Post-Buckling of Stiffened Composite Structures with Delaminations
N. Kontis1, R. Butler2 and G. W. Hunt3 University of Bath, Bath, BA2 7AY, U.K.
The current study explores both experimentally and theoretically the effect of embedded delaminations on the strength and stability of a thin-walled laminated channel-section strut. The objective is to identify the influence of structural coupling in the delaminated region and the effect of the delamination itself on post-buckling behavior, in order to enhance efficiency and safety requirements. A range of channel-section struts has been manufactured with a variety of delamination sizes, locations and through-thickness positions and tested under a uni-axial compressive load. Geometry selection of the channel-section has been conducted according to post-buckling requirements, by extending the Shanley model approach to laminated structures. Pre and post-buckling paths are modeled with a bi-linear approximation that satisfies initial requirements and instability load levels to produce a simple design formula. The experimental results are compared to linear and non-linear finite element results. It is revealed that according to the position of the embedded delamination a reduction in strength may be observed and may also be responsible for the stability or otherwise of the post-buckling path.
Nomenclature Aij = elements of the extensional stiffness matrix of the laminate �
sk = extensional stiffness of the skin laminate �st = extensional stiffness of the stiffener laminate
b = width of the skin of the channel cross-section strut d = diameter of circular/semi-circular delamination h = height of the stiffener of the channel cross-section strut ksk = stiffness of the skin kst = stiffness of the stiffeners L = length of the channel section strut n = delamination level P = total load applied psk = load in the skin pst = load in the stiffeners t = thickness y’ = distance from the neutral axis �
I. Introduction OMPOSITE stiffened structures have been extensively used in primary aerospace applications, due to their efficient stiffness/weight, strength/weight ratios and the ability to tailor properties. Despite their apparent
superiority to metals, the susceptibility of composites to damage has proved to be a critical factor. A common
1 PhD student, Department of Mechanical Engineering 2 Senior Lecturer, Head of Composites Research Unit, Department of Mechanical Engineering 3 Professor, Department of Mechanical Engineering
C
47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Confere1 - 4 May 2006, Newport, Rhode Island
example in practise is the barely visible impact damage (BVID) caused by a low-energy impact event. Such events can generate inter-laminar defects (delaminations), which under compressive loading can reduce the strength and stability of the structure. There is a need therefore to establish the influence of such damage on composite stiffened structures. Current practise allows for delamination induced strength reduction by applying strain limits to the material that ensure the delamination does not produce failure before ultimate levels of loading. However these limits are established via coupon tests, which focus on the scale of the delamination only and do not account for interaction with the scale of the structure itself. Simplified plate and strut models have demonstrated that two configurations dominate the post-buckling behavior; the “opening mode” and “closing mode”1,2. The opening mode corresponds to the process under which the laminates move in different directions, see Fig. 1(a), and is considered to be the most dangerous as far as propagation and fatigue3 is considered. The reverse happens in the closing mode Fig. 1(b), although the individuality of the delaminated parts is still maintained. It has been shown that a switch from closing to the opening occurs at a “critical delamination depth”. Furthermore the resulting post-buckling path may be stable, unstable or neutral, depending on the delamination geometry and its distance from that “critical” depth. However the coupling phenomena displayed by stiffened structures may alter the response predicted by simple plate and strut modelling. Certain delaminations may cause a drop in load carrying capacity at the structural scale, although in some situations they may have little if any debilitating effect.
Fig. 1. Modes of buckling for delamination: (a) opening mode, delamination just above critical depth and (b) closing mode, delamination below critical depth.
An experimental investigation4 on composite stiffened panels has demonstrated that artificially induced delaminations may account for a 12% reduction in the strength of the structure, due to accelerated damage growth. Catastrophic failure is observed, caused by detachment of the stiffeners soon after the critical load. Furthermore, numerical work5 has been developed, based on fracture mechanics, capable of predicting the initial instability of stiffened plates. However the influence of structural coupling effects in the delaminated region and the post-buckling response of the stiffened structure, still remains unidentified. The present work investigates experimentally and theoretically the effect of embedded delaminations on the strength and stability of a thin-walled channel-section laminated strut. The objective is to evaluate the influence of structural coupling on post-buckling behavior of delaminated structures, in order to enhance efficiency and safety requirements. To achieve this, a range of channel-section struts has been manufactured with a variety of sizes, locations and through-thickness positions of delamination. The channel geometry is selected according to a simple model based on Shanley’s strut formulation, previously applied to metallic panels6,7, that has been modified in order to describe de-stiffening of the structure according to a post-buckling requirement.
II. Channel Section Strut Design It is well documented that analytical modeling of the non-linear post-buckling behaviour of stiffened structures may prove to be a complicated task, which nevertheless can be simplified by delineating the underlying phenomena. Single or more degree of freedom Shanley models have extensively been used in order to achieve this. The current analysis however, draws on two single degree of freedom spring-system models; a full stiffness model and a reduced stiffness model. These models assume that all plate segments of the structure under investigation maintain their individuality and each plate (or a combination of identical plates) is represented by a single spring. Furthermore the stiffness of the reduced stiffness model is selected according to an assumed stiffness reduction � and a given factor of post-buckling strength � . Here the reduced (or post-buckled) stiffness represents the stiffness of the channel
(a)
(b)
3
following initial buckling of the web of the channel, subsequently referred to as the skin. Failure of the channel is assumed to occur when the flanges, referred to as the stiffeners, reach their buckling load. Typically, buckling analysis of thin walled structures involves two approaches8; the first considers that the whole structure becomes unstable, satisfying continuity conditions at the intersections, while the second regards every plate segment as individual and rotationally restrained by stiffer adjoining plate members. In the current analysis each plate is considered individual and does not take into account any interaction or restraint between them. Furthermore initial imperfection, shear deformation and fillet radius effects at the intersection are ignored. It has been shown that for intermediate channel section strut lengths, imperfection of the overall type, which is considered to be typical in laminated structures, has a minor effect9; while ignoring the radius effect may lead to a minor underestimation of the buckling load10. In constant thickness channel-section struts local and stiffener buckling are indistinguishable11. However after initial buckling, the deformation of the skin plate and the distortion of the stiffeners results in further reduction in stiffness. As a result an overall mode is initiated in the skin and stiffeners, causing the wavelength to increase. When the wavelength has increased enough, the stiffeners have insufficient post-buckling reserve of strength and behave as a plate with one free and three simple-supported edges with no warping restraint. As a consequence Euler buckling is observed. Therefore stiffener buckling can be taken as the ultimate load, after which the structure buckles in an Euler mode. As a result the following design formula that enables geometry selection according to post-buckling requirements is produced. For the channel-section strut both full and reduced stiffness models will be represented by two springs, in which one represents the skin stiffness ksk the other the stiffeners kst as illustrated in Fig. 2. The stiffness of the plate segments for a balanced symmetric laminate under pure in-plane axial stress are,
sksk Abk = and stst Ahk 2= (1)
where sk
skA
AAA �
��
����
�−=
22
212
11 , st
stA
AAA �
��
�
�−=
22
212
11 and ijA are the elements of the extensional stiffness matrix of the
skin and stiffener laminates, respectively.
Fig. 2. Modified Shanley Model concept for the channel-section strut. (a) Isometric view of the channel thickness t and (b) equivalent Shanley spring model.
(a)
(b)
4
The total load P is applied on the neutral axis, and any rotation either before or after initial buckling at the ends is prevented. This arrangement is, for example, representative of a long wing cover panel7, which resists the applied bending moment via a compressive force applied to the neutral axis of the panel both before and after initial buckling. Before initial buckling the force and moment equilibrium in the spring system give,
stsk ppP += (2)
and 2
'h
pp
py
skst
sk ����
����
+= . (3)
Applying Hooke’s Law for linear elastic springs, Eqs. (1) and (2) become, ( )xkkP stsk += (4)
and 2
'h
kk
ky
skst
sk ����
�
+= (5)
where x is axial strain. Equating Eqs. (3) and (5) and substituting in Eq. (2) we obtain,
sksk
skst pk
kkP ��
�����
+= (6)
and stst
sksk p
k
kp ���
�����
= . (7)
The above equations reveal the force distribution in the skin and stiffeners of the channel-section strut, while Eq. (4) describes the linear function of the total load applied, represented by line OA in Fig. 3, according to strain x of the spring system. At a strain xsk, skin buckling occurs and the structure reduces in stiffness. Therefore a reduced stiffness model is considered where the equivalent spring of the skin now has a reduced stiffness,
sksk Abk η=' (8)
where � is the assumed stiffness reduction factor, corresponding to a buckled plate, simply-supported along the unloaded edges. The reduced stiffness model is equivalent to a Shanley hypothetical column6, where the slope of the reduced stiffness system is equivalent to the summation of the full stiffener stiffness kst and the reduced skin stiffness ksk’ , see region AB of Fig. 3. In order to describe this region it is necessary to understand the hypothetical region DA, which represents the load-strain relationship of the reduced stiffness system prior to skin buckling at Psk. At D there is no load in the skin spring, and the stiffener spring carries a tensile force Pst’ , as an effect of a negative x, as defined in the spring system, and taking into account the initial assumption of plate independence. Since the stiffness of the stiffeners spring does not change, it must be extended by –x0, see region OD, which in conjunction with the reduced stiffness of the skin spring, produce the hypothetical region DA. Negative x corresponds to tensile strain. From the plot it is clearly shown that the function of the reduced stiffness model is, 0' PppP stsk ++= (9)
or ( ) 0'' xkxkkP skstsk ++= (10)
5
Where P0 is the total applied force on the reduced stiffness model at x=0. At strain xsk the skin buckles and we can equate Eqs. (4) and (10). Hence we obtain the following relationship,
skxxη
η−= 10 (11)
Fig. 3. Plot of end-shortening strain against total load and individual skin and stiffener load of full stiffness (OA) and reduced stiffness model (AB).
Assuming that the load is applied once more at the new neutral axis, restricting rotation, Eq. (7) can be rewritten as
stst
sksk p
k
kp ���
�����
='
(12)
Substituting Eqs. (12) and (11) in Eq. (9) we obtain the following relationship
( ) skskstst
skst xkpk
kkP η−+���
��
+= 1
' (13)
For the original model when the load applied is equal to the skin buckling load, P=Psk, then the load applied on the skin is equal to the local skin buckling load, psk=psk,cr and Eq. (6) becomes
crsksk
skstsk p
k
kkP ,
�� �
����
+= (14)
When the load applied P becomes equal to the stiffener buckling load, P=Pst, then the load applied on the stiffener pst is equal to the local stiffener buckling load, pst,cr and Eq. (13) becomes
6
( ) crskcrstst
skstst pp
k
kkP ,, 1
'η−+��
�
����
�+
= (15)
However from the initial assumption that stiffener buckling precipitates failure of the channel-section, and that this should occur at factor � above Psk, we have Pst= � Psk. Substituting Eq. (14) in Eq. (15),
( ) crskcrstst
skstcrsk
sk
skst ppk
kkp
k
kk,,, 1
'ηα −+��
�
�
�+
=���
�
�+
(16)
or ( )
stsksk
stskst
crsk
crst
kkk
kkk
p
p
+−++
=2
2
,
, 1
ηηαα
(17)
Assuming that the skin plate is a simply supported plate with straight unloaded edges and the stiffener is pinned at the skin edges, the following standard equations for initial buckling are assumed,
h
Dp crst
66,
24= (18)
and ( )[ ]66122211
2
, 222 DDDDb
p crsk ++= π (19)
For a constant thickness laminated structure
st=
sk ; substituting the latter and the above equations in Eq. (17) we
obtain the following design formula,
( ) ( )[ ] −�
�
���
�
++−+−�
�
���
�
+�
�
���
�
h
bDDDD
h
bD
h
bD 66122211
22
66
3
66 22211212 ηαπη
( )[ ] 0222 661222112 =++ DDDDαπ (20)
Solving the cubic function for given values of � and � we obtain three real roots, two negative and one positive. The solution, which gives the required breadth to height ratio b/h for the channel, is the real positive root.
III. Manufacturing and Experimental Testing With the aid of Eq. (20), the channel-section strut was designed so that stiffener buckling occurs 50% above initial skin buckling i.e. � =1.5, with an assumed skin stiffness reduction ratio of � =0.512,13. The skin width b=67mm was fixed due to use of an existing lamination mould. The laminate thickness was then selected to be t=2mm with a stacking sequence [45/-45/0/90]s in order to achieve initial skin buckling in the region of 40 kN at material strain of approximately 4000 micro strain, using the material properties given below. The breadth to width ratio derived was b/h=3.55 according to which the height of the stiffeners was h=19mm. In addition the length of the strut selected was L=250mm, which was small enough to eliminate any interaction between local and overall buckling at initial buckling. For the design of the delaminated structures, circular and semicircular delaminations were considered in the skin and the stiffeners respectively, as shown in Figs. 4(a) and (b), at various levels; the size of which was chosen according to the following criteria;
1) the delamination size d should be large enough to be realizable experimentally by the inclusion of PTFE patches; 2) the delamination size should be approximately equal to the equivalent size resulting from low velocity impact; 3) delamination buckling at any level (through-thickness) should be below the initial local buckling load of the undamaged structure.
7
The initial investigation was conducted with the aid of the linear elastic finite strip package VICONOPT14. Uniform end-strain in the channel-section strut was assumed in the pre-buckled state, which allowed the individual parts of the delamination, i.e. top and bottom delaminated plates, to be investigated separately by imposing corresponding strain resultants. Circular and semi-circular boundary conditions were created in order to represent the delamination geometry in the skin and stiffener respectively. Note that for the results of section 4, a 2nd level delamination in the skin and stiffener is referred to as n=2 (skin) and n=2 (stiffener) respectively and defines the delamination between the second and third layer, where the first layer forms the inner surface of the channel section.
Fig. 4. (a) Plan-view with circular delamination in the skin and, (b) side-view with semicircular delamination in the stiffener.
The laminated channel-section struts have been manufactured using a high-strength carbon-fibre unidirectional prepreg with ply thickness tply=0.25mm and material properties E11=138 GPa, E22=9.6 GPa, G12=4.6 GPa and � =0.3. All the components have been produced by a hand lay-up laminating process and were cured within an autoclave in a controlled pressure and temperature environment. PTFE film of 0.1 mm thickness and diameter d=27 mm was embedded between the layers in order to produce the required delamination. Ultrasonic C-scans were incorporated to investigate the quality of the de-bonding area and the manufacturing process. The channel-section struts were strain-gauged at various structural locations and LVDT’s were used to monitor the out-of-plane deflection of the delaminated parts. Furthermore the specimens were potted in epoxy resin in order to achieve the required clamped-end boundary conditions and were subjected to compressive loading until failure in a 100 kN Instron tension-compression testing machine.
IV. Analytical and Experimental Results The initial mode shape and critical load for the pre-buckling analysis have been obtained by solving the linear eigenvalue problem with the aid of the finite element and finite strip software, ABAQUS15 and VICONOPT, respectively. ABAQUS was also used in order to obtain the post-buckling solution utilizing the modified Riks method. In contrast to the undamaged model, the damaged models consist of three distinct regions; the un-delaminated structure and the upper and lower delaminated plates. In ABAQUS the delaminated plates are modelled as two overlaying layers with identical mesh patterns, whose reference planes were offset at their neutral axes. The utilization of the offset is imperative in order to generate a realistic response. In contrast, if the load transferred through the nodal connection were applied to the mid-plane of the delaminated parts, because of the non-symmetric lay-ups bending-compression coupling would occur. The structure was modelled using the ABAQUS 8-noded quadratic shell element S8R. To prevent inter-penetration of the delaminated parts, contact elements were applied at the interface. Appropriate displacement constraints were also applied at the intersections of the delaminated parts to
(a)
(b)
8
satisfy compatibility conditions. The ends of the structure were constrained to represent clamped conditions imposed by a large stiffened plate. Finally, for the post-buckling analysis two imperfections were considered: an overall imperfection, in the form of a single half-wave of amplitude 0.15mm (resulting in skin subjected to increased compression) and for the delaminated struts a local imperfection in the form of the first buckling mode, e.g. see Fig. 6(b), with an amplitude equal to 10% of the thickness of the embedded PTFE film between the layers. The local imperfection is always applied to the outer surface of the laminated structure, to represent the actual inclusion of the PTFE during the lamination process. (It should be noted that for larger amplitudes of local imperfection, inter-penetration of the delaminated layers is observed at the beginning of analysis.) Linear eigenvalue analyses in both finite element ABAQUS and finite strip VICONOPT predicted that the undamaged-channel of the chosen geometry will buckle into three half-wavelengths with a good agreement between ABAQUS, VICONOPT and experimental results, Fig. 5(a), (b) and (c) and Table 1. In addition the results for the delaminated channel-section strut obtained by VICONOPT are compared with those from ABAQUS in Table 1 and Fig. 6.
(a) (b) (c)
Fig 5. Initial buckling modes of undamaged channel. (a) ABAQUS plot, (b) VICONOPT contour plot with stiffeners flattened-out and (c) experimental mode.
(a) (b)
Fig 6. (a) VICONOPT and (b) ABAQUS contour plot of initial buckling of the thin delaminated plate.
9
Table 1 Results of ABAQUS and VICONOPT linear eigenvalue analysis and experimental testing of the undelaminated and delaminated channel-section strut
ABAQUS non-linear analysis was conducted for the undamaged channel and the channel with a n=2 (skin) and n=6 (skin) delamination. Comparisons of ABAQUS modes with the experimental mode shapes are shown in Fig. 7, while the post-buckling stiffness of the undamaged channel and the delaminated structures, derived from the non-linear analysis, are compared in Fig. 8. The Shanley model post-buckling stiffness is also compared in the latter figure for different values of skin stiffness reduction parameter � . The comparison of the experimental axial stiffness of the undamaged and damaged channels is represented individually in Fig. 9. The quality of the numerical analysis is inspected by comparison of back to back strain in the middle of the skin of the ABAQUS and experimental results for the undamaged channel, see Fig. 10, and comparison of back to back strain in the middle of the skin and left stiffener of the ABAQUS and experimental results for the damaged channel with 2nd level delamination in the skin in Figs. 11 and 12 respectively. Finally the experimental back to back strain in the middle of the skin and stiffener for the damaged channel with a 2nd level delamination in the stiffener is represented in Fig. 13 and the equivalent mode shape is shown in Fig. 14.
(a) (b) (c) (d) (e)
Fig. 7. Plan views of ABAQUS and experimental mode shapes.(a) and (b) show the undamaged channel and (c)-(e) show the channel with skin (n=2) delamination; (c) and (e) are initial post-buckling modes and (d) is the unstable state.
10
0
10
20
30
40
50
60
70
0 0.5 1 1.5 2 2.5 3 3.5
End-Shortening (mm)
Loa
d (
kN)
Fig. 8. Comparison of ABAQUS axial stiffness and limit loads of the un-delaminated, n=2 (skin) and n=6 (skin) delaminated channel-section strut and Shanley undelaminated model.
0
10
20
30
40
50
60
-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
End-Shortening (mm)
Load
(kN
)
Fig. 9. Comparison of experimental axial stiffness for the delaminated channels with n=2 delamination in the skin and stiffener and the undamaged channel n=0.
11
0
10
20
30
40
50
60
70
-8000 -6000 -4000 -2000 0 2000 4000 6000
Strain (micro)
Load
(kN
)
Fig. 10. Comparison of back-to-back strain in the middle of the skin of the ABAQUS and experimental results for the undamaged channel.
0
10
20
30
40
50
60
-14000 -12000 -10000 -8000 -6000 -4000 -2000 0
Strain (micro)
Load
(kN
)
Fig. 11. Comparison of experimental and ABAQUS strain 5mm below the middle of the skin for the damaged channel containing a 2nd level (n=2) delamination in the skin.
Fig. 12. Comparison of ABAQUS and experimental back-to-back strain in the middle edge of the left stiffener for the damaged channel containing a 2nd level delamination (n=2) in the skin.
0
5
10
15
20
25
30
35
40
45
50
-7000 -6000 -5000 -4000 -3000 -2000 -1000 0 1000
Strain (micro)
Load
(kN
)
Fig. 13. Experimental back-to-back strain in the middle of the skin and left stiffener for the damaged channel with a second level delamination in the right stiffener.
13
Fig 14. Post-buckling mode shape of the damaged channel containing a 2nd level (n=2) delamination in the right stiffener.
V. Discussion of Results The comparison of the Shanley model developed in the current analysis with ABAQUS non-linear analysis for the undamaged channel, Fig. 8, demonstrates that after initial instability the former is not able to evaluate the post-buckling stiffness. Even though the limit load predicted by the Shanley model is in good agreement with ABAQUS, the structure follows a less stiff post-buckling path. This is mainly attributed to the fact that initially it was assumed that the loss of stiffness in the structure is exclusively attributed to the loss in stiffness in the skin, while the stiffeners distorted in a half-wavelength equivalent to the skin width without any significant stiffness loss. Furthermore under the assumption that the stiffeners form into a single mode wave, referred to as stiffener buckling, the channel reaches ultimate loading or global buckling occurs. However it is evident from Fig. 8 that the initial stiffness reduction of the channel is not entirely due to stiffness loss in the skin corresponding to a plate with unloaded straight edges (� =0.5) or to the lower bound value for a plate with edges with in plane traction (� =0.33)12. Nevertheless the design formula produced can be utilized as a rough approximation of the breadth to height ratio of the skin and stiffeners, according to a desirable post-buckling strength. Initial buckling of the undamaged channel, which Figs. 9 and 10 show occurring at about 41kN, is in a mode producing increased compression on the inner surface of the channel skin before a sudden jump to a reversal of this mode, as illustrated in Fig. 10, occurs at 43kN. This mode jump is not captured by ABAQUS which indicates that initial buckling is in the mode with increased tension on the inner skin surface. However, it should be noted that following the test there was evidence that a small amount of restraint was suddenly lost at one of the channel ends during the test. This is thought to have occurred soon after initial buckling and may account for the experimental mode jump. Overall buckling, in which the skin forms the tension side of a clamped-clamped Euler strut, occurs at about 45kN in the experimental test compared with the FE predicted value of 48kN, see Fig. 10. Failure of the experimental specimen occurred at 48kN. This is well below the 60kN predicted by the Shanley model reinforcing the fact that for this case the simple model, which delineates skin and stiffener buckling, cannot capture the highly coupled behaviour once post-buckling has reached an advanced stage. Failure is also well below the ABAQUS prediction of 59kN showing that in practice the material fails due to excessive levels of strain well before the theoretical collapse occurs. For the damaged channel containing a 2nd level delamination in the skin, Fig. 12 reveals that as for the undamaged channel the strains in the stiffeners are reversed at a load value of 43 kN in a stable manner, while the skin strain increases at a constant rate. Failure is observed around 50 kN suggesting that the delaminated structure has post-buckling strength similar to the undamaged structure. In addition Fig. 9 demonstrates that the structure follows a slightly stiffer post-buckling path. However both experimental and analytical results show that the delaminated structure will never achieve an overall Euler-type mode since at 50 kN a new unstable equilibrium is reached and the structure reaches a limit load just before material failure. This phenomenon is attributed to the fact that the mode shape has suddenly progressed from between three and four into three half-waves. This behaviour was also seen in the ABAQUS simulation where the mode suddenly jumped from the shape shown in Fig. 7(c) to that of (d) after the limit load of 49 kN was reached. The fact that an Euler mode is never achieved means that the Shanley
14
model which is based on separate skin and stiffener buckling may form a closer representation in this case. (For example if the skin is assumed to lose practically all of its stiffness after buckling (e.g � =0.1) a limit load of 54 kN is obtained using the Shanley model, see Fig. 8.) In comparison, ABAQUS results for the channel with delamination position n=6 (skin) shown in Fig. 8 indicate little variation from the undamaged mode shape and stiffness, and a slight decrease in initial buckling load. With sufficient material strength, the structure would actually form an Euler-type mode. The significant difference between the two damaged counterparts is that in the case with a delamination n=2 (skin) a closing mode of the delaminated parts is observed while, according to ABAQUS, in the n=6 (skin) case an opening mode occurs. From the latter it is suggested that when a closing mode occurs the post-buckling strength could be significantly reduced while an opening one can, for this structure, be considered safe. This is a direct result of the coupling between the level of laminate and structure and contradicts earlier work1, which did not consider structural effects. Both closing and opening modes however, are influenced by through thickness position and the structural location of the delamination, as well as the overall and local imperfection. Simulation work has demonstrated that it is of particular importance which of the overall or local imperfection is bigger as their relative size will eventually drive the mode shape achieved in post-buckling. Finally the damaged channel containing a 2nd level delamination in the stiffener demonstrated that once more, the embedded delamination had an insignificant effect on initial buckling, which took place in the experimental test around 40 kN, see Fig. 13. The mode shown in Fig. 14 is the result of asymmetric behaviour that produces a kink in the right flange where the delamination is positioned. As a result excessive strain is developed in the right stiffener, leading to a failure load of 45 kN. Fig. 9 demonstrates that the channel has reduced post-buckling stiffness which in combination with the lower failure load can be considered a dangerous situation from the point of view of structural integrity.
VI. Conclusions and Future Work The objective of the analysis was to investigate the effect of embedded delaminations on the strength and stability of a thin-walled laminated channel-section strut in order to understand their influence on skin-stiffened structures. The cross-sectional geometry of the channel was derived according to a design formulated by extending the Shanley model to laminated composite structures, with initial buckling occurring at a high level of strain. Even though the model appeared to be in good agreement with numerical analysis, with respect to initial buckling and failure load (as defined in the analysis), it produced a stiffer post-buckling path. Experimental and numerical work conducted for a damaged channel with an embedded delamination in the middle of the skin demonstrated that according to the through thickness position of the delamination a different post-buckling path and potential strength might be observed. In particular a comparison between results for a delamination close to the inner surface of the channel and those for one close to the outer surface showed that the former, despite following a stiffer post-buckling path, reached an unstable limit load where structural failure occurred both in the experimental and numerical analysis. In contrast, numerical analysis for the outer delamination reveals that if the structure had sufficient material strength, it would eventually form an Euler-type mode equivalent to that of the undamaged structure with small variations in ultimate strength and stiffness. The phenomenon is attributed to the fact that in the structure with an inner surface level delamination a closing mode of the delaminated parts is observed while in the channel with outer delamination an opening mode occurs. Therefore it is concluded that when a closing mode occurs the post-buckling strength could be significantly reduced while an opening one could be considered safe. Furthermore experimental evidence on a damaged channel containing an inner surface delamination in the stiffener, forming a closing mode, demonstrated that further reduction in stiffness and post-buckling strength occurred suggesting that the latter location and through thickness position could be considered as the most dangerous.
Further experimental and numerical analysis will be conducted in order to investigate the behavior of the damaged channel with embedded delaminations in varying through thickness positions. Having gained insight in the local and structural interaction of the delaminated layers and the structure, further experimental work will be conducted in channel-section struts with barely visible impact damage (BVID), which represents a more realistic damage event in stiffened structures.
15
Acknowledgments The authors would like to thank Dr. Martin Gaitonde, Dr. Nihong Yang and Mr. Phillip Brown (Airbus UK Ltd)
and Mr. Robert Lewis (Hexcel Composites Ltd.) for their valuable assistance. This project is sponsored by Airbus UK Ltd, Hexcel Composites Ltd and the Department of Mechanical Engineering, University of Bath.
References 1Hunt, G.W., Hu, B., Butler, R., Almond, D.P and Wright, J.E., “Nonlinear modeling of delaminated struts”,
AIAA Journal, Vol. 42, No. 11, 2004, pp. 2364-2372. 2Hu, B., Butler, R., Almond, D.P., and Hunt, G.W., “Post-buckling and fatigue limit of artificially delaminated
composites”, Proceedings of 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, Palm Springs, CA, Paper No. AIAA-2004-1568 (April 2004).
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