-off. Fig. 1presents the geometry of a ply drop-off. The laminate tapersfrom a thick section to a thin section as a result of the plydrop-off. Plies may be inserted, on the other hand, at accessholes, lightening holes and at joints or connections tostrengthen them. In all these applications use of ply drop-off results in significant saving in material and therefore, it iscost effective. However, ply drop-off causes a discontinuitywithin the laminate and therefore, it introduces structuraldifficulties like stress concentration at the drop station.This leads to failure of the components through delamina-tion and/or failure of resin. The formation of interlaminar

stresses at the drop-off may initiate failure long before theultimate load carrying capacity of the laminate is reached.Hence, the potential benefits in dropping plies may becompromised through a reduction of the strength of thelaminate. Hence, the dropping-off of plies has to be donein a manner that does not affect the strength of the laminatesto a great extent.

Ply drop-off in laminated composites has been identifiedas a stress riser from the very beginning [1,7]. Since the lastdecade several experimental and analytical studies havebeen reported regarding various aspects of this problem. Adetailed review is available in Ref. [5]. However, it is notfeasible for a designer to carry out a detailed analysis of theentire structure. Therefore, it is necessary to develop a fewthumb rules that the designers can follow to avoid prematurefailure. In the absence of a systematic study such rules haveremained conservative and they constrain the design. More-over, they lead to higher consumption of material leading tohigher cost and weight. In this paper, the influence of threemost important design parameters, viz. number of pliesdropped-off at a single station, distance between successivedrop-offs (stagger distance) and lay-up of a dropped subla-minate, on the strength of the laminate has been studied. Thepaper also discusses the variation of major interlaminarstresses at the drop locations of the laminates.

Dropping-off of plies introduces out-of-plane (interlami-nar) stresses in addition to in-plane stresses. However, theinterlaminar stresses are predominant only in the vicinity ofthe drop-off and at locations away from the drop-off theybecome negligible. Hence, it is possible to neglect the inter-laminar stresses at locations away from the drop-off and

they can be included only in the vicinity of the drop-off.Thus, it is possible to conduct aglobalanalysis of the wholestructure and a detailedlocal analysis in the region of thedrop-off. Such a global–local approach results in consider-able saving in computational effort. In the present study theglobal analysis has been carried out by a special ply drop-offelement. The ply drop-off element is capable of incorporat-ing closely spaced ply drop-offs within the element. Subse-quently, the detailed analysis (local analysis) has beencarried out at the drop location. The above ply drop-offanalysis has already been validated for various ply drop-off configurations. The details of the analysis and its valida-tion are discussed in separate papers [3,6]. We shall use themethodology in analyzing a number of tapered laminates.

2. Failure criterion for delamination

Failure in a composite laminate may occur in differentways:

• fiber fracture;• matrix cracking;• delamination.

Out of the three modes delamination requires special treat-ment. Delamination in laminated composites is an out-of-plane failure mode caused primarily by interlaminar stres-ses. The contribution of in-plane stresses to delamination isinsignificant. Therefore, failure criterion that is used for in-plane failure analysis cannot be used for delaminationanalysis. Hence, the delamination criterion is often devel-oped from the three-dimensional form of failure criterion.This is achieved by considering only the terms that corre-spond to the out-of-plane stress components. The form offailure criterion used here is developed by modifying thethree-dimensional Tsai–Wu failure criterion [4].

According to the Tsai–Wu strength criterion the failuresurface in stress-space is expressed as

Fis i 1 Fijs is j � 1 �1�whereFi andFij �ij � 1;2; 3;…;6� are strength coefficientsdetermined from lamina strengths. The Tsai–Wu failurecriterion has been modified in such a way that the resultingdelamination criterion contains only the terms correspond-ing to the major interlaminar stresses. These stresses include

the normal stress (s3) and shear stress (t13), which arecomputed along the principal material directions (1,2,3).The other interlaminar shear stress (t23) is not consideredin the criterion. This is because the magnitude of shear stress(t23) at the drop-off was negligible when the dimension ofthe laminate alongy is several times higher than its dimen-sion alongx. The interlaminar normal stress (s3) and shearstress (t13) have been computed at different locations of asection at the center of the dropped laminate.

Since delamination is an out-of-plane phenomenon and is

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Fig. 1. A typical ply drop-off.

Fig. 2. Geometry and global mesh of the laminates. (a) nD� 1 (not toscale); (b) nD� 2 (not to scale); (c) nD� 3 (not to scale); (d) globalmesh for all laminates.

independent of the in-plane stress state, the in-plane stressesmay be uncoupled from the out-of-plane stresses. Further, ifstrength is considered to be independent of the sign of theshear stresses all components containing a linear shear stressterm may be excluded. It may be noted that the interlaminarshear stresss4 has been assumed negligible and therefore,the above criterion reduces to

F3s3 1 F33s23 1 F55s

25 � 1: �2�

The stressess3 ands5 in Eq. (2) are stresses computed inthe principal material directions. The criterion in Eq. (2) canbe rewritten as

F3s3 1 F33s23 1 F55t

213 � 1: �3�

The above delamination criterion is used in the presentinvestigation. It consists of terms corresponding to the inter-laminar normal stress (s3) and the interlaminar shear stress(t13). The criterion in Eq. (3) can easily be incorporated intothe analysis of tapered laminates.

In order to predict the failure of resin pocket and resinlayers the maximum stress criterion has been used. Sinceresin has isotropic properties the principal normal stress iscompared with the allowable resin strength to determine thefailure of resin:

snormal $ sallowable: �4�The criterion in Eq. (4) has been computed for the elementsin the resin regions. It is assumed that the resin elastic prop-erties are linear until failure. It may be noted that sometimesthe resin may fail much before the delamination in thelaminate.

3. Parametric studies

The effect of three parameters on the strength of thelaminates has been studied. The first one is to observe theeffect of dropping many plies off at a single station. When itis necessary to achieve the tapering within a small span ofthe structure it may be required to drop a large number ofplies at a single station. The drop-off causes a large step atthat point. The step size is changed by varying the numberof dropped plies in the laminate. The effect of this aspect onthe strength is examined.

The formation of a large step can be avoided by distribut-ing the drop-off over a number of stations. The distancebetween two successive drop-offs (stagger distance) isselected in such a way that the tapering is smooth. Thedrop-offs cannot be too close as it may lead to a large stressconcentration. On the other hand, it may not be possible tokeep a large stagger distance when the span of the structureis restricted. Though there are thumb rules for providingstagger distance (typically 6 mm), they proved conservativeaccording to this study. In the present work the staggerdistance has been studied as the second parameter.

The third parameter considered is the lay-up of the

dropped plies. In practice 0, 90 and458 plies are usedfor laminate construction. The stiffness contribution ofthese plies are different along the laminate’s axis. Whenthese plies are dropped at different locations in a laminate,the rise of out-of-plane stresses at each drop-off may bedifferent according to the stiffness of the dropped plies. Itis necessary to reduce stress concentration at the drop-off toprevent an early delamination failure. This is possiblethrough a careful selection of lay-up of dropped plies atdifferent drop locations. In order to establish design guide-lines on this aspect a detailed investigation has been carriedout in the present study.

The study on the above parameters has been carried outthrough extensive numerical experimentation. Laminates ofvarious geometrical and lay-up configurations have beenused for this purpose. The global–local approach [3] hasbeen used for analyzing the laminates. A thin long laminatehas been chosen (Fig. 3). The location of the ply drop-offwas kept away from both the load and constraint locations toavoid their interference. A global model of the entirestructure (Fig. 3d) has been analyzed with an in-planetensile loading at the tip and constrained displacement atthe root. The laminate is assumed to be linearly elastic untilthe point of delamination. The material used in these exam-ples is T300/5208 Graphite–Epoxy unidirectional tape. Thetriangular pocket generated due to drop-off is filled withresin. The properties of the laminate and resin materialshave been presented in Table 1. The region of the drop-offhas been modeled in detail for failure analysis. The responseof the global model has been used as input for the localmodel.

3.1. Effect of dropping plies at a single station

To study this effect three symmetrically dropped lami-nates with plies dropped at a single station have beenconsidered. The geometry and lay-up of these laminatesare shown in Fig. 2a–c. In all the laminates, the lay-up ofthe core and covering laminates has been kept the same.However, the number of plies dropped has been variedfrom 1 to 3. The dropped sublaminate consists of only 08plies. In general, the lay-ups of the laminates can be repre-sented as (45/245/0/90/0nD/90/0/245/45)s where nD is thenumber of plies dropped and s signifies symmetric laminate.As nD increases from 1 to 3 the thickness of the thicksection (H) increases; however, the thickness of the thinsection (h) is the same in all the laminates. The thickness(H) of each laminate at the thick end is indicated in Fig. 2a–c. The triangular wedge created by the tapering is filled withresin. Resin layers are also assumed as shown in the Fig.3a–c. The laminates are fixed at the thick end and free at thethin end.

The global–local analysis is then carried out with eachlaminate. A typical global finite element mesh used for theanalysis is shown in Fig. 2c. The local finite element meshes

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used for laminates with nD� 1 and 2 are shown in Fig. 3aand b, respectively.

From the local analysis the interlaminar normal stress(s z) and interlaminar shear stress (t xz) have been computed.In order to understand the effect of dropping many plies offat a station the interlaminar stresses along the laminate hasbeen presented here. The normalized stresses for laminateswith nD� 1 and 2 are shown in Fig. 4. Kemp and Johnson[2] have studied the distribution of interlaminar normalstress and shear stress for the laminate with nD� 3. Acomparison of the results has been presented elsewhere[3]. The correlation between the two results is very good.

The stresses peak at two points — the drop-off and thethin end of the transition region. At the drop-off the normalstress (s z) is compressive as shown by the negative sign.The stress changes its sign and becomes tensile and peaks atthe thin end of the transition. Fig. 5 presents the direction ofnormal stress at the key locations. As a tensile force isapplied at the thin end of the laminate, the tendency of thecore and the covering sublaminates is to come closer andresults in compressive normal stress at the drop-off. While atthe tip, the tendency of the normal stress is to split the coreand covering sublaminates resulting in a tensiles z. If theapplied stress is compressive the stress pattern will betensile at the drop-off and compressive at the thin end.

It can be seen from Fig. 4a that the compressives z ismaximum at the junction between the covering and droppedsublaminates. The absolute maximum normal stress is at thetop of the dropped sublaminate. The interlaminar shearstress (t xz) shows peak values at the drop-off at the topand bottom interfaces of dropped sublaminates. The value

of shear stress at the thin end of the transition region is lessthan that at the drop-off. The shear stress value is positive atthe top of the dropped plies and it is negative in the bottominterface. The interlaminar shear stress is developed becauseof the relative deformation between the dropped plies andthe core and the covering sublaminates. The core and thecovering sublaminates tend to slide over the droppedsublaminate as shown in Fig. 6.

In order to get a better understanding of the effect ofdropping plies at a station on the magnitude and distributionof the normal and shear stresses the peak values and the rateof decay of these stresses in the thick section of the lami-nates is presented in Table 2. The stresses correspond to a tipload of 0.28 kN/mm. The rate of decay gives an idea of howquickly the stress reduces as we go away from the drop-off.

The maximum normal stress and shear stress increaseas the number of plies dropped (nD) increases. The rateof decay is higher for the normal stress (s z) than thatfor shear stress (t xz). The concentration of normal stressis high only near the drop-off. The shear stress,however, reduces slowly. It is also observed that the

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Table 1Material properties

Material T300/5208Graphite Epoxy tape

Resin

Elastic moduli (MPa) E1 � 138× 103; E2 �E3 � 9:65× 103

3.45× 103

Poisson’s ratio n12 � n13 � 0:3; n23 � 0:6 0.36Shear moduli (MPa) G12 � G13 � 5:516× 103;

G23 � 4:14× 103

Fig. 3. Local finite element mesh for laminate. (a) nD� 1; (b) nD� 2.

concentration of interlaminar stresses near the drop-offincreases with the number of dropped plies at astation. However, when the number of plies droppedat a station reduces then the stresses spread over alarger area and that reduces the risk of a sudden failureat the drop-off.

Failure analysis has been conducted on the threelaminates using the failure criteria discussed in Eqs. (3)and (4). The tip strain at delamination initiation has beenobtained from the failure analysis. The location of failure isalso identified. The strain at delamination computed for thethree laminates is shown in Fig. 7. The strain at failure is

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Fig. 5. Formation of positive and negative normal stress. Fig. 6. Formation of positive and negative shear stress at the interfaces.

Fig. 4. Stresses at the drop-off. (a) Normal stress (nD� 1); (b) shear stress (nD� 1); (c) normal stress (nD� 2); (d) shear stress (nD� 2).

highest for the laminate with one ply dropped and itdecreases as the number of plies dropped increases. Thefailure strain has dropped by 7.5% when two plies aredropped in place of one ply and it drops by 12.3% whenthree plies are dropped instead. The strain at failuredecreases at a decreasing rate with an increase in thenumber of plies dropped. Therefore, the number of pliesdropped must be decided on the basis of the reductionin strength that the designer can accept. For the leastreduction only one ply should be dropped at a timewitha staggering. However, if higher reduction in strength canbe allowed more plies can be dropped at a station. Thereduced drop-off strength should not be less than that ofthe thin end.

3.2. Distance between successive drop-offs (staggerdistance)

A gradual drop-off by introducing many drop-off stationsis preferable to a single drop-off station. The selection ofminimum distance between successive drop-offs (staggerdistance) is important to achieve the desired tapering. Acommon thumb rule is to keep a staggered distance of6 mm. In order to examine this rule a laminate having twodrop-off stations has been studied. The geometry of thelaminate model used is shown in Fig. 8a. The laminate is

symmetric about thex–y plane. In Fig. 8a the two stationsare situated at a distance ‘dd’ (stagger distance) apart. Theeffect of distancedd on the laminate’s strength is studiedhere. For this purpose failure analysis is carried out on thelaminate for different values ofdd. The value ofdd is variedby gradually shifting the location of the second drop-offtowards the first drop off. The first drop station has beenkept stationary. When the two drop stations are away fromeach other the zones of stress concentration do not interact.Therefore, they behave as independent drop-offs withoutinfluencing the failure loads of one another. This distanceforms the upper bound of the study. Any stagger distancebeyond this value does not influence failure load at all. Asthe stagger distance reduces the zones of stress concentra-tion start interacting. Therefore, the failure load of the lami-nate reduces gradually. In this study the effect of such aninteraction is examined until the lower bound of thestagger distance is reached. The aspect ratio of thetransition region is 1:3. Therefore, the lower bound ofstagger distance is three times the thickness of drop-off

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Table 2Effect of nD on the peak value and distribution of interlaminar stresses

nD Normal stress (s z) Shear stress (t xz)

Peak value( × 1022) (N/mm2)

Rate of decay (× 1022)(N/mm2/mm)

Peak value( × 1022) (N/mm2)

Rate of decay (× 1022)(N/mm2/mm)

1 21.22 3.00 21.23 1.052 21.78 3.50 21.66 1.163 22.19 5.40 21.97 1.47

Fig. 7. Strain at failure for various nD values. Fig. 8. Geometry of drop-offs. (a) Staggered drop-offs; (b) lower bound.

(Fig. 8b). The staggered distancedd has been reducedfrom 3.048 mm (12 times the drop thickness) to 0.762 mm(three times the drop thickness). Eight stagger distancesbetween these two bounds have also been examined. Thelengths of thick and thin sections have been kept constantfor all values ofdd.

The triangular pockets created at each station by thecovering plies and dropped plies have been filled withresin. Two plies are dropped at each drop station of thelaminate. The thickness of each ply is 0.127 mm. The thick-ness of the symmetric half at the thick end is 1.01 mm andthat at the thin end is 0.508 mm.

The main purpose of this study is to determine the mini-mum stagger distance between two drop-offs so that asmooth transfer can take place. The minimum staggerdistance for various stacking sequences can be different.Therefore, six combinations of stacking sequence havebeen analyzed. The lay-up of laminate in different sets ispresented in Table 3. In each set the lay-ups of two droppedsublaminates have been varied. The orientations of droppedplies at the first and second drop stations have been speci-fied by subscripts D1 and D2, respectively. The lay-ups ofthe covering and core sublaminates have not been changed.Fig. 8 may be referred along with Table 3 to understand thelay-up sequence in each set.

A typical finite element mesh for local analysis is shownin Fig. 9. The distance of the thick end from the root of thelaminate is 4.572 mm and is constant for all values ofdd.The local mesh is fine near the two drop-offs and otherlocations the mesh is coarse. Both eight-noded quadrilateraland six-noded triangular isoparametric elements have beenused. The largest finite element model has 624 elements and1851 nodes.

3.3. Effect of stagger distance on interlaminar stresses

In order to study the effect of stagger distance, the varia-tion of interlaminar normal stress and shear stress have beenstudied at the first and second drop-offs of the laminate (Fig.10). The stresses have been computed just below the firstand the second drop-offs, i.e. atz� 0:505 and 0.250 mm,respectively. Fig. 10a–f shows the normal stress and shearstress at the first drop-off for sets 1–6. The stresses for twoextreme values of staggered distances, i.e. 0.762 and2.540 mm are plotted. All other values of staggereddistances lie between these two bounds. Stagger distancesbeyond 2.540 mm can be considered independent ofdrop-offs.

The shear stress remains largely unaffected by the staggerdistance. Towards the lower bound the shear stress ismarginally higher. The normal stresses fordd� 0:762 and2.54 mm have little difference at the drop-off station�d �0�: As we proceed towards the tip of the second drop-off, thestresses, however, diverge. This is due to the interactionbetween the two drop-offs for smalldds. The largest stressesare obtained in case of set 3 that has the severest drop-offwith two 08 plies dropped at both stations. The normal stres-ses in case of set 5 that has two 08 plies dropped at thesecond station, are also of comparable magnitude.

The divergence in stresses is most drastic in case of sets 5and 6, where 0/0 and 90/90 layers are dropped subsequently.However, the largest stresses for the twodds were close toeach other; indicating that the failure stresses are not verysensitive to stagger distance. Therefore, there is a scope forconsidering the smallest possible stagger distance as threetimes the thickness of the dropped sublaminate. However,this point must be confirmed by studying the second drop-off in the laminate. The normal and shear stress variation atthe second drop-off for all the six sets have been presented inFig. 11a–f. The stresses at the second drop-off are substan-tially higher than the first drop-off. The effect of seconddrop-off is more severe because the laminate is thinner atthat location than at the first drop-off. Only in set 6, whichhas the (0/0)D1/(90/90)D2 dropping sequence, the stresses atthe second drop-off are lower than that at the first drop-off.This is due to lower stiffness of the 90/90 sublaminate.

The stresses are very close to one another for the wholerange ofdds. The divergence ins z at the thin end of thedrop-off in case ofdd� 0:762 mm is absent here as

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Table 3Lay-up sequence for various sets

Set number Lay-up sequence�u1=u2=�u3=u4�D1=�u5=u6�D2=u7=u8�

1 (145/245/(0/90)D1/(90/0)D2/245/145)s2 (145/245/(90/90)D1/(90/90)D2/245/145)s3 (145/245/(0/0)D1/(0/0)D2/245/145)s4 (145/245/(145/245)D1/(245/145)D2/245/145)s5 (145/245/(90/90)D1/(0/0)D2/245/145)s6 (145/245/(0/0)D1/(90/90)D2/245/145)s

Fig. 9. Local modeling of the laminate with two drop-offs.

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Fig. 10. Stress variation at the first drop. (a) Set 1; (b) set 2; (c) set 3; (d) set 4; (e) set 5; (f) set 6.

Fig. 11. Stress variation at second drop-off. (a) Set 1; (b) set 2; (c) set 3; (d) set 4; (e) set 5; (f) set 6.

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Fig. 11. (continued)

there is no drop-off ahead of this one. That reinforces theobservation that stagger distance can be kept as low as threetimes the sublaminate thickness.

The normal and shear stresses reached maximum valuesin case of sets 1 and 3. In both these cases 08 plies have beendropped in the sublaminate. A 0/90 drop-off has marginallyless stress than a 0/0 drop-off. This demonstrates that theorientation of dropped plies is a very important factor. 08plies lead to the largest stresses and^45 and 90 plies haveconsiderably less severe effects. Therefore as far as possiblethe^45 and 90 plies should be dropped. We shall investi-gate the drop-off of 08 plies now.

We shall observe the stresses in sets 5 and 6 to devise astrategy for drop-off of 08 plies. In sets 5 and 6 the same setof plies, 0/0/90/90, has been dropped. In set 6 the 08 plies aredropped at the first drop-off and in set 5 they are dropped atthe second drop-off. The stresses in set 6 are much lowerthan that in set 5. The relatively thicker laminate at the firstdrop-off is responsible for a better performance of set 6.From this experiment we can conclude that, when we designa staggered drop-off the plies should be dropped in decreas-ing order of their stiffnesses, i.e. stiffest plies should bedropped first and the softest plies last. We shall observethe effect of these stresses on the failure load of thelaminate.

The tip load intensity at the onset of failure for the lami-nates in sets 1–6 for the whole range of stagger distancesfrom 0.762 to 2.54 mm is plotted in Fig. 12. It can be seenthat set 2 has the largest failure load. This set has only 90plies dropped. These plies are softest and they have the leasteffect on the failure loads. Set 4 is next in failure loads. Thisset has 45 plies dropped and they are next in the order ofsoftness. Therefore, the failure load is second to the 90 plies.The^45 layers also exhibit sensitivity to stagger distance.The failure load reduces sharply as the stagger distancereduces from eight times the dropped laminate thickness.

Therefore, higher stagger distance is necessary when^45laminates are dropped. Higher interlaminar shear stresses incase of^45 laminates is responsible for the reduction offailure load. Sets 1, 3, 5 and 6 are not sensitive to staggerdistance. Set 6 has a higher failure load than sets 1, 3 and 5.The difference between set 6 and set 5 is that in case ofset 6 the stiffer plies (0) are dropped first. This observationreinforces our guideline that the plies should bedropped in decreasing order of their stiffnesses. Set 1 thathas a (0/90) combination at both drop-offs has not generatedfailure load higher than that of set 5. Therefore, it can beconcluded that, combining plies of different orientations atthe drop-off may not be effective in increasing the laminatestrength.

4. Design guidelines

We shall summarize the design observations we havemade in this paper:

• When in a laminate a large number of plies are to bedropped, the drop-offs may be staggered with the numberof plies dropped at any location kept to a minimum. Thestrength of the drop-off should not be below the thin endstrength. The strength of the drop-off reduces decreas-ingly with the increase in number of plies dropped.

• The stagger distance can be kept as low as three times thethickness of the drop-off. The stagger distance should beat least eight times the thickness of the drop-off when 45plies are dropped.

• The plies should be dropped in decreasing order of theirstiffness. The stiffest plies (0 plies) should be dropped atthe thick end and the softest plies (90 plies) should bedropped at the thin end. This ensures a smooth transfer ofload and reduces stress concentration.

• The strength of the laminate does not improve if, to tailorstiffness, plies of different orientations are dropped at thesame station. For example, a 0/90 drop-off and a 0/0drop-off have similar strength reduction. Therefore,combination sublaminates (e.g. 0/90), if dropped shouldbe dropped at the thick end.

Acknowledgements

This work has been supported by ARDB, Ministry ofDefence, Government of India, under the grant no. Aero/RD-134/100/10/93-94/779. The authors gratefully acknowl-edge help by Mr V. Ravindra in preparing the manuscript.

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Fig. 12. Tip loads at the onset of delamination.

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